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Mathematics


Mapping
For
Instruction
Grade Five
Prince William County Public Schools
June 2003 (Updated July 2009)
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Prince William County Mathematics
Vision Statement
The Prince William County Schools’ mathematics program
promotes an environment in which students develop a
comprehensive and enduring understanding of the concepts of
mathematics. Students learn to effectively apply these concepts
and use a variety of problem solving strategies. The program
nurtures a productive disposition toward mathematics,
challenges all learners, and supports further investigations in
this field.
Updated July 2009
Grade 5 Curriculum Map—page 2
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Sample Assessment Techniques
Category
Techniques
Information Provided
Observations
Anecdotal Records
Conferences
Checklists
immediate evaluation and feedback of learning, focus on specific learner expectations, social
skills and behaviors, teamwork, interactions, knowledge into context, levels of understanding,
relationships, attitude, oral language skills, listening skills, analysis, real-life application,
process, procedures, equipment handling
Journals
Journal
Personal Response Journal
Dialogue Journals
Reflective Interactive On-line Journals
understanding, written ability, conventions, organizations, pre and post comparisons, feedback
to teachers, personal connections, social skills, connection to concepts in literature,
understanding of story elements, internalization of literature, personal experience, goal setting,
understanding process, affective mode, background knowledge
Tests and Quizzes
Multiple Choice
True/False
Short Answer
pre and post test of knowledge, content mastery, ability to make inferences, recall, recognition,
memorization, content, problem solving process, summative information
Performance Tasks
Simulations
Multimedia Productions
Demonstrations
Presentations
Lab Experiments
Drama/Music/Dance
Investigations
Data Analysis
Mathematical Models
Computer Software Demonstrations
creativity, understanding, end product, public speaking and performing, group work,
organization skills, application of skills to new situations, reasoning skills, analysis, real-life
application, process, procedures, equipment handling
Written Projects
Laboratory Reports
Research Papers
Essays
Brochures
Word Puzzles
Proposals
Articles/Stories/Scripts
logical organization, hypothesis, comprehension, following directions, writing skills, use of
logic, interpersonal relations, expression, vocabulary, style, understanding of different writing
structures/genres, research skills, evaluations, summative, initiative
Oral Projects
Retelling
Debates
Interviewing
Questions/Responses
Audio Tapes
Teaching a Lesson
comprehension, synthesis, paraphrasing, speaking and listening skills, substantiation of
positions, development of counter argument, reasoning, assessment of background knowledge,
perspective, organization, decision making skills, personal information, attitude, synthesizing,
analyzing, memorization, interpretation, composure, confidence.
Visual Projects
Story Boards
Illustrations
Advertisements
Multimedia Projects
Science Fair Displays
Collages/Maps/Designs
Photographs
Models
Scrapbooks
Work Samples
assessment of background knowledge, comprehension, organization, creativity, growth and
maturity level, depth of conceptualization, good for non-readers or early readers, application,
synthesis, process, application of knowledge and skills, equipment use, decision making
Updated July 2009
Paper and Pencil
Matching
Extended Response
Grade 5 Curriculum Map—page 3
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Number and Number Sense
Mathematics instruction in grades 4 and 5 should continue to foster the development of number sense,
especially with decimals and fractions. Students with good number sense understand the meaning of numbers,
develop multiple relationships and representations among numbers, and recognize the relative magnitude of
numbers. They should learn the relative effect of operating on whole numbers, fractions, and decimals and
learn how to use mathematical symbols and language to represent problem situations. Number and operation
sense continues to be the cornerstone of the curriculum.
The focus of instruction at grades 4 and 5 allows students to investigate and develop an understanding of
number sense by modeling numbers, using different representations (e.g., physical materials, diagrams,
mathematical symbols, and word names). Students should develop strategies for reading, writing, and judging
the size of whole numbers, fractions, and decimals by comparing them, using a variety of models and
1
benchmarks as referents (e.g., 2 or 0.5). Students should apply their knowledge of number and number sense
to investigate and solve problems.
Updated July 2009
Grade 5 Curriculum Map—page 4
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Number and Number Sense
SOL Reporting Category
Number and Number Sense
Concept
Place Value
PWC Grade Level Objective 5.1
Virginia SOL 5.1
5.1A The student will read, write, and identify
the place values of decimals through
thousandths.
5.1B The student will round decimals to the
nearest tenth or hundredth place.
5.1C The student will compare the value of
two decimals through thousandths using the
symbols >, <, or =.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

How are reading, writing, and place values of decimals similar to/different from reading,
writing, and place values of whole numbers?
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How is rounding numbers with decimal places similar to/different from rounding whole
numbers?
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What are the effects of multiplying or dividing a given number (whole number and or decimal
number) by a multiple of ten?
Understanding the Objective (Teacher Notes)
The structure of the base-ten number system is based upon a simple pattern of tens, where each
place is ten times the value of the place to its right. This is known as a ten-to-one place value
relationship. The term, decimal number, refers to a number based upon a system of tens. Thus,
whole numbers are decimal numbers; however, when we typically use the term decimal number or
simply decimal, we are referring to a decimal fraction. A decimal point separates the whole number
places from the places less than one (fractional places). Place values extend infinitely in two
directions from the decimal point.
Decimals may be written in a variety of forms:
 Standard: 26.537
 Written: twenty-six and five hundred thirty-seven thousandths
 Expanded: (2 x 10) + (6 x 1) + (5 x 0.1) + (3 x 0.01) + (7 x 0.001); or 20 + 6 + 0.5 + 0.03 +
0.007; or 2 tens + 6 ones + 5 tenths + 3 hundredths + 7 thousandths
 It is important for students to also write decimals in their equivalent fractional form.
Students should model decimal numbers using different representations for a whole, e.g., base-ten
blocks (cube, flat, rod – where any of these could represent one whole), decimal squares, circle,
meter stick, and money. Concrete and pictorial representations help students extend their
understanding of the ten-structure of whole numbers into an understanding of decimal fractions.
Decimal models help students develop their own number-sense-based procedures for reading,
writing, comparing, ordering, and rounding decimal numbers. Understanding where to place given
decimal numbers between anchor numbers on a number line is a prerequisite to ordering and
rounding decimal numbers. Reading, writing, comparing, ordering, and rounding decimal numbers
should be related to real-world situations.
All students should:

Understand the place value structure of
decimals and use this structure to read,
write, and compare decimals.

Understand that decimal numbers can be
rounded to estimate when exact numbers are
not needed for the situation at hand.

Understand that decimals are rounded in a
way that is similar to the way that whole
numbers are rounded.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Identify the place values for each digit in
decimals through thousandths.

Read decimal numbers through thousandths
from written word names or place value
format.

Write decimal numbers through thousandths
from written word names or from decimal
numbers presented orally.

Round decimal numbers to the nearest
tenths or hundredths.

Identify the symbols for “is greater than,”
“is less than,” and “is equal to”.

Compare the value of two decimal numbers
through thousandths, using the symbols >,
<, and =.
Students should understand that the way decimals are read is related to the meaning of the numbers.
For example, 14.638 is read as “fourteen and six-hundred thirty-eight thousandths.” Students who
flexibly understand that 638 thousandths is equivalent to 6 tenths + 3 hundredths + 8 thousandths
(or 6 tenths + 38 thousandths) can understand why, when reading decimal numbers, the decimal
fraction is named by the smallest place.
continued
Updated July 2009
Grade 5 Curriculum Map—page 5
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Number and Number Sense
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Number and Number Sense
Subject:
5.1 continued
Likewise, students with strong place value understanding will apply reasoning when comparing and
ordering numbers rather than relying on a rote procedure for lining up numbers (for example,
recognizing that 14.638 <14.7 because 7 tenths are greater than 638 thousandths).
To round a number means to substitute a “nice” number that is close to the actual number so that
computation or comparison may be more easily done. Emphasis should be on understanding the
rounding concept, not on memorization of a procedure. Students should develop their own
PWC Grade Level Objective 5.1
procedures for rounding instead of memorizing a given procedure without understanding. For
Virginia SOL 5.1
example, students who have learned rote procedures for rounding whole numbers will have
5.1A The student will read, write, and identify difficulty understanding why 14.638 rounded to the nearest tenth is 14.6 rather than 14.600.
the place values of decimals through
Students should pair models with symbolic notation when exploring strategies for rounding.
thousandths.
Emphasis should be on understanding the rounding concept as practical real-life application.
Concept
Place Value
5.1B The student will round decimals to the
nearest tenth or hundredth place.
5.1C The student will compare the value of
two decimals through thousandths using the
symbols >, <, or =.
Updated July 2009
Computations on a calculator may provide a context for exploring the rounding of decimals. For
example, in investigating the conversion of 1/6 to a decimal fraction, students encounter the
repeating decimal 0.16666… (Note: Some calculators will round the final digit of a repeating
decimal, while others will truncate it.) Mathematically, a repeating decimal is denoted with a
superscript line over the repeating digit(s), technically known as a vinculum. Students may also
express the decimal equivalent as a rounded number. Thus, to the nearest hundredth, 1/6 is
equivalent to 0.17; to the nearest thousandth it would be 0.167. Students may use calculators to
build an understanding of the effects of multiplying or dividing numbers by powers of ten. This
understanding is useful for computation and estimation with whole numbers and decimals and for
conversions within the metric system of measurement. It also builds a foundation for the future
understanding of scientific notation.
Year: Revised 2009
Essential Understanding
All students should:

Understand the place value structure of
decimals and use this structure to read,
write, and compare decimals.

Understand that decimal numbers can be
rounded to estimate when exact numbers are
not needed for the situation at hand.

Understand that decimals are rounded in a
way that is similar to the way that whole
numbers are rounded.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Identify the place values for each digit in
decimals through thousandths.

Read decimal numbers through thousandths
from written word names or place value
format.

Write decimal numbers through thousandths
from written word names or from decimal
numbers presented orally.

Round decimal numbers to the nearest
tenths or hundredths.

Identify the symbols for “is greater than,”
“is less than,” and “is equal to”.

Compare the value of two decimal numbers
through thousandths, using the symbols >,
<, and =.
Grade 5 Curriculum Map—page 6
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Number and Number Sense
SOL Reporting Category
Number and Number Sense
Concept
Place Value
PWC Grade Level Objective 5.1
Virginia SOL 5.1
School: _____________________________
Subject:
Resources
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Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 6 and Ten-Minute
Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Number and Operations in Grades 3-5, NCTM
Nimble with Numbers Grade 4-5 & 5-6 by Leah Childs and Laura Choate
Number Sense Grades 4-6 by McIntosh and others – Section 3 and 5
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004): Building on Numbers
You Know, Scott Foresman
The Super Source books - Base 10 Blocks, ETA/ Cuisenaire
Math: A Way of Thinking
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html
Updated July 2009
Grade 5 Curriculum Map—page 7
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Number and Number Sense
SOL Reporting Category
Number and Number Sense
Concept
Order and Compare Fractions
PWC Grade Level Objective 5.2
Virginia SOL 5.2
5.2A The student will recognize and name
commonly used fractions (halves, fourths,
fifths, eighths, and tenths) in their equivalent
decimal form and vice versa.
5.2B The student will order a given set of
fractions and decimals from least to greatest.
Fractions will include like and unlike
denominators limited to 12 or less and mixed
numbers.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

When is it appropriate to use fractions? …decimals?

How can a fraction represent division?

What models and relationships help us name commonly-used fractions and mixed numbers in
their equivalent decimal forms and vice versa?

How can we use landmarks (benchmarks) and the number line to help us order a set of fractions
and decimals?
All students should:

Understand the relationship between
commonly used fractions and their decimal
form.

Understand that fractions and decimals can
be ordered from least to greatest.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation
to:

Represent fractions for halves, fourths,
fifths, eighths, and tenths in their
equivalent decimal forms.

Represent decimals in their equivalent
fraction form of half, fourth, fifth, eighth,
and tenth.

Determine equivalent relationships
between decimals and fractions with
Fractions have multiple meanings and interpretations. In Grade 5 students expand their understanding
denominators up to 12.
of fractions. They work more extensively with fractions greater than one whole. They also extend

Order a given set of no more than five
their understanding of fractions as divisions of whole numbers. In division, they may express
numbers written as fractions and mixed
quotients as mixed numbers, rather than indicating remainders. The concept of fractions as
numbers, with denominators of 12 or less,
representing division is the foundation for the computational procedure for converting fractions to
and decimals from least to greatest.
their equivalent decimal forms. Students should focus on finding equivalent decimals of familiar
fractions such as halves, fourths, fifths, eighths, and tenths. In middle school, students will learn to
use fractions as multiplicative operators and as ratios.
Understanding the Objective (Teacher Notes)
Fractions, decimals, and percents* can all be used to represent numbers less than one and numbers
between whole numbers; they can also be alternative representations for whole numbers. The form
used depends largely on context and convention. For example, fractions are most often used to
describe objects or groups we split up and for certain measurements, particularly those expressed in
U.S. Customary units; decimals are used to express baseball statistics and certain measurements such
as odometer readings, rainfall amounts, money, and most measurement expressed in metric units;
percentages are used to describe test scores, parts of large groups, weather probabilities, sale prices,
and expressions of increase and decrease. The fractions and decimals students study in Grade 5 are
all rational numbers; that is, they can all be expressed as a ratio of integers.
Decimal numbers are another way of writing fractions. Decimals and fractions represent the same
relationships; however they are presented in two different formats. Decimal numbers rely on place to
represent the value of the denominator. Models help students concretely relate fractions to decimals,
e.g., 10 x 10 grids, meter sticks, number lines, decimal squares, circles, and money.
Region models (fraction squares and rectangles, 10x10 grids, decimal squares, and fractions circles)
illustrate fractional or decimal parts as fair shares of a certain whole; as such, they place fraction and
decimals in a specific visual context. Rotational models such as the clock face help students visualize
relationships among halves, thirds, fourths, sixths, and twelfths; visual analogies between the clock
face and circle graph (pie chart) help students connect common fractions to their percent and decimal
equivalents. Linear/measurement models (fraction bars, percent equivalent strips, and number lines)
aid students in comparing rational numbers and in adding and subtracting them.
continued
Updated July 2009
Grade 5 Curriculum Map—page 8
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Number and Number Sense
Concept
Order and Compare Fractions
PWC Grade Level Objective 5.2
Virginia SOL 5.2
5.2A The student will recognize and name
commonly used fractions (halves, fourths,
fifths, eighths, and tenths) in their equivalent
decimal form and vice versa.
5.2B The student will order a given set of
fractions and decimals from least to greatest.
Fractions will include like and unlike
denominators limited to 12 or less and mixed
numbers.
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Number and Number Sense
Subject:
Year: Revised 2009
Essential Understanding
5.2 continued
The number line representation is a particularly critical one, because it helps students see fractions
and decimals as numbers that are part of the number system. This is an important concept because, ½
in a context may be one of two equal parts of a region or group of things, and ½ of a pizza represents
a larger visual area than ½ of a typical cookie; however, ½ as a number always has the same
relationship to the other numbers in our number system. ½ of 12 is always 6, no matter what the 12
represents. The number line helps students visualize fractions and decimals as relationships.
By studying multiple representations for fractions and decimals, students develop mental images of
landmark (benchmark) fractions and decimals and enhance their understanding of the complex
meanings of rational numbers.
Using money as an analog, student can easily establish landmark (benchmark) equivalents for
fractions and decimals (e.g., 1/10 as 0.1, ¼ as 0.25, ½ as 0.5 or 0.50, ¾ as 0.75, 1¼ as 1.25, etc.).
Exploring approximate decimal/fraction equivalents builds number sense for rational numbers; for
example, 0.215 is close to 0.2 or 1/5; 6.56 is close to 6.5 or 6 ½. Although students will learn to
change a fraction to its decimal equivalent by dividing the numerator of a fraction by its
denominator, they should also use reasoning from benchmark equivalents to find decimal
equivalents. For example, since 3/8 is halfway between ½ and ¼, its decimal equivalent is halfway
between 0.50 and 0.25, or 0.385 (38½ hundredths).
Students will develop number sense for the relative magnitude of rational numbers by using
reasoning to order given sets of numbers written as fractions, mixed numbers, and decimals on a
classroom number line marked with decimal/fraction equivalents of landmark/benchmark numbers.
(Sets of numbers should be limited to no more than five numbers written as fractions or mixed
numbers with like or unlike denominators of 12 or less and decimals through thousandths.) Students
with strong number sense for rational numbers – conceptual thought patterns for comparing fractions
– and place value understanding for decimals will find it unnecessary to change all the numbers to
one form (fractions) or another (decimals) to order a given set of numbers written as fractions, mixed
numbers, and decimals.
All students should:

Understand the relationship between
commonly used fractions and their decimal
form.

Understand that fractions and decimals can
be ordered from least to greatest.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation
to:

Represent fractions for halves, fourths,
fifths, eighths, and tenths in their
equivalent decimal forms.

Represent decimals in their equivalent
fraction form of half, fourth, fifth, eighth,
and tenth.

Determine equivalent relationships
between decimals and fractions with
denominators up to 12.

Order a given set of no more than five
numbers written as fractions and mixed
numbers, with denominators of 12 or less,
and decimals from least to greatest.
continued
Updated July 2009
Grade 5 Curriculum Map—page 9
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Number and Number Sense
Subject:
Essential Understanding
5.2 continued
All students should:

Understand the relationship between
commonly used fractions and their decimal
form.

Understand that fractions and decimals can
be ordered from least to greatest.
Conceptual Thought Patterns for Comparison of Fractions:
SOL Reporting Category
Number and Number Sense

PWC Grade Level Objective 5.2
Virginia SOL 5.2
5.2A The student will recognize and name
commonly used fractions (halves, fourths,
fifths, eighths, and tenths) in their equivalent
decimal form and vice versa.
5.2B The student will order a given set of
fractions and decimals from least to greatest.
Fractions will include like and unlike
denominators limited to 12 or less and mixed
numbers.
Different number of the same-sized parts
3
Concept
Order and Compare Fractions
<
5
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5
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4
5
Same number of parts of different sizes
3
>
3
Visualize:
8
4
More or less than one-half or one whole, e.g.
4
> 2
7
5
Distance from one-half or one whole
9 > 3
4
10
Year: Revised 2009
is a little more than 1
2
7
but
2
is a little less than 1
5
2
Visualize:
9 is 1 away from one whole
10
10
but 3 is 1 away from one whole.
4
4
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation
to:

Represent fractions for halves, fourths,
fifths, eighths, and tenths in their
equivalent decimal forms.

Represent decimals in their equivalent
fraction form of half, fourth, fifth, eighth,
and tenth.

Determine equivalent relationships
between decimals and fractions with
denominators up to 12.

Order a given set of no more than five
numbers written as fractions and mixed
numbers, with denominators of 12 or less,
and decimals from least to greatest.
*NOTE: In this country, the study of percent has traditionally been delayed until students have
worked extensively with fractions and decimals. Research on student understanding of rational
numbers, however, supports learning about the different forms of rational numbers together to build
deeper, more connected conceptual understanding.
For example, associating 20% with 1/5 and 0.2 and examining contexts in which they might be used
help students recognize that these values have the same relationship to the other numbers in our
number system. For example, 1/5 of 10 or 0.2 x 10 or 20% of 10) will always equal 2, no matter what
the 10 represents. Similarly, students benefit from investigating contexts in which they use rational
numbers to describe numbers greater than one. Students who have internalized the notion that a
fraction represents a part of a whole, may have difficulty conceptualizing the meaning of fraction
operations (e.g., 6 ½ x ¾). Likewise, if students conceptualize percentages only as a way of
expressing parts of 100, they may have difficulty understanding percents over 100%. Associating
alternative representations for quantities (e.g., 1 ¼ , 5/4, 1.75, and 175%) and thinking about
contexts in which they may be used allows students to develop the deeper understanding of rational
numbers necessary for future study of computation with rational numbers and ratio.
Updated July 2009
Grade 5 Curriculum Map—page 10
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Number and Number Sense
SOL Reporting Category
Number and Number Sense
Concept
Order and Compare Fractions
PWC Grade Level Objective 5.2
Virginia SOL 5.2
School: _____________________________
Subject:
Resources
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Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Units 4 & 6 and Ten-Minute
Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Number and Operations in Grades 3-5, NCTM
Nimble with Numbers Grades 4-5 & 5-6 by Leah Childs and Laura Choate
Number Sense Grade 4-6 by McIntosh and others – Section 3
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004)—Name That Portion, Scott
Foresman
The Super Source books- Pattern Blocks, Color Tiles, Snap Cubes and Cuisenaire
Rods
Lessons for Introducing Fractions, Grade 4-5 by Marilyn Burns
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne H.
Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and Jayne
Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html
Updated July 2009
Grade 5 Curriculum Map—page 11
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Computation and Estimation
Computation and estimation in grades 4 and 5 should focus on developing fluency in multiplication and division with whole
numbers and should begin to extend students’ understanding of these operations to working with fractions and decimals.
Instruction should focus on computation activities that enable students to model, explain, and develop reasonable proficiency
with basic facts and algorithms. These proficiencies are often developed as a result of investigations and opportunities to
develop algorithms. Additionally, opportunities to develop and use visual models, benchmarks, and equivalents, to add and
subtract with common fractions, and to develop computational procedures for the addition and subtraction of decimals are a
priority for instruction in these grades.
Students should develop an understanding of how whole numbers, fractions, and decimals are written and modeled; an
understanding of the meaning of multiplication and division, including multiple representations (e.g., multiplication as repeated
addition or as an array); an ability to identify and use relationships between operations to solve problems (e.g., multiplication as
the inverse of division); and the ability to use (not identify) properties of operations to solve problems [e.g., 7  28 is equivalent
to (7  20) + (7  8), or (7  30) – (7  2)].
Students should develop computational estimation strategies based on an understanding of number concepts, properties, and
relationships. Practice should include estimation of sums and differences of common fractions and decimals, using benchmarks
2 1
1
(e.g., 5 + 3 must be less than 1 because both fractions are less than 2 ). Using estimation, students should develop strategies to
recognize the reasonableness of their computations.
Additionally, students should enhance their ability to select an appropriate problem-solving method from among estimation,
mental math, paper-and-pencil algorithms, and the use of calculators and computers. With activities that challenge students to
use this knowledge and these skills to solve problems in many contexts, students develop the foundation to ensure success and
achievement in higher mathematics.
Updated July 2009
Grade 5 Curriculum Map—page 12
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
SOL Reporting Category
Computation and Estimation
Concept
Whole Numbers:
Addition and Subtraction
Multiplication and Division
Grade Level Objective 5.3
5.3A The student will create and solve
problems involving addition and subtraction
of whole numbers using paper and pencil,
estimation, mental computation, and
calculators.
5.3B The student will create and solve
problems involving multiplication and
division of whole numbers using paper and
pencil, estimation, mental computation, and
calculators.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

How are the four basic operations related to one another?

What situations call for the computation of sums? ... differences? …products? …quotients?

How do we determine whether it is more appropriate to estimate the solutions to problems than
to compute them? What determines a reasonable estimation for a given situation?

How is estimation used to check the reasonableness of the computation involved in solving a
problem?

How can we use place value understandings, the relationships between operations, and the
properties of numbers to devise efficient estimation and computation strategies for addition
and subtraction?

How can we use place value understandings, the relationships between operations, and the
properties of numbers to devise efficient estimation and computation strategies for
multiplication and division?

When is it advantageous to use an alternative computation strategy? …a traditional
computational algorithm?
Understanding the Objective (Teacher Notes)
A major goal of problem-solving instruction is to enable students to develop strategies to solve
problems. Strategies include but are not limited to using manipulatives, making an organized list or
table, using trial and error, drawing a diagram, looking for a pattern, and acting out a problem.
An example of an approach to solving problems is Polya’s four-step plan shown below. Specific
steps are not to be taught. They are to be used as an organizer for planning and problem solving.
Polya’s steps:
Virginia SOL 5.3
Understand — Retell the problem, read it twice, take notes, study the charts or diagrams, look
The student will create and solve problems
up words and symbols that are new.
involving addition, subtraction,
Plan — Decide what operation(s) to use and what sequence of steps to use to solve the
multiplication, and division of whole numbers
problem.
using paper and pencil, estimation, mental
Solve — Follow the plan and work accurately. If the first attempt doesn’t work, try another
computation, and calculators.
plan.
Look back — Does the answer make sense?
All students should:

Select appropriate methods and tools for
computing with whole numbers from among
paper and pencil, estimation, mental
computation, and calculators according to
the context and nature of the computation.

Understand the meaning of mathematical
operations and how these operations relate
to one another when creating and solving
word problems.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Create problems involving the operations of
addition, subtraction, multiplication, and
division of whole numbers, using real life
situations.

Estimate the sum, difference, product, and
quotient of whole number computation.

Solve problems involving addition,
subtraction, multiplication, and division of
whole numbers, using paper and pencil,
mental computation, and calculators, where:
 sums and differences, and products will
not exceed five digits.
 multipliers will not exceed two digits.
 divisors will not exceed two digits.
 dividends will not exceed four digits.
An estimate is a number close to an exact amount. An estimate tells about how much or about how
many. Estimation is useful for approximations in everyday situations. (Do I have enough money?
How big a turkey will I need for 12 people?) Estimation also provides a tool for judging the
reasonableness of calculator, mental, and paper-pencil computation.
continued
Updated July 2009
Grade 5 Curriculum Map—page 13
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
Year: Revised 2009
Essential Understanding
5.3 continued
All students should:

Select appropriate methods and tools for
An estimate produces answers that are “good enough” for the purpose. The situation determines
computing with whole numbers from among
what we need to know and, thus, the strategy we use for estimation. Consider the sum: $349.29 +
paper and pencil, estimation, mental
SOL Reporting Category
Computation and Estimation
$85. 99 + $175.25. For the three prices, the question, “About how much?” is very different from, “Is
computation, and calculators according to
it more than $600?” Students should consider the context when deciding what estimation strategy to
the context and nature of the computation.
use.
They
should
be
able
to
explain
and
justify
their
strategy
and
describe
the
closeness
of
their

Understand the meaning of mathematical
Concept
estimate.
Different
strategies
for
estimation
include
rounding,
compatible
numbers,
front-end
operations and how these operations relate
Whole Numbers:
estimation, and compensation.
to one another when creating and solving
Addition and Subtraction
word problems.
Multiplication and Division
Compatible numbers are numbers that are easy to work with mentally. For example, 52 + 74 can be
estimated using 50 + 75. The product 291 x 27 is close to 300 x 25. The quotient 4929 ÷ 26 is close The student will use problem solving,
Grade Level Objective 5.3
to 4800 ÷ 24 or 5000 ÷ 25.
mathematical communication, mathematical
5.3A The student will create and solve
reasoning, connections, and representation to:
problems involving addition and subtraction
Front-end
or
leading
digit
estimation
is
useful
when
totaling
many
large
numbers,
e.g.
the
number

Create problems involving the operations of
of whole numbers using paper and pencil,
of people who attended football games in a season. Front-end estimation of sums always gives a
addition, subtraction, multiplication, and
estimation, mental computation, and
sum
less
than
the
actual
sum;
however,
the
estimate
can
be
adjusted
or
refined
so
it
is
closer
to
the
division of whole numbers, using real life
calculators.
actual sum. For example, 9,162 + 5, 643 + 6,636 could be estimated using 9,000 + 5,000 + 6,000.
situations.
(To refine the estimate, one might glance down the hundreds in each number and see that the

Estimate the sum, difference, product, and
5.3B The student will create and solve
estimate
could
be
increased
by
1,000.)
quotient of whole number computation.
problems involving multiplication and

Solve problems involving addition,
division of whole numbers using paper and
Compensation
is
a
strategy
shoppers
may
use
when
mentally
estimating
a
total
purchase
amount.
subtraction, multiplication, and division of
pencil, estimation, mental computation, and
For example, $2.38 + $5.22 + $0.39 may be estimated as $2 + $5 + $1 (where the $1 represents an
whole numbers, using paper and pencil,
calculators.
approximation of the accumulated cent amounts: .38 + .22 + .39)
mental computation, and calculators, where:
 sums and differences, and products will
Virginia SOL 5.3
Rounding to a given place is another method of estimation.
not exceed five digits.
The student will create and solve problems

multipliers will not exceed two digits.
involving addition, subtraction,
≈
4,000
 divisors will not exceed two digits.
multiplication, and division of whole numbers 3,654
5,421
≈
5,000
 dividends will not exceed four digits.
using paper and pencil, estimation, mental
+2,793
≈ +3,000
computation, and calculators.
12,000
Computational strategies for addition, subtraction, multiplication, and division develop from deep
experience-based understandings of place value, the characteristics of and relationships among the
four basic operations, and the properties of numbers. Computational fluency is supported by fluency
with basic number combinations. Fluency with number combinations (“facts”) develops from a firm
and flexible understanding of number and operation and a focus on thinking.
continued
Updated July 2009
Grade 5 Curriculum Map—page 14
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Computation and Estimation
Concept
Whole Numbers:
Addition and Subtraction
Multiplication and Division
Fifth Grade Mathematics
Year: Revised 2009
Essential Questions
Understanding the Objective
Essential Understanding
More than one efficient and accurate algorithm exists for each of the operations. Preferred
algorithms vary across cultures. Students are computationally fluent when they show flexibility in
the computational methods they choose, are able to explain those methods, and produce answers
accurately and efficiently. Students should develop the flexibility to select computational
procedures appropriate to particular situations. For certain situations, an algorithm may provide the
most efficient route to solution; for others, alternative strategies may be more efficient.
All students should:

Select appropriate methods and tools for
computing with whole numbers from among
paper and pencil, estimation, mental
computation, and calculators according to
the context and nature of the computation.

Understand the meaning of mathematical
operations and how these operations relate
to one another when creating and solving
word problems.
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
5.3 continued
For example, to compute 2983 + 4729, a student might use the following mental strategy involving
the commutative property, approximation, and compensation: 4729 + 3000 = 7729; 7729 - 20 =
7709; 7709 + 3 = 7712.
The student will use problem solving,
mathematical communication, mathematical
Similarly, in a situation requiring students to find the difference between 4002 and 1786, the student reasoning, connections, and representation to:
might apply his/her understandings of equality and the operation of subtraction to create an

Create problems involving the operations of
equivalent problem which can be computed without regrouping: 3999 – 1783 = 2216. (In this case
addition, subtraction, multiplication, and
the student knew that the difference would remain the same if both the minuend and subtrahend
division of whole numbers, using real life
were decreased by the same amount.)
situations.

Estimate the sum, difference, product, and
5.3B The student will create and solve
Another
student
might
apply
an
understanding
of
inverse
operations
and
an
open
number
line
to
quotient of whole number computation.
problems involving multiplication and
compute the difference 4002 – 1786:

Solve problems involving addition,
division of whole numbers using paper and
subtraction, multiplication, and division of
pencil, estimation, mental computation, and
+4
+10
+200
+2000
+2 = 2,216
whole numbers, using paper and pencil,
calculators.
mental computation, and calculators, where:
1786 1790
1800
2000
4000
4002
 sums and differences, and products will
Virginia SOL 5.3
not exceed five digits.
The student will create and solve problems
Students will benefit from decomposing factors and applying the distributive property to compute

multipliers will not exceed two digits.
involving addition, subtraction,
products. Often called the partial product method, this strategy can be modeled with base ten
 divisors will not exceed two digits.
multiplication, and division of whole numbers materials and provides a foundation for multiplying polynomials in algebra. For example, 24 x 13 =
 dividends will not exceed four digits.
using paper and pencil, estimation, mental
(20 + 4) x (10 + 3) may be shown using partial products, or, using a matrix, as shown below:
computation, and calculators.
24
x 13
20
4
20 x 10 
200
10
200
40
240
20 x 3 
60
3
60
12
+ 72
4 x 10 
40
312
4x3 
12
312
Grade Level Objective 5.3
5.3A The student will create and solve
problems involving addition and subtraction
of whole numbers using paper and pencil,
estimation, mental computation, and
calculators.
Students should be able to explain how this application of the distributive property underlies the
U.S. traditional multiplication algorithm.
continued
Updated July 2009
Grade 5 Curriculum Map—page 15
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Computation and Estimation
Concept
Whole Numbers:
Addition and Subtraction
Multiplication and Division
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
5.3 continued
Students should not limit themselves to multiplication algorithms when computing products,
however. Depending upon the problem, other strategies may be more efficient. Some can enable
mental computation. For example, using doubling and halving twice:
75 x 16 = 150 x 8 = 300 x 4 = 1200.
Students will also benefit from the exploration of number-sense based strategies for computing
quotients.
Year: Revised 2009
Essential Understanding
All students should:

Select appropriate methods and tools for
computing with whole numbers from among
paper and pencil, estimation, mental
computation, and calculators according to
the context and nature of the computation.

Understand the meaning of mathematical
operations and how these operations relate
to one another when creating and solving
word problems.
As students develop computational methods, they should be encouraged to share and explain them
to their peers. As students share their computational strategies with their classmates, they test,
refine, and solidify their own thinking, learn from one another, and analyze the efficiency of various The student will use problem solving,
Grade Level Objective 5.3
mathematical communication, mathematical
approaches.
5.3A The student will create and solve
reasoning, connections, and representation to:
problems involving addition and subtraction
A certain amount of practice is necessary to develop fluency with computational strategies for

Create problems involving the operations of
of whole numbers using paper and pencil,
multi-digit numbers; however, the practice must be meaningful, engaging and purposeful if students
addition, subtraction, multiplication, and
estimation, mental computation, and
are to develop fluency in computation. Calculators are appropriate tools for solving problems with
division of whole numbers, using real life
calculators.
large numbers. Using calculators during problem solving changes the focus from the steps in the
situations.
computational algorithm to the process for solving the problem.

Estimate the sum, difference, product, and
5.3B The student will create and solve
quotient of whole number computation.
problems involving multiplication and

Solve problems involving addition,
division of whole numbers using paper and
subtraction, multiplication, and division of
pencil, estimation, mental computation, and
whole numbers, using paper and pencil,
calculators.
mental computation, and calculators, where:
 sums and differences, and products will
Virginia SOL 5.3
not exceed five digits.
The student will create and solve problems

multipliers will not exceed two digits.
involving addition, subtraction,
 divisors will not exceed two digits.
multiplication, and division of whole numbers
 dividends will not exceed four digits.
using paper and pencil, estimation, mental
computation, and calculators.
Updated July 2009
Grade 5 Curriculum Map—page 16
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
SOL Reporting Category
Computation and Estimation
Concept
Whole Numbers:
Addition and Subtraction
Multiplication and Division
Grade Level Objective 5.3
Virginia SOL 5.3
School: _____________________________
Subject:
Resources





















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Units 1, 3, & 7 and TenMinute Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Number and Operations in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004): Building on Numbers
You Know, Scott Foresman
Nimble with Numbers Grades 4-5 & 5-6 by Leah Childs and Laura Choate
Number Sense Grade 4-6 by McIntosh and others
Fundamentals Levels 4-5 & 5-6, Origo
The Good Time Math Event Book by Marilyn Burns
Lessons for Extending Multiplication, Grades 4-5, by Maryann Wickett and
Marilyn Burns
Lessons for Introducing Division, Grades 3-4 by Maryann Wickett, Susan
Ohanian, and Marilyn Burns
Lessons for Extending Division, Grades 4-5 by Maryann Wickett and
Marilyn Burns
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Save the Best For Last,” What's Next?" Volume 1

"Peddle the Metal,” Hardhatting in a Geo-World

"Magic Multiplication,” What's Next? Volume 1

"From Fractions to Decimals,” What's Next? Volume 2
Updated July 2009
Grade 5 Curriculum Map—page 17
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
SOL Reporting Category
Computation and Estimation
Concept
Addition, Subtraction, and Multiplication of
Decimals
PWC Grade Level Objective 5.4
5.4A The student will find the sum and
difference of two numbers expressed as
decimals through thousandths using an
appropriate method of calculation including
paper and pencil, estimation, mental
computation, and calculators.
5.4B The student will find the product of two
numbers expressed as decimals through
thousandths using an appropriate method of
calculation including paper and pencil,
estimation, mental computation, and
calculators.
Virginia SOL 5.4
The student will find the sum, difference, and
product of two numbers expressed as
decimals through thousandths, using an
appropriate method of calculation, including
paper and pencil, estimation, mental
computation, and calculators.
Updated July 2009
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Essential Questions

What situations require the addition or subtraction of decimal numbers?

How are operations with decimals similar to or different from those used with whole numbers?

How can we use models and pictures to demonstrate why multiplication of two numbers does
not always result in a larger product?

What strategies can be developed to estimate and compute sums, differences and products of
numbers expressed as decimals?

How can estimation skills and computational strategies/algorithms reinforce one another?
Understanding the Objective (Teacher Notes)
The understanding that a decimal is part of a whole is critical to the computation of decimals. Place
value of decimal numbers must be developed for students to understand computation with decimals.
Year: Revised 2009
Essential Understanding
All students should:

Use the same procedures developed for
whole number computation and apply them
to decimal place values, giving attention to
the placement of the decimal point in the
solution.

Select appropriate methods and tools for
computing with decimals numbers from
among paper and pencil, estimation, mental
computation, and calculators according to
the context and nature of the computation.
The student will use problem solving,
mathematical communication, mathematical
Strategies for whole number computation may be applied with attention to decimal place values. It
is important, however, that the “rules” for placement of the decimal point should be derived through reasoning, connections, and representation to:

Determine an appropriate method of
number sense rather than rote memorization.
calculation to find the sum, difference, and
product of two numbers expressed as
Decimal computation, particularly multiplication and division by decimal numbers, requires that
decimals through thousandths selecting
students understand decimal quantities and the conceptual meaning of each operation. (For
from among paper and pencil, estimation,
example, students cannot rely on the naive conceptions that multiplication makes larger and
mental computation, and calculators.
division makes smaller.) Estimation and models play critical roles in developing understanding for

Connect decimals to metric system.
decimal computation. Symbolic manipulation (procedural algorithms) should not be emphasized

Estimate the sum, difference, and product of
until conceptual understanding is established.
two numbers expressed as decimals through
thousandths.
Addition and subtraction of decimals should be explored using a variety of models (e.g., 10 x 10
Find the sum, difference, and product of two
grids, number lines, and money) in the context of realistic problems. Students should both solve and 
numbers expressed as decimals through
create problems involving decimals.
thousandths, using paper and pencil.

Find the sum, difference, and product of two
Decimal sums and differences may be estimated using rounding and approximate fractional
numbers expressed as decimals through
equivalents; for example, 3.712 + 1.4 is close to 4 + 1, or 5; more precisely, 3.712 + 1.4 is close to
thousandths, using mental computation.
3 ¾ + 1 ½ , or 5 ¼. The context of the problem should guide the selection of method and the

Find the sum, difference, and product of two
precision needed.
numbers expressed as decimals through
thousandths, using calculators.
Area models can be used to demonstrate that the product of decimals is dependent upon the two

Use estimation to check the reasonableness
factors being multiplied.
of the sum, difference, and product.
Factors
Product
tenths x tenths
= hundredths
tenths x hundredths
= thousandths
hundredths x hundredths = ten-thousandths
tenths x thousandths
= ten-thousandths
continued
Grade 5 Curriculum Map—page 18
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Computation and Estimation
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
5.4 continued
Models demonstrate visually why the product of two numbers may be smaller than the factors. For
example, 0.5 x 0.4 may be thought of as 1/2 of 4/10 or 4/10 of 1/2.
0.5 x 0.4 = 0.2 can be modeled by shading 0.5 horizontally on a 10 x 10 grid, and shading 0.4
vertically. The overlap represents the product 0.20 or 0.2.
Concept
Addition, Subtraction, and Multiplication of
Decimals
PWC Grade Level Objective 5.4
5.4A The student will find the sum and
difference of two numbers expressed as
decimals through thousandths using an
appropriate method of calculation including
paper and pencil, estimation, mental
computation, and calculators.
5.4B The student will find the product of two
numbers expressed as decimals through
thousandths using an appropriate method of
calculation including paper and pencil,
estimation, mental computation, and
calculators.
Similar models can be drawn on grid paper to represent the multiplication of mixed decimals (e.g.,
1.3 x 2.7).
Virginia SOL 5.4
The student will find the sum, difference, and
product of two numbers expressed as
decimals through thousandths, using an
appropriate method of calculation, including
paper and pencil, estimation, mental
computation, and calculators.
In cases where an exact product is not required, the product of decimals may be estimated using
strategies for multiplying whole numbers, such as front-end and compatible numbers, or rounding.
In each case the student needs to determine where to place the decimal point to ensure that the
product is reasonable.
Updated July 2009
The traditional algorithm for computation of decimal products is similar to the procedure developed
for whole number computation, giving attention to the placement of the decimal point in the
solution. Estimation should be used to determine placement of the decimal point.
Year: Revised 2009
Essential Understanding
All students should:

Use the same procedures developed for
whole number computation and apply them
to decimal place values, giving attention to
the placement of the decimal point in the
solution.

Select appropriate methods and tools for
computing with decimals numbers from
among paper and pencil, estimation, mental
computation, and calculators according to
the context and nature of the computation.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Determine an appropriate method of
calculation to find the sum, difference, and
product of two numbers expressed as
decimals through thousandths selecting
from among paper and pencil, estimation,
mental computation, and calculators.

Connect decimals to metric system.

Estimate the sum, difference, and product of
two numbers expressed as decimals through
thousandths.

Find the sum, difference, and product of two
numbers expressed as decimals through
thousandths, using paper and pencil.

Find the sum, difference, and product of two
numbers expressed as decimals through
thousandths, using mental computation.

Find the sum, difference, and product of two
numbers expressed as decimals through
thousandths, using calculators.

Use estimation to check the reasonableness
of the sum, difference, and product.
Grade 5 Curriculum Map—page 19
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
SOL Reporting Category
Computation and Estimation
Concept
Addition, Subtraction, and Multiplication
of Decimals
PWC Grade Level Objective 5.4
Virginia SOL 5.4
School: _____________________________
Subject:
Resources

















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Units 6 and 7 and TenMinute Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Investigations in Number, Data, and Space (2004)-- Name That Portion, Scott
Foresman
Elementary and Middle School Mathematics by John Van de Walle
Nimble with Numbers Grades 5-6 by Leah Childs and Laura Choate
Number Sense Grades 4-6 by McIntosh and others
Fundamentals Levels 4-5 & 5-6, Origo
Lessons for Decimals and Percents by Carrie DeFrancisco and Marilyn Burns
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html
Updated July 2009
Grade 5 Curriculum Map—page 20
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Computation and Estimation
Concept
Division with Remainders
PWC Grade Level Objective 5.5
5.5A The student will estimate the quotient of
two whole numbers when given a dividend of
four digits or fewer and a divisor of two digits
or fewer.
5.5B The student will determine the quotient
and remainder of two whole numbers when
given a dividend of four digits or fewer and a
divisor of two digits or fewer.
5.5C The student will create and solve reallife problems using division of whole
numbers.
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
Year: Revised 2009
Essential Understanding
Essential Questions

How can the relationship between multiplication and division be used to estimate and solve
problems involving division situations?

How can place value understandings and number sense be used to devise strategies for
estimating and computing quotients with one- and two-digit divisors?

How can estimates be used to check the reasonableness of answers?

How is the U.S. traditional division algorithm similar to and different from other strategies for
dividing whole numbers?

How does the problem situation determine how to represent a remainder?
All students should:

Understand the various meanings of
division and its effect on whole numbers.

Understand horizontal, computational, and
fractional representations of division. For
example:
Understanding the Objective (Teacher Notes)
Division is the operation of making equal parts (groups or shares).



There are two situations that require division:

Finding the size of the group when you know the original amount and the number of groups.
This is technically referred to as partitive division, because the action involved is one of
dividing or partitioning a set into a predetermined number of groups. In a card game, dealing
cards equally to each player is a partitive division process.

Finding the number of groups when you know the original amount and the size of the group.
This is called quotitive or measurement division. Repeated subtraction models the quotitive
division process.
dividend  divisor = quotient 8  2 = 4
quotient
divisor ) dividend
dividend
divisor
= quotient
4
2)8
8
4
2
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation
to:

Estimate the quotient of two whole
numbers, when given a dividend of four
digits or fewer and a divisor of two digits
Both partitive and quotitive division situations may be modeled with base-ten or set
Virginia SOL 5.5
or fewer.
manipulatives. Modeling division problems using area (array) and set models helps students
The student, given a dividend of four digits or

Determine the quotient (no remainder) of
conceptualize the nature of the remainder with respect to the other terms used in division.
fewer and a divisor of two digits or fewer,
two whole numbers when given a dividend
will find the quotient and remainder.
of four digits or fewer and a divisor of two
Division is the inverse of multiplication; therefore, multiplication and division are inverse operations.
digits or fewer.
Terms used in division include:

Determine the quotient and remainder of
two whole numbers when given a dividend

divisor - the quantity by which another quantity is to be divided
of four digits or fewer and a divisor of two

dividend - the quantity to be divided
digits or fewer.

quotient - the result of division

Use estimation to check the reasonableness
of the quotient.
Representations for division include:

horizontal form:

computational form:

fractional form:
dividend  divisor = quotient
quotient
divisor )dividend
dividend = quotient
divisor
continued
Updated July 2009
Grade 5 Curriculum Map—page 21
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Computation and Estimation
Concept
Division with Remainders
PWC Grade Level Objective 5.5
5.5A The student will estimate the quotient of
two whole numbers when given a dividend of
four digits or fewer and a divisor of two digits
or fewer.
5.5B The student will determine the quotient
and remainder of two whole numbers when
given a dividend of four digits or fewer and a
divisor of two digits or fewer.
5.5C The student will create and solve reallife problems using division of whole
numbers.
Virginia SOL 5.5
The student, given a dividend of four digits or
fewer and a divisor of two digits or fewer,
will find the quotient and remainder.
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
Essential Understanding
5.5 continued
Students should recognize dividends and divisors given in various forms (horizontal, computational,
and fractional) and be able to write one way given another. They also need to investigate and model
the relationships among factors and product in multiplication and the divisor, quotient, and dividend
in division. Area (array) models can be used to demonstrate these relationships.
All students should:

Understand the various meanings of
division and its effect on whole numbers.

Understand horizontal, computational, and
fractional representations of division. For
example:
By exploring the effects of multiplying or dividing the numerator and denominator of fractions by the 
same number, students build an understanding of equivalence in fractions. Students can apply the
same principle to the dividend and divisor in division situations and understand that the quotient is
not changed in value. This idea can be applied as a strategy to simplify computation; for example:

270 ÷ 45
= 30 ÷ 5
= 6

(270 and 45 are both divided by 9)
Students should have opportunities to explore various strategies for division with one- and two-digit
divisors. Students’ understanding of the concept of division is strengthened by investigating various
approaches such as repeated multiplication and subtraction as well as traditional algorithms.
More than one efficient and accurate algorithm exists for division with multi-digit dividends, and
preferred algorithms vary across cultures. Facility with “partial quotient” algorithms, which preserve
the place value of all numbers, provides a foundation for students’ understanding of the traditional
U.S. long division algorithm. (It is particularly “forgiving” for division with multi-digit divisors.)
Partial Quotient
Algorithm
72) 2531
- 2160
371
- 360
11
30
+ 5
35 R11
Less Sophisticated
Partial Quotient
Algorithm
72) 2531
- 720
10
1811
- 720
10
1091
- 720
10
371
- 360 + 5
11
35 R11
Year: Revised 2009
Traditional U.S.
Algorithm
35 R11
72) 2531
216
371
360
11
dividend  divisor = quotient 8  2 = 4
quotient
divisor ) dividend
dividend
divisor
= quotient
4
2)8
8
4
2
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation
to:

Estimate the quotient of two whole
numbers, when given a dividend of four
digits or fewer and a divisor of two digits
or fewer.

Determine the quotient (no remainder) of
two whole numbers when given a dividend
of four digits or fewer and a divisor of two
digits or fewer.

Determine the quotient and remainder of
two whole numbers when given a dividend
of four digits or fewer and a divisor of two
digits or fewer.

Use estimation to check the reasonableness
of the quotient.
continued
Updated July 2009
Grade 5 Curriculum Map—page 22
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Computation and Estimation
Concept
Division with Remainders
PWC Grade Level Objective 5.5
5.5A The student will estimate the quotient of
two whole numbers when given a dividend of
four digits or fewer and a divisor of two digits
or fewer.
5.5B The student will determine the quotient
and remainder of two whole numbers when
given a dividend of four digits or fewer and a
divisor of two digits or fewer.
5.5C The student will create and solve reallife problems using division of whole
numbers.
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
Year: Revised 2009
Essential Understanding
5.5 continued
All students should:

Understand the various meanings of
Students are computationally fluent when they show flexibility in the computational methods they
division and its effect on whole numbers.
choose, are able to explain those methods, and produce answers accurately and efficiently.

Understand horizontal, computational, and
fractional representations of division. For
A remainder is an amount left over once the division is complete. Remainders occur when the divisor
example:
is not a factor of the dividend. Students need experience exploring the meaning of remainders in
various problem contexts and determining when it makes sense to express remainders as whole

dividend  divisor = quotient 8  2 = 4
numbers, express them as fractions, ignore them, or to round up to the next whole number. For
example, the following problems illustrate how the remainder to the same division computation
quotient
4
problem means different things in different situations.

divisor ) dividend
2)8


If 39 people are going to Kings Dominion and each van holds 6 people, how many vans are
needed for the trip? 7 vans are needed. (The remainder tells us we need a 7th van.)
If a ribbon is 39 in. long and Sue needs 6-inch pieces to put on each nametag, how many
nametags can she complete? 6 nametags can be completed. (The remainder tells us there
will be ribbon left over, but not enough for another nametag.)
If 39 cookies are divided evenly between 6 children, how many will each get? 6 ½ cookies
each (Because cookies can be divided, the remainder can be represented as a fractional part
in the quotient.)

dividend
divisor
= quotient
8
4
2
The student will use problem solving,
mathematical communication, mathematical

reasoning, connections, and representation
to:

Estimate the quotient of two whole
numbers, when given a dividend of four
An estimate produces answers that are “good enough” for the purpose. The situation determines what
digits or fewer and a divisor of two digits
Virginia SOL 5.5
we need to know and, thus, influences the strategy we select for estimation. The specific quantities in
or fewer.
The student, given a dividend of four digits or the dividend and divisor may also influence our approach to estimation. Strategies for estimating

Determine the quotient (no remainder) of
fewer and a divisor of two digits or fewer,
quotients may include using front-end digits, rounding, compatible numbers, and chunking. Terms
two whole numbers when given a dividend
will find the quotient and remainder.
such as "closer to," "between," and "more than" are used to define the estimated quotients.
of four digits or fewer and a divisor of two
digits or fewer.
For example, for 329  8, rounding the dividend to the nearest compatible number (320) is more

Determine the quotient and remainder of
two whole numbers when given a dividend
manageable than rounding to the nearest ten (330) or hundred (300). 320 8 = 40. We know that the
of four digits or fewer and a divisor of two
estimated quotient 40 is slightly less than the actual quotient, because 320 is less than 329.
digits or fewer.

Use estimation to check the reasonableness
of the quotient.
Updated July 2009
Grade 5 Curriculum Map—page 23
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
SOL Reporting Category
Computation and Estimation
Concept
Division with Remainders
PWC Grade Level Objective 5.5
Virginia SOL 5.5
School: _____________________________
Subject:
Resources




















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Units 1 and 7 and TenMinute Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Number and Operations in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
The Super Source books – Color Tiles, Base 10
Investigations in Number, Data, and Space (2004) – Building on Numbers
You Know, Scott Foresman
Nimble with Numbers Grades 5-6 by Leah Childs and Laura Choate
Number Sense Grades 4-6 by McIntosh and others
The Good Time Math Event Book by Marilyn Burns
Math Works, Math Sequence, Exploring Mathematics by Jean Show
Lessons for Extending Division, Grades 4-5, by Maryann Wickett and
Marilyn Burns
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Save the Best For Last,” What’s Next? Volume 1

"Peddle the Metal,” Hardhatting in a Geo-World

"Magic Multiplication,” What's Next? Volume 1

"From Fractions to Decimals,” What's Next? Volume 2

"Let's Recycle,” Overhead and Underfoot
Updated July 2009
Grade 5 Curriculum Map—page 24
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Computation and Estimation
Concept
Division With Decimals
PWC Grade Level Objective 5.6
5.6A The student will find the quotient of a
problem with a dividend expressed as a
decimal through thousandths and a single
digit divisor (whole number).
5.6B The student will estimate solutions to
division problems involving decimals.
Virginia SOL 5.6
The student, given a dividend expressed as a
decimal through thousandths and a single
digit divisor, will find the quotient.
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
Essential Questions

How can estimation of quotients aid in finding the quotients of problems with dividends
expressed as decimals?

How are division algorithms used when dividing decimal numbers?
Understanding the Objective (Teacher Notes)
Please refer to Objective 5.5 Teacher Notes for discussion of division concepts.
Both the fair share and measurement (repeated subtraction) concepts of decimal division can be
modeled using base-ten manipulatives (e.g., base-ten blocks or play money). The traditional
division algorithm is based upon the action of repeated subtraction.
Understanding the place value of each digit in a decimal number is critical to understanding the
division of decimals. Estimation should be used to establish the approximate magnitude of the
quotient and, thus, the correct placement of the decimal point in the quotient, as well as to check
the reasonableness of the computed quotient.
Long division algorithms for division of decimals are similar to the procedure students developed
for whole number computation. As in the multiplication algorithm with decimal numbers, it is
reasonable to perform the division computation as if all numbers were whole numbers and then
place the decimal point based on the estimated quotient. Long division is the one traditional
algorithm that starts with the left-hand place or larger pieces. The understanding of a decimal as
part of a whole is critical to computation of decimals.
Year: Revised 2009
Essential Understanding
All students should:

Use the procedures developed for whole
number division and apply these procedures
to decimal place values, giving attention to
the placement of the decimal point in the
solution.

Be able to correctly place the decimal in a
quotient based on estimation.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Determine the quotient, given a dividend
expressed as a decimal through tenthousandths (and no annexing of zeros
during the division process) with a singledigit divisor. All dividends should be evenly
divisible by the divisor.

Use estimation to check the reasonableness
of the quotient.
By exploring the effects of multiplying or dividing the numerator and denominator of fractions by
the same number, students build an understanding of equivalence in fractions. Students can apply
the same principle to the dividend and divisor in division situations and understand that the quotient
is not changed in value. This understanding underlies the “moving the decimal point” algorithm for
finding quotients with decimal divisors; for example:
6.7^ ) 45.93^
Updated July 2009
Moving the decimal point one place to the right in both divisor and
dividend is actually the result of multiplying both divisor and
dividend by 10, thus creating an equivalent problem.
Grade 5 Curriculum Map—page 25
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
CMS Unit of Study
Decimals
SOL Reporting Category
Computation and Estimation
Concept
Division With Decimals
PWC Grade Level Objective 5.6
Virginia SOL 5.6
School: _____________________________
Subject:
Resources



















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 7 and Ten-Minute
Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Investigations in Number, Data, and Space (2004) – Name That Portion, Scott
Foresman
Nimble with Numbers Grades 5-6 by Leah Childs and Laura Choate
The Super Source books – Base 10 Blocks
Elementary and Middle School Mathematics by John Van deWalle
Exploring Mathematics by Jean Show
The Good Time Math Event Book by Marilyn Burns
Hands on Math by Bill Linderman
Lessons for Decimals and Percents, Grades 5-6 by Carrie De Francisco and
Marilyn Burns
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html
Updated July 2009
Grade 5 Curriculum Map—page 26
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
SOL Reporting Category
Computation and Estimation
Concept
Addition and Subtraction of Fractions
PWC Grade Level Objective 5.7
5.7A The student will add and subtract with
fractions and mixed numerals, with and
without regrouping, and express answers in
simplest form. Problems will include like and
unlike denominators limited to 12 or less.
5.7B The student will rename improper
fractions as mixed numbers and vice versa.
Virginia SOL 5.7
The student will add and subtract with
fractions and mixed numbers, with and
without regrouping, and express answers in
simplest form. Problems will include like and
unlike denominators, limited to 12 or less.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

What does it mean to “simplify” a fraction, and why is it important?

How can models be used to explain why the value of a fraction isn’t changed when the
numerator and denominator are multiplied or divided by the same number?

How can we use models to devise strategies for renaming improper fractions as mixed
numbers and vice versa?

How is the understanding of multiples and factors useful in renaming/simplifying fractions and
mixed numbers?

How can we use mental models, benchmarks, and approximate decimal equivalents to estimate
sums and differences of fractions?

What strategies can be developed to compute sums and differences with fractions and mixed
numbers?

Why is it necessary to rename fractions to have common denominators when using the
traditional algorithm for adding and subtracting fractions?
All students should:

Develop and use strategies to estimate and
compute addition and subtraction of
fractions.

Understand the concept of least common
multiple and least common denominator.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Add and subtract fractions having like and
unlike denominators. Denominators should
be limited to 12 or less, and answers should
be expressed in simplest form.

Add and subtract with mixed numerals
having like and unlike denominators, with
Understanding the Objective (Teacher Notes)
and without regrouping. Denominators
A fraction is in simplest form when its numerator and denominator have no common factors other
should be limited to 12 or less, and answers
than 1. The term “simplify” should be used instead of “reduce.” When we rename a fraction in
should be expressed in simplest form.
6
3
lowest terms, it does not become smaller as the word “reduce” implies:
and are equivalent.

Use estimation to check the reasonableness
8
4
of the sum or difference.

Rename fractions with unlike denominators
A fraction with a numerator greater than (or equal to) its denominator represents a number greater
of 12 or less before addition and subtraction
than (or equal to) 1. The term “improper fraction” is misleading. There are instances when a
is performed.
fraction greater than 1 is preferable to its mixed number equivalent.

Rename improper fractions as mixed
numbers and vice versa.
A fraction represents a quantity in relation to some unit as a whole. In their early work with

Find a common denominator before adding
fractions, students focused on fractions which represented parts of one whole. As students work
and subtracting fractions with unlike
with fractions greater than one (“improper fractions”) and mixed numbers, they need to expand their
denominators by finding the least common
understanding of fractions to include the possibility of representing more than one whole. Students
multiple of the denominators.
should use fraction models and number lines to convert mixed numbers to fractions greater than one 
Express a fraction in simplest form (i.e.
(improper fractions) and vice versa in order to derive the generalizations for procedures. Mixed
rename in lowest terms) by dividing the
numbers may be expressed as a fraction by multiplying the whole number by the denominator and
numerator and denominator by their greatest
adding the numerator (keeping the same denominator). Improper fractions may be expressed as
common factor.
mixed numbers by dividing the numerator by the denominator and using the remainder to write the
fractional part of the quotient.
Writing a fraction in simplest form (renaming a fraction such as 10/8 as another fraction 5/4) is not
the same as changing an improper fraction to a mixed number. For example, both 5 and its
4
equivalent 1 1 are in simplest form, but 10/8 is not in simplest form.
4
continued
Updated July 2009
Grade 5 Curriculum Map—page 27
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
Essential Understanding
5.7 continued
Fractions having like denominators are said to have common denominators.
SOL Reporting Category
Computation and Estimation
Concept
Addition and Subtraction of Fractions
PWC Grade Level Objective 5.7
5.7A The student will add and subtract with
fractions and mixed numerals, with and
without regrouping, and express answers in
simplest form. Problems will include like and
unlike denominators limited to 12 or less.
5.7B The student will rename improper
fractions as mixed numbers and vice versa.
Virginia SOL 5.7
The student will add and subtract with
fractions and mixed numbers, with and
without regrouping, and express answers in
simplest form. Problems will include like and
unlike denominators, limited to 12 or less.
Year: Revised 2009
Equivalent fractions name the same part of a whole. By exploring the effects of multiplying or
dividing the numerator and denominator of a fraction by the same non-zero number and modeling
the resulting fraction, students build an understanding of equivalence. This is reinforced by using a
calculator to derive the decimal equivalents. Students build understandings and skills crucial to
future success in algebra when they investigate and develop conjectures about the relationships of
multiples and factors, prime and composite numbers, and even and odd numbers on the process of
simplifying fractions. (For example, a fraction is in simplest form if both numbers are prime or one
is prime and the other is not a multiple of the prime; a fraction cannot be in simplest form if both
numbers are even; etc.)
By using the multiplication chart to note the relationships among equivalent fractions, students can
begin to identify and understand the concepts of least common multiple (LCM), least common
denominator (LCD), and greatest common factor (GCF). They can justify the generalizations that
the least common multiple (LCM) of the unlike denominators is the least common denominator
(LCD) of the fractions and that a fraction may be expressed in simplest form by dividing both the
numerator and denominator by their greatest common factor (GCF).
Students should investigate addition and subtraction of fractions and mixed numbers using a
variety of concrete and pictorial models; e.g., fraction circles or squares, fraction strips, rulers,
unifix cubes, pattern blocks, egg cartons, number lines, and grid paper. Emphasis should be on
conceptual development of fraction operations. Students should derive generalizations from their
concrete and pictorial experiences rather than practicing and memorizing rote procedures. If
students are focused on rules alone, what appears to be mastery of addition and subtraction of
fractions in the short term is quickly lost (particularly when students begin to multiply and divide
fractions). Presenting computation in the context of interesting story problems helps students keep
the part-whole relationship in perspective and use number sense in the estimation and solution of
sums and differences with fractions.
Students with a good foundation with fraction concepts should be able to add and subtract fractions
having like denominators immediately. Often writing the problem in words may help build
understanding, e.g. one fifth plus two fifths is ___ fifths. It should not be necessary to memorize the
traditional rule, “When adding or subtracting fractions having like denominators, add the
numerators and use the same denominators.”
All students should:

Develop and use strategies to estimate and
compute addition and subtraction of
fractions.

Understand the concept of least common
multiple and least common denominator.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Add and subtract fractions having like and
unlike denominators. Denominators should
be limited to 12 or less, and answers should
be expressed in simplest form.

Add and subtract with mixed numerals
having like and unlike denominators, with
and without regrouping. Denominators
should be limited to 12 or less, and answers
should be expressed in simplest form.

Use estimation to check the reasonableness
of the sum or difference.

Rename fractions with unlike denominators
of 12 or less before addition and subtraction
is performed.

Rename improper fractions as mixed
numbers and vice versa.

Find a common denominator before adding
and subtracting fractions with unlike
denominators by finding the least common
multiple of the denominators.

Express a fraction in simplest form (i.e.
rename in lowest terms) by dividing the
numerator and denominator by their greatest
common factor.
continued
Updated July 2009
Grade 5 Curriculum Map—page 28
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Computation and Estimation
Concept
Addition and Subtraction of Fractions
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
Essential Understanding
5.7 continued
The traditional rule for adding and subtracting fractions with unlike denominators is, “When adding
or subtracting fractions having unlike denominators, rewrite them as fractions with common
denominators.” Although it is true that to use the traditional algorithm, you must first find the
common denominator, it is not always necessary to find common denominators to add or subtract
fractions with unlike denominators, especially when a student is able to visualize the fractions.
PWC Grade Level Objective 5.7
5.7A The student will add and subtract with
fractions and mixed numerals, with and
without regrouping, and express answers in
simplest form. Problems will include like and
unlike denominators limited to 12 or less.
7
1
For example: to add 3 + 5
one student may compute it mentally in the following steps:
8
2
1
1 3
1
7
3 + 5 = 8. I can decompose
into +
and add the
to the to make 1 whole.
2
8 8
8
8
3
7
1
3
Then I add 8 + 1 +
to find the total. So, 3 + 5
= 9 .
8
8
2
8
5.7B The student will rename improper
fractions as mixed numbers and vice versa.
Another student may arrive at the sum by visualizing or drawing an open number line:
Virginia SOL 5.7
The student will add and subtract with
fractions and mixed numbers, with and
without regrouping, and express answers in
simplest form. Problems will include like and
unlike denominators, limited to 12 or less.
Problem: 3
7
1
+5
8
2
Think:
+5
7
8
(start)
3
+
8
7
8
1
4
1
3 1
= , so
+ =
2
8
8
8
2
1
8
+
9
Year: Revised 2009
3
8
3
8
(end, solution)
9
All students should:

Develop and use strategies to estimate and
compute addition and subtraction of
fractions.

Understand the concept of least common
multiple and least common denominator.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Add and subtract fractions having like and
unlike denominators. Denominators should
be limited to 12 or less, and answers should
be expressed in simplest form.

Add and subtract with mixed numerals
having like and unlike denominators, with
and without regrouping. Denominators
should be limited to 12 or less, and answers
should be expressed in simplest form.

Use estimation to check the reasonableness
of the sum or difference.

Rename fractions with unlike denominators
of 12 or less before addition and subtraction
is performed.

Rename improper fractions as mixed
numbers and vice versa.

Find a common denominator before adding
and subtracting fractions with unlike
denominators by finding the least common
multiple of the denominators.

Express a fraction in simplest form (i.e.
rename in lowest terms) by dividing the
numerator and denominator by their greatest
common factor.
continued
Updated July 2009
Grade 5 Curriculum Map—page 29
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Computation and Estimation
Concept
Addition and Subtraction of Fractions
PWC Grade Level Objective 5.7
5.7A The student will add and subtract with
fractions and mixed numerals, with and
without regrouping, and express answers in
simplest form. Problems will include like and
unlike denominators limited to 12 or less.
Likewise, to subtract mixed numbers, a student may visualize equivalents and use the inverse
operation, addition:
Problem: 8
Virginia SOL 5.7
The student will add and subtract with
fractions and mixed numbers, with and
without regrouping, and express answers in
simplest form. Problems will include like and
unlike denominators, limited to 12 or less.
Updated July 2009
1
5
-2
4
6
+
10
12
(start)
Visualize: clock face or egg carton, so
2
12
+5
3
+
8
3
12
=
Year: Revised 2009
Essential Understanding
5.7 continued
2
5.7B The student will rename improper
fractions as mixed numbers and vice versa.
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
1 3
5 10
=
;
=
4 12 6 12
5
12
(solution)
5
3
12
(end)
8
By grounding students’ experiences in adding and subtracting fractions and mixed numbers with
problems involving fractions they have modeled extensively and can visualize (halves, thirds,
fourths, sixths, eighths, twelfths or halves, fifths, and tenths), students can come to recognize that
they are, in fact, using fraction equivalents with common denominators in their own procedures.
By comparing their solution strategies to the procedural algorithms for addition and subtraction of
fractions and mixed numbers, they can develop an enduring understanding that will enhance
procedural fluency.
All students should:

Develop and use strategies to estimate and
compute addition and subtraction of
fractions.

Understand the concept of least common
multiple and least common denominator.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Add and subtract fractions having like and
unlike denominators. Denominators should
be limited to 12 or less, and answers should
be expressed in simplest form.

Add and subtract with mixed numerals
having like and unlike denominators, with
and without regrouping. Denominators
should be limited to 12 or less, and answers
should be expressed in simplest form.

Use estimation to check the reasonableness
of the sum or difference.

Rename fractions with unlike denominators
of 12 or less before addition and subtraction
is performed.

Rename improper fractions as mixed
numbers and vice versa.

Find a common denominator before adding
and subtracting fractions with unlike
denominators by finding the least common
multiple of the denominators.

Express a fraction in simplest form (i.e.
rename in lowest terms) by dividing the
numerator and denominator by their greatest
common factor.
Grade 5 Curriculum Map—page 30
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Computation and Estimation
Subject:
Resources

Concept
Addition and Subtraction of Fractions







PWC Grade Level Objective 5.7
Virginia SOL 5.7


SOL Reporting Category
Computation and Estimation
School: _____________________________











Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 4, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Number and Operations in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004) – Name that Portion, Scott
Foresman
Nimble with Numbers Grades 4-5 & 5-6 by Leah Childs and Laura Choate
Number Sense Grades 4-6 by McIntosh and others
Fundamentals Level 4-5, Origo
The Super Source books – Base 10 Blocks
Weaving Your Way From Arithmetic to Mathematics With Manipulatives,
Mary Laycock and Peggy McClean, p. 81-87.
Lessons for Introducing Fractions, Grade 4-5 by Marilyn Burns
Lessons for Extending Fractions, Grade 5 by Marilyn Burns
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"A Major Concept in Mathematics, Part IV,” AIMS Magazine, Volume 3
Issue 9

"Missing By a Fraction,” What's Next?
Volume 3

"Odd Denominators,” What’s Next? Volume 2
Updated July 2009
Grade 5 Curriculum Map—page 31
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Measurement
Students in grades 4 and 5 should be actively involved in measurement activities that require a dynamic interaction between
students and their environment. Students can see the usefulness of measurement if classroom experiences focus on measuring
objects and estimating measurements. Textbook experiences cannot substitute for activities that utilize measurement to answer
questions about real problems.
The approximate nature of measurement deserves repeated attention at this level. It is important to begin to establish some
benchmarks by which to estimate or judge the size of objects. The intent is for students to make “ballpark” comparisons and not
to memorize conversion factors between U.S. Customary and metric units. To fully understand these ballpark comparisons,
students must be actively engaged in the process of measurement.
Students use standard and nonstandard, age-appropriate tools to measure objects. Students also use age-appropriate language of
mathematics to verbalize the measurements of length, weight/mass, liquid volume, area, temperature, and time.
The focus of instruction should be an active exploration of the real world in order to apply concepts from the two systems of
measurement (metric and U.S. Customary), to measure perimeter, weight/mass, liquid volume/capacity, area, temperature, and
time. Students continue to enhance their understanding of measurement by using appropriate tools such as rulers, balances,
clocks, and thermometers. The process of measuring is identical for any attribute (i.e., length, weight/mass, liquid
volume/capacity, area): choose a unit, compare that unit to the object, and report the number of units .
Updated July 2009
Grade 5 Curriculum Map—page 32
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Perimeter and Area
PWC Grade Level Objective 5.8
5.8A The student will measure and find the
perimeter of a polygon.
5.8B The student will describe and determine
the area of a square, rectangle and right
triangle given the appropriate measures.
Virginia SOL 5.8
The student will describe and determine the
perimeter of a polygon and the area of a
square, rectangle, and right triangle, given the
appropriate measures.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions
All students should:

Understand the concept of perimeter and

What real-life situations use perimeter? …area?
area.

How are the areas of rectangles (including squares) and triangles related?
Understand and use appropriate units of

How can models be used to develop strategies/formulas for computing the perimeters and areas 
measure for perimeter and area.
of squares, rectangles, and triangles?
Understanding the Objective (Teacher Notes)
Perimeter is the distance around the edge of a shape or closed figure. The perimeter of any straightsided shape can be found by adding the lengths of the sides.
Area is the amount of space occupied by a two-dimensional shape. Area is typically measured by
the number of square units needed to cover a surface or figure.
Students should practice associating the words perimeter and area with real-life examples of the
concepts. They should be actively involved in measuring perimeter and area in order to fully
understand the concepts involved and the associated relationships between measures and units.
Students should attach the appropriate unit of measure to the perimeter (e.g., centimeters, meters,
inches, feet, yards, or units) and to the area (e.g., square centimeters, square inches, square feet,
square yards, or square units).
Area and perimeter (and their associated units of measurement) are often confused by students,
particularly if they are taught how to calculate perimeter and area before they have developed
conceptual clarity about each idea. Through investigating questions such as, Do figures with the
same area have the same perimeter? and Will figures with the same perimeter have the same area?,
students will observe interesting relationships among dimensions, perimeter, and area.
(Corresponding relationships exist in three-dimensions in the measurement of volume.)
Students should investigate, using manipulatives (e.g., paper folding, graph paper, geoboards) the
physical relationships among rectangles, squares, and right triangles, in order to develop formulas
for their areas. The area of rectangular regions serves as the foundation upon which formulas for
areas of other geometric figures are based.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Determine the perimeter of polygons, with
or without diagrams, when:
 The lengths of all sides of the polygons
(excluding rectangles and squares) are
given.
 The length and width of a rectangle are
given.
 The length of a side of a square is given.

Determine the area of a square, with or
without diagrams, when the length of a side
is given.

Determine the area of a rectangle, with or
without diagrams, when the length and
width are given.

Determine the area of a right triangle, with
or without diagrams, when the base and the
height are given.

Determine the perimeter of a polygon and
area of a square, rectangle, and triangle
(following the parameters listed above),
using only whole number measurements
given in metric and U.S. Customary units,
and record the solution with the appropriate
unit of measure, e.g., 24 square inches.
continued
Updated July 2009
Grade 5 Curriculum Map—page 33
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Measurement and Geometry
Concept
Perimeter and Area
PWC Grade Level Objective 5.8
5.8A The student will measure and find the
perimeter of a polygon.
5.8B The student will describe and determine
the area of a square, rectangle and right
triangle given the appropriate measures.
Virginia SOL 5.8
The student will describe and determine the
perimeter of a polygon and the area of a
square, rectangle, and right triangle, given the
appropriate measures.
Updated July 2009
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Measurement
Subject:
5.8 continued
It is beneficial for students to associate the area representation of products in multiplication
(rectangular arrays) with the concept of area. They will readily recognize:
Area of a rectangle = length x width OR Area of a rectangle = base x height
Using the formula A = base x height makes it easier to relate the formula for the area of a rectangle
to the formula for area of a triangle, including those without right angles.
Year: Revised 2009
Essential Understanding
All students should:

Understand the concept of perimeter and
area.

Understand and use appropriate units of
measure for perimeter and area.
The student will use problem solving,
mathematical communication, mathematical
Students should consider the relationship between squares and rectangles and the formulas for their reasoning, connections, and representation to:
areas. Since a square is a rectangle with equal length and width, the rectangle formulas also apply.

Determine the perimeter of polygons, with
or without diagrams, when:
Area of a square = side x side OR length x width OR base x height
 The lengths of all sides of the polygons
(excluding rectangles and squares) are
Working with right triangles, students may suggest the formula:
given.
 The length and width of a rectangle are
1
1
given.
Area of a right triangle =
length x width
OR
base x height
2
2
 The length of a side of a square is given.

Determine the area of a square, with or
Although PWC/SOL 5.8 does not require students to find the areas of non-right triangles, it is useful
without diagrams, when the length of a side
to have the students use paper models to explore briefly the reason for the more universal formula.
is given.

Determine the area of a rectangle, with or
1
without diagrams, when the length and
Area of a triangle =
base x height
2
width are given.
For example, consider the obtuse triangle ABC:

Determine the area of a right triangle, with
or without diagrams, when the base and the
B
height are given.

Determine the perimeter of a polygon and
x
y
area of a square, rectangle, and triangle
(following the parameters listed above),
using only whole number measurements
given in metric and U.S. Customary units,
A
C
D
and record the solution with the appropriate
unit of measure, e.g., 24 square inches.
Using grid paper or cut paper copies of the triangle, students can prove that the area of the triangle
is one-half the area of the corresponding rectangle with base AC and its side or height, the length of
the perpendicular line segment BD.
1
Thus the formula Area of a triangle =
base x height applies to all triangles.
2
Grade 5 Curriculum Map—page 34
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Perimeter and Area
PWC Grade Level Objective 5.8
Virginia SOL 5.8
School: _____________________________
Subject:
Resources















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 5, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Measurement in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004) – Measurement
Benchmarks, Scott Foresman
The Super Source books – Pattern Blocks, Tangrams and Geoboards
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Can You Tell?” Floaters and Sinkers

"Wreck-Tangles,” Hardhatting in a Geo-World

"Paper Pinchers,” Hardhatting in a Geo-World

"Playground Geometry,” Hardhatting in a Geo-World

"The Chocolate Cake Challenge,” Historical Connections in Mathematics,
Volume II
Updated July 2009
Grade 5 Curriculum Map—page 35
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Geometry
SOL Reporting Category
Measurement and Geometry
Concept
Circles
PWC Grade Level Objective 5.9
5.9A The student will identify and describe
the diameter, radius, chord, and
circumference of a circle.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Essential Questions

How can models be used to derive definitions for the diameter, radius, chord, and
circumference of a circle?

How can models be used to demonstrate the relationships among the radius, diameter, and
circumference of a circle?
Understanding the Objective (Teacher Notes)
A circle is a set of points on a flat surface (plane) that are the same distance (equidistant) from a
given point called the center. The center is not part of the circle itself.
A chord is a line segment connecting any two points on a circle.
A diameter is a chord that goes through the center of a circle.
5.9B The student will describe the
relationships among the radius, diameter, and
circumference of a circle.
A radius is a segment from the center of a circle to any point on the circle. Two radii end-to-end
form a diameter of a circle. Thus, the diameter is twice the size of the radius, and the radius is half
of the diameter.
Virginia SOL 5.9
The student will identify and describe the
diameter, radius, chord, and circumference of
a circle.
Circumference is the distance around (or perimeter) of a circle.
Students should investigate to find the relationships among radius, diameter, and circumference.
For example, using paper circles of various sizes, students can fold them to find chords, diameters,
and radii; they can prove that a radius is half its corresponding diameter and contrast the
characteristics of the diameter with non-diameter chords.
By collecting and analyzing measurement data for each circle – radius, diameter, and approximate
circumference – they can deduce that a circle’s circumference is always slightly longer than three
times its diameter. This relationship is fundamental to the understanding of the mathematical
constant, pi (π), and the formulas for circumference and area of circles and surface area and volume
of spheres and solids with circular bases.
Updated July 2009
Year: Revised 2009
Essential Understanding
All students should:

Identify the parts of a circle.

Understand that the circumference is the
distance around the circle.

Understand the relationship between the
measure of diameter and radius and the
relationship between the measure of radius
and circumference.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Describe the diameter, radius, chord, and
circumference of a circle.

Describe the relationship between
diameter and radius.
diameter and circumference.
radius and circumference.

Identify the diameter, radius, chord, and
circumference of a given circle.

Solve real-life problems involving the
relationship between the radius and the
diameter.

Solve real-life problems involving the
approximate relationship between the
circumference and the diameter (i.e. the
circumference is about 3 times the
diameter).
Grade 5 Curriculum Map—page 36
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Geometry
SOL Reporting Category
Measurement and Geometry
Concept
Circles
PWC Grade Level Objective 5.9
Virginia SOL 5.9
School: _____________________________
Subject:
Resources

















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
The Super Source books – Tangrams
Exploring Mathematics by Jean Shaw (Grades 4-6)
Hands on Math by Bill Linderman
Mathematics: A Way of Thinking by Robert Baratta-Lorton
Tables, Charts, and Graphs by Milliken Math
Creative Constructions, Grades 5 and Up by Seymour-Schadler
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Circle Sighs,” Hardhatting in a Geo-World

"Playground Geometry,” Hardhatting in a Geo-World

"Finding Pi,” Historical Connections in Mathematics, Volume II
Updated July 2009
Grade 5 Curriculum Map—page 37
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Perimeter, Area and Volume
PWC Grade Level Objective 5.10
Virginia SOL 5.10
The student will differentiate among
perimeter, area, and volume and identify
whether the application of the concept of
perimeter, area, or volume is appropriate for a
given situation.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Essential Questions

How are two dimensional and three dimensional figures related? How do these relationships
help us differentiate the ideas of perimeter, area, and volume?

What problem situations require us to find the perimeter? …the area? …the volume?
Understanding the Objective (Teacher Notes)
Perimeter is the distance around an object. It is a measure of length and is a measurement in onedimension and is measured in linear units: millimeter, centimeter, inch, foot, mile, etc.
Year: Revised 2009
Essential Understanding
All students should:

Understand the concept of perimeter, area,
and volume.

Understand and use appropriate units of
measure for perimeter, area, and volume.

Understand the difference between using
perimeter, area, and volume in a given
situation.
Area is the number of square units needed to cover a surface. Area is a measurement of the space
occupied by two-dimensional shapes or regions. Area is measured in square units: square meters,
square inches, square kilometers, etc.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Differentiate between the concepts of area,
Volume is a measure of capacity. Volume and capacity both refer to measures of three-dimensional
perimeter, and volume.
regions but the specific terms are typically used in different contexts and with different units of

Describe real life situations where area,
measurement. Volume usually refers to the amount of space an object takes up. Volume is measured
perimeter, and volume are appropriate
in cubic units; e.g., cubic centimeters, cubic inches, cubic feet, etc. Units of volume are based on
measures to use.
linear measures. Capacity usually refers to the amount that a container will hold. Units of capacity 
Identify whether the application of the
include milliliter, liter, cup, pint, quart, gallon, etc. In casual use, the terms volume and capacity are
concept of perimeter, area or volume is
often used interchangeably. It is interesting that, in the metric system, volume and capacity are
appropriate for a given situation.
related by the relationship: 1 milliliter(ml) of a substance occupies the space of 1 cubic centimeter
(cc).
In middle school, students will investigate the concept of surface area, the area required to cover
the surface of a solid or three-dimensional shape. It is the sum of the areas of all the surfaces of the
solid or three-dimensional shape or simply, the area of the net of the shape.
Hands–on investigations with rectangular prisms created from nets of square centimeter (or square
inch) grid paper may help students differentiate the concepts of perimeter, area, and volume. For
example, they can count or measure the perimeter and count or calculate the areas of faces; they can
assemble the nets and fill them with centimeter (or inch) cubes to find the volume. They can
develop an awareness of the relationships among the linear dimensions, area, and volume that will
help them transition to more formal work with volume in later grades. (Sometimes students are
confused by the use of cubic units to describe the volume of irregular or curved shapes. They can
use the concept of equal capacity to overcome this confusion.)
Students should associate their hands-on experiences with real-life situations in which area,
perimeter, and volume are appropriate. They should be able to justify their choices orally or in
writing.
Updated July 2009
Grade 5 Curriculum Map—page 38
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Perimeter, Area and Volume
PWC Grade Level Objective 5.10
Virginia SOL 5.10
School: _____________________________
Subject:
Resources
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Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Units 2 & 5, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Measurement in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004) – Measurement
Benchmarks, Picturing Polygons, Containers and Cubes, Scott Foresman
The Super Source books – Pattern Blocks, Tangrams and Geoboards
Sizing Up Measurement, Grades 3 – 5 by Chris Confer
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Can You Tell?” Floaters and Sinkers
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"Wreck-Tangles,” Hardhatting in a Geo-World

"Paper Pinchers,” Hardhatting in a Geo-World

"Playground Geometry,” Hardhatting in a Geo-World

"The Chocolate Cake Challenge,” Historical Connections in Mathematics,
Volume II
Updated July 2009
Grade 5 Curriculum Map—page 39
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Linear Measurement
PWC Grade Level Objective 5.11
5.11A The student will choose an appropriate
measuring device and unit of measure and
solve linear measurement and area problems
in real life situations.
5.11B The student will estimate and solve
linear measurement problems to parts of an
inch (1/2, 1/4, 1/8), inches, feet, yards, miles,
millimeters, centimeters, meters, and
kilometers.
5.11C The student will describe numerical
relationships between units of measure within
the same measurement system such as one
inch is 1/12 of a foot.
Virginia SOL 5.11
The student will choose an appropriate
measuring device and unit of measure to solve
problems involving measurement of

Length - part of an inch (1/2, 1/4, and
1/8), inches, feet, yards, miles,
millimeters, centimeters, meters, and
kilometers

Area - square units
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

Why are all measurements of length approximations?

What tools are used in linear measurement? How does one determine which is appropriate to
use?

How does the selection of an appropriate unit of measurement and measurement tool affect the
precision of the solution to problems involving linear measurement?
All students should:

Understand how to select a measuring
device and unit of measure to solve
problems involving measurement.

Understand and use appropriate units of
measure for area.
Understanding the Objective (Teacher Notes)
Measurement Background Information
The process of measuring something consists of three main steps: First, select an attribute that can be
measured; second, choose an appropriate unit of measurement; finally, determine the number of
units, usually by using a measurement tool.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation
to:

Solve problems involving measurement by
selecting an appropriate measuring device
and a U.S. Customary or metric unit of
measure for:
Length—part of an inch (1/2,1/4,1/8,),
inches, feet, yards, miles, millimeters,
centimeters, meters, and kilometers
Area - square units

Recognize equivalent linear measures in
different units within the same system.
In order for students to learn to use measurement concepts, tools, and units effectively, instruction
must require students to solve problems by performing real measurements with various measurement
tools.
Estimation is a critical part of measurement. Estimating helps students internalize measurement
concepts. Benchmarks can help students develop a working familiarity with various measurement
units and their relationships.
“Measurements” are different from “counts”. Counts provide discrete data and can be exact, but
measurements provide continuous data (that is, each unit of measure can be repeatedly subdivided
into smaller subunits.), and thus, all measurements are approximate. Precision and accuracy are
different. The precision of measurement depends upon the size of the smallest measuring unit.
Accuracy refers to how correctly a measurement has been made.
The metric system was developed by the French Academy of Sciences in 1791 in order to provide an
internationally accepted system of measurement for commerce and science. The metric system has
evolved technically into the International System of Units (SI), with units defined in terms of
scientific formulas and natural constants. The metric system defines standard units for length, mass,
temperature, area, and volume. Larger and smaller units are obtained by multiplying or dividing the
standard units by powers of ten. The principles of our base ten numeration system are utilized in the
relationships among metric units.
Linear Measurement
Length is the distance along a line or figure from one point to another. The term length can be used
inclusively to represent any linear measure (height, width, thickness, perimeter, etc.), or it can denote
one specific dimension of length (as in length versus width). This ambiguity may create some
confusion, particularly for ELL students.
continued
Updated July 2009
Grade 5 Curriculum Map—page 40
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Linear Measurement
PWC Grade Level Objective 5.11
5.11A The student will choose an appropriate
measuring device and unit of measure and
solve linear measurement and area problems
in real life situations.
5.11B The student will estimate and solve
linear measurement problems to parts of an
inch (1/2, 1/4, 1/8), inches, feet, yards, miles,
millimeters, centimeters, meters, and
kilometers.
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Essential Understanding
5.11 ABC continued
All students should:

Understand how to select a measuring
U.S. Customary units include inches, feet, yards and miles. Appropriate measuring devices include
device and unit of measure to solve
customary rulers and yardsticks. When measuring with U.S. Customary units, students should be able
problems involving measurement.
to measure to the nearest part of an inch (1/2, 1/4, 1/8), foot, yard, or mile. They should also

Understand and use appropriate units of
recognize equivalent linear measures within the Customary system. Students should be able to
measure for area.
convert between inches, feet, yards and miles.
The student will use problem solving,
Metric units include millimeters, centimeters, meters and kilometers. Appropriate measuring devices mathematical communication, mathematical
include centimeter rulers and meter sticks. They should also recognize equivalent linear measures
reasoning, connections, and representation
within the Metric system, and be able to convert between millimeters, centimeters, meters, and
to:
kilometers.

Solve problems involving measurement by
selecting an appropriate measuring device
Today’s students have fewer measurement experiences in the home than did many of their parents.
and a U.S. Customary or metric unit of
Practical experiences measuring the length of familiar objects and applying linear measurement in
measure for:
school projects and scientific investigations will help students understand the effects of the selection
Length—part of an inch (1/2,1/4,1/8,),
of measurement units and measurement tools on precision as well as facilitate their ability to measure
inches, feet, yards, miles, millimeters,
accurately, establish benchmarks, and estimate length.
centimeters, meters, and kilometers
Area - square units
Area is the number of square units needed to cover a surface or figure. Students should understand

Recognize equivalent linear measures in
the relationship between linear dimension and the calculation of area.
different units within the same system.
5.11C The student will describe numerical
relationships between units of measure within
the same measurement system such as one
inch is 1/12 of a foot.
Virginia SOL 5.11
The student will choose an appropriate
measuring device and unit of measure to solve
problems involving measurement of

Length - part of an inch (1/2, 1/4, and
1/8), inches, feet, yards, miles,
millimeters, centimeters, meters, and
kilometers

Area - square units
Updated July 2009
Grade 5 Curriculum Map—page 41
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Linear Measurement
PWC Grade Level Objective 5.11
Virginia SOL 5.11
School: _____________________________
Subject:
Resources
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Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Units 5 & 8, Pearson
(area)
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Measurement in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004) – Measurement
Benchmarks, Scott Foresman
The Super Source books – Pattern Blocks, Tangrams and Geoboards
Exploring Mathematics by Jean Shaw
Hands on Math by Bill Linderman
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Inside Out,” Critters

"Growing Pains,” Critters

"How Do You Measure Up?” From Head to Toe

Primarily Bears: "Guessing Jars," " Area Forms on Geo boards," 10/88;
"Area Sand Perimeter on Geo boards," 9/88, "Paper Penny Boxes," 1/92,
"Paper Pinchers," 7-8/94.
Updated July 2009
Grade 5 Curriculum Map—page 42
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Measurement of Mass / Weight
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Essential Questions

Why are all measurements of weight and mass approximations?

What tools are used to measure weight and mass? How does one determine which is
appropriate to use?

How does the selection of an appropriate unit of measurement and measurement tool affect the
precision of the solution to problems involving measurement of weight and mass?
Understanding the Objective (Teacher Notes)
PWC Grade Level Objective 5.11
Please refer to the Measurement Background Information provided in Objective 5.11 A & B.
5.11D The student will choose an appropriate
measuring device and unit of measure to solve Measurement of Weight and Mass
weight and mass problems in real life
situations.
Weight and mass are different, although in casual use, the terms are often used interchangeably.
Mass is the amount of matter in an object. Weight is determined by the pull of gravity on the mass
5.11E The student will estimate and solve
of an object. The mass of an object remains the same regardless of its location. The weight of an
weight and mass problems to include ounces, object changes depending on the gravitational pull at its location. In everyday life, most people are
pounds, tons, grams, and kilograms.
actually interested in determining an object’s mass, although they use the term weight e.g., “How
much does it weigh?” versus “What is its mass?”
5.11F The student will describe numerical
relationships between units of measure within U.S. Customary units used in the measurement of weight and mass are ounces, pounds, and tons.
the same measurement system such as an
Metric units for weight and mass are grams and kilograms.
ounce is 1/16 of a pound.
Balances are the appropriate scientific tools to measure mass. Counter-weight scales (such as those
Virginia SOL 5.11
used in doctors’ offices) operate on the principle of the balance. Spring scales (like those in the
The student will choose an appropriate
produce section of the grocery store) actually measure weight, since they depend upon the pull of
measuring device and unit of measure to solve gravity.
problems involving measurement of

weight and mass – ounces, pounds, tons, Students should be able to convert between ounces, pounds, and tons, and between grams and
grams, and kilograms
kilograms.
Year: Revised 2009
Essential Understanding
All students should:

Understand how to select a measuring
device and unit of measure to solve
problems involving measurement.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Solve problems involving measurement by
selecting an appropriate measuring device
and a U.S. Customary or metric unit of
measure for:
Weight - ounces, pounds, and tons
Mass - grams and kilograms
 Recognize equivalent measures of weight
given in different units within the same
system.
Today’s students have fewer measurement experiences in the home than did many of their parents.
Practical experiences measuring the mass/weight of familiar objects and in school projects and
scientific investigations will help students understand the effects of the selection of measurement
units and measurement tools on precision as well as facilitate their ability to measure accurately,
establish benchmarks, and estimate weight/mass.
Updated July 2009
Grade 5 Curriculum Map—page 43
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Measurement of
Mass/Weight
PWC Grade Level
Objective 5.11
School: _____________________________
Subject:
Resources
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
Virginia SOL 5.11



Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Measurement in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004) – Measurement
Benchmarks, Scott Foresman
The Super Source books – Pattern Blocks, Tangrams and Geoboards
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Icebergs,” Down to Earth

"By Golly, By Gum,” Jaw Breakers and Heart Thumpers

Primarily Bears: "Guessing Jars," " Area Forms on Geo boards," 10/88;
"Area Sand Perimeter on Geo boards," 9/88, "Paper Penny Boxes," 1/92,
"Paper Pinchers," 7-8/94.
Updated July 2009
Grade 5 Curriculum Map—page 44
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions
All students should:

Why are all measurements of liquid volume and capacity approximations?

Understand how to select a measuring

What tools are used to measure liquid volume and capacity? How does one determine which is
device and unit of measure to solve
appropriate to use?
problems involving measurement.
SOL Reporting Category

How does the selection of an appropriate unit of measurement and measurement tool affect the 
Measurement and Geometry
Recognize equivalent measures of capacity
precision of the solution to problems involving measurement of liquid volume and capacity?
given in different units within the same
system.
Concept
Understanding the Objective (Teacher Notes)
Measurement of Capacity and Liquid
Volume
The student will use problem solving,
Please refer to the Measurement Background Information provided in Objective 5.11 A & B.
mathematical communication, mathematical
PWC Grade Level Objective 5.11
reasoning, connections, and representation to:
5.11 G The student will choose an appropriate

Solve problems involving measurement by
Liquid volume is a measure of capacity. Capacity is the amount something can hold. The U.S.
measuring device and unit of measure to solve customary units of liquid volume include the following: cup (C), pint (pt.), quart (qt.), and gallon
selecting an appropriate measuring device
capacity and liquid volume problems in real
and a U.S Customary or metric unit of
(gal.). The metric units of liquid volume include the following: milliliter (ml) and liter (l).
life situations.
measure for:
Liquid volume-cups, pints, quarts,
Tools to measure liquid volume include measuring cups and containers of specific capacities (e.g.,
5.11 H The student will estimate and solve
gallons, millimeters, and liters
pint, quart, gallon), beakers, and graduated cylinders. A general rule of thumb is that a narrower,
capacity and liquid volume problems to
taller container gives more precision than a wider, shorter container. This is why the graduated
include cups, pints, quarts, gallons, milliliters, cylinder or syringe is used in scientific work.
and liters.
PWC Curriculum Strand
Measurement
5.11 I The student will describe numerical
relationships between units of measure within
the same measurement system such as a cup
is 1/4th of a quart.
Virginia SOL 5.11
The student will choose an appropriate
measuring device and unit of measure to solve
problems involving measurement of

liquid volume – cups, pints, quarts,
gallons, milliliters, and liters
Updated July 2009
Today’s students have fewer measurement experiences in the home than did many of their parents.
Students should have experiences measuring liquid volume of everyday objects, using metric and
U.S. Customary units, including cups, pints, quarts, gallons, milliliters, and liters, and record the
liquid volume using the appropriate unit of measure, e.g., 24 gallons. Students should be able to
convert from one unit to another within each system.
Practicing measurement as part of school projects and scientific investigations will help students
understand the effects of the selection of measurement units and measurement tools on precision as
well as facilitate their ability to measure accurately, establish benchmarks, and estimate liquid
volume/capacity.
Grade 5 Curriculum Map—page 45
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Measurement of Capacity
and Liquid Volume
PWC Grade Level Objective 5.11
Virginia SOL 5.11
School: _____________________________
Subject:
Resources
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












Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004) – Measurement
Benchmarks, Scott Foresman
Navigating through Measurement in Grades 3-5, NCTM
The Super Source books – Pattern Blocks, Tangrams and Geoboards
Exploring Mathematics by Jean Shaw
Real World Math by Edupress
Hands on Math by Bill Linderman
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Metric Mania,” Historical Connections in Mathematics, Volume III

"Can You Tell?” Floaters and Sinkers

Primarily Bears: "Guessing Jars," " Area Forms on Geo boards," 10/88;
"Area Sand Perimeter on Geo boards," 9/88, "Paper Penny Boxes," 1/92,
"Paper Pinchers," 7-8/94.
Updated July 2009
Grade 5 Curriculum Map—page 46
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Measurement of Temperature
PWC Grade Level Objective 5.11
5.11 J The student will choose an appropriate
measuring device and unit of measure to solve
problems involving measurement of
temperature in Celsius and Fahrenheit.
5.11 K The student will estimate the
conversion of temperature in Celsius units to
Fahrenheit units and vise versa, relative to
familiar situations. (Water freezes at 0ºC and
32ºF, water boils at 100ºC and 212ºF, normal
body temperature is about 37ºC and 98.6ºF).
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Essential Questions

Why are all temperature measurements approximations?

What tools are used to measure temperature? How does one determine which is appropriate to
use?

What is the relationship between Celsius and Fahrenheit units?

What are some benchmark temperature measurements in Celsius and Fahrenheit?

How does the selection of appropriate measurement tool and temperature scale affect the
precision of the solution to problems involving measurement of temperature?
Understanding the Objective (Teacher Notes)
A thermometer is a tool used to measure temperature. A scale marked in degrees Fahrenheit or
degrees Celsius indicates temperature. The U.S. Customary unit of measurement is degree
Fahrenheit (°F); the metric unit of measurement is degree Celsius (°C).
Students should measure temperature in a variety of situations, using various types of thermometers
in each scale. Reading temperatures using both a Fahrenheit thermometer and a Celsius
thermometer at the same time reinforces the concept that measurements given in Celsius and
Fahrenheit are two different representations for the same temperature.
Making sense of temperature measurements in Celsius and Fahrenheit is an essential life skill,
requiring reasonable estimates of what the measurements mean in each scale. When students
internalize benchmark Celsius and Fahrenheit temperatures relative to familiar situations, they can
Virginia SOL 5.11
use them to estimate the conversion of Celsius and Fahrenheit units:
The student will choose an appropriate

Water freezes at 0oC and 32oF
measuring device and unit of measure to solve 
Water boils at 100oC and 212oF
problems involving measurement of

Normal body temperature is about 37oC and 98.6oF
Temperature - Celsius and Fahrenheit

Average room temperature is about 20° C and 68° F.
units.
Problems also will include estimating the
Students should be able to look at the range of the temperature scale on a given thermometer and
conversion of Celsius and Fahrenheit units
assess whether it is an appropriate tool for a specific task. (For example, a thermometer designed for
relative to familiar situations (water freezes at weather measurement will not have the range necessary to measure cooking temperatures.)
0C and 32F, water boils at 100C and at
212F, normal body temperature is about
37C and 98.6F).
Updated July 2009
Year: Revised 2009
Essential Understanding
All students should:

Understand how to select a measuring
device and unit of measure to solve
problems involving measurement.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Solve problems involving measurement by
selecting an appropriate measuring device
and a U.S. Customary or metric unit of
measure for temperature: Celsius and
Fahrenheit units.
Estimate the conversion of Celsius and
Fahrenheit units relative to familiar situations:

Water freezes at 0oC and 32oF

Water boils at 100oC and 212oF

Normal body temperature is about 37oC and
98.6oF
Grade 5 Curriculum Map—page 47
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Measurement of Temperature
Updated July 2009
Subject:
Resources








PWC Grade Level Objective 5.11
Virginia SOL 5.11
School: _____________________________
Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Measurement in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004) – Measurement
Benchmarks, Scott Foresman
The Super Source books – Pattern Blocks, Tangrams and Geoboards
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"White Rain,” Our Wonderful World

"When You're Hot, You're Hot,” Down to Earth

"Temp-Rate,” Down to Earth

"Salty Change,” Down to Earth

"Cool It,” Math + Science: A Solution

"Hot Stuff,” Math + Science: A Solution
Grade 5 Curriculum Map—page 48
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Essential Questions

What is meant by elapsed time?

How can elapsed time be determined in hours and minutes within a 24-hour period?

In what everyday situations do we determine elapsed time?
Year: Revised 2009
Essential Understanding
All students should:

Understand the “counting on” strategy for
determining elapsed time in hours and
minutes.
Understanding the Objective (Teacher Notes)
Concept
Measurement of Time
PWC Grade Level Objective 5.12
Virginia SOL 5.12
The student will determine an amount of
elapsed time in hours and minutes within a
24-hour period.
Time can be thought of as the duration of an event. Elapsed time is the amount of time that has
passed between two given events. Measuring elapsed time in hours and minutes in a 24-hour period
requires knowledge of the relationship of minutes to hours and the process of changing minutes
greater than 60 to hours and minutes.
Elapsed time can be found by counting on from the beginning time to the finishing time. Using or
visualizing a clock face may help students apply this strategy.

Count the number of whole hours between the beginning time and the finishing time.

Count the remaining minutes.

Add the hours and minutes.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Determine elapsed time in hours and
minutes within a 24-hour period.

Investigate everyday applications of elapsed
time in a 24- hour period, such as, hours and
minutes spent in school for a day.
For example, to find the elapsed time between 10:15 a.m. and 1:25 p.m., count on as follows:

from 10:15 a.m. to 1:15 p.m., count 3 hours;

from 1:15 p.m. to 1:25 p.m., count 10 minutes; and then

add 3 hours to 10 minutes to find the total elapsed time of 3 hours and 10 minutes.
Updated July 2009
Grade 5 Curriculum Map—page 49
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Measurement
SOL Reporting Category
Measurement and Geometry
Concept
Measurement of Time
PWC Grade Level Objective 5.12
Virginia SOL 5.12
Updated July 2009
School: _____________________________
Subject:
Resources







Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004) – Measurement
Benchmarks, Scott Foresman
The Super Source books – Pattern Blocks, Tangrams and Geoboards
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Around the Clock,” AIMS Magazine, Volume 8 Issue 3

"Singing Bears,” Popping with Power
Grade 5 Curriculum Map—page 50
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Geometry
The study of geometry helps students represent and make sense of the world. At the fourth- and fifth-grade levels, reasoning skills
typically grow rapidly, and these skills enable students to investigate geometric problems of increasing complexity and to study how
geometric terms relate to geometric properties. Students develop knowledge about how geometric shapes relate to each other and begin to
use mathematical reasoning to analyze and justify properties and relationships among shapes.
Students discover these relationships by constructing, drawing, measuring, comparing, and classifying geometric shapes. Investigations
should include explorations with everyday objects and other physical materials. Exercises that ask students to visualize, draw, and
compare shapes will help them not only to develop an understanding of the relationships, but to develop their spatial sense as well.
Discussing ideas, conjecturing, and testing hypotheses precede the development of more formal summary statements. In the process,
definitions become meaningful, relationships among figures are understood, and students are prepared to use these ideas to develop
informal arguments.
Students investigate, identify, and draw representations and describe the relationships between and among points, lines, line segments,
rays, and angles. Students apply generalizations about lines, angles, and triangles to develop understanding about congruence, other lines
such as parallel and perpendicular ones, and classifications of triangles. Students also explore coordinate geometry, using the coordinate
plane to describe points in the first quadrant.
The van Hiele theory of geometric understanding describes how students learn geometry and provides a framework for structuring student
experiences that should lead to conceptual growth and understanding.
 Level 0: Pre-recognition. Geometric figures are not recognized. For example, students cannot differentiate between threesided and four-sided polygons.
 Level 1: Visualization. Geometric figures are recognized as entities, without any awareness of parts of figures or relationships
between components of a figure. Students should recognize and name figures and distinguish a given figure from others that
look somewhat the same. (This is the expected level of student performance during grades K and 1.)
 Level 2: Analysis. Properties are perceived but are isolated and unrelated. Students should recognize and name properties of
geometric figures. (Students are expected to transition to this level during grades 2 and 3.)
 Level 3: Abstraction. Definitions are meaningful, with relationships being perceived between properties and between figures.
Logical implications and class inclusions are understood, but the role and significance of deduction is not understood. (Students
should transition to this level during grades 5 and 6 and fully attain it before taking Algebra.)
Updated July 2009
Grade 5 Curriculum Map—page 51
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Geometry
SOL Reporting Category
Measurement and Geometry
Concept
Angles and Triangles
PWC Grade Level Objective 5.13
Virginia SOL 5.13
The student will measure and draw right,
acute, and obtuse angles and triangles using
appropriate tools.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

How does angle measurement differ from linear measurement?

How can visualizing a circle divided into wedges help us think about angle measurement?

How is a protractor or an angle ruler used to measure and construct angles and triangles?
All students should:

Understand how to measure and draw acute,
right, and obtuse angles.
Understanding the Objective (Teacher Notes)
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Identify the appropriate tools, e.g.,
protractor and straightedge or angle ruler, as
well as available computer software, used to
measure and draw angles and triangles.

Draw right, acute, and obtuse angles, using
appropriate tools.

Measure right, acute, and obtuse angles with
a protractor or angle ruler and identify their
measure in degrees.

Measure the angles of right, acute, and
obtuse triangles with a protractor or angle
ruler and identify their measure in degrees.
The word angle comes from the Latin word angulus, meaning "a corner". Angles are formed by two
rays with a common endpoint, called the vertex. It is helpful to think of an angle as a rotation (turn)
around a vertex; the size of the angle is a measure of how far one ray is turned from the other ray
(or in geometric figures, how far one side is turned from the other side).
Angles are typically measured in degrees*. A degree is 1/360 of a complete rotation. The division
of one complete rotation (a circle) into 360 degrees can be traced back to the ancient Babylonians
and was probably based on ancient calendars which used 360 as the number of days in a year.
Another reason for choosing the number 360 may be that it is readily divisible; 360 has 24 divisors
(factors).
Students need to understand the attribute of angle measure as a measure of rotation. A common
confusion among students is to think that angle measurement is linear and determined by the length
of the rays or of the “spread” between the rays. In geometry, the concept of similarity requires
students to understand that corresponding angle measurements in two polygons may be equal
despite the fact that the corresponding sides may differ in size.
Angles may be measured using a tool marked in degrees. Protractors and angle rulers are tools used
to measure and construct angles. A protractor is a circular or semicircular tool usually ruled into
units of measurement – degrees – based on the division of one complete rotation into 360°.
Protractors have existed since ancient times. Most angle rulers are devices with two arms which
pivot around a point in the center of a 360° (circular) protractor. The arms of most angle rulers are
also marked in linear measurements (inches and centimeters), thus allowing the student to measure
and construct polygons with specific side lengths, as well as measure and construct angles. There
are many types of protractors and angle rulers.
A right angle is an angle that forms a square corner. A right angle measures exactly 90. Before
measuring an angle, students should first compare it to a right angle to decide if the measure of the
angle is less than or greater than 90°. Recognizing immediately whether an angle’s measure is
greater or less than 90° is critical to selecting the correct scale (inner or outer) on the protractor.
continued
Updated July 2009
Grade 5 Curriculum Map—page 52
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Measurement and Geometry
Concept
Angles and Triangles
PWC Grade Level Objective 5.13
Virginia SOL 5.13
The student will measure and draw right,
acute, and obtuse angles and triangles using
appropriate tools.
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Geometry
Subject:
5.13 continued
Students should be able to estimate an angle’s measure by comparing it to known landmarks.
Knowing that a right angle measures 90 degrees and a straight angle, 180 degrees, students can use
these angles as landmarks to find angles of 30, 45, and 60, 120 and 150 degrees and estimate others
in between these measures. Using fraction circle manipulatives or graphics to represent angles of
different magnitudes helps students visualize and internalize the concept of 360 degrees as one full
rotation and estimate the measure of angles. For example, in a circle divided into fourths, each
wedge represents an angle of 90°; in a circle divided into sixths, each wedge represents an angle of
60°; in a circle divided into twelfths, each wedge represents and angle of 30°. Experience with
Turtle graphics in the Logo computer language (or one of its variants) provides functional practice
with angle measure and enables students to internalize angle benchmarks/landmarks.
Students should understand how to work with a protractor or angle ruler, as well as available
computer software, to measure and draw angles and triangles.
*Note: It is interesting to note that the degree is not the only unit of measure for angles. The radian
is often the preferred unit of angle measure in higher mathematics; the angular mil and its variants
are used in military applications. Computer games require very fast computations to depict a threedimensional virtual world; thus, in these situations, a binary 256 degree system is often used, and
the unit of angle measure is the brad or binary radian. When the degree, as a unit of angle measure,
does not provide sufficient precision, as in astronomy and navigation (latitude and longitude on
Earth), the degree is subdivided into 60 minutes of arc, and the minute into 60 seconds of arc.
Updated July 2009
Year: Revised 2009
Essential Understanding
All students should:

Understand how to measure and draw acute,
right, and obtuse angles.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Identify the appropriate tools, e.g.,
protractor and straightedge or angle ruler, as
well as available computer software, used to
measure and draw angles and triangles.

Draw right, acute, and obtuse angles, using
appropriate tools.

Measure right, acute, and obtuse angles with
a protractor or angle ruler and identify their
measure in degrees.

Measure the angles of right, acute, and
obtuse triangles with a protractor or angle
ruler and identify their measure in degrees.
Grade 5 Curriculum Map—page 53
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Geometry
SOL Reporting Category
Measurement and Geometry
Concept
Angles and Triangles
Subject:
Resources








PWC Grade Level Objective 5.13
Virginia SOL 5.13
School: _____________________________










Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 5, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Measurement in Grades 3-5, NCTM
Investigations in Number, Data, and Space (2004)- Picturing Polygons, Scott
Foresman
The Super Source books – Cuisenaire Rods, Color Tiles, Pattern Blocks and
Geoboards
Elementary and Middle School Mathematics by John Van deWalle
The Super Source books – Cuisenaire Rods and Geoboards
Hands on Math by Bill Linderman
Mathematics: A Way of Thinking by Robert Baratta-Lorton
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html
Updated July 2009
Grade 5 Curriculum Map—page 54
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Geometry
SOL Reporting Category
Measurement and Geometry
Concept
Angles and Triangles
PWC Grade Level Objective 5.14
Virginia SOL 5.14
The student will classify angles and triangles
as right, acute, or obtuse.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

How is the size of an angle related to rotation?

How are angle properties used to classify angles?

How are angle properties used to classify triangles?

How do line relationships affect angle relationships in triangles?
All students should:

Understand how to identify angles as right,
acute, and obtuse.

Understand how to classify angles as right,
acute, and obtuse.

Understand how to identify a triangle as
Understanding the Objective (Teacher Notes)
either acute, right, or obtuse.

Understand that triangles can be classified
Angles are formed by two rays with a common endpoint, called the vertex. Angles occur where lines
by the size of their angles.
and/or line segments intersect. The symbol  is used to indicate an angle.
An angle is a rotation (turn) around a point, and the size of the angle is a measure of how far one ray The student will use problem solving,
is turned from the other ray (or in geometric figures, how far one side is turned from the other side). mathematical communication, mathematical
Angles are measured in degrees. A degree is 1/360 of a complete rotation.
reasoning, connections, and representation to:

Identify angles as right, acute, and obtuse.
Angles are classified according to their measures as right, acute, obtuse, or straight angles. A right

Classify angles as right, acute, and obtuse.
angle is an angle that forms a square corner. A right angle measures exactly 90°. An acute angle

Identify triangles as right, acute, and obtuse.
forms an angle less than a right angle. An acute angle measures greater than 0° and less than 90°.

Classify triangles as right, acute, and obtuse.
An obtuse angle forms an angle greater than a right angle. An obtuse angle measures greater than
90° and less than 180° (a straight angle).
Geometric figures are identified and classified by their attributes, and many figures can be classified
in more than one way. Triangles are typically classified in one of two ways: according to the
measure of their angles or according to the measure of their sides.
Triangles are classified as right, acute or obtuse according to the size of their angles. A right
triangle has one right angle and two acute angles. An obtuse triangle has one obtuse angle and two
acute angles. An acute triangle has three acute angles.
Students should investigate the relationships between sides and angles in triangles using geoboards,
paper strips, drawings, etc. They should be able to answer questions such as the following: Can a
right triangle have an obtuse angle? Why or why not? Can an obtuse triangle have more than one
obtuse angle? Why or why not? What type of angles are the two angles other than the right angle in
a right triangle? What type of angles are the two angles other than the obtuse angle in an obtuse
triangle?
Triangles are classified as equilateral, scalene, or isosceles according to the length of their sides. An
equilateral triangle has three equal sides (all sides congruent). In a scalene triangle the three sides
are three different lengths (no sides congruent). An isosceles triangle has at least two equal
(congruent) sides.
Updated July 2009
Grade 5 Curriculum Map—page 55
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Geometry
SOL Reporting Category
Measurement and Geometry
Concept
Angles and Triangles
PWC Grade Level Objective 5.14
Virginia SOL 5.14
School: _____________________________
Subject:
Resources






















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 5, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Measurement in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004) – Picturing Polygons,
Scott Foresman
The Super Source books – Geoboards
Hands on Math by Bill Linderman
Creative Constructions, Grades 5 and Up by Seymour-Schadler
Exploring Mathematics by Jean Shaw (G 4-6)
Hands on Math by Bill Linderman
Computer Software: Geometric Concepts
Mathematics: A Way of Thinking by Robert Baratta-Lorton
Creative Constructions, Grades 5 and Up by Seymour-Schadler
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html
Updated July 2009
Grade 5 Curriculum Map—page 56
MAPPING FOR INSTRUCTION
Teacher: _____________________________
SOL Reporting Category
Measurement and Geometry
Concept
Plane Figures
PWC Grade Level Objective 5.15
Virginia SOL 5.15
5.15A The student will recognize, identify,
describe, and analyze the properties of twodimensional (plane) figures (square,
rectangle, triangle, parallelogram, rhombus,
kite, and trapezoid) in order to develop
definitions of these figures.
5.15B The student will identify and explore
congruent, non-congruent, and similar figures.
5.15C The student will investigate and
describe the results of combining and
subdividing shapes.
5.15D The student will identify and describe a
line of symmetry.
5.15E The student will recognize the images
of figures resulting from geometric
transformations such as translations (slides),
reflections (flips), or rotations (turns).
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Geometry
School: _____________________________
Year: Revised 2009
Essential Understanding
Essential Questions

How can relationships among angles and sides be used to distinguish and define twodimensional figures?

How do relationships among angles and sides determine congruent, non-congruent, and similar
figures?

How can we predict and explain the results of combining and dividing shapes into other shapes?

What are regular polygons?

What are the characteristics of a line of symmetry in a two-dimensional shape? How do lines of
symmetry relate to regular polygons?

How does geometric transformation (translation, reflection, or rotation) represent the movement
of an object or figure?
Understanding the Objective (Teacher Notes)
A polygon is a two-dimensional (plane) closed geometric figure, which has straight (line segment)
sides. A polygon is equiangular if its interior angles are congruent (having exactly the same size and
shape); it is equilateral if all of its sides are congruent. A polygon is regular if its sides are congruent
and its interior angles are congruent. A line of symmetry of a regular polygon divides the figure into
two congruent halves that are mirror images of each other.
A triangle is a polygon with three sides. Triangles may be classified according to the measure of the
angles: right, acute, or obtuse. Triangles may also be classified according to the measure of the sides:
scalene (no sides congruent), isosceles (at least two sides congruent) and equilateral (all sides
congruent).
A quadrilateral is a polygon with four sides.

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Properties of a parallelogram include:

A diagonal (a segment that connects two vertices but is not a side of the polygon)
divides the parallelogram into two congruent triangles.

The opposite sides of a parallelogram are congruent. The opposite angles of a
parallelogram are congruent. The diagonals of a parallelogram bisect each other.

A rectangle is a parallelogram with four right angles. Since a rectangle is a parallelogram,
the rectangle has the same properties of a parallelogram.

A square is a rectangle with four congruent sides. Since a square is a rectangle, the square
has all the properties of the rectangle and the parallelogram.

A rhombus is a parallelogram with four congruent sides. Opposite angles of a rhombus are
congruent. Since a rhombus is a parallelogram, the rhombus has all the properties of a
parallelogram.
All students should:

Understand that plane figures are unique in
their defining properties and symmetry.

Understand that simple plane figures can
be combined to make more complicated
figures and that complicated figures can be
subdivided into simple plane figures.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation
to:

Recognize and identify the properties of
squares, rectangles, triangles,
parallelograms, rhombi, kites and
trapezoids.

Describe the properties of squares,
rectangles, triangles, parallelograms,
rhombi, kites and trapezoids.

Analyze the properties of squares,
rectangles, triangles, parallelograms,
rhombi, kites and trapezoids.

Identify congruent, non-congruent, and
similar figures.

Describe the results of combining and
subdividing shapes.

Identify and describe a line of symmetry.

Recognize the image of figures resulting
from geometric transformations such as
translations, reflections, or rotations.
continued
Updated July 2009
Grade 5 Curriculum Map—page 57
MAPPING FOR INSTRUCTION
Teacher: _____________________________
SOL Reporting Category
Measurement and Geometry
PWC Grade Level Objective 5.15
Virginia SOL 5.15
5.15A The student will recognize, identify,
describe, and analyze the properties of twodimensional (plane) figures (square,
rectangle, triangle, parallelogram, rhombus,
kite, and trapezoid) in order to develop
definitions of these figures.
5.15B The student will identify and explore
congruent, non-congruent, and similar figures.
5.15C The student will investigate and
describe the results of combining and
subdividing shapes.
5.15D The student will identify and describe a
line of symmetry.
5.15E The student will recognize the images
of figures resulting from geometric
transformations such as translations (slides),
reflections (flips), or rotations (turns).
Fifth Grade Mathematics
5.15 continued

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are
called bases and the non-parallel sides are called legs. If the legs have the same length then
the trapezoid is an isosceles trapezoid.

Concept
Plane Figures
Subject:
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Geometry
School: _____________________________
A kite is a quadrilateral with two distinct pairs of adjacent congruent sides.
Year: Revised 2009
Essential Understanding
All students should:

Understand that plane figures are unique in
their defining properties and symmetry.

Understand that simple plane figures can
be combined to make more complicated
figures and that complicated figures can be
subdivided into simple plane figures.
A line of symmetry is a line that divides a figure into congruent halves, each of which is the reflection
image of the other. Figures and pictures can possess more than one line of symmetry. It is important
for teachers to recognize that line symmetry is not the only type of symmetry. Although students in
Virginia do not study rotational symmetry in mathematics until middle school, they may encounter it
in art class, and some students may intuitively recognize it in a figure such as the parallelogram
below. It would be important to note that such a figure does not have line symmetry, but we would be
incorrect, and possibly creating a misconception, if we stated that it is not symmetrical.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation
to:

Recognize and identify the properties of
squares, rectangles, triangles,
parallelograms, rhombi, kites and
trapezoids.

Describe the properties of squares,
In geometry, a transformation can change the position, size, or shape of a figure. The original figure
rectangles, triangles, parallelograms,
is called the “object’ and the result is the “image.” The geometric transformations studied in
rhombi, kites and trapezoids.
elementary school involve change in position. A translation (slide) is a transformation in which an

Analyze the properties of squares,
image is formed by moving every point on a figure the same distance in the same direction. A
rectangles, triangles, parallelograms,
reflection (flip) is a transformation in which a figure is flipped over a line called the line of reflection.
rhombi, kites and trapezoids.
All corresponding points in the image and pre-image are equidistant from the line of reflection. A

Identify congruent, non-congruent, and
rotation (turn) is a transformation in which the image is formed by turning its pre-image about a
similar figures.
point.

Describe the results of combining and
subdividing shapes.
Figures are said to be congruent if they have exactly the same size and shape; that is, their

Identify and describe a line of symmetry.
corresponding angles and sides have the same measures. Change in position (translation, rotation, or

Recognize the image of figures resulting
reflection) does not alter the congruence of figures.
from geometric transformations such as
translations, reflections, or rotations.
Figures are said to be similar if they have exactly the same shape, but not necessarily the same size;
that is, their corresponding angles must be congruent, but their corresponding sides must only be
proportional.
Two or more figures can be combined to form a new shape. Students should be able to identify the
figures that have been combined. Likewise, a polygon may be subdivided into two or more figures.
Students should understand how to divide a polygon into familiar figures.
The Van Hiele theory of geometric understanding (see Geometry strand introduction) describes how
students learn geometry and provides a framework for structuring student experiences that should
lead to conceptual growth and understanding.
Updated July 2009
Grade 5 Curriculum Map—page 58
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Geometry
SOL Reporting Category
Measurement and Geometry
Concept
Plane Figures
PWC Grade Level Objective 5.15
Virginia SOL 5.15
School: _____________________________
Subject:
Resources
















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 5 and Ten-Minute
Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Geometry in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004)- Picturing Polygons, Scott
Foresman
The Super Source books – Cuisenaire Rods, Color Tiles, Pattern Blocks and
Geoboards
Pieces and Patterns, AIMS, Available through county library
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

Back Talk [geometric shapes], AIMS Magazine, Volume 10 Issue 9

“Phone Home,” Out of This World

“Slice Me Twice,” Hardhatting in a Geo-World
Updated July 2009
Grade 5 Curriculum Map—page 59
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Geometry
SOL Reporting Category
Measurement and Geometry
Concept
Three-Dimensional Figures
PWC Grade Level Objective 5.16
Virginia SOL 5.16
The student will identify, compare, and
analyze properties of three-dimensional
(solid) geometric shapes (cylinders, cones,
cubes, square pyramids, and rectangular
prisms).
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

What are three-dimensional shapes?

What is a face? an edge? a vertex?

What is a rectangular prism?

How can the properties of geometric figures be used to define and classify them?

How do the properties of “related” two-dimensional (plane) and three-dimensional (solid)
figures compare (e.g., circle-sphere, square-cube, rectangle-rectangular prism)?
Understanding the Objective (Teacher Notes)
Three-dimensional shapes are solid figures, or simply solids. Solids enclose a region of space.
Solids are classified by the types of surfaces they have. These surfaces may be flat, curved, or both.
A sphere is a solid with all of the points on its surface the same distance from the center.
A cylinder is a solid bounded by two congruent and parallel circular regions joined by a curved
surface whose cross-section perpendicular to the axis is always a circle congruent to the bases.
A cone is a solid bounded by a circular base and a curved surface with one vertex.
A solid bounded by polygons is known as a polyhedron. A prism is a polyhedron for which the top
and bottom faces (known as the bases) are congruent polygons, and all other faces (known as the
lateral faces) are parallelograms. Technically, when the lateral faces are rectangles, the shape is
known as a right prism, because the lateral faces meet the sides of the base at right angles. The
prism definitions used in the 5th Grade SOL are actually definitions for specific right prisms. A
prism is described by the shape of its base. For instance, a rectangular prism has bases that are
rectangles, a triangular prism has bases that are triangles, and a pentagonal prism has bases that are
pentagons.
All students should:

Understand that solid figures are unique in
their defining properties.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Identify properties of three-dimensional
(solid) geometric shapes (cylinders, cones,
cubes, square pyramids, and rectangular
prisms).

Analyze and compare properties of threedimensional (solid) geometric shapes
(cylinders, cones, cubes, square pyramids,
and rectangular prisms).

Visualize, describe, and make models of
rectangular prisms in the terms of the
number and shape of faces, edges, and
vertices.

Interpret two-dimensional representations of
three-dimensional objects (rectangular
prisms) and draw patterns of faces for a
solid that, when cut and folded, will make a
model of the solid.

Given a pictorial representation of a model,
identify the number of faces, vertices, and
edges.
A rectangular solid or rectangular prism is a three-dimensional figure in which all six faces are
rectangles with three pairs of parallel, congruent opposite faces. A rectangular prism in which all
the faces are congruent squares is a cube.
A cube is a solid with six congruent square faces. A cube has 6 faces and 12 edges; every edge is
the same length. Since a square is an equilateral rectangle, a cube is a special instance of a
rectangular prism.
continued
Updated July 2009
Grade 5 Curriculum Map—page 60
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Measurement and Geometry
Concept
Three-Dimensional Figures
PWC Grade Level Objective 5.16
Virginia SOL 5.16
The student will identify, compare, and
analyze properties of three-dimensional
(solid) geometric shapes (cylinders, cones,
cubes, square pyramids, and rectangular
prisms).
Updated July 2009
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Geometry
Subject:
5.16 continued
A pyramid is a polyhedron formed by connecting a polygonal base and a point not in the plane,
called the apex. Each base edge and apex forms a triangle. A pyramid is described by the shape of
its base. For instance, a triangular pyramid has a base which is a triangle, a pentagonal prism has a
base that is a pentagon, and a square pyramid has a base that is a square.
A square pyramid is a polyhedron whose base is a square and whose other faces are triangles that
share a common vertex.
By handling solid figures and/or assembling polyhedra from their nets, students are able to apply
the vocabulary of geometry to describe, compare and contrast three dimensional figures. The Van
Hiele theory of geometric understanding (see Geometry strand introduction) describes how students
learn geometry and provides a framework for structuring student experiences that should lead to
conceptual growth and understanding.
Year: Revised 2009
Essential Understanding
All students should:

Understand that solid figures are unique in
their defining properties.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Identify properties of three-dimensional
(solid) geometric shapes (cylinders, cones,
cubes, square pyramids, and rectangular
prisms).

Analyze and compare properties of threedimensional (solid) geometric shapes
(cylinders, cones, cubes, square pyramids,
and rectangular prisms).

Visualize, describe, and make models of
rectangular prisms in the terms of the
number and shape of faces, edges, and
vertices.

Interpret two-dimensional representations of
three-dimensional objects (rectangular
prisms) and draw patterns of faces for a
solid that, when cut and folded, will make a
model of the solid.

Given a pictorial representation of a model,
identify the number of faces, vertices, and
edges.
Grade 5 Curriculum Map—page 61
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Geometry
SOL Reporting Category
Measurement and Geometry
Concept
Three-Dimensional Figures
PWC Grade Level Objective 5.16
Virginia SOL 5.16
School: _____________________________
Subject:
Resources


















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 2, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Geometry in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004)- Picturing Polygons, Scott
Foresman
The Super Source books – Cuisenaire Rods, Color Tiles, Pattern Blocks and
Geoboards
Investigations in Number, Data, and Space (2004)- Picturing Polygons, Scott
Foresman
Investigations in Number, Data, and Space (2004) - Cubes and Containers,
Scott Foresman
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
Virginia Department of Education Website – Geometry:
http://www.pen.k12.va.us/VDOE/Instruction/Elem_M/geo_elem.html
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

“Diabolical Cube,” AIMS Magazine, Volume 8 Issue 2

“Cutting Corners,” AIMS Magazine, Volume 8 Issue 2

“Straws Take a Stand,” Hardhatting in a Geo-World

“Polyhedron Models,” Soap Films and Bubbles
Updated July 2009
Grade 5 Curriculum Map—page 62
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Probability and Statistics
Students entering grades 4 and 5 have explored the concepts of chance and are able to determine possible outcomes of given
events. Students have utilized a variety of random generator tools, including random number generators (number cubes),
spinners, and two-sided counters. In game situations, students are able to predict whether the game is fair or not fair.
Furthermore, students are able to identify events as likely or unlikely to happen. Thus the focus of instruction at grades 4 and 5
is to deepen their understanding of the concepts of probability by
 developing the continuum of terms to include impossible, unlikely, equally likely, possible, and certain;
 offering opportunities to set up models simulating real-life events;
 engaging students in activities to enhance their understanding of fairness; and
 engaging students in activities imbued with a spirit of investigation and exploration and providing students with
opportunities to use manipulatives.
The focus of statistics instruction is to assist students with further development and investigation of data-collection strategies.
Students should continue to focus on:
 posing questions;
 collecting data and organizing this data into meaningful graphs, charts, and diagrams based on issues relating to real-world
experiences;
 interpreting the data presented by these graphs;
 answering descriptive questions (“How many?” “How much?”) from the data displays;
 identifying and justifying comparisons (“Which is the most?” “Which is the least?” “Which is the same?” “Which is
different?” “How much more is ______ than _____?”) about the information;
 comparing their initial predictions to the actual results; and
 writing a few sentences to communicate to others their analysis and interpretation of the data.
Through a study of probability and statistics, students develop a real appreciation of data-analysis methods as powerful means
for decision making.
Updated July 2009
Grade 5 Curriculum Map—page 63
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
SOL Reporting Category
Probability and Statistics
Concept
Probability
PWC Grade Level Objective 5.17
Virginia SOL 5.17
5.17A The student will solve problems
involving the probability of a single event by
using tree diagrams or by constructing a
sample space representing all possible results.
5.17B The student will predict the probability
of the outcome of a simple experiment,
representing it with fractions or decimals
from 0 to 1, and test this prediction.
5.17C The student will create a problem
statement involving probability based on
information from a given problem situation.
Students will not be required to solve the
problem created.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

How is the probability of an event determined and described?

How are experimental and theoretical probability related?

How can we organize information in a sample space to show all of the possible combinations
(outcomes)?

What types of real-world situations involve probability?
Understanding the Objective (Teacher Notes)
Probability – the mathematics of chance – is an important topic in mathematics because the
probability of particular events happening - or not happening - can be important to us in the real
world. Probability helps people analyze games, lotteries, sports, traffic patterns, weather, insurance,
business and many other aspects of life. Why might a TV weather forecast predict a 60% chance of
rain? The forecast was calculated by meteorologists who looked at all other days in their historical
database that had the same weather characteristics (temperature, pressure, humidity, etc.) and
determined that on 60% of similar days in the past, it rained. Scientists and engineers use probability
to model events that they cannot (or would not want to) actually create such as the likelihood of a
plane crash after a wing design change, accidents resulting from a proposed traffic pattern, or a
building collapse in high winds.
Young children have difficulty distinguishing between certain events and likely events (and similarly
between unlikely and impossible events). Elementary students should have opportunities to describe
the degree of likelihood of an event occurring in informal terms, e.g., impossible, unlikely, as likely
as unlikely or equally likely, likely, and certain. Activities should include real-life examples as well
as probability experiments.
Mathematically, probability expresses the chance that something will occur. The “somethings” that
might occur are referred to as events.
For any event such as flipping a coin, the equally likely things that can happen are called outcomes;
e.g., there are two equally likely outcomes when flipping a coin: the coin can land heads up, or the
coin can land tails up.
A sample space represents all possible outcomes of an experiment. The sample space may be
organized in a list, chart, or tree diagram.
All students should:

Understand and apply basic concepts of
probability to make predictions of
outcomes of simple experiments.

Understand that a sample space represents
all possible outcomes of an experiment.

Understand that the measure of the
likelihood of an event can be represented
by a number from 0 to 1.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation
to:
 Construct a sample space using a tree
diagram to identify all possible outcomes of
a single event.
 Construct a sample space using a list or
chart to represent all possible outcomes of a
single event.
 Determine the probability of a single event
where the total number of possible
outcomes is 12 or less.
 Determine the outcome of an event that is
impossible to occur (0) or certain to occur
(1), when the number of possible outcomes
is 12 or less.
 Create a problem statement involving
probability based on information from a
given problem situation. Students will not
be expected to solve the problem.
Tree diagrams are drawn to show all of the possible combinations (outcomes) in a sample space.
The counting principle tells how to find the number of outcomes when there is more than one way
to put things together. For example, how many different outfit combinations can you make from 2
shirts (red and blue) and 3 pants (black, white, khaki)? The sample space displayed in a tree diagram
would show that there are 2  3 = 6 (counting principle) outfit combinations: red-black; red-white;
red-khaki; blue-black; blue-white; blue-khaki).
continued
Updated July 2009
Grade 5 Curriculum Map—page 64
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Probability and Statistics
The tree diagram below shows all combinations for 2 shirts (red and blue) and 3 pants (black, white,
and khaki):
Shirts
red
Pants
Possible Outcomes
black
red shirt with black pants
white
red shirt with white pants
khaki
red shirt with khaki pants
Year: Revised 2009
Essential Understanding
5.17 continued
Concept
Probability
PWC Grade Level Objective 5.17
Virginia SOL 5.17
5.17A The student will solve problems
involving the probability of a single event by
using tree diagrams or by constructing a
sample space representing all possible results.
5.17B The student will predict the probability
of the outcome of a simple experiment,
representing it with fractions or decimals
from 0 to 1, and test this prediction.
5.17C The student will create a problem
statement involving probability based on
information from a given problem situation.
Students will not be required to solve the
problem created.
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
Subject:
All students should:

Understand and apply basic concepts of
probability to make predictions of
outcomes of simple experiments.

Understand that a sample space represents
all possible outcomes of an experiment.

Understand that the measure of the
likelihood of an event can be represented
by a number from 0 to 1.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation
black
blue shirt with black pants
to:
blue
white
blue shirt with white pants
 Construct a sample space using a tree
diagram to identify all possible outcomes of
khaki
blue shirt with khaki pants
a single event.
 Construct a sample space using a list or
chart to represent all possible outcomes of a
The probability of an event occurring is the ratio of that specific outcome (“favorable” outcome) to
single event.
the total number of possible outcomes. If all the outcomes of an event are equally likely to occur,
 Determine the probability of a single event
the probability of the event =
where the total number of possible
outcomes is 12 or less.
number of favorable outcomes
 Determine the outcome of an event that is
total number of possible outcomes
impossible to occur (0) or certain to occur
(1), when the number of possible outcomes
The probability of an event occurring can be represented along a continuum from impossible to
is 12 or less.
certain. Numerically, the probability of an event occurring is represented by a ratio between 0 and 1.  Create a problem statement involving
An event that is impossible has a probability of 0; e.g., the probability that the month of April has 31
probability based on information from a
days. An event that is certain has a probability of 1; e.g., the probability that the sun will rise
given problem situation. Students will not
tomorrow morning. If two outcomes are equally probable, each outcome has a probability of ½ (or
be expected to solve the problem.
0.5 or 50%). Most events have a probability between zero and one.
0%
25%
50%
75%
100%
0
0.25
0.5
0.75
1
0
¼
½
¾
1
|____________|____________|____________|____________|
impossible ↔ less likely ↔ equally likely ↔ more likely ↔ certain
continued
Updated July 2009
Grade 5 Curriculum Map—page 65
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
SOL Reporting Category
Probability and Statistics
Concept
Probability
PWC Grade Level Objective 5.17
Virginia SOL 5.17
5.17A The student will solve problems
involving the probability of a single event by
using tree diagrams or by constructing a
sample space representing all possible results.
5.17B The student will predict the probability
of the outcome of a simple experiment,
representing it with fractions or decimals
from 0 to 1, and test this prediction.
5.17C The student will create a problem
statement involving probability based on
information from a given problem situation.
Students will not be required to solve the
problem created.
Updated July 2009
5.17 continued
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Essential Understanding
All students should:

Understand and apply basic concepts of
probability to make predictions of
outcomes of simple experiments.

Understand that a sample space represents
all possible outcomes of an experiment.

Understand that the measure of the
likelihood of an event can be represented
by a number from 0 to 1.
Probability expresses the chance that an event will happen over the long term. When a probability
experiment has very few trials, the results can be misleading. The more times an experiment is
repeated, the closer the experimental probability comes to the theoretical probability, e.g., a coin
lands heads up ½ of the time. It is challenging for students (and some adults) to understand that a
mathematical probability is a prediction of the outcomes over a long period and not the prediction of
the outcome for any single instance. Another common misconception (especially among gamblers) is
the belief that chance has “memory”, for example, believing that if you haven’t rolled doubles for a
long time, you are likely to roll a double very soon. Students tend to have difficulty with the concept
of randomness. Our English language uses the term “luck” to represent both random good fortune (as
in “the luck of the draw”) and the non-scientific predetermination of superstitious luck.
The student will use problem solving,
mathematical communication, mathematical
To better understand the likelihood of events, students need to investigate probability first-hand. A
reasoning, connections, and representation
spirit of investigation and experimentation should permeate probability instruction, where students
to:
are actively engaged in exploration and data collection and have opportunities to experience the
 Construct a sample space using a tree
difference between theoretical and experimental probability.
diagram to identify all possible outcomes of
a single event.
Students should be able to write a problem statement for a situation involving probability. For
 Construct a sample space using a list or
example, given a spinner with eight equal-size sections, three of which are red, three green, and two
chart to represent all possible outcomes of a
yellow, students should understand that the spinner is equally likely to land on any one of the
single event.
sections. They could write a problem statement such as, “What is the probability that the spinner will  Determine the probability of a single event
land on green?”
where the total number of possible
outcomes is 12 or less.
 Determine the outcome of an event that is
impossible to occur (0) or certain to occur
(1), when the number of possible outcomes
is 12 or less.
 Create a problem statement involving
probability based on information from a
given problem situation. Students will not
be expected to solve the problem.
Grade 5 Curriculum Map—page 66
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
Subject:
Resources

Concept
Probability







PWC Grade Level Objective 5.17
Virginia SOL 5.17


SOL Reporting Category
Probability and Statistics
School: _____________________________







Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 9, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Data Analysis and Probability in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004)- Between Never and
Always, Scott Foresman
The Super Source books –Color Tiles and Pattern Blocks
Virginia Department of Education Website – Probability & Statistics:
http://www.pen.k12.va.us/VDOE/Instruction/Elem_M/geo_elem.html
Math: A Way of Thinking, Robert Baratta-Lorton
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Unique U,” Math + Science: A Solution

"See How They Roll,” Pieces and Patterns

"Pascal Wins the World Series,” What's Next? Volume 1

"Sum Will Sum Won't,” What's Next? Volume 1

"What's Happening?” What's Next? Volume 2

"Flip for It,” What's Next? Volume 2
Updated July 2009
Grade 5 Curriculum Map—page 67
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
SOL Reporting Category
Probability and Statistics
Concept
Statistics
School: _____________________________
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Essential Questions
Understanding the Objective
Essential Understanding
Essential Questions

What relationships are represented by tables, line plots, stem-and-leaf plots, bar graphs, and line
graphs?

How can we determine which data display is appropriate for a given set of data?

How is a stem-and-leaf plot created and interpreted?

How do the selections of the sample, method of data collection, and way in which data are
displayed influence conclusions about the data?
All students should:

Understand that bar graphs compare
categorical data; stem-and-leaf plots list data
in a meaningful array; and line graphs show
changes over time.

Understand how to propose and justify
conclusions and predictions that are based on
displays of data.
Understanding the Objective (Teacher Notes)
Data is the collective term for pieces of information. (Datum refers to a single piece of data from a
data set.) Data are collected in a context and for some purpose. Data analysis helps describe data,
recognize patterns or trends, and make predictions.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Collect data using observations (e.g.,
weather), measurement (e.g., shoe size),
Statistical investigations should be active, with students formulating questions about something in
surveys (e.g., favorite television show), or
their environment and finding quantitative ways to answer the questions. Investigations involving
experiments (e.g., plant growth).
real-world data should occur frequently, and data can be collected though brief class surveys or

Organize the data into a chart or table.
through more extended projects that can take many days. Data analysis should include opportunities

Construct bar graphs, labeling one axis with
to describe the data, recognize patterns or trends, and make predictions. The emphasis in all work
Virginia SOL 5.18
equal whole number or decimal increments
with data and statistics should be on the analysis and the communication of the analysis, rather than
The student will, given a problem situation,
and the other axis with attributes of the topic
on a single correct answer.
(categorical data: skiing, basketball, ice
collect, organize, and display a set of
hockey, skating, sledding as the categories of
numerical data in a variety of forms, using bar
By carrying out complete investigations – formulating questions, predicting answers to questions
their favorite winter sports). Bar graphs will
graphs, stem-and-leaf plots, and line graphs,
under investigation, collecting and representing data, analyzing and drawing conclusions, and
have no more than six categories.
to draw conclusions and make predictions.
evaluating whether the data answer the questions – students gain an understanding of data analysis as 
Display data in line graphs, bar graphs, or
a tool.
stem-and-leaf plots.

Construct line graphs, labeling the vertical
Graphical displays of data convey information visually, in compact form. Not all graphs are
axis with equal whole number, decimal or
appropriate for a given data set. Through experiences displaying and analyzing data in a variety of
fractional increments and the horizontal axis
graphical representations, students learn to select an appropriate representation.
with continuous data often related to time,
e.g., hours, days, months, years, age. Line
graphs will have no more than six identified
Categorical data summarize responses with respect to a given category; categorical data are often,
points along a continuum for continuous data
though not always, words. Questions generating categorical data often start with “Which” or “What”:
e.g., decades: 1950, 1960,1970,1980, 1990,
In which months were the greatest number of classmates born? What are the favorite flavors of ice
2000.
cream? What kinds of pets do our classmates have? What is your favorite number? The only method

Construct a stem-and-leaf plot to organize
for comparing categorical data is to describe the number of data points in each category. Because
and display data, where the stem is listed in
frequency counts can be made for the values of each category, students often confuse numerical and
ascending order and the leaves are in
categorical data. Categorical data can be organized in tallies and tables and are often represented in
ascending order, with or without commas
pictographs, circle graphs (pie charts), and bar graphs.
between leaves.

Title the given graph, or identify the title.
Bar graphs should be used to compare counts of different categories (categorical data). One bar is

Interpret the data to compare the answer to
used for each category with the length of the bar representing the count for that category. There is
the prediction.
space before, between, and after the bars. The width of the bars should be approximately equal to the 
Write a few sentences to describe the
width of the spaces before, between, and after the bars.
interpretation of the data.
continued
PWC Grade Level Objective 5.18
5.18A The student will collect, organize, and
display a set of numerical data in a variety of
forms given a problem situation.
5.18B The student will use tables, bar graphs,
stem-and-leaf plots, and line graphs to draw
conclusions and make predictions.
Updated July 2009
Grade 5 Curriculum Map—page 68
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Probability and Statistics
Concept
Statistics
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
Subject:
5.18 continued
The axis displaying the scale representing the count for the categories should extend one increment
above the greatest recorded piece of data. Fifth-grade students should collect data that is recorded in
increments of multiples of whole numbers, decimals, and fractions. Each axis should be labeled and
the graph should have a title. Using grid paper ensures more accurate graphs. Double bar graphs are
often used to compare two sets of data. In double bar graphs, the two bars for each category are
colored (or hatched) differently and positioned without space in between the pair. A key is used to
identify the data sets.
Year: Revised 2009
Essential Understanding
All students should:

Understand that bar graphs compare
categorical data; stem-and-leaf plots list data
in a meaningful array; and line graphs show
changes over time.

Understand how to propose and justify
conclusions and predictions that are based on
displays of data.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Collect data using observations (e.g.,
weather), measurement (e.g., shoe size),
surveys (e.g., favorite television show), or
Numerical data are values that are quantities that can be ordered, such as counts, measurements and
experiments (e.g., plant growth).
ratings. Questions generating numerical data often ask “How much” or “How many”; for example:

Organize the data into a chart or table.
How many children are in each family? How tall are the basketball players on the team? How many 
Construct bar graphs, labeling one axis with
Virginia SOL 5.18
minutes do students watch TV each night? On a scale of 1 (low) to 5 (high), how would you rate your
equal whole number or decimal increments
The student will, given a problem situation,
love of pizza? Because numerical data involve a range of values, data spread or concentration can be
and the other axis with attributes of the topic
(categorical data: skiing, basketball, ice
collect, organize, and display a set of
described and statistics can be calculated.
hockey, skating, sledding as the categories of
numerical data in a variety of forms, using bar
their favorite winter sports). Bar graphs will
graphs, stem-and-leaf plots, and line graphs,
Numerical data can be discrete or continuous. Discrete data are data for which only certain values
have no more than six categories.
to draw conclusions and make predictions.
are possible. For instance, the number of children in a family or the number of letters in a name
Display data in line graphs, bar graphs, or
generates discrete data because each data point must be a whole number. Continuous data occur on a 
stem-and-leaf plots.
continuous scale. For any two specified values of continuous data, there are always values in

Construct line graphs, labeling the vertical
between. For instance, the time it takes to walk to school or the height of basketball players are
axis with equal whole number, decimal or
continuous data. Line plots, stem-and-leaf plots, histograms, box-and-whisker plots, and line graphs
fractional increments and the horizontal axis
are useful representations of numerical data.
with continuous data often related to time,
e.g., hours, days, months, years, age. Line
Line graphs are used to represent a specific type of continuous data which records how one variable
graphs will have no more than six identified
changes in relationship to another variable over time. By looking at a single-line graph, you can
points along a continuum for continuous data
determine whether the variable is increasing, decreasing or staying the same over time.
e.g., decades: 1950, 1960,1970,1980, 1990,
2000.
The values along the horizontal axis represent continuous data on a given variable, usually some

Construct a stem-and-leaf plot to organize
measure of time (e.g., time in years, months, or days). The vertical axis represents the variable being
and display data, where the stem is listed in
measured over time (the dependent variable). The values along the vertical axis are the scale and
ascending order and the leaves are in
represent the frequency with which those values occur in the data set. The values should represent
ascending order, with or without commas
equal increments of multiples of whole numbers, fractions, or decimals depending upon the data
between leaves.

Title the given graph, or identify the title.
being collected. The scale should extend one increment above the greatest recorded piece of data.

Interpret the data to compare the answer to
Each axis of a line graph should be labeled, and the graph should have a title.
the prediction.
continued

Write a few sentences to describe the
interpretation of the data.
PWC Grade Level Objective 5.18
5.18A The student will collect, organize, and
display a set of numerical data in a variety of
forms given a problem situation.
5.18B The student will use tables, bar graphs,
stem-and-leaf plots, and line graphs to draw
conclusions and make predictions.
Updated July 2009
Students should write statements representing their analysis and interpretation of the characteristics
of the data in the bar graph, e.g., similarities and differences, least and greatest, the categories, total
number of responses.
Grade 5 Curriculum Map—page 69
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Probability and Statistics
Concept
Statistics
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
Subject:
5.18 continued
Multiple line graphs enable the display and comparison of two or more quantities that are increasing
or decreasing over time.
Students should generate statements representing an analysis and interpretation of the characteristics
of the data in the graph; e.g., similarities and differences, mode, least and greatest, and trends.
Year: Revised 2009
Essential Understanding
All students should:

Understand that bar graphs compare
categorical data; stem-and-leaf plots list
data in a meaningful array; and line graphs
show changes over time.

Understand how to propose and justify
conclusions and predictions that are based
on displays of data.
Line plots (introduced to students in Grade 3) are used to show the frequency with which each value
of data appears and provide a visual representation of the shape of the data.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:
Stem-and-leaf plots also allow the listing of the exact value of each piece of data in a meaningful

Collect data using observations (e.g.,
array, but they do so in a more compact fashion which groups data within specific sub-ranges. Stem
weather), measurement (e.g., shoe size),
and leaf plots provide a quick way to see the shape of the data set – the range of the data, where
surveys (e.g., favorite television show), or
values are spread or concentrated, the minimum and maximum values, and outliers (atypical data
experiments (e.g., plant growth).
points). As with line plots, the statistics – range, median, and mode – can be “seen” in the stem-and- 
Organize the data into a chart or table.
leaf plot, and mean, median, mode, and range can be calculated from the data display.

Construct bar graphs, labeling one axis with
Virginia SOL 5.18
equal whole number or decimal increments
From this data of students’ ages: 13, 10, 14, 12,
From this data of precipitation in one decade: 212,
The student will, given a problem situation,
and the other axis with attributes of the
14, 9, 23, 13, 13, 21
233, 239, 240, 248, 236, 236, 230, 235, 240
topic (categorical data: skiing, basketball,
collect, organize, and display a set of
ice hockey, skating, sledding as the
numerical data in a variety of forms, using bar
Ages of Students Taking Piano Lessons
Number of Days with Precipitation
categories of their favorite winter sports).
graphs, stem-and-leaf plots, and line graphs,
Stem Leaves
Stem
Leaves
Bar graphs will have no more than six
to draw conclusions and make predictions.
0
9
21
2
categories.
1
0233344
22

Display data in line graphs, bar graphs, or
23
035669
2
13
stem-and-leaf plots.
24
008

Construct line graphs, labeling the vertical
axis with equal whole number, decimal or
Stem and leaf plots are best used to display small data sets; e.g., data covering a range of 25 numbers.
fractional increments and the horizontal
To create a stem-and-leaf plot, organize the data from least to greatest. Each value should be
axis with continuous data often related to
separated into a stem and a leaf, e.g., two-digit numbers are separated into stems (tens) and leaves
time, e.g., hours, days, months, years, age.
(ones). The stems are listed vertically from least to greatest with a line to their right. The leaves are
Line graphs will have no more than six
listed horizontally, also from least to greatest, and can be separated by spaces or commas. Every
identified points along a continuum for
value is recorded, regardless of the number of repeats. The plot should be titled, and a key is often
continuous data e.g., decades: 1950,
included to explain how to read the plot. Double stem and leaf plots (back-to-back stem and leaf
1960,1970,1980, 1990, 2000.
plots) can be used to compare two sets of data.

Construct a stem-and-leaf plot to organize
and display data, where the stem is listed in
From each data display, students should be able to discuss what information it does and does not
ascending order and the leaves are in
provide about the topic or question and what conclusions can and cannot be drawn. In addition to
ascending order, with or without commas
creating and interpreting their own data representations, students should analyze and critique the
between leaves.
interpretations made by others. One way to accomplish this is to select from among given choices an 
Title the given graph, or identify the title.
appropriate analysis of the data presented in a bar graph, line graph, or stem-and-leaf plot. For

Interpret the data to compare the answer to
example, given a line graph showing the number of in-line skaters (in millions) in the U.S. over the
the prediction.
time period 1980-2000 in five-year intervals, students could select the correct answer response that

Write a few sentences to describe the
relates to the graph, such as, the greatest increase in number of in-line skaters occurred between
interpretation of the data.
1990-1995.
PWC Grade Level Objective 5.18
5.18A The student will collect, organize, and
display a set of numerical data in a variety of
forms given a problem situation.
5.18B The student will use tables, bar graphs,
stem-and-leaf plots, and line graphs to draw
conclusions and make predictions.
Updated July 2009
Grade 5 Curriculum Map—page 70
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
SOL Reporting Category
Probability and Statistics
Concept
Statistics
PWC Grade Level Objective 5.18
Virginia SOL 5.18
School: _____________________________
Subject:
Resources






















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Units 8 and 9 and TenMinute Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Data Analysis and Probability in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004)- Between Never and
Always, Data: Kids, Cats and Ads, Scott Foresman
Number Sense Grades 4-6 by McIntosh et al
The The Super Source books–Color Tiles and Pattern Blocks
Virginia Department of Education Website – Probability & Statistics
http://www.pen.k12.va.us/VDOE/Instruction/Elem_M/geo_elem.html
Exploring Probability by Newman T. Etal
About Teaching Mathematics, a K-8 Resource by Marilyn Burns
Tables, Charts, and Graphs by Milliken Publications
Used Numbers: Real Data in the Classroom by R. Corwin and S. Friel
Statistics: Middles, Means, and In-Betweens, Grades 5 and 6
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"Seeds From Fruit,” The Budding Botanist
Updated July 2009
Grade 5 Curriculum Map—page 71
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
SOL Reporting Category
Probability and Statistics
Concept
Measures of Central Tendency
PWC Grade Level Objective 5.19
Virginia SOL 5.19
The student will find the mean, median,
mode, and range of a set of data.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

How do statistics—mean, median, mode, and range—provide numeric pictures of the shape of
data?

How are mean, median, and mode similar? How are they different? What are the
advantages/disadvantages of each for describing a data set?

How are the mean, median, mode, and range of a set of data computed? How can they be
determined from various data displays: Tables? Graphs? Stem-and-leaf plots?
All students should:
Understand how to determine the mean,
median, mode, and range of a set of data.

Understand that the mean is the numerical
average of a data set; the median is the
number in the middle of a set of data; the
mode is the piece of data that occurs most
often; and the range is the spread of a set of
data.

Understanding the Objective (Teacher Notes)
The branch of mathematics concerned with collecting, recording, representing, interpreting, and
analyzing large amounts of data is known as statistics. Statistics are used to describe numerical data. The student will use problem solving,
A statistic is a numerical value, calculated from a data set, which characterizes or describes that
mathematical communication, mathematical
data set in a specific way. All statistics should be interpreted in the context of the shape of the data. reasoning, connections, and representation to:

Calculate the mean of a group of numbers
Mean, median, mode, and range are four of the various statistics that can be use to analyze and
representing data from a given context.
describe numerical data (although categorical data can also have a mode). The mean, median, and

Determine the median of a group of
mode are all types of averages or measures of central tendency. An average is a single number that
numbers representing data from a given
is descriptive of what is “typical” in a larger collection of numbers. In everyday use, most people
context.
associate the term “average” with the arithmetic mean, but it is important to understand that median

Determine the mode of a group of numbers
and mode are also averages.
representing data from a given context.

Determine the range of a group of numbers
representing data from a given context.
The mean or arithmetic mean is the numerical average of the data set found by adding all the values
in the set and dividing by the number of values. The mean can be thought of as a statistic describing
a “fair share.” To understand the concept of the mean, students should experience the mean as a
“leveling” of the data or as the “balance point” of a set of numbers. The mean can be thought about
as a point on the number line where the data on either side of the point are balanced. As an
indicator of center, the mean may be skewed by outliers in the data set.
The median is the piece of data that is in the middle of the set of data arranged in order; i.e., the
median divides the pieces of data in two equal parts. If the data set contains an even number of
numbers, the median is the number halfway between the two central values. The median in not
necessarily the number in the middle of the range. In their analyses, statisticians frequently “slice”
or segment the data in order to look more closely at parts and see patterns; the median is the first
slice. The median is a stable measure of center and is not as easily affected by outliers as is the
mean. Thus, for large data sets (e.g., census data), the median often provides a more reliable
average (or measure of center) than the mean.
continued
Updated July 2009
Grade 5 Curriculum Map—page 72
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Probability and Statistics
Concept
Measures of Central Tendency
PWC Grade Level Objective 5.19
Virginia SOL 5.19
The student will find the mean, median,
mode, and range of a set of data.
Updated July 2009
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
Subject:
Year: Revised 2009
Essential Understanding
5.19 continued
All students should:
Understand how to determine the mean,
The mode is the piece of data that occurs most frequently. There may be one, more than one, or no
median, mode, and range of a set of data.
mode in a data set. The mode is deceptively easy to identify; however, its value lies in considering it 
Understand that the mean is the numerical
in relation to the entire data set. For example, is the mode part of a representative “clump” of data or
average of a data set; the median is the
is it simply the number that occurs most frequently in a set of data more evenly spread out?
number in the middle of a set of data; the
mode is the piece of data that occurs most
The range is a measure of variation. The range describes the spread of a set of data and is
often; and the range is the spread of a set of
determined by subtracting the smallest number (minimum) in the data from the largest number
data.
(maximum) in the data. The range should always be considered in the context of the entire data set.
For example, do the minimum or maximum values represent outliers? (An outlier is an unusual
The student will use problem solving,
value – usually one that has a much higher or lower value than others in a data set.) Students should mathematical communication, mathematical
analyze whether/why outliers may represent errors or special circumstances. They should consider
reasoning, connections, and representation to:
how outliers may affect other statistics such as the mean.

Calculate the mean of a group of numbers
representing data from a given context.
Students need to learn more than how to identify the mean, median, mode, and range in a data set

Determine the median of a group of
and from graphic representations of data sets. They need to build an understanding of what the
numbers representing data from a given
numbers tell them about the data, and they need to interpret those values in the context of the shape
context.
of the data and other characteristics of the data including the real-life context from which it was

Determine the mode of a group of numbers
collected.
representing data from a given context.

Determine the range of a group of numbers
representing data from a given context.

Grade 5 Curriculum Map—page 73
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Probability and Statistics
SOL Reporting Category
Probability and Statistics
Concept
Measures of Central Tendency
PWC Grade Level Objective 5.19
Virginia SOL 5.19
School: _____________________________
Subject:
Resources

















Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 9 and Ten-Minute
Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Data Analysis and Probability in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Investigations in Number, Data, and Space (2004)- Data: Kids, Cats and Ads
Scott Foresman
Tables, Charts, and Graphs by Milliken Publications
Used Numbers: Real Data in the Classroom by R. Corwin and S. Friel
Statistics: Middles, Means, and In-Betweens, Grades 5 and 6
About Teaching Mathematics: A K – 8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Good Questions for Math Teaching: Why Ask Them and What to Ask, K - 6
by Peter Sullivan and Pat Lilburn
Classroom Discussion: Using Math Talk to Help Students Learn, Grades 1 - 6
by Suzanne H. Chapin, Catherine O’Conner, and Nancy Canavan Anderson
Math for All: Differentiating Instruction, Grades 3 – 5 by Linda Dacey and
Jayne Bamford Lynch
Supporting English Language Learners in Math Class, Grades 3 – 5 by Rusty
Bresser, Kathy Melanese, and Christine Sphar
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"The Penny Sort and Nickel Dates,” Math + Science: A Solution

"Can You Planet?” Out of This World
Updated July 2009
Grade 5 Curriculum Map—page 74
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Subject:
Fifth Grade Mathematics
Year: Revised 2009
Patterns, Functions, and Algebra
Students entering grades 4 and 5 have had opportunities to identify patterns within the context of the school curriculum and in
their daily lives, and they can make predictions about them. They have had opportunities to use informal language to describe
the changes within a pattern and to compare two patterns. Students have also begun to work with the concept of a variable by
describing mathematical relationships in open number sentences, and they have begun to solve simple equations with one
unknown.
The focus of instruction is to help students develop a solid use of patterning as a problem-solving tool. At this level, patterns are
represented and modeled in a variety of ways, including numeric, geometric, graphic, and algebraic formats. Students develop
strategies for organizing information more easily to understand various types of patterns and functional relationships. They
analyze the structure of patterns by exploring and describing patterns that involve change, and they begin to generalize these
patterns. By analyzing mathematical situations and models, students begin to represent these, using symbols and variables to
write “rules” for patterns, to describe relationships and algebraic properties, and to represent unknown quantities.
Updated July 2009
Grade 5 Curriculum Map—page 75
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Patterns, Functions, and Algebra
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Patterns, Functions, and Algebra
Subject:
Essential Understanding
Essential Questions

How can a pattern be identified, described, and represented?

What is the relationship between patterns and functions (“rules”)?

How can pattern identification be used to solve problems?
Understanding the Objective (Teacher Notes)
Concept
Patterns
PWC Grade Level Objective 5.20
Virginia SOL 5.20
The student will analyze the structure of
numerical and geometric patterns (how they
change or grow) and express the relationship
using words, tables, graphs, or mathematical
sentences. Concrete materials and calculators
will be used.
Year: Revised 2009
A logical pattern is a predictable sequence of elements (sounds, colors, letters, words, numbers,
objects, geometric shapes, etc.). Logical patterns occur regularly in mathematics. A pattern is
recognized by detecting its underlying structure or rule. In patterns represented with geometric
shapes, students must often recognize transformations of a figure, particularly, rotation or reflection.
Rotation (turn) is the action of turning a figure around a point or a vertex; and reflection (flip) is the
result of a figure flipped over a line. Identifying and extending patterns is an important process in
algebraic thinking. There are an infinite number of patterns.
The same pattern can be found in many different forms. Reproduction of a given pattern in a
different representation, using symbols and objects, lays the foundation for writing the relationship
symbolically or algebraically.
All students should:

Understand that patterns and functions can
be represented in many ways and described
using words, tables, graphs, and symbols.

Understand the structure of a pattern and
how it grows or changes.

Understand that mathematical relationships
exist in patterns.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Describe numerical and geometric patterns
formed by using concrete materials and
calculators.

Express the relationship found in numerical
and geometric patterns, using words, tables,
graphs or a mathematical sentence.
Patterns can be described verbally, numerically, and generalized symbolically using variables.
Organizing data into tables and recognizing patterns in the data is one strategy for problem solving.
Tables of values should be analyzed for a pattern to determine what element comes next.
The simplest types of patterns are repeating patterns. In each case, students need to identify the
basic unit (or core) of the pattern and repeat it. Opportunities to create, recognize, describe, and
extend repeating patterns are essential to the primary school experience.
Non-repeating patterns are more difficult for students to understand than repeating patterns as they
must not only determine what comes next, but they must also begin the process of generalization.
Growing patterns are non-repeating patterns that involve a progression from step to step. (These
are technically termed sequences, and the steps are also referred to as terms or stages.)
Sample growing patterns (where one variable changes in the basic unit):
ABAABAAABAAAAB
123112311123
▄ ▲▲▄ ▄ ▲▲▄ ▄ ▄ ▲▲
Students may confuse the term “growing” with “increasing”. They should also encounter growing
patterns that “diminish” such as 220, 200, 180, 160, …
continued
Updated July 2009
Grade 5 Curriculum Map—page 76
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
SOL Reporting Category
Patterns, Functions, and Algebra
Concept
Patterns
PWC Grade Level Objective 5.20
Virginia SOL 5.20
The student will analyze the structure of
numerical and geometric patterns (how they
change or grow) and express the relationship
using words, tables, graphs, or mathematical
sentences. Concrete materials and calculators
will be used.
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Patterns, Functions, and Algebra
Subject:
Essential Understanding
5.20 continued
Growing patterns (sequences) built with objects or geometric shapes also have a numeric
component (the number of shapes or objects in each step). Students need experiences recognizing,
creating and extending growing patterns in both numeric and geometric formats. Representing
growing patterns in tables provides an informal entry point to the concept of function, as students
extend the pattern and describe the “rule”.
Pattern

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



?
Year: Revised 2009
All students should:

Understand that patterns and functions can
be represented in many ways and described
using words, tables, graphs, and symbols.

Understand the structure of a pattern and
how it grows or changes.

Understand that mathematical relationships
exist in patterns.
Rule: ?
Step
Number
1
5
2
7
3
9
4
?
The numerical relationships represented in the function table can also be graphed on coordinate
graphs (using ordered pairs representing the step (x) and the number (y), and the pattern or rule can
be deduced from the shape of the graph. Tables and graphs are important tools in solving problems
and making predictions in situations involving change.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Describe numerical and geometric patterns
formed by using concrete materials and
calculators.

Express the relationship found in numerical
and geometric patterns, using words, tables,
graphs or a mathematical sentence.
In some growing patterns called arithmetic sequences (or linear sequences), students must
determine the difference, called the “common difference,” between each succeeding number in
order to determine what is added to each previous number to obtain the next number. Sample
arithmetic patterns include 6, 9, 12, 15, 18, … ; and 5, 7, 9, 11, 13, ... (The sequence 1, 2, 4, 7, 11,
16, … is a numeric pattern, but is not an arithmetic sequence because there is no common
difference. The difference increases by 1 each time.)
In some other growing patterns called geometric sequences (or exponential sequences), students
must determine what each number is multiplied by to obtain the next number in the geometric
sequence. This multiplier is called the “common ratio.” Sample geometric number patterns include:
2, 4, 8, 16, 32, ...; 1, 5, 25, 125, 625, ...; and 80, 20, 5, 1.25, .... In describing a sequence, the use of
the term “geometric” refers to the way the numeric pattern changes, not to the format of the pattern
(e.g., a pattern made with geometric shapes).
Other patterns are formed by specific sequences. The sequence of triangular numbers is the sum of
consecutive positive integers: 1, 3, 6, 10, … (1 = 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10, …).
The sequence of perfect squares can be represented pictorially (with arrays) and numerically: 1, 4,
9, 16, 25, … (1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16, 5 x 5 = 25, …)
It is not advisable to name these specific types of sequences with elementary students. Both
arithmetic and geometric sequences can be represented with shapes as well as numbers.
Updated July 2009
Grade 5 Curriculum Map—page 77
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Patterns, Functions, and Algebra
SOL Reporting Category
Patterns, Functions, and Algebra
Concept
Patterns
PWC Grade Level Objective 5.20
Virginia SOL 5.20
School: _____________________________
Subject:
Resources














Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 8, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Algebra in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Number Sense Grades 4-6 by McIntosh et al
The Super Source books – Snap Cubes
About Teaching Mathematics, a K-8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Discovery in Mathematics by Robert B. Davis
Family Math by Jean Kerr Stenmark, Virginia Thompson, and Ruth Cossey
Lessons for Algebraic Thinking, Grades 3- 5 by Maryann Wickett, Katharine
Kharas, and Marilyn Burns
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html
Updated July 2009
Grade 5 Curriculum Map—page 78
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Patterns, Functions, and Algebra
SOL Reporting Category
Patterns, Functions, and Algebra
Concept
Variables
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Essential Questions

What is a variable? …a variable expression? …an open sentence?

How is the “equal sign” in an open sentence (equation) like the fulcrum of a balance scale?

How can we use variable expressions and equations (open sentences) to represent problem
situations?

How can we write problem situations to represent given variable expressions and equations
(open sentences)?
Understanding the Objective (Teacher Notes)
PWC Grade Level Objective 5.21
5.21A The student will use a variable to
represent a given verbal quantitative
expression involving one operation. Describe
how the variable is used in the given verbal
quantitative expression.
Algebra is a tool that can make communicating mathematical ideas and solving mathematical
problems easier. If mathematics is viewed as a language, then algebra can be viewed as the
shorthand of mathematics.
Year: Revised 2009
Essential Understanding
All students should:

Understand that a variable is a symbol that
can stand for any member of a set of
numbers.

Understand that a variable expression is a
variable or combination of variables,
numbers, and symbols that represents a
mathematical relationship.

Understand that verbal quantitative
expressions can be translated to variable
expressions.

Understand that an open sentence is a
mathematical sentence with a variable.
A variable is a symbol that can stand for any one of a set of numbers or other objects. A variable is
a quantity that can change. Any letter can stand for a variable.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:
5.21B The student will write an open sentence An expression is like a phrase because it has no equal sign. An expression can be a number (e.g., 7),

Describe the concept of a variable
using a variable to represent a given
a variable (e.g., x), or show an arithmetic operation involving a number and a variable (e.g., 7 + x).
(presented as boxes, letters, or the other
mathematical relationship. Describe how the
To represent a verbal quantitative expression involving one operation, write an expression that
symbols) as a representation of an unknown
variable is used to represent a given
describes what is going on. Use numbers when they are known; use variables when the numbers are
quantity.
mathematical relationship.
unknown. For example, where b is a variable standing for the number of cookies in one box of

Use a variable expression to represent a
cookies ,“a full box of cookies and 4 extra” may be represented as b + 4, “three full boxes of
given verbal expression, involving one
Virginia SOL 5.21
cookies” as 3b, and “a full box of cookies shared among 4 people” as b/4. Expressions with
operation e.g., 5 more than a number can be
The student will
variables are known as variable expressions.
represented by x + 5.
a) Investigate and describe the concept of

Write an open sentence with addition,
variable;
An equation is a mathematical sentence that describes a relationship between two mathematical
subtraction, multiplication, or division using
b) Use a variable expression to represent a
expressions or ideas. An open sentence is a mathematical sentence (equation) with a variable. It
a variable to represent a missing number.
given verbal quantitative expression involving contains an equals (=) sign. For example, where b is a variable standing for the number of cookies
one operation; and
in one box of cookies, “one full box of cookies and 4 extra are 24 cookies” may be represented as b
c) Write an open sentence to represent a given + 4 = 24, and “three full boxes of cookies are 60 cookies” as 3b = 60.
mathematical relationship using a variable.
The concept of equality is not as straightforward or simple as it might appear. Students often view
the equal sign as a signal to carry out the specified operation(s). They must understand, instead, that
the equal sign signifies that the quantity on each side of the equation is the same. To understand
equality in a manner that will support algebraic thinking, students must interpret expressions such as
6 + 2, 42 x 8, and a + b as single quantities.
At this level, discuss how the “x” symbol used to represent multiplication, can often be confused
with a variable x. Students can minimize this confusion using parentheses [e.g., 4(x) = 20], or a
small raised dot to represent multiplication.
Updated July 2009
Grade 5 Curriculum Map—page 79
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Patterns, Functions, and Algebra
SOL Reporting Category
Patterns, Functions, and Algebra
Concept
Variables
PWC Grade Level Objective 5.21
Virginia SOL 5.21
School: _____________________________
Subject:
Resources













Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 8, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Algebra in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Ideas From the Arithmetic Teacher, Elementary by George Immerzeel and
Melvin Thomas
Activities from the Mathematics Teacher, NCTM
Think About It by Marcy Cook
About Teaching Mathematics, a K-8 Resource by Marilyn Burns
Lessons for Algebraic Thinking, Grades 3- 5 by Maryann Wickett, Katharine
Kharas, and Marilyn Burns
Virginia Department of Education Website – Patterns, Functions & Algebra
http://k12.va.us/VDOE/Instruction/Elem_M/mathtrain.html
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"The Big Banana Peel,” Math + Science: A Solution
Updated July 2009
Grade 5 Curriculum Map—page 80
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Patterns, Functions, and Algebra
SOL Reporting Category
Patterns, Functions, and Algebra
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Essential Questions

What is an open sentence?

How are problem situations represented by open sentences?

How can open sentences represent problem situations?
Year: Revised 2009
Essential Understanding
All students should:

Understand that an open sentence is a
mathematical sentence with a variable.

Understand that problem situations can be
expressed as open sentences.
Understanding the Objective (Teacher Notes)
Concept
Problem Solving with Variables
PWC Grade Level Objective 5.22
The student will create and solve problem
situations based on a given open sentence
using a single variable.
Virginia SOL 5.22
The student will create a problem situation
based on a given open sentence using a single
variable.
Updated July 2009
Please refer to the Teacher Notes for Objective 5.21.
An open sentence is a mathematical sentence (equation) containing a variable. It also contains an
equals (=) sign, e.g., b + 3 = 23. It represents the solution to a word problem; e.g.: How many
cookies are in a box if the box plus three more equals 23? In this equation, b stands for the number
of cookies in one box. The variable (number of cookies in one box) could be represented by any
letter; thus, the open sentence for this problem could also be represented as n + 3 = 23 or x + 3 = 23,
etc.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Create and write a word problem to match a
given open sentence with a single variable
and one operation.
By using story problems and numerical sentences, students begin to explore forming equations and
representing quantities using variables.
Grade 5 Curriculum Map—page 81
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective







SOL Reporting Category
Patterns, Functions, and Algebra
Concept
Problem Solving with Variables
PWC Grade Level Objective
Subject:
Resources

PWC Curriculum Strand
Patterns, Functions, and Algebra
School: _____________________________
5.22






Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Units 7 and 8 and TenMinute Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Algebra in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Ideas From the Arithmetic Teacher, Elementary by George Immerzeel and
Melvin Thomas
Activities from the Mathematics Teacher, NCTM
Think About It, Marcy Cook
About Teaching Mathematics, a K-8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Lessons for Algebraic Thinking, Grades 3- 5 by Maryann Wickett, Katharine
Kharas, and Marilyn Burns
Virginia Department of Education Website – Patterns, Functions & Algebra:
http://www..k12.va.us/VDOE/Instruction/Elem_M/mathtrain.html
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"The Big Banana Peel,” Math + Science: A Solution
Updated July 2009
Grade 5 Curriculum Map—page 82
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective
PWC Curriculum Strand
Patterns, Functions, and Algebra
SOL Reporting Category
Patterns, Functions, and Algebra
Concept
Coordinate graphing
PWC Grade Level Objective 5.23
The student will identify the ordered pair for a
point on a graph, and locate the point for an
ordered pair in the first quadrant of a
coordinate plane or the x- or y-axis.
School: _____________________________
Subject:
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Year: Revised 2009
Essential Understanding
Essential Questions

What is an ordered pair?

How can ordered pairs be used to read and describe the locations of points and objects on a
coordinate plane?

How are graphs of ordered pairs used to represent relationships?
Understanding the Objective (Teacher Notes)
Coordinate systems are used in the disciplines of mathematics, science, social studies, and
geography to determine locations, distances, and positions. In mathematics a coordinate system is a
reference system for locating and graphing points. In two dimensions, a coordinate system usually
consists of a horizontal axis and a vertical axis, which intersect at the origin. Each point in the plane
is located by its horizontal distance and vertical distance from the origin. These distances, or
coordinates, form an ordered pair of numbers.
A coordinate plane is a way to precisely locate points in a plane. To draw a coordinate plane, draw
a horizontal number line, called the x-axis, and a vertical number line called the y-axis, which
intersect in a right angle at 0 (the origin) on each number line.
Any point on a coordinate plane may be named with two numbers. These two numbers are the
Cartesian coordinates (ordered pair). The pair is always named in order; first x, (the location along
the x-axis), then y, (the location along the y-axis), and is represented (x, y).
All students should:
 Understand how to use two numbers to
name points on a coordinate grid.
 Understand that a pair of numbers on a
coordinate plane corresponds to only one
point on the grid.
 Understand how to write and graph ordered
pairs from a table to related values.
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Identify the ordered pair for a point in the
first quadrants of a coordinate plane.

Locate points on a coordinate grid, given the
order pair. Write correct notation for
ordered pairs.

Locate points on a coordinate grid, given a
table of values.
The x-axis and the y-axis divide the coordinate plane into four sections called quadrants. The
coordinates are plotted in relationship to the x- and y-axes. When both numbers in an ordered pair
(x, y) are positive, the ordered pair is in the first quadrant. Students’ first experiences with
coordinate graphing utilize the first quadrant of the coordinate plane.
When plotting a point, start at (0, 0), the origin, and let the x-coordinate tell you how far to move
horizontally. Then, let the y-coordinate tell you how far to move vertically. If the x-coordinate is
positive, you move to the right on the horizontal axis. If the y-coordinate is positive, you move up
on the vertical axis.
continued
Updated July 2009
Grade 5 Curriculum Map—page 83
MAPPING FOR INSTRUCTION
Teacher: _____________________________
School: _____________________________
Fifth Grade Mathematics
Essential Questions
Understanding the Objective
Curriculum Objective
PWC Curriculum Strand
Patterns, Functions, and Algebra
Subject:
5.23 continued
Concept
Coordinate graphing
A table of values is an organized way to list related ordered pairs. The related ordered pairs are
formed according to a “rule” or a relationship. As students explore patterns that involve a
progression from step to step, the students not only extend the pattern but also look for a
generalization or an algebraic relationship that will tell them what the pattern will be at any point
along the way (rule for the pattern). This exploration provides the foundation for determining a
functional relationship.
PWC Grade Level Objective 5.23
The student will identify the ordered pair for a
point on a graph, and locate the point for an
ordered pair in the first quadrant of a
coordinate plane or the x- or y-axis.
Event
1
2
3
4
SOL Reporting Category
Patterns, Functions, and Algebra
Year: Revised 2009
Essential Understanding
All students should:
 Understand how to use two numbers to
name points on a coordinate grid.
 Understand that a pair of numbers on a
coordinate plane corresponds to only one
point on the grid.
 Write and graph ordered pairs from a table
of values.
# of counters
2
4
6
8
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representation to:

Locate points on a coordinate grid, given a
table of values.

Identify the ordered pair for a point in the
Students can extend their work with patterns and functions to explore how a graph, a description,
first quadrants of a coordinate plane.
and an expression/equation can represent a functional relationship.

Locate points on a coordinate grid, given the
order pair. Write correct notation for
ordered pairs.
The coordinate view of shape offers another way to understand certain properties of shapes, changes
in position (transformations), and how they appear or change in size (visualization). Transformation
of shapes (translations – slides, reflections – flips, and rotations – turns) can be described in terms
of coordinates, allowing for digital manipulation of shapes. Computer animation applies a
combination of coordinate geometry and algebra.
Coordinate geometry also provides a way to determine relationships among lengths, areas, and
volumes. In algebra, coordinate graphing provides an analytic view of the concept of slope and of
perpendicular and parallel relationships.
Updated July 2009
Grade 5 Curriculum Map—page 84
MAPPING FOR INSTRUCTION
Teacher: _____________________________
Curriculum Objective







SOL Reporting Category
Patterns, Functions, and Algebra
Concept
Problem Solving with Variables
PWC Grade Level Objective
Subject:
Resources

PWC Curriculum Strand
Patterns, Functions, and Algebra
School: _____________________________
5.23






Fifth Grade Mathematics
Year: Revised 2009
Teacher Notes
Investigations in Number, Data, and Space (2008)—Unit 8 and Ten-Minute
Math, Pearson
PWC Mathematics Web Site: http://www.pwcsmath.com
Curriculum and Evaluation Standards, NCTM, 1989
Principals and Standards for School Mathematics, NCTM, 2000
Addenda Series, Grade 5, NCTM
Navigating through Measurement in Grades 3-5, NCTM
Elementary and Middle School Mathematics by John Van deWalle
Ideas From the Arithmetic Teacher, Elementary by George Immerzeel and
Melvin Thomas
Activities from the Mathematics Teacher, NCTM
Think About It, Marcy Cook
About Teaching Mathematics, a K-8 Resource by Marilyn Burns
Math Matters: Understanding the Math You Teach, Grades K – 8 by Suzanne
H. Chapin
Lessons for Algebraic Thinking, Grades 3- 5 by Maryann Wickett, Katharine
Kharas, and Marilyn Burns
Virginia Department of Education Website – Patterns, Functions & Algebra:
http://www..k12.va.us/VDOE/Instruction/Elem_M/mathtrain.html
AIMS Activities:

Virginia SOL Correlations to AIMS Activities:
http://www.aimsedu.org/statedocs/virginia/virginia.html

"The Big Banana Peel,” Math + Science: A Solution
Updated July 2009
Grade 5 Curriculum Map—page 85