Download Additional Problems 4-6 - Math Professional Development

Additional
Problems
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 73
(2008 DRAFT)
74 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems Using Literature
The Greedy Triangle by Marilyn Burns7
Annotation: The greedy triangle loves being busy! Each day it
supports bridges, makes music, holds up roofs, and much,
much more. For most shapes this lifestyle would be fulfilling.
But not for the greedy triangle. Always eager to try new
things, the triangle decides it is time to add another line and
angle to its shape. Unfortunately this new shape isn't a
perfect fit either. Thus begins a succession of new shapes
until the greedy triangle no longer knows which side is up!8
Just One More Side
Curriculum Correlation
Grade 5
Shape and Space (Measurement), Specific Outcome 1:
 Identify 90º angles.
[ME, V]
Shape and Space (3-D Objects and 2-D Shapes), Specific Outcome 6:
 6. Describe and provide examples of edges and faces of 3-D objects, and sides of
2-D shapes that are:
 parallel
 intersecting
 perpendicular
 vertical
 horizontal.
[C, CN, R, T, V]
[ICT: C6–2.2, P5–2.3]
Grade 6
Shape and Space (Measurement), Specific Outcome 1:
 Demonstrate an understanding of angles by:
 identifying examples of angles in the environment
 classifying angles according to their measure
 estimating the measure of angles, using 45°, 90° and 180° as reference angles
 determining angle measures in degrees
 drawing and labelling angles when the measure is specified.
[C, CN, ME, V]
7. M. Burns, The Greedy Triangle (New York, NY: Scholastic Press, 1995).
8. TheTeachersCorner.net, “Math Worksheets and Lesson Plans–The Greedy Triangle,” 2008,
http://www.theteacherscorner.net/lesson-plans/math/geometry/triangle.htm (Accessed January 22, 2008).
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 75
(2008 DRAFT)
Materials
 toothpicks
Activity
Challenge pairs of students to explore creating and recording shapes (pentagons,
hexagons, heptagons, octagons), using 5, 6, 7 or 8 toothpicks. Students can build their
shapes on grid paper, to assist with alignment of the toothpicks, to create both regular
and irregular polygons. Students can record their shapes by gluing the toothpicks
directly on the grid paper. Have students cut around each shape.
Grade 5
Challenge students to sort their shapes, based on the sides of their shapes that are
perpendicular, parallel, include both perpendicular and parallel sides, or have neither
perpendicular or parallel sides.
Grade 6
Challenge students to sort their shapes, based on the interior angles of their shapes
(acute, obtuse, right).
Extension: Students try to sort their shapes by types of angles and by number of types
of angles; e.g.,
Acute
Right
Obtuse
Three or less
Four
Five
Six
Seven or more
Have students try to see any patterns in how the shapes are sorted. Ask them what
generalizations can be made about shapes that have:
 acute angles
 more than one obtuse angle
 right angles
 any relationship between angles and shapes.
76 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Creating and Sorting Quadrilaterals
This activity is also found in the Additional Problems section. It is included here with
adaptations to reflect the context of The Greedy Triangle.
Curriculum Correlation
Grade 5
Shape and Space (3-D Objects and 2-D Shapes), Specific Outcome 7:
 Identify and sort quadrilaterals, including:
 rectangles
 squares
 trapezoids
 parallelograms
 rhombuses
according to their attributes.
[C, R, V]
Grade 6
Shape and Space (3-D Objects and 2-D Shapes), Specific Outcome 5:
 Describe and compare the sides and angles of regular and irregular polygons.
[C, PS, R, V]
Materials
 geoboards and elastic bands
 dot paper cut into 3 x 3 dot sections
Activity
Have students mark off their geoboard to recognize a 3 dot x 3 dot area. Read The
Greedy Triangle, stopping after reading about the favourite thing the triangle liked to do
as a triangle. Have students create a triangle on their geoboard. In a group of three,
students share their triangle with the other group members. Ask them, “What is the
same about your triangles? What is different?”
Continue reading and stop after the shapeshifter changes the triangle to a quadrilateral.
Ask students, “What if the shapeshifter did not change the triangle into a regular
quadrilateral (square)? What other quadrilateral shapes could the triangle have been
changed into?”
Working with the small group, each student experiments with different ways to make
four-sided figures (quadrilaterals) on their geoboards. Each time a student in the group
creates a quadrilateral that has not been created before, he or she records the
quadrilateral on one of the group’s dot papers. After making each shape, the student
checks the group’s dot paper to see if that shape has already been created. Students
rotate the geoboard or the dot paper recording to compare for congruency.
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 77
(2008 DRAFT)
There are 16 possible quadrilaterals in a 3 x 3 area; these are included below for
teacher reference.
Share with students, “As the shapeshifter keeps track of all the shapes he has created,
consider the four-sided shapes you have found as photographs of the shapeshifter’s
work. If the shapeshifter was to sort his photographs as he hangs them on the wall, how
might he sort them?”
Ask students to use the photographs and examine the properties of the different shapes
they found. Challenge groups to sort their photographs into groups to hang on the wall.
Have the class discuss the different categories they used for sorting; e.g.,
 shapes with equal side lengths
 shape name (trapezoid, square, rectangle)
 convex or concave shapes
 shapes with acute angles.
Extension—The Shapeshifter’s Secret Shape
Share with students, “The shapeshifter and his friends like to play games when they are
together. One of their favorite games is called The Shapeshifter’s Secret Shape.”
Working in groups, one student creates one of the 16 figures on their geoboard and
conceals it from all group members. The group members ask questions, one at a time,
to determine the secret shape that has been created. The group can use the
photographs (16 figures on dot paper) and eliminate the shapes that are excluded each
time a question is answered; e.g., a group member might ask, “Does the shape have a
right angle?” If the answer is yes, all other figures can be removed from the group.
Questions continue until students determine the exact shape that has been created.
The student then reveals the geoboard to the group. A new group member would create
the next secret shape.
Extension—Grade 6
Have students look closely at the shapes that have been made. Have them explain how
the sides and angles of the shapes created could be the same or different.
78 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Just One More Triangle
Curriculum Correlation
Grade 4
Shape and Space (Transformations), Specific Outcome 5:
 Demonstrate an understanding of congruency, concretely and pictorially.
[CN, R, V]
Materials
 small green triangle pattern blocks
 triangle grid paper
Activity
Ask students, “Suppose the greedy triangle did not ask for one more side and one more
angle but asked for one more triangle? What are the different shapes that could be
created with 2, 3, 4, 5 or 6 triangles?”
Provide students with 6 green triangle pattern blocks and triangle grid paper. Challenge
them to see how many different shapes can be made with two triangles. Have students
share the different shapes that have been created.
Challenge students, working as partners or in small groups, to see how many different
shapes can be made with a certain number of triangles. Shapes that are flips
(reflections) or turns (rotations) of other shapes are not considered different. Ask
students to choose a number of triangles to work with (3, 4, 5 or 6) or assign groups a
number to work with. By having students working with different numbers, differentiation
for ability levels and increased student ownership of the problem can be addressed.
Good group discussions will emerge as students decide if shapes really are the same or
different from one another and if all shapes have been found.
Shape Up!
Curriculum Correlation
Grade 3
Shape and Space (3-D Objects and 2-D Shapes), Specific Outcome 6:
 Describe 3-D objects according to the shape of the faces and the number of edges
and vertices.
[C, CN, PS, R, V]
Grade 4
Shape and Space (3-D Objects and 2-D Shapes), Specific Outcome 4:
 Describe and construct right rectangular and right triangular prisms.
[C, CN, R, V]
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 79
(2008 DRAFT)
Materials
 straws or toothpicks
 modelling clay
Activity
Share with students, “The greedy triangle is a 2-D shape. If the greedy triangle was a
stamp of one face of a 3-D object, what object could it be?”
Using toothpicks or straws and modelling clay, have students experiment with 3-D
skeletons that would have one face as the greedy triangle. Have them try to create
skeletons that would reflect the greedy triangle at different stages of the story; e.g., as a
triangle, quadrilateral, pentagon, hexagon.
Note: 3-D shapes do not need to be regular polyhedra; e.g., one face could be a
quadrilateral (as the greedy triangle) but other faces could be different shapes
(triangular, pentagonal) that would create some very interesting skeletons.
Symmetry Search
Curriculum Correlation
Grade 4
Shape and Space (Transformations), Specific Outcome 3:
 Demonstrate an understanding of line symmetry by:
 identifying symmetrical 2-D shapes
 creating symmetrical 2-D shapes
 drawing one or more lines of symmetry in a 2-D shape.
[C, CN, V]
Materials
 mira or mirrors
 photocopies of greedy triangle shapes; e.g., triangle, quadrilateral, pentagon,
hexagon, dodecagon
Activity
Share with students, “If the shape of the greedy triangle could be folded to create 2
identical halves, where are the possible fold lines for each shape?”
Using the shapes into which the greedy triangle transformed, have students find lines of
symmetry in each of the shapes. Have students use the mira or mirror to check their
initial predictions before drawing the lines of symmetry on each shape.
Ask students, “How many different ways could we fold the shape to create 2 identical
halves? Are there any shapes that could not be folded to create 2 identical halves?”
80 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Shape Party
Curriculum Correlation
Grade 4
Shape and Space (Measurement), Specific Outcome 3:
 Demonstrate an understanding of area of regular and irregular 2-D shapes by:
 recognizing that area is measured in square units
 selecting and justifying referents for the units cm2 or m2
 estimating area, using referents for cm2 or m2
 determining and recording area (cm2 or m2)
 constructing different rectangles for a given area (cm2 or m2) in order to
demonstrate that many different rectangles may have the same area.
[C, CN, ME, PS, R, V]
Materials
 pattern blocks
 triangle grid paper
Activity
Share with students, “All of the greedy triangle’s friends had a party for him when he
was back in shape! All of his friends were at the party, including the hexagon, the
trapezoid triplets, the blue parallelogram and the triangle twins. The shapeshifter came
with his camera to get lots of pictures of the greedy triangle and his friends. The
shapeshifter was curious about how he could get a photograph of the greedy triangle
with his friends in different arrangements.”
Have students select the pattern blocks that represent the greedy triangle and his
friends who were at the party. Have students arrange all the partygoers, with each
shape touching another with one edge, so the shapeshifter can take his photographs.
Share with students, “What is the area and perimeter of the shape created by the
greedy triangle and his friends? If the friends created another shape for another
photograph, would the shape have the same area and perimeter?”
Have the students try at least 3 shapes for the shapeshifter’s photographs. Ask them to
determine the area and perimeter of each shape and organize the shapes from least to
greatest area or perimeter.
Share with students, “What can you tell others about the shapes you created? Is there
more than one shape with the same perimeter? Is the area different in any of the new
shapes? Why or why not?”
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 81
(2008 DRAFT)
Mingle at the Shape Party
Curriculum Correlation
Grade 5
Shape and Space (Transformations), Specific Outcome 8:
 Identify and describe a single transformation, including a translation, rotation and
reflection of 2-D shapes.
[C, T, V]
[ICT: C6–2.1]
Grade 6
Shape and Space (Transformations), Specific Outcome 6:
 Perform a combination of translations, rotations and/or reflections on a single 2-D
shape, with and without technology, and draw and describe the image.
[C, CN, PS, T, V]
Materials
 triangle grid paper
 pattern blocks
Share with students, “All of the greedy triangle’s friends had a party for him when he
was back in shape! All of his friends were at the party, including the hexagon, the
trapezoid triplets, the blue parallelogram and the triangle twins. The shapeshifter came
with his camera to get lots of pictures of the greedy triangle and his friends. During the
course of the evening, the shapeshifter’s photographs showed each partygoer speaking
to somebody different at the party.”
Have students consider that a piece of triangle grid paper represents the party room.
Ask them to record each guest that was at the party, except for one of their choice,
around the edges of the party room. Have students draw a map of the route that their
chosen partygoer took to talk to each friend. Students should:
 draw their chosen partygoer in position
 explain the movement of the partygoer, using the terms translation, rotation and
reflection, as he or she moves from one friend to another.
82 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Alexander, Who Used to Be Rich Last Sunday by Judith Viorst
Annotation: Last Sunday, Alexander’s grandparents gave him a
dollar – and he was rich. There were so many things that he
could do with all of that money! He could buy as much gum as
he wanted, or even a walkie-talkie, if he saved enough. But
somehow the money began to disappear.9
Which Would Be Greater? 10
Curriculum Correlation
Grade 4
Number, Specific Outcome 1:
 Represent and describe whole numbers to 10 000, pictorially and symbolically.
[C, CN, V]
Number, Specific Outcome 2:
 Compare and order numbers to 10 000.
[C, CN, V]
Number, Specific Outcome 6:
 Demonstrate an understanding of multiplication (2- or 3-digit by 1-digit) to solve
problems by:
 using personal strategies for multiplication with and without concrete materials
 using arrays to represent multiplication
 connecting concrete representations to symbolic representations
 estimating products
 applying the distributive property.
[C, CN, ME, PS, R, V]
Number, Specific Outcome 7:
 Demonstrate an understanding of division (1-digit divisor and up to 2-digit dividend)
to solve problems by:
 using personal strategies for dividing with and without concrete materials
 estimating quotients
 relating division to multiplication.
[C, CN, ME, PS, R, V]
9. Reproduced with permission (pending) from J. Viorst, Alexander, Who Used to Be Rich Last Sunday (New York,
NY: Aladdin Paperbacks, 1978).
10. Adapted with permission (pending) from M. Ellis, C. Yeh and S. Stump, “Problem Solvers: Height in Coins and
Solutions to the Making Brownies Problem,“ Teaching Children Mathematics 14, 3 (2007), pp. 170–175.
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 83
(2008 DRAFT)
Number, Specific Outcome 11:
 Demonstrate an understanding of addition and subtraction of decimals (limited to
hundredths) by:
 using personal strategies to determine sums and differences
 estimating sums and differences
 using mental mathematics strategies
to solve problems.
[C, ME, PS, R, V]
Grade 5
Number, Specific Outcome 1:
 Represent and describe whole numbers to 1 000 000.
[C, CN, V, T]
[ICT: C6–2.2]
Number, Specific Outcome 2:
 Use estimation strategies, including:
 front-end rounding
 compensation
 compatible numbers
in problem-solving contexts.
[C, CN, ME, PS, R, V]
Number, Specific Outcome 4:
 Apply mental mathematics strategies for multiplication, such as:
 annexing then adding zero
 halving and doubling
 using the distributive property.
[C, CN, ME, R, V]
Number, Specific Outcome 5:
 Demonstrate, with and without concrete materials, an understanding of multiplication
(2-digit by 2-digit) to solve problems.
[C, CN, PS, V]
Number, Specific Outcome 6;
 Demonstrate, with and without concrete materials, an understanding of division
(3-digit by 1-digit), and interpret remainders to solve problems.
[C, CN, ME, PS, R, V]
Grade 6
Number, Specific Outcome 2:
 Solve problems involving whole numbers and decimal numbers.
[ME, PS, T]
[ICT: C6–2.4]
Number, Specific Outcome 8:
84 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
 Demonstrate an understanding of multiplication and division of decimals (1-digit
whole number multipliers and 1-digit natural number divisors).
[C, CN, ME, PS, R, V]
Materials
 coins or coin stamps
 string
 calculators
 bathroom scale
Activity
Alexander’s grandmother and grandfather heard about Alexander’s adventures with his
one dollar. On their next visit, they have promised to give him a choice of one of the
following, rather than just giving him a dollar.
Share with students, “If you were Alexander, which would you rather have?”
 The value of quarters arranged on a flat surface in a line to represent your height.
 The value of nickels stacked vertically (on top of one another) to represent your
height.
 Your weight in dimes (1 dime/kg).
 The area of your footprint in loonies.
Handfuls of Coins
Curriculum Correlation
Grade 4
Number, Specific Outcome 1:
 Represent and describe whole numbers to 10 000, pictorially and symbolically.
[C, CN, V]
Number, Specific Outcome 11:
 Demonstrate an understanding of addition and subtraction of decimals (limited to
hundredths) by:
 using personal strategies to determine sums and differences
 estimating sums and differences
 using mental mathematics strategies
to solve problems.
[C, ME, PS, R, V]
Grade 5
Number, Specific Outcome 1:
 Represent and describe whole numbers to 1 000 000.
[C, CN, V, T]
[ICT: C6–2.2]
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 85
(2008 DRAFT)
Number, Specific Outcome 4:
 Apply mental mathematics strategies for multiplication, such as:
 annexing then adding zero
 halving and doubling
 using the distributive property.
[C, CN, ME, R, V]
Number, Specific Outcome 2:
 Demonstrate, with and without concrete materials, an understanding of multiplication
(2-digit by 2-digit) to solve problems.
[C, CN, PS, V]
Grade 6
Number, Specific Outcome 2:
 Solve problems involving whole numbers and decimal numbers.
[ME, PS, T]
[ICT: C6–2.4]
Number, Specific Outcome 5:
 Demonstrate an understanding of ratio, concretely, pictorially and symbolically.
[C, CN, PS, R, V]
Number, Specific Outcome 8:
 Demonstrate an understanding of multiplication and division of decimals (1-digit
whole number multipliers and 1-digit natural number divisors).
[C, CN, ME, PS, R, V]
Materials
 bucket of coins (pennies, nickels, dimes, quarters)
 calculators
Activity
Share with students, “Alexander’s brothers have put all of their coins into a big dish in
their bedroom. His brothers bet that he could not take a handful of coins that would total
more than $5. Would this be a good bet for Alexander to take with his brothers?”
Have students calculate, with a partner, how much money they can grab in one handful
of coins from the bucket.
Extension—Grade 6
Share with students, “Alexander’s brothers said that they would be surprised if
Alexander was able to grab coins that had a part/whole ratio of 4:10.”
Ask students what ratios of coins they think they have in their handfuls. Ask students if
any of their coins have the part/whole ratio of 4:10. Have them discuss how they might
identify the amount of each coin that is in their handfuls, using ratios.
86 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Pocket Change
Curriculum Correlation
Grade 4
Number, Specific Outcome 1:
 Represent and describe whole numbers to 10 000, pictorially and symbolically.
[C, CN, V]
Number, Specific Outcome 11:
 Demonstrate an understanding of addition and subtraction of decimals (limited to
hundredths) by:
 using personal strategies to determine sums and differences
 estimating sums and differences
 using mental mathematics strategies
to solve problems.
[C, ME, PS, R, V]
Materials
 coins, coin stamps or paper coins
 calculators
Activity
Suppose Alexander has 6 coins in his pocket. Ask students what different coin
combinations and values of money Alexander might have in his pocket.
Confusing Combinations
Curriculum Correlation
Grade 4
Number, Specific Outcome 1:
 Represent and describe whole numbers to 10 000, pictorially and symbolically.
[C, CN, V]
Number, Specific Outcome 2:
 Compare and order numbers to 10 000.
[C, CN, V]
Number, Specific Outcome 5:
 Describe and apply mental mathematics strategies, such as:
 skip counting from a known fact
 using doubling or halving
 using doubling or halving and adding or subtracting one more group
 using patterns in the 9s facts
 using repeated doubling to determine basic multiplication facts to 9 × 9 and
related division facts.
[C, CN, ME, R]
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 87
(2008 DRAFT)
Number, Specific Outcome 6:
 Represent and describe decimals (tenths and hundredths), concretely, pictorially
and symbolically.
[C, CN, R, V]
Number, Specific Outcome 11:
 Demonstrate an understanding of addition and subtraction of decimals (limited to
hundredths) by:
 using personal strategies to determine sums and differences
 estimating sums and differences
 using mental mathematics strategies
to solve problems.
[C, ME, PS, R, V]
Grade 5
Number, Specific Outcome 3:
 Apply mental mathematics strategies and number properties, such as:
 skip counting from a known fact
 using doubling or halving
 using patterns in the 9s facts
 using repeated doubling or halving
to determine, with fluency, answers for basic multiplication facts to 81 and related
division facts.
[C, CN, ME, R, V]
Number, Specific Outcome 5:
 Demonstrate, with and without concrete materials, an understanding of multiplication
(2-digit by 2-digit) to solve problems.
[C, CN, PS, V]
Number, Specific Outcome 8:
 Describe and represent decimals (tenths, hundredths, thousandths), concretely,
pictorially and symbolically.
[C, CN, R, V]
Number, Specific Outcome 10:
 Compare and order decimals (to thousandths) by using:
 benchmarks
 place value
 equivalent decimals.
[C, CN, R, V]
Number, Specific Outcome 11:
 Demonstrate an understanding of addition and subtraction of decimals (limited to
thousandths).
[C, CN, PS, R, V]
88 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Grade 6
Number, Specific Outcome 1:
 Demonstrate an understanding of place value, including numbers that are:
 greater than one million
 less than one thousandth.
[C, CN, R, T]
Number, Specific Outcome 2:
 Solve problems involving whole numbers and decimal numbers.
[ME, PS, T]
[ICT: C6–2.4]
Number, Specific Outcome 8:
 Demonstrate an understanding of multiplication and division of decimals (1-digit
whole number multipliers and 1-digit natural number divisors).
[C, CN, ME, PS, R, V]
Materials
 coins (real or play money)
 calculators
Activity
The story begins with Alexander saying he thinks it is unfair that his brother Anthony
has 2 dollars, 3 quarters, 1 dime, 7 nickels and 18 pennies. Ask students to discuss if
Alexander was confused about how many coins Anthony had. Ask them what difference
would it make if Anthony had 7 dollars, 18 quarters, 3 dimes, 1 nickel and 2 pennies.
Have students work with a partner to find out the different:
 arrangements of the numbers 1, 2, 3, 7 and 18 with the coins (dollars, quarters,
dimes, nickels and pennies)
 values the arrangements would represent.
Ask students to address the following questions.
 How might you record your information?
 How will you know if you found all the combinations?
 Which arrangement has the greatest value? The least?
 What is the difference between the greatest and the least value?
 What arrangement has a value that is in-between the greatest and least amounts?
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 89
(2008 DRAFT)
Pickpockets!
Curriculum Correlation
Grade 4
Statistics and Probability (Data Analysis), Specific Outcome 1:
 Demonstrate an understanding of many-to-one correspondence.
[C, R, T, V]
[ICT: C6–2.2, C6–2.3]
Statistics and Probability (Data Analysis), Specific Outcome 2:
 Construct and interpret pictographs and bar graphs involving many-to-one
correspondence to draw conclusions.
[C, PS, R, V]
Grade 5
Statistics and Probability (Data Analysis), Specific Outcome 2:
 Construct and interpret double bar graphs to draw conclusions.
[C, PS, R, T, V]
[ICT: C6–2.2, P5–2.3]
Grade 6
Statistics and Probability (Data Analysis), Specific Outcome 2:
 Select, justify and use appropriate methods of collecting data, including:
 questionnaires
 experiments
 databases
 electronic media.
[C, CN, PS, R, T]
[ICT: C4–2.2, C6–2.2, C7–2.1, P2–2.1, P2–2.2]
Statistics and Probability (Data Analysis), Specific Outcome 3:
 Graph collected data, and analyze the graph to solve problems.
[C, CN, PS, R, T]
[ICT: C6–2.5, C7–2.1, P2–2.1, P2–2.2]
Activity
Have students discuss how much money, in coins, they think adults carry in their
pockets or wallets. Have students work in small groups to survey adults at home, in the
school or at the grocery store.
Have students design a bar graph or histogram to show the range of values of coins
from the adults surveyed. Have them design two questions that can be answered by
others when looking at the data on their graphs.
90 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Grade 4 adaptations: When creating a bar graph or pictograph to represent the data,
ensure students use many-to-one correspondence.
Grade 5 adaptations: Have students create double bar graphs that compare
information between men and women or pockets and wallets.
Grade 6 adaptations: Have students display their recording methods from their data
collection and their graphs. Have them discuss and compare the various methods used
for collecting data.
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 91
(2008 DRAFT)
92 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Grade 4
Road Trips
Curriculum Correlation
Grade 4
Number, Specific Outcome 5:
 Describe and apply mental mathematics strategies, such as:
 skip counting from a known fact
 using doubling or halving
 using doubling or halving and adding or subtracting one more group
 using patterns in the 9s facts
 using repeated doubling
to determine basic multiplication facts to 9 × 9 and related division facts.
[C, CN, ME, R]
Number, Specific Outcome 6:
 Demonstrate an understanding of multiplication (2- or 3-digit by 1-digit) to solve
problems by:
 using personal strategies for multiplication with and without concrete materials
 using arrays to represent multiplication
 connecting concrete representations to symbolic representations
 estimating products
 applying the distributive property.
[C, CN, ME, PS, R, V]
Activity
In this problem, students will calculate distances that could be travelled and determine
possible destinations, based on mileage.
A hybrid vehicle is any vehicle that combines two or more sources of power; e.g., a
moped is a hybrid as it combines the power of a gasoline engine with the pedal power
of the rider. A hybrid car can run on a gas engine only, an electric motor only or both at
the same time—two sources of power. It gets better mileage than a car that only runs
on gasoline and it does not cause as much pollution.
According to Ford.ca, a Ford Escape Hybrid uses 5.7 L/100 km (litres per 100
kilometres) in the city and 6.7 L/100 km on the highway.
 How far could you drive in the city if you had 10 litres of gas? How far could you
drive on the highway if you had 10 litres of gas? Explain your thinking.
 If you took a road trip from Edmonton and used about 25 litres of gas, what
location(s) might you end at? Explain your thinking.
 How much gas would you need to drive from Edmonton to Vancouver? Explain your
thinking.
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 93
(2008 DRAFT)

Determine a destination you might like to drive to. Calculate how much gas you
would need to drive to your destination. How long might it take for you to get
there? Explain your thinking.
Patterns Galore!11
Curriculum Correlation
Grade 4
Patterns and Relations (Patterns), Specific Outcome 1:
 Identify and describe patterns found in tables and charts.
[C, CN, PS, V]
Patterns and Relations (Patterns), Specific Outcome 3:
 Represent, describe and extend patterns and relationships, using charts and tables,
to solve problems.
[C, CN, PS, R, V]
Patterns and Relations (Patterns), Specific Outcome 4:
 Identify and explain mathematical relationships, using charts and diagrams, to solve
problems.
[CN, PS, R, V]
In this problem, students will explore patterns in charts.
Activity
Take a look at the 0–99 chart shown in figure 1. No, don’t just glance at it—really look
at it. It is full of fascinating patterns. You already know some of them. For example,
every other column contains all even numbers. Also, if you move down one row, the
tens digit increases by one. You probably know a few other patterns. Your job is to find
the less obvious patterns. For example, look at diagonals; start from the middle and
work your way out; consider the digits of numbers as two separate numbers; or look at
smaller portions of the chart. Be creative!
Next comes the really exciting part of the problem. Try the same exercise using the
triangular 0–99 chart pictured in figure 2. Can you find similar patterns? Is the
reasoning the same? Is it easier to find patterns, or more difficult? You can find one very
nice pattern by looking down the left side of the triangle. Do you see it? Why does that
happen? Does the pattern tell you anything about the nature of those numbers?
11. Reproduced with permission (pending) from K. Jeon, J. Bishop and B. Britton, “Problem Solvers: Nine Jumping
Numbers and Solutions to the Patterns Galore!, Problem,” Teaching Children Mathematics 13, 6 (2007),
pp. 330–335 (see Teacher Articles).
94 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Finally, try to make up a 0–99 chart in another shape. Do you find more patterns in your
shape than you did in the triangle? How about the square? List as many of the patterns
as you can, and then try to explain them.
Variations: This problem can be modified for younger students by having them create
patterns instead of finding and explaining them. For example, have students find the
pattern that is created when they count by twos or by threes. After they understand what
is meant by a number pattern on the 0–99 chart, they may be able to find more patterns
without help from the teacher. The problem does not really need modifying for older
students. It essentially modifies itself by the level of sophistication of the patterns that
students find.
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 95
(2008 DRAFT)
96 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Grade 5
Risky Allowance12
Curriculum Correlation
Grade 5
Number, Specific Outcome 3:
 Apply mental mathematics strategies and number properties, such as:
 skip counting from a known fact
 using doubling or halving
 using patterns in the 9s facts
 using repeated doubling or halving
to determine, with fluency, answers for basic multiplication facts to 81 and related
division facts.
[C, CN, ME, R, V]
Number, Specific Outcome 4:
 Apply mental mathematics strategies for multiplication, such as:
 annexing then adding zero
 halving and doubling
 using the distributive property.
[C, CN, ME, R, V]
Grade 6
Number, Specific Outcome 2:
 Solve problems involving whole numbers and decimal numbers.
[ME, PS, T]
Statistics and Probability (Chance and Uncertainty), Specific Outcome 6:
 Demonstrate an understanding of probability by:
 identifying all possible outcomes of a probability experiment
 differentiating between experimental and theoretical probability
 determining the theoretical probability of outcomes in a probability experiment
 determining the experimental probability of outcomes in a probability experiment
 comparing experimental results with the theoretical probability for an experiment.
[C, ME, PS, T]
Activity
In this problem, students will experiment with different allowance payment structures
and justify which plan should be used.
12. Reproduced with permission (pending) from B. Britton and C. Tayeh, “Problem Solvers: Risky Allowance,”
Teaching Children Mathematics 11, 8 (2005), pp. 422–424 (see Teacher Articles).
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 97
(2008 DRAFT)
Matthew has a dilemma. Normally his weekly allowance is $2. His family has proposed,
however, that for the next 12 weeks he pick one of the following allowance plans.13

Plan A: Receive $2 per week.

Plan B: Receive 1 cent for the first week, 2 cents for the second week, 4 cents for
the third week, and so on, with the amount doubling each week for the 12 weeks.

Plan C: Use a spinner to determine the allowance for each week.

Plan D: Flip a coin each week to determine the allowance. Heads, you get no
allowance that week. Tails, you get $7.50.
1. What allowance plan should Matthew use for the next 12 weeks?
2. Would you use a different plan if it were for 16 weeks?
3. Is there a certain number of weeks for which Plan A would be best? Plan B? Plan C?
Plan D?
4. Are there other risky allowance plans that you would like to consider for your
allowance?
Creating and Sorting Quadrilaterals
Curriculum Correlation
Grade 5
Shape and Space (3-D Objects and 2-D Shapes), Specific Outcome 7:
 Identify and sort quadrilaterals, including:
 rectangles
 squares
 trapezoids
 parallelograms
 rhombuses
according to their attributes.
[C, R, V]
Materials
 geoboards and elastics
 dot paper cut into 3 x 3 dot sections
13. See solutions in B. Britton and C. Tayeh, “Problem Solvers: Counting Counts! and Solutions to the Risky
Allowance Problem,” Teaching Children Mathematics 12, 18 (2006), pp. 423–427 (see Teacher Articles).
98 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Activity
In this problem, students will explore properties of quadrilaterals. Students will be
involved in visualizing and testing for congruency.
Have students mark off their geoboard to recognize a 3 dot x 3 dot area for them to
work in. Working with a partner or in a small group, each student experiments with
different ways to make four-sided figures (quadrilaterals) on their geoboards. Each time
a student in the group creates a quadrilateral that has not been created before, he or
she records the quadrilateral on one of the group’s dot papers. After making each
shape, the student checks the group’s dot paper to see if that shape has already been
created. This may involve rotating the geoboard or the dot paper recording to compare
for congruency.
There are 16 possible quadrilaterals in a 3 x 3 area; these are included below for
teacher reference.
After groups have found all 16 quadrilaterals, ask students to examine the properties of
the different shapes they found. Challenge groups to sort their figures into groups. Have
the class discuss the different categories they used for sorting; e.g.,
 shapes with equal side lengths
 shape name (trapezoid, square, rectangle)
 convex or concave shapes
 shapes with acute angles.
Extension
Working in groups, one student creates one of the 16 figures on their geoboard and
conceals it from all group members. The group members ask questions, one at a time,
to determine the secret shape that has been created. The group can use the 16 figures
on dot paper and eliminate the shapes that are excluded each time a question is
answered; e.g., a group member might ask, “Does the shape have a right angle?” If the
answers is yes, all other figures can be removed from the group. Questions continue
until students determine the exact shape that has been created. The student then
reveals the geoboard to the group. A new group member would create the next secret
shape.
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 99
(2008 DRAFT)
Grade 6
Calories From Fat14
Curriculum Correlation
Grade 6
Number, Specific Outcome 2:
 Solve problems involving whole numbers and decimal numbers.
[ME, PS, T]
Number, Specific Outcome 6:
 Demonstrate an understanding of percent (limited to whole numbers), concretely,
pictorially and symbolically.
[C, CN, PS, R, V]
Activity
In this problem, students will categorize foods based on fat content and relate their
findings to diet recommendations.
The labels on canned and packaged food contain important information about the
nutritional content of a single serving. These labels indicate the total calories and fat
grams in one serving. Health Canada recommends less than 64g of fat per day.

Using information from a variety of food labels, sort the foods according to the
percent of daily value for fat: little fat (<10%); some fat (10–25%); fat (25–90%);
much fat (>90%). What do you notice about the data you gathered?

Keep track of the total grams of fat in your diet for one week. Find your mean fat
gram intake per day. How do your personal results compare to the Health
Canada recommendations? How do your personal results compare to your
classmates?
Up, Up, and Away!15
Curriculum Correlation
Grade 6
Number, Specific Outcome 2:
 Solve problems involving whole numbers and decimal numbers.
[ME, PS, T]
14. Adapted with permission (pending) from Linda Chick et al, “Math by the Month: A Healthy Start,” Teaching
Children Mathematics 14, 1 (2007), p. 33.
15. Reproduced with permission (pending) from Linda Chick et al, “Math by the Month: Inventors and Their
Inventions,” Teaching Children Mathematics 14, 2 (2007), pp. 96–98.
100 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Activity
In this problem, students will determine a rate of travel. Students will apply information
from a historic flight and compare that information with current information on today’s
aircraft statistics.
On December 17, 1903, the brothers Orville and Wilbur Wright stunned the world with
their first successful powered flight, which traveled 260 m in 59 seconds. At this rate,
how long would it have taken this airplane to travel 1 kilometre? To travel 10
kilometres? How far could it have traveled in 1 hour? In 12 hours? Compare this
information with information about today’s airplanes.
What’s in a Name?
Curriculum Correlation
Grade 6
Number, Specific Outcome 2:
 Solve problems involving whole numbers and decimal numbers.
[ME, PS, T]
Shape and Space (Measurement), Specific Outcome 1:
 Demonstrate an understanding of angles by:
 identifying examples of angles in the environment
 classifying angles according to their measure
 estimating the measure of angles, using 45°, 90° and 180° as reference angles
 determining angle measures in degrees
 drawing and labelling angles when the measure is specified.
[C, CN, ME, V]
Materials
 letter stencils (available at dollar stores or stationery shops)
Activity
In this problem, students will use letter stencils to write their names and determine the
number and types of angles in the letters in their name. For this activity, students need
to be aware of the terms acute, right and obtuse angles.
Write a word on the board, e.g., MATH, or point out a word on a bulletin board in the
classroom. Ask students to identify the different angles they see in each letter; i.e.,
MATH
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada
Additional Problems / 101
(2008 DRAFT)
M has 3 acute angles (identified by the circles)
A has 5 angles, 3 acute angles (circled) and 2 obtuse angles (jagged circles)
T has 2 right angles (squared)
H has 4 right angles (squared).
In groups, have students use a letter stencil to write out a given word; e.g., TEACHER.
As a group, students identify the number of acute, right and obtuse angles. Ask a group
to share their results and see if all groups agree; i.e.,
T has 2 right angles
E has 4 right angles
A has 5 angles—3 acute and 2 obtuse
C has 0 angles
H has 4 right angles
E has 4 right angles
R has 5 angles—3 right, 1 obtuse and 1 acute.
Introduce a numeric point value for each type of angle; e.g.,
 acute angles = 0.5 point
 right angles = 1 point
 obtuse angles = 1.5 points.
How many points would the word MATH have (12 points)? How many points does the
word TEACHER have (23.5 points)?
Have students stencil their first name on a piece of paper. Ensure that students lift the
stencil and complete the lines in each letter to have full letters. Have each student
calculate the number of angles and the value of their name. Compare results of angles
and point values with a partner. Which student in class do they think would have the
least/greatest amount of angles? Which student in class do they think would have the
least/greatest point value? (These are often tricky because it is not always the student
that has the longest or shortest name. Letters, e.g., s, c, j, do not have any angles or
point values.)
In partners, challenge students to find a word that has:
 less angles than either of their names
 a lesser point value than either of their names
 more angles than either of their names
 a greater point value than either of their names
 an amount of angles between the amount of angles in each of their names
 a point value between the point values of their names
 less angles than one partner’s name but a greater point value
 more angles than one partner’s name but a lesser point value
 a target point value; e.g., 15 points.
102 / Additional Problems
(2008 DRAFT)
Teaching through Problem Solving, 4–6
 Alberta Education, Alberta, Canada