Download Wake County Sample Math Unit #1

Daily/Weekly Instructional Guides
Content Area/Subject: Math
Grade: 7th Grade
SCoS Objective(s):
7.NS.1a Describe situations in which opposite
quantities combine to make zero.
7.NS.1d Apply properties of operations as strategies to
add and subtract rational numbers.
7.EE.1 Apply operations to add, subtract, factor, and
expand linear expressions with rational coefficients.
Day(s) of Instruction: Day 17
Date:
Allocated Time for Instruction: 45 to 90
minutes depending on school
Connections to EOG/EOC/Assessment:
Use the Hands-On Equations Learning
System to solve:
4x + (-x) = 15
A. x = 3
7.EE.2 Understand that rewriting an expression in
different, yet equivalent, forms in a problem can show
how the quantities in it are related.
B. x = 4
C. x = 5
7.EE.3 Solve multi-step real-life and mathematical
problems posed with positive and negative rational
numbers in any form (whole numbers, fractions, and
decimals). Apply properties of operations to calculate
with numbers in any form; convert between forms as
appropriate; and assess the reasonableness of answers
using mental computation and estimation strategies.
D. x = 15
The answer is C.
MP.1 Make sense of problems and persevere in solving
them.
MP.4 Model with mathematics.
MP.6 Attend to precision.
National Objective(s):
Learner Objective(s): As a result in learning, students
should be able to… use the Hands-On Equations
Learning System to introduce negative integers and the
addition and subtraction of integers. Students will also
learn another legal move to solve equations that involves
adding the same amount to each side of an equation to
help simplify sides.
Language Objective(s):
C&I/MS Math/Fall 2011
Topic(s):
Use the Hands-On Equations Learning System to:
 Introduce the green number cubes (negative
integers).
 Introduce a legal move that allows students to
solve equations by adding the same value
number cube to both sides of a balanced system.
Prerequisite skills/knowledge:
 Ability to substitute to determine whether a given
number makes an equation true. (6.EE.5)
 Understand that variables represent numbers
(6.EE.6)
 Ability to set-up and solve one-step equations
(6.EE.7)
 Ability to add, subtract, multiply, and divide
positive and negative integers.
 Understand the idea of opposites (7.NS.1b)
Real World importance of objective(s) – Essential Questions
Why do students need to know or be able to do this?


How do you solve equations that have negative numbers involved?
What are some real-world problems that can be solved using equations? Do any real-world
examples involve negative variables? Integers?
Definitions of Critical Vocabulary and Underlying Concepts
rational numbers
a number expressible in the form a/b or –a/b for some fraction a/b.
The rational numbers include the integers.
integers
a number expressible in the form a or –a for some whole number a.
constant
a number that does not change.
expression
a mathematical phrase that contains operations, numbers, and/or
variables.
evaluate
To find the value of a numerical or algebraic expression.
equivalent expressions
expressions having the same value.
equation
a mathematical sentence that shows that two expressions are
equivalent.
additive inverses
two numbers whose sum is 0 are additive inverses of one another.
additive identity property of
The property that states the sum of zero and any number is that
zero
number.
addition property of opposites
the property that states that the sum of a number and its opposite
equals zero.
subtraction property of equality the property that states that if you subtract the same number from
both sides of an equation, the new equation will have the same
solution.
distributive property
the property that states if you multiply a sum by a number, you will
C&I/MS Math/Fall 2011
get the same result if you multiply each addend by that number and
then add the products.
Introduction/Focus/Anticipatory Activities/Guiding questions/Connections to what students
already know through learning, culture, or experience:
 Ask students to figure out the answer to the following equation through guess and check:
o _____ + (-2) = 10
o What would you fill in the blank with to make a true statement? What integer rules did
you use?

Materials that each student will need:
o
o
o
o
o
Eight Blue Pawns
Eight White Pawns
Four red number cubes, numbered 0-5 or 5-10
Four white number cubes, numbered 0-5 or 5-10
A laminated balance sheet
Initial Instructional Strategies (research-based teaching strategies):
A. Direct Instruction – What the Teacher does:
Today’s lesson utilizes the Hands-On Equations Learning System.
 Things to know when using resources provided by Hands-On Equations:
- Seventh grade objectives do not cover solving equations with variables on both sides,
which is why many lessons in Level II and III will be skipped. It is critical in any of the
lessons that are utilized that you are selective in picking problems that do not contain
variables on both sides.
- Please always refer to the county-provided Hands-On Equations Learning System
resources that are located at your school for additional examples, comments, consistent
language, and teaching strategies. Examples provided in this lesson are original
examples and are not found in the Hands-On Equations resources.
- For today, students should not be required to write down any steps, other than their
checks. They will need a pencil to show how to solve pictorially.
Source: Henry Borenson, Ed.D. The Hands-On Equations Learning System. Pennsylvania: Berenson
and Associates, Inc., 2008. Print.
 All students will need a copy of the Hands-On Equations Student Packet (Same
Packet as day 15)
7ccss_HoE-Student
Packet_ed_d15-18.docx

All teachers will need a copy of the Hands-On-Equations Teacher Packet (Same
Packet as day 15)
C&I/MS Math/Fall 2011
7ccss_HoE-Teacher
Solutions to Student Packet_ed_d15-19.docx
Topic 1: Using Level III, Lesson 17, students will work with adding and subtracting positive and
negative whole numbers using the number cubes. Although they will have already completed the
integers unit and should have a solid foundation with these concepts, it is critical that they understand
how to use the number cubes to complete addition and subtraction. Students need to know that the
green number cubes represent negatives, so a 3 red-number cube (3) and a 3 green number cube (-3)
are additive inverses (negatives) and their sum will be 0.
 If necessary, use opposite number cubes to demonstrate that these are additive inverses.
Direct Instruction Examples:
 Example : (-4) + (-5) =
 To set-up, students should put a 4 green-number cube and a 5 green-number cube on the left
side. Walk around to ensure that students have set this up properly.
 Students already learned in previous lessons that they could combine red-number cubes into
the sum of the numbers in the same color number cube. They should extend this
information to this example and see that a 4 green-number cube and a 5 green-number cube
is the same thing as a 9 green-number cube.
 So (-4) + (-5) = -9

Example : (-4) + 6 =
 Students will probably be able to answer this easily due to the integers unit, but it is helpful
for them to see how to utilize the number cubes. Ask them to break the 6 red-number cube
into two number cubes, one being a 4 and the other a 2.
 Additive inverses (opposites) allows them to see that the 4 red-number cube and then 4
green-number cube will cancel.
 (-4) + 4 + 2. They can remove this pair of opposites.
 They will be left with 2 (2 red-number cube).

Example : (-4) – 7 =
 Again, students will probably be able to apply rules that they learned in the integers unit to
solve this, but encourage them to demonstrate the rules they know using the number cubes.
 To set-up, students should put a 4 green-number cube on their balance board. They aren’t
able to take a 7 red-number cube away because it does not exist yet, therefore, they should
hopefully begin discussing what they can do using rules that they’ve learned in Level II.
 Students should remember their ability to add a convenient zero pair in the set-up process
(not considered a LEGAL MOVE since it is completed in the set-up) to add both a positive
7 and negative 7 (zero pair, sum is 0) to the left hand side. They can then take away the 7
red-number cube.
o This will leave a 4-green number cube and a 7-green number cube on the balance.
So, (-4) – 7 = -11
C&I/MS Math/Fall 2011
Expression:
-6 – (-4)
Example 1:
Lesson 17
Example 2:
Lesson 17
Example 3:
Lesson 17
12 – (-3)
-5 – (-9)
Solution/Check:
-2
15
4
(add convenient zero pair of -4
and 4 so that you have -9 to take
off during set-up)
Topic 2: Using Level III, Lesson 18, students will learn another a legal move where they can add the
same number cube to both sides of the balance to cancel out constants next to it and isolate the variable.
Direct Instruction Examples:
 Example : x + (-4) = 6
 To set-up for this example, students will put a blue pawn and a 4 green-number cube on
the left and a 6 red-number cube on the right.
 Remind students that the goal is to isolate the variable. Ask students if they can think of
any ways to get rid of the 4 green-number cube (-4). Once students realize that they can
cancel it out with a 4 red-number cube, the discussion should proceed into how to
remain balanced. In the 6th grade, students should have learned that to solve equations,
what you do to one side you must do to the other. So, if they add a 4 red-number cube
to one side of the balance scale, they must also add it to the other. This is a LEGAL
MOVE.
 Their balance board should now look like:
 x + (-4) + (4) = 6 + 4
 The (-4) and (4) are opposites and can be taken away from the left side since
their sum is zero.
 So, x = 10
Equation:
Example 4:
Lesson 18
x + 5 = -9
Solution/Check:
x = -14
(put a 5 green-number cube on both sides to cancel
out the 5 red-number cube on the left)
Example 5:
Lesson 18
x+6=2
x = -4
(put a 6 green-number cube on both sides to cancel
out the 6 red-number cube on the left. Students will
then need to use their knowledge from lesson 17 to
simplify a 2 red-number cube and a 6 green-number
cube.)
C&I/MS Math/Fall 2011

Example : 3x + 8 = -4
 To set-up for this example, students will put three blue pawns and a 8 red-number on the
left of the balance scale. They will put a 4 green-number cube on the right.
 To isolate the variable, the 8 must be removed first by giving both sides a 8 greennumber cube. The 8-red and 8 green-number cubes can be removed from the left-side
since they are additive inverses and their sum is zero.
 3x + 8 + (-8) = -4 + (-8)
 3x = -12
 x = -4
Equation:
Example 6:
Lesson 18
4x + (-5) = -1
Solution/Check:
x=1
(put a 5 red-number cube on both sides to cancel out the 5
green-number cube on the left)
Example 7:
Lesson 18
2x + (-4) = 10
x=7
(put a 4 red-number cube on both sides to cancel out the 4
green-number cube on the left.
Topic 3: Review how to move from the pawns and number cubes to a pictorial representation using
paper and pencil. Use lesson 25 for additional assistance, but know that lesson 25 will contain
advanced problems that have not been introduced to students. Basic components of pictorial
representations as recommended by Dr. Borenson:
- Draw the balance by using one long horizontal line for the scale and a short vertical line that
divides the scale in half.
- Shaded triangles will represent the blue pawns.
- Unshaded triangles will represent the white pawns.
- Boxed numbers represent the red (positive) number cubes.
- Circled numbers represent the green (negative) number cubes.
- Students will circle pairs of additive inverses of triangles that they want to remove from a side
and use arrows pointing away from the scale to reference the removal.
- Students will cross of boxes/circles (number cubes) when they are eliminated or changed
on either side of the scale. They will rewrite a new number cube if a constant is left on one
side. (The program uses arrows to show when you take pawns from both sides. Since we are
refraining from using variables on both sides in these lessons that may not be necessary to
show).
Direct Instruction Examples:
 Example : 2x + (-4) = 6
 To set-up, students will draw 2 shaded triangles and a circled 4 on the left side of their
drawn scale balance. They will put a boxed 6-number cube on the right.
 They will then add a 4-boxed number cube on both sides (legal move). Typically, the
newly added pieces to both sides are drawn in a row above the original set-up picture.
C&I/MS Math/Fall 2011



Students should cross off the boxed 4-number cube and the circled 4-number cube (both
on the left). They can combine the boxed 6-number cube and the boxed 4-number cube
to a boxed 10-number cube if they wish, but it isn’t necessary in this example.
Then will then be left with 2x = 10. So, x = 5
Check: 5 + 5 + (-4) = 6
 6=6
Equation:
Example 8:
Solution/
Check:
x = -1
6x + 10 = 4
-Set-up: 6 shaded triangles and a boxed 10
-6 shaded triangles, a boxed 10, a circled 10
| boxed 4
| boxed 4, a circled 10
-cross of boxed 10 and circled 10 on the left.
You are left with:
6 shaded triangles | boxed 4, a circled 10
6 shaded triangles | Circled 6
x = -1
**Challenge- Use examples with white pawn.
B. Questions to Promote Higher Level and Critical Thinking (i.e., Socratic):
 Does x always have to be a positive number?
 If x is a negative number, then what would –x equal?
 x and –x are opposites with a sum of 0. What is their quotient?
Instructional Resources (specific to each objective):
 Primary
o The Hands-On Learning System by Henry Borenson, Ed.D

Supplemental
o Holt textbook- Selected sections from Chapters 2, 3, and 11.
Guided Practice Strategies or Activities:
 Guided Practice examples and solutions are provided after each direct instruction example.
You can have students complete these after each example OR wait until the end to complete
them all at once. Solutions have been provided in the direct instruction section.
 Hands-On Equations Practice & Homework Set
7ccss_HoE_Pracitce&
Homework_ed_d17.docx
C&I/MS Math/Fall 2011
Formative Assessment (Include context and accuracy percentage)
Informal - Skill/Knowledge Acquisition (Ex. Teacher monitor, analyze student questions)
 Peer Assessment- Each student had the opportunity to complete the same problems and
assist students when there was a lack of understanding. Have students peer assess using a
few guiding sentence fragments, such as:
 You did these really well…
 You could have…
 Next time you need to focus on…
Formal - Skill/Knowledge Mastery (ex. Open-ended quiz, prompted written response)
 Transfer and Apply
o On an index card, students write down concepts learned from the class on one side;
on the other side, they provide an application of each concept (you may wish to
provide an example that they complete.
Differentiated Instructional Strategies/Modifications:
 Students can use the National Library of Virtual Manipulatives for another way to assist them in
completing equations.
o http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?open=instructions&from=ca
tegory_g_3_t_2.html
 Copies of all notes and classwork examples.
 Students can check evaluated expressions (equations) using the TI-73
Re-teaching Strategies or Activities:
 Students should use the Hands-On materials to assist them through the homework questions.
Enrichment:
 Students should complete the class work/homework worksheet and provide reasons (vocabulary
words) for being able to complete each move.
 Write it out! Instead of representing solutions pictorially, have students write what they would
do solve the problems. Have them include properties used when solving for an additional
challenge.
Independent Practice: (ex. WebQuest, textbook problem, interview, etc…)
 Additional Examples from Classwork Sheets provided with each class set of the Hands-On
Equations Learning System. These examples will not have a set-up that involves variables on
both sides:
o Lesson 17: #1, 2, 3, 4
o Lesson 18: #1, 2, 3, 4, 10
Comments /Notes:

C&I/MS Math/Fall 2011