Grade 7 Mathematics Unit 6 Equations - ed.gov.nl.ca · Unit 6: Equations Grade 7 Math Curriculum Guide 183 Unit 6 Overview Introduction Students will focus on developing skills and

Grade 7 Mathematics Curriculum Outcomes 181

Outcomes with Achievement Indicators

Unit 6

Grade 7 Mathematics

Unit 6

Equations

Estimated Time: 20 Hours

[C] Communication [PS] Problem Solving

[CN] Connections [R] Reasoning

[ME] Mental Mathematics [T] Technology

and Estimation [V] Visualization



Unit 6

Unit 6: Equations

Grade 7 Math Curriculum Guide 183

Unit 6 Overview

Introduction

Students will focus on developing skills and knowledge necessary for understanding how to solve

equations using a variety of methods. The big ideas in this unit are:

• An equation states a relationship between two expressions; specifically, that the two expressions

are equal.

• Preservation of equality is at the core of solving equations.

• An equation can be solved by systematic trial, using a two-pan balance model, using algebra tiles, or solved symbolically by using algebraic techniques.

• Equations can be used to model and solve problems.

Context The students will begin to solve equations using systematic trial and inspection. The students will often

know the solution to an equation instantly. However, they will be asked to explain their reasoning before

they move on to solving equations with two-pan balance models and algebra tiles. Students will solve

equations that involve positive and negative integers and they will solve equations that are limited to no

more than two steps. Ultimately students will apply algebraic techniques, requiring the use of preservation of equality, in order to solve equations.

Why are these concepts important?

Developing a good understanding of solving equations will permit students to:

• Become good problem solvers. Students will be able to decide on an appropriate method for problem solving and determine if their answer makes sense.

• Be able to manipulate formulas using algebra and know how to verify answers when studying

subjects like chemistry, physics, and calculus to name a few.

“It is hard to convince a high-school student that he will encounter a lot of problems more difficult than

those of algebra and geometry.”

Edgar Watson Howe (1853-1937)

Strand: Patterns and Relations (Variables and Equations)



Unit 6

General Outcome: Represent algebraic expressions in

multiple ways.

Specific Outcome

It is expected that students will: 7PR4. Explain the difference

between an expression and

an equation.

[C, CN]

(Cont’d)

Achievement Indicators

Elaborations: Suggested Learning and Teaching Strategies

It is assumed that students can:

• recognize patterns in a table of values

• write a pattern rule for a number pattern

• use a pattern rule to find the value of a given term

This outcome was introduced in Unit 1 (see achievement

indicators 7PR4.2, 7PR4.3, and 7PR4.4). Identifying the

difference between an algebraic expression and an equation

can now be further developed.

Recall that an algebraic equation is a mathematical statement

that two expressions are equal. In an equation such as 2a + 5 =

11, we are searching for one input value, or value that can be

substituted for a, that would produce the desired output value

of 11.

Students should now be exposed to expressions where the

constant term is negative, e.g.

4x – 7 is equivalent to 4x + -7, thus the constant term is -7.

7PR4.6 Provide an

example of an expression

and an equation, and

explain how they are

similar and different.




Unit 6


multiple ways.

Suggested Assessment Strategies

Paper & Pencil

1. Which are expressions? Which are equations? How are they

similar? How are they different?

A. 2 – x

B. 5v = 20

C. 43

=h

D. w + 7

2. Does the algebra tile diagram below model an expression or an

equation? Explain.

3. Below are three algebraic expressions and/or equations.

4p + 5 = 55

4p – 5 = 55

4p – 5

A. Which are equations and which are expressions? Explain

why.

B. List ways in which they are similar and ways in which

they differ.

4. Have students complete concept maps for expressions and

equations such as:

Sample Responses

Equation

Essential Characteristics

Non-Essential Characteristics

Examples Non-ExamplesEquation

Essential Characteristics

Non-Essential Characteristics

Examples Non-Examples

= sign

Two expr

essions eq

ual

Two constant terms

More than one operation

2x = 6 5x 2x - 1

44 + 3 = 7

Variable

4 < 6

3x +

4 = 7

y = 2

Resources/Notes

Math Makes Sense 7

Lesson 6.1 Unit 6: Equations

TR: ProGuide, pp. 4–9

Master 6.9, 6.18

CD-ROM Unit 6 Masters

ST: pp. 220–225

Practice and HW Book

pp. 132–134




Unit 6

General Outcome: Represent algebraic expressions in multiple

ways.

Specific Outcome

It is expected that students will:

7PR7. Model and solve,

concretely, pictorially and

symbolically, problems that

can be represented by linear

equations of the form:

• ax + b = c

• ax = b

• 0x

= ≠b, aa

where a, b and c are whole

numbers.

[CN, PS, R, V]



Note: Only whole numbers should be used for a, b, and c.

Refer to the Achievement indicator 7PR4.4 in Unit 1 for a

discussion of systematic trial (i.e. guess and check) to solve

equations.

Another commonly used concrete model for equations is to use

a two-pan balance approach.

Example: Solve the equation 2x + 1 = 5 using guess and check:

2(3) + 1

5Too

Heavy!

2(1) + 1

5

Too

Light!

2(2) + 1 5

Balance!

Students will need to recall, from Unit 1, how to write an

algebraic equation from a number sentence. Refer to Student

Text pages 221-223.

7PR7.2 Solve a given

linear equation by

inspection and by

systematic trial.




Unit 6


multiple ways.


Paper and Pencil

1. A hockey school charges $80 per day to use the facility plus

$20 per play per day for food, use of equipment and lessons. A

team raised $320 for a one-day practice.

A. Write an equation you can solve to find the number of

athletes that can attend the hockey school?

B. Solve the equation by inspection, then by systematic trial.

Which method was easier, and why?

2. The formula for the area of a triangle is 2÷×= hbA .

Find all the possible whole number values for b and h that will

result in an area of 72 cm2.

Journal/Interview

1. Ryan was given the equation 2275 =+d and asked to solve for

d. He indicated that d = 15, but was told that his answer was

incorrect. Explain what his misconception was and how you

would help him to correctly solve for d.

2. When solving 36244 =+d , Sarah chose 3 for her first value for

d and Billy chose 6. Which number is the better choice, and

why?

Informal Observation

1. Play ‘I Have, Who Has’. See Teacher Resource Master 6.6a and

6.6b.

2. Play ‘Equation Concentration’. See Teacher Resource Master

6.7a and 6.7b.

Resources/Notes

Math Makes Sense 7

Lesson 6.1

(continued)




Unit 6


multiple ways.

Specific Outcome


7PR3. Demonstrate an

understanding of

preservation of equality by:

• modelling preservation of

equality, concretely,

pictorially and

symbolically

• applying preservation of

equality to solve equations.

[C, CN, PS, R, V]


Students should use concrete materials to investigate the

process of solving equations. Refer to Outcome 7PR7 in Unit 1

for discussion.

When solving linear equations, the idea is to isolate the

variable while preserving equality at each step of the process.

To move from the concrete stage to the symbolic stage,

students should record each step of a concretely modelled

process in symbolic form, e.g.

Concrete Representation Symbolic Representation

2x + 1 = 5

Remove a unit tile from each

side:

2x + 1 – 1 = 5 – 1

Simplify:

2x = 4

Since we have two x tiles, we

separate both sides into two equal groups.

Each x-tile is paired with 2

unit tiles. Therefore, the solution is: x = 2

One other approach for solving equations symbolically might

be to revisit the skills of writing related equations learned in

primary and elementary grades. For example:

• 3 + 2 = 5 A related equation that isolates the 3 is 3 = 5 – 2.

• 3 × 2 = 6 A related equation that isolates the 3 is 6

32

= .

• 4(3) + 1 = 13. A related equation that isolates the 3 is

13 13

4

−= .

• Similarly, when writing 2N + 1 = 201, a related equation

that isolates the N is 201 1

2N

−= . Therefore, we can

calculate that the input value must have been 100.




Unit 6


multiple ways.


Resources/Notes




Unit 6


multiple ways.

Specific Outcome



understanding of




pictorially and

symbolically



[C, CN, PS, R, V]

(Cont’d)



Note: Students should consider in advance what might be a

reasonable solution, and be aware that once they acquire a

solution, it can be checked for accuracy by substitution into the

original equation.

Build understanding of equality by using number sentences to

explore what must be done to preserve equality when one side

is changed. Balance scales can be used to help illustrate an

equality and then to connect the concrete to the pictorial and

symbolic representations. Consider:

246 ++ 34 ×

Since 246 ++ = 34 × , the pans are balanced. Ask students to

consider what would happen if a number, such as 5, is added to

the left pan only (the pan tips to the left). Discuss why this

happens (the left side is greater than the right side) and what

must be done in order to rebalance the pans (add 5 to the right

side). Students should come to realize that what is done to one

side must also be done to the other in order to preserve

equality. Demonstrate similar examples using each of the four

operations.

7PR3.1 Model the

preservation of equality

for each of the four

operations, using concrete

materials or pictorial

representations; explain

the process orally; and

record the process

symbolically.

7PR3.2 Write equivalent

forms of a given equation

by applying the

preservation of equality,

and verify, using concrete

materials; e.g., 3b = 12 is

the same as 3b + 5 = 12 +

5 or 2r = 7 is the same as

3(2r) = 3(7).




Unit 6


multiple ways.


Paper and Pencil

1. Have students write the equation based on the balance scale

model (all pieces are positive). Then solve the equation both

pictorially and symbolically to show the connections between

the two.

2. A. Write two equations equivalent to 3 1 5n + =

B. Use the balance scales below to illustrate your equations

Interview

1. Consider:

6 2× 10 4+

A. Are the pans balanced? How do you know?

B. How can you balance the pans?

2. Consider: 4 3 9+ − 6 4 4− −

A. Are the pans balanced? How do you know?

B. What would happen if you add 5 to the right hand side?

C. How can you rebalance the pans to preserve the equality?

Resources/Notes

This outcome is covered

throughout:

Lesson 6.2

Lesson 6.3

Lesson 6.4

Lesson 6.5




Unit 6


multiple ways.

Specific Outcome







• ax + b = c

• ax = b

• ,x

b aa

= ≠ 0


numbers.

[CN, PS, R, V]

(Cont’d)



A two-pan balance can be used to model and visually represent

equations of the form ax b+ = c and ax b= .

Consider the following: 14 22x+ =

Many students will immediately arrive at the value of the

unknown. However, it is important for students to recognize

what will happen if a mass is removed from one side of the

balance only and what they must do to compensate for this.

This will help develop the method for solving an equation

algebraically (Lesson 6.4).

We can verify the solution by replacing the unknown mass

with 8g.

Check:

Left Pan: 14g + 8g = 22g

Right Pan: 22g So, the solution is correct!

Students are required to substitute their answer for the variable

and check to make sure that it makes the equation true.

To verify that x=7 is a solution to 4426 =+x ,

Left side: 2)7(6 + Right side :44

= 242 +

= 44

Since the left side equals the right side, x=7 is correct.

7PR7.3 Draw a visual

representation of the steps

used to solve a given

linear equation.

7PR7.5 Verify the

solution to a given linear

equation, using concrete

materials and diagrams.

14g ? 22g

7PR7.6 Substitute a

possible solution for the

variable in a given linear

equation into the original

linear equation to verify

the equality.


problem, using a linear

equation, and record the

process.




Unit 6


ways.


Paper and Pencil

1. Find the values of the unknown mass on each balance scale.

Sketch the steps you use.

A.

B.

2. A. Sketch balance scales to represent each equation

B. Solve each equation. Verify the solution.

i. 182 =y

ii. 1723 =+n

3. Solving Equations:

A. Write a problem that can be solved using the equation

123 =+x .

B. How would your problem change if the equation was

123 =x ?

C. What new problem can you write for 123

x= ?

D. Solve each equation in parts A,B, and C. Show the steps

you followed.

4. Write an equation for each sentence. Solve each equation, and

verify you answer.

A. The cost shared by 5 people amounts to $35 each.

B. There are 38 boys. This is 6 more than double the number

of girls.

C. Sixty centimetres is one half of Bob’s height

5. Show whether or not 7=x is the solution to each equation.

A. 486 =x

B. 2023 =+x

Resources/Notes

Math Makes Sense 7


TR: ProGuide, pp. 10–14

Master 6.10, 6.19


ST: pp. 226–230

Practice and HW Book

pp. 135–137

Note: This is continued

throughout Lesson 6.4 and

Lesson 6.5.

w 16g w 4g 12g 8g

15g 15g 20g x 10g




Unit 6


multiple ways.

Specific Outcome







• ax + b = c

• ax = b

• ,x

b aa

= ≠ 0


numbers.

[CN, PS, R, V]

(Cont’d)



When students solve a linear equation symbolically

(algebraically), it is important to visualize the balance scale

model. In order to preserve the equality, whatever is done to

the left pan of the balance must be done to the right pan. The

same is true for an algebraic equation; always perform the

same operation on both sides of the equation.

1952 =+n

To isolate 2n, subtract 5 from each side.

519552 −=−+n

142 =n

Divide each side by 2, 2

14

2

2=

n

7=n

Students can verify the solution by substituting n = 7

into 1952 =+n . Since the left side equals the right side, n = 7

is the correct solution.

n n 5g

n 19g n 5g

5g 14g

n n 7g 7g




linear equation. (cont’d)

7PR7.5 Verify the




(cont’d)

7PR7.6 Substitute a





the equality. (cont’d)




process. (cont’d)




Unit 6


ways.


Paper and Pencil

1. Write an equation for each situation. Solve each equation, and

verify your answer.

A. The cost shared by 5 people amounts to $35 each.

B. There are 38 boys. This is 6 more than double the number

of girls.

C. Sixty centimetres is one half of Bob’s height.

2. Show whether or not 7=x is the solution to each equation.

A. 486 =x

B. 17

=x

C. 2023 =+x

Resources/Notes

Math Makes Sense 7

Lesson 6.2

(continued)

Note: This is continued

throughout Lesson 6.4 and

Lesson 6.5.




Unit 6


multiple ways.

Specific Outcome







• ax + b = c

• ax = b

• ,x

b aa

= ≠ 0


numbers.

[CN, PS, R, V]

(Cont’d)



It will be necessary, however, to model equations of the form

xb

a= , 0a ≠ and to supplement the student text exercises with

examples of this type.

Example: Laurie has 1

3 of a chocolate bar. It weighs 5g. She

wants to know how much a whole chocolate bar weighs. Write

an equation to represent this situation and then solve the

equation using a visual representation. Verify your answer.

Solution:

In order to solve this problem, students will need to think

about how many pieces Laurie will need to make a whole

chocolate bar. She knows:

She needs 3 pieces to make a whole, so she can draw:

By combining these pieces to form a whole bar…

…she concludes that one bar is 15g.

Verify: Left Pan: 33

bb= ÷ Right Pan : 5g

= 15 3÷

5g= So the solution is correct!

Balance scales reinforce the idea of the equality on two sides.

If this is well understood, teachers may also wish to use

algebra tiles to represent these types of equations.




linear equation. (cont’d)

7PR7.5 Verify the




(cont’d)

7PR7.6 Substitute a





the equality. (cont’d)




process. (cont’d)




Unit 6


multiple ways.


Paper and Pencil

1. Brigitte is solving the equation 68

f= . This is her solution:

108

8 10 88

2

f

f

f

=

− = −

=

A. Is her solution correct or incorrect? Draw a visual to

demonstrate how you know.

B. If you think her solution is incorrect, what would you

change to solve the equation?

2. A clothing store is having a sale. Jacob pays $19 for two shirts

and a pair of sunglasses. The sunglasses cost $5.

A. Write an equation that represents the situation.

B. Draw a model to represent the equation.

C. Use the model to determine how much does Jacob pay for

each shirt?

D. Verify your answer.

Resources/Notes

This outcome is covered

throughout:

Lesson 6.1

Lesson 6.2

Lesson 6.4

Lesson 6.5




Unit 6


multiple ways.

Specific Outcome

It is expected that students will: 7PR6. Model and solve,



can be represented by one-

step linear equations of the

form x + a = b, where a and

b are integers.

[CN, PS, R, V]



Note: When solving linear equations that require

multiplication or division, only whole numbers should be used

as these operations with integers will be addressed in grade 8.

Consider the sentence:

Three less than a number is -9. Students should be able to

write an equation for the number sentence, and then solve

using algebra tiles. (In the diagram below, represents a

negative, represents a positive.)

Example: 93 −=−x

To model this equation, students need to recall that subtracting

3 is equivalent to adding negative 3.

To isolate the variable tile, add 3 positive tiles to the left side

to make zero pairs. Add 3 positive tiles to the right side to

preserve equality. Remove the zero pairs from both sides.

The tiles show that 6−=x

Students can verify the solution by replacing x, the variable

tile, with 6 negative tiles. They can also verify by replacing x

with -6 in the equation.

Refer to student text page 231-234 for relevant examples.

7PR6.1 Represent a

given problem with a

linear equation; and

solve the equation, using

concrete models, e.g.,

counters, integer tiles.



equation.


representation of the

steps required to solve a

given linear equation.

7PR6.4 Verify the




7PR6.5 Substitute a



equation into the

original linear equation

to verify the equality.




Unit 6


multiple ways.


Paper and Pencil

1. Solve each of the equations using algebra tiles. Sketch your

steps. Verify your solution.

A. 43 =−n

B. 21 −=+h

C. 62 −= y

D. 14 −=−w

2. Algebra Tiles:

A. Write an equation you can use to solve each problem.

B. Use algebra tiles to solve each equation. Sketch your

steps.

C. Verify your solution.

i. The temperature dropped C°5 to C°− 2 . What was the

original temperature?

ii. Frank is 9 years old. He is 4 years older than Joe. How

old is Joe?

iii. Susan checked out books from the library. She

returned 4 books, and she still has 3 books at home.

How many books did she borrow?

3. Which of the following equations is 2−=x a solution?

A. 53 −=−x

B. 31 =+x

C. 12 =+x

D. 13 =+x

Resources/Notes

Math Makes Sense 7

Lesson 6.3

Lesson 6.4


TR: ProGuide,

pp. 15–19

pp. 21–23

pp. 24–28

Master 6.11, 6.20

Master 6.12, 6.21

Master 6.13, 6.22

PM 30


ST: pp. 231–235

ST: pp. 237–239

ST: pp. 240–244

Practice and Homework

Book

pp. 138–140

pp. 141–144

pp. 145–147




Unit 6


multiple ways.

Specific Outcome



understanding of




pictorially and

symbolically



[C, CN, PS, R, V]

(Cont’d)



Students should now be able to move away from the use of

diagrams and concrete materials when solving an equation for

a variable. Students should be able to apply preservation of

equality to solve equations algebraically.

713 =+y

17113 −=−+y

63 =y

3

6

3

3=

y

2=y


problem by applying

preservation of equality.




Unit 6


multiple ways.


Paper and Pencil

1. Solve the following equations:

A. 243 =x

B. 77

=x

C. 3156 =+x

D. 198 =−x

E. 37 −=+x

2. Are the following algebraic equations solved correctly?

Explain.

A. 23 −=−f

3233 −−=−−f

5−=f

B. 1242 =+w

412442 −=−+w

82 =w

28 ×=w

16=w

3. The table shows the relationship between the number of riders

on a tour bus and the cost of providing boxed lunches.

Customers 1 2 3 4 5

Cost ($) 4.25 8.50 12.75 17.00 21.25…

A. Ask students to explain how the lunch cost is related to the

number of riders.

B. Have them write an equation for finding the lunch cost (l)

for the number of customers (n).

C. Ask them to use the equation to find the cost of lunch if

there were 25 people on the tour.

D. Ask how many people were on the bus if the tour-bus

leader spent $89.25 on lunch.

Resources/Notes

Math Makes Sense 7

Lesson 6.4

Lesson 6.5

(continued)




Unit 6

Documents

Grade 7 Mathematics Unit 6 Equations - ed.gov.nl.ca · Unit 6: Equations Grade 7 Math Curriculum Guide 183 Unit 6 Overview Introduction Students will focus on developing skills and