G8

Math Task

Unit 4 · Lesson 12: Linear Equations

Directions: Solve and interpret.
3. You need to buy wrapping paper for the gift box pictured. You need to find the height of
the box. You know the length is 12 inches and the width is 8 inches and the total surface
area of the box is 752 square inches if there is no overlapping of the wrapping paper.
(Surface area is the total amount of paper needed to cover the box.)



8
12

Solution:
Let = height of box
2(12 ) + 2(8 ) + 2(12 ∙ 8) = 752 sq in
24 + 16 + 192 = 752
40 + 192 = 752
40 = 560
= 14 in

Copyright © Swun Math 396Grade 8 Unit 4 Lesson 12 T TE

Homework Name: ___________________________
Date: ___________________________
Unit 4 · Lesson 12: Linear Equations

Directions: Solve and interpret.

1. Consider Problem #3 from the task you completed. What would be a good visual
representation of the box with each side shown separately? Explain why this might be
helpful when writing the equation for the surface area of the box?

2. A gift box needs to be filled with jellybeans. The volume of the box is 229.5 cubic inches.
The length of the box is 6 inches and the width of the box is 4.5 inches. Draw a picture
and label what you know about the box. Define a variable for the missing dimension and
write and equation to find it. What is the height of the box?

3. You need to buy tissue paper sheets to cushion gifts being placed inside a box. The box is
12 inches wide by 16 inches long. The area of the tissue paper sheet is based on twice the
length times the width of the box. Find the area of a tissue paper sheet.

4. Challenge Question: There is an equal amount of tissue paper that hangs over each side of
the box. How much extra per side? Draw and label the tissue paper sheet using dotted
lines to show the extra tissue that is wider than the box. Use x for each extra amount on
the width. Can you think of another way to represent an equation for the area of the
tissue paper using x for the extra paper flaps? What is the width of each flap?

397 Copyright © Swun Math Grade 8 Unit 4 Lesson 12 T TE

Unit 4 ∙ Constructed Response Name: ____________________________
Linear Equations Date: ______________________________

Task 1: Equations & Expressions with the Student Journal Pages
Distributive Property
217-224

Directions: Read and solve each problem.

1. Evaluate each pair of expressions and decide whether the distributive property has
been used correctly.

a. Place a check mark (✓) in the “Yes” or “No” column to show your answer.

Yes No

b. Rewrite the incorrect expressions to show the correct use of the distributive
property.

Copyright © Swun Math 398Grade 8 Unit 4 Constructed Response TE

Unit 4 ∙ Constructed Response
Linear Equations

2. Jian solved this equation. He cleared the decimals by multiplying both sides by 100
using the distributive property.

Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:

a. Is Jian’s work correct? If not, find and fix his error. Explain your thinking.

b. Explain what happens to an equation when both sides are multiplied by a power of
ten.

399 Copyright © Swun Math Grade 8 Unit 4 Constructed Response TE

Unit 4 ∙ Constructed Response
Linear Equations

3. Maritza correctly solved this equation. She rewrote the equation to have common
denominators, and then solved it. Her work is in Column A.

a. In Column B, solve the same equation by clearing the fractions using the distributive
property.

Column A Column B

Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
Step 7:
Step 8:

b. Explain how using the distributive property is more efficient than Maritza’s process.

Copyright © Swun Math 400Grade 8 Unit 4 Constructed Response TE

Unit 4 ∙ Constructed Response Name: ____________________________
Linear Equations Date: ______________________________

Task 2: Solve Equations with One or More Solutions

Directions: Read and solve each problem.
4. Mario incorrectly solved this equation on a test. This is his work:

Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:

a. Describe Mario’s error.

b. Solve the equation correctly. Show work.

401 Copyright © Swun Math Grade 8 Unit 4 Constructed Response TE

Unit 4 ∙ Constructed Response
Linear Equations

5. Decide if each equation has no solution, one solution or an infinite number of
solutions. Write the letter of each equation in the correct category in the table.

Show your work in detailed and organized steps.

No Solution One Solution Infinite Solutions

A B
C D

A B

CD

Copyright © Swun Math 402Grade 8 Unit 4 Constructed Response TE

Unit 4 ∙ Constructed Response
Linear Equations

6. Write one integer from the answer bank to complete each indicated equation. Each
integer may be used more than once in each equation.

Prove your answer is correct by showing detailed and organized work.

a. No solution ANSWER BANK

-2
-1
0
1
2

b. One solution

c. Infinite solutions

403 Copyright © Swun Math Grade 8 Unit 4 Constructed Response TE

Unit 4 ∙ Constructed Response Name: ____________________________
Linear Equations Date: ______________________________

Task 3: Real-World Application: Write & Solve Equations

Directions: Read and solve each problem.

7. Susan placed a picture in a frame. The picture is x inches wide. The picture’s length

is 2 inches greater than the width. The frame is 1 inch wide on all sides. The
perimeter of the frame is 32 inches.

a. Write an expression that represents the width of the frame. Write another
expression that represents the length of the frame.

Width: ________________
Length: _______________

b. Write and solve an equation to represent the width of the picture.

Equation: _______________ 404Grade 8 Unit 4 Constructed Response TE
Answer: _______________
c. What is the length of the picture?

Answer: _______________

Copyright © Swun Math

Unit 4 ∙ Constructed Response
Linear Equations

8. Josh wants to earn $1367.50 to purchase a laptop. He was offered a job cleaning
offices for ten days. If he takes the job, he will earn a flat rate of $96.50 each day,
plus $5.75 for each window he washes. He must wash the same number of windows
each day. He wants to know how many windows he will need to wash every day to
earn enough money for the laptop.

a. Write an equation that will help Josh find the number of windows he needs to wash.

Equation: _______________

b. Solve the equation.

Answer: _______________

c. If Josh decides only to work 8 days, how many windows would he need to wash to
meet his goal? (Round to the nearest whole number). Write an equation and solve.

Answer: _______________

405 Copyright © Swun Math Grade 8 Unit 4 Constructed Response TE

Unit 4∙ Constructed Response Unit 1 ∙ Constructed
Linear Equations ResponseSCORING RUBRIC

TASK 1: Equations & Expressions with the Distributive Property

# Answer/Exemplar Standard Points

1a. YN 8.EE.7 1

✓

✓

✓
✓

1b. 8—6(2x—5) and 8– 12x +30 8.EE.7b 1
8.EE.7b 1
2a. Yes, Jian’s work is accurate. I checked his work by solving the 8.EE.7b 1
original equation with his solution, x = 1.
8.EE.7b 1
2b. When both sides of an equation are multiplied by a power of ten
(or any other value), the equation remains balanced because the
same change is made to both sides.

3a.

3b. I multiplied each term on both sides of the equation by 12, which is 8.EE.7b 1
the greatest common multiple of each denominator. This allowed
me to solve the problem more efficiently than Maritza because I
eliminated the fractions. I solve the problem in fewer steps and
without having to do so much work with adding, subtracting,
multiplying, and dividing fractions. For me, it’s easier to do
mental math working with whole numbers than with fractions.

Copyright © Swun Math 406Grade 8 Unit 4 Constructed Response Rubric TE

Unit 4∙ Constructed Response Unit 1 ∙ Constructed
Linear Equations ResponseSCORING RUBRIC

TASK 2: Solve Equations with One or More Solutions Standard Points
8.EE.7ab 1
# Answer/Exemplar
8.EE.7ab 1
4a. To move from Step 4 to Step 5, Mario needed to subtract 6 from 8.EE.7a 1
both sides of the equation. Instead, she added 6 to the right side.

4b. f= ─1; student shows accurate work

5a.

No One Infinite

B C A
D

5b. A B 8.EE.7a 1

Infinite Solutions No Solution
C D

One Solution One Solution

6a. Answers will vary. Sample response: 2, ─2 8.EE.7a 1
8.EE.7a 1
6b. Answers will vary. Sample response: 0, ─1 8.EE.7a 1

6c. Answers will vary. Sample response: 2, 2

407 Copyright © Swun Math Grade 8 Unit 4 Constructed Response Rubric TE

Unit 4∙ Constructed Response Unit 1 ∙ Constructed
Linear Equations ResponseSCORING RUBRIC

TASK 3: Real-World Application: Write & Solve Equations

# Answer/Exemplar Standard Points
8.EE.7b 1
7a. Width= x + 2
Length = x + 4 or (x +2)+ 2

7b. Equation: 2(x + 2) + 2(x + 4) = 36 8.EE.7b 1

Answer: 6 inches

7c. 8 inches 8.EE.7b 1

8a. 8.EE.7b 1
8b. 7 windows 8.EE.7b 1
8c. 13 windows 1
8.EE.7b

Copyright © Swun Math 408Grade 8 Unit 4 Constructed Response Rubric TE

NUnotites1 ∙ Constructed Response
The Number System — Part 1

409 Copyright © Swun Math Grade 8 Unit 4 Constructed Response Rubric TE

Glossary

410

Grade 8

Glossary

Term Definition Example Unit

additive a number and its opposite —3 + 3 = 0 4
inverse whose sum is 0; also called a 10
zero pair
adjacent 4
angles angles that share a side and 4
vertex; angles don’t overlap 10

algebra tiles tools or drawings used to 10

represent integers and 4
variables in expressions and
equations

algebraic an expression using variables, (9n + 1) ÷ 4
expression
integers and operations; does
not have an equal sign

alternate pairs of angles on opposite

exterior sides of the transversal, but
angles outside the two lines

alternate pairs of angles on opposite

interior sides of the transversal, but
angles inside the two lines

area model a visual model using area to
multiply two quantities

411 Copyright © Swun Math Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

base the repeated factor when 5³; 5 is the base 2
written next to the exponent 9

center of the fixed point on the plane 4
10
dilation which all points are made
11
(dilation point) larger or smaller from 9

coefficient a number used to multiply a Center of dilation= (0,0)
variable Scale factor= ½

7m; 7 is the coefficient

complementary two angles that add up to 90°
angles

composite figures made by combining
figures
two or more three-
dimensional figures.

compound when two or more

transformation transformations are

performed on a figure

Copyright © Swun Math 412Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

cone a three-dimensional figure 11
with a circular base connected
to a point with a curved side 9

congruent figures with exactly the same ∆ ABC ≅ ∆PQR
figures shape and same size

constant a number on its own in an 2a + 5; 5 is the constant 4
expression or equation

converse of the if a2 + b2 = c2, then the triangle If a2 + b2 < c2, then the triangle is obtuse. 5
If a2 + b2 > c2, then the triangle is acute.
Pythagorean is a right triangle
Theorem

coordinates a pair of numbers used to (x, y) 6
locate a point on a coordinate
plane: written in order of x
(horizontal value), y (vertical
value); also called an ordered
pair

corresponding matching angles in similar In transversals, corresponding 9, 10
angles are in matching corners
angles figures; are congruent

corresponding matching sides in similar AC ≅ PR 9, 10

sides figures; are proportional

413 Copyright © Swun Math Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

cross a method to compare fractions 9
multiplication in which you multiply the
numerator of one fraction by
the denominator of the other

cube the result of multiplying a 2 cubed = 23 = 8 2
cube root base by itself three times 2

cylinder a special value that, when 11
used in multiplication three
dependent times, gives the original
variable number

dilation a three-dimensional figure with
circular bases and a curved area
for the side

a variable whose value is d = 60t 7
based on the value of another
variable in the equation; the y- Distance( d ) is a dependent
axis variable because the distance
depends on time spent traveling.

to make a figure larger or 9
smaller

elimination uses addition & subtraction to 8
method
eliminate a variable in order 414Grade 8 Cumulative Glossary TE
to solve for the remaining

variable

Copyright © Swun Math

Grade 8 • Glossary

Term Definition Example Unit

elimination uses multiplication to create 8
method terms that are opposites; then
(multiplication) equations are added to 4
eliminate one variable and 2
the other variable can be 2
found 4
10
equation a number sentence that 12 + 2 = 20 — 6
2
shows equal values on both 12
sides of an equal sign

exponent indicates the number of times 5³; 3 is the exponent
the base is multiplied by 5 · 5 · 5 = 125
exponential itself; also known as power “5 to the power of 3”
form
expressions written with 23
exponents instead of 2 · 2 · 2 or 8

expression a number phrase that 9+4

represents a value with no
equal sign

exterior angle the angle between any side of a

shape, and a line extended
from the next side

factors numbers that are multiplied 5 • 3 = 15
frequency to create a product 5 and 3 are the factors

the number of times a result
occurs

415 Copyright © Swun Math Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

function a relation where each input is 7
related to exactly one output

function describes a function; the symbol y= 2x + 1 is the same as 7
notation f(x) replaces y and is read as “the f(x) = 2x +1 7

function rule value of f of x” 11
11
an equation which shows the
relationship between the
input (x) and the output (y)
values

height of cone the perpendicular distance

from the vertex to the center of
the circular base

hemisphere half of a sphere

hypotenuse the longest side in a right 7

triangle; the side opposite the
right angle

image a figure after a transformation 9
2
increase the size increases as the power 5 → 25 → 125
exponentially 5→ 5² → 5³ 416Grade 8 Cumulative Glossary TE
increases

Copyright © Swun Math

Grade 8 • Glossary

Term Definition Example Unit

independent a variable whose value does d = 60t 7
variable not depend on another
variable in the equation; the x Time ( t ) is an independent variable
infinite axis because the passage of time does not
solutions depend on distance.

initial value an equation that, when solved, 5x + 3 = 1 + 5x + 2 4
3=3
yields a statement that is
always true

the first value of the The initial value of the graph is (0, 60) 7
dependent variable (y); same
as the

y-intercept

input the x-value in an ordered pair 7
or table
1
integer the set of all counting ...─3, ─2, ─1, 0, 1, 2, 3… 10

numbers, including 0 and their
opposites

interior angle an angle inside a shape

inverse operations that undo each addition ↔ subtraction 4
operations other; the opposite operation multiplication ↔ division

417 Copyright © Swun Math Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

irrational the set of all numbers that 0.101101110… , 1
number cannot be written as a ratio; it 2
cannot be written as a simple 5
irrational fraction because the decimal is
solution non-repeating and non- 4
terminating 12
legs
solutions to equations that 7

cannot be written as a simple
fraction

the sides that form the right
triangle

like terms terms that have the same 4a + 5 + 2a – 3
variable, power, or constant 5b2 – 6b + 3b2

line of best fit a straight line drawn through
the center of data points on a
scatter plot; a trend line

linear function a function whose graph is a y = x—2
straight line

Copyright © Swun Math 418Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

multiplicative a number and its reciprocal 2·½=1 4
inverse whose product is 1 2

negative indicates the number of times a—n = 1 7
exponent to divide by the base an
4
5—3 = 1 7
53

1 ÷ 5 ÷ 5 ÷ 5 = 0.008

negative slope is decreasing; as x increases, y
decreases

no solution an equation that, when solved, 2=3

nonlinear yields a statement that is not
function true

a function whose graph is not a y = x2 + 2
straight line

observed data data that has been collected 12
and plotted on a graph 12
7
outliers a data point which “lies 35, 39, 40, 40, 41, 81
(81 is an outlier)
outside” most of the other
values in a set of data

output the y-value in an ordered pair or
table

419 Copyright © Swun Math Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

perfect cube the result after multiplying an 8, 27, 64... 2
integer three times by itself; 1
also called a cubed number 7

perfect square the result after multiplying an 4, 9, 16, 25, … 7

integer by itself; also called a
squared number

polynomial an expression with many 0 = ─6a + 12b + 6

terms; can have constants,
variables and exponents that

can be combined

positive slope is increasing; as x increases, y
increases

power indicates the number of times 5³; 3 is the exponent 2
5 · 5 · 5 = 125 3
the base is multiplied by itself; “5 to the power of 3” 9
also known as the exponent
102 means 10 × 10 = 100
powers of ten ten multiplied by itself the

number of times indicated by
the exponent

pre-image the original figure prior to a
transformation

properties of mathematical rules to apply additive inverse 4
multiplicative inverse
equality when solving linear equations

Copyright © Swun Math 420Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

properties of mathematical rules to apply Power to Power (Multiply) 2
exponents when solving equations with (an)m = anm
exponents 6
Multiply (Like Bases) (Add) 5
am · an = am+n

Power of a Product (Keep)
anbn = (ab)n

Division (Like Bases) (Subtract)

an = an—m
am

proportional a relationship in which the
relationship
ratio between two variables is
constant

Pythagorean the idea that, in any right a2 + b2 = c2
Theorem triangle, the sum of the squares
of the two shorter sides is
equivalent to the square of the
hypotenuse

rate of change same as slope, but the term is 7
used more often in real-world 6
problems

ratio a comparison of two quantities The ratio of circles to stars:
½ or 1:2 or 1 to 2

rational whole numbers, fractions, and x 1
number y
decimals represented as a ratio
of two integers; can be (where x and y are integers & y ≠ 0)

represented in fractional form

421 Copyright © Swun Math Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

rational solutions to equations that are y2 =9→ 2 y2 = 2 9 =3 2
solution integers or fractions 1
2
real numbers the set of numbers that ½, 9

includes both rational and 7
irrational numbers 12
1
reciprocal what to multiply a value by to 4—5 is the reciprocal of 45 5
reflection get 1; multiplicative inverse 9

a flip; a transformation across a
line creating a mirror image

relation a set of input and output values, {(1, 2), (3, 6), (5, 10)}

relative usually represented by ordered
frequency pairs

the number of times a result
happens divided by all possible
results

repeating a decimal number that has 0.444 = 4 = 0.4
decimal digits that repeat indefinitely; 9
the repeating digit is marked

with a bar notation

right triangle A triangle with one right angle
(90°) and two acute angles

rotation a turn; a figure rotated clockwise

or counterclockwise around an
angle or center point

Copyright © Swun Math 422Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

scale factor (k) the ratio of the side lengths 9
between the pre-image and
image of a dilation; also called 12
dilation factor

Center of dilation= (0,0)
Scale factor= ½

scatter plot a graph showing the (positive,

negative, or no) relationship
between two data sets

scientific a method to write very large or 99 = 9.9 × 101 3
notation very small numbers as the 9, 10
product of a number between 1
6, 7
and 10, and a power of 10

similar figures have same shape, but different
sizes; congruent angles and
proportional sides

slope (m) the steepness of a straight line;

also called rate of change;
calculated using rise over run
(change of y, over change of x)
or slope-intercept form

423 Copyright © Swun Math Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

slope-intercept a common way of writing a 6, 8,
12
form linear equation so that the
8
slope and y-intercept are
11
immediately apparent
1
solution to a system can have three 1
systems of possible solutions: 6
equations • no solution (parallel lines)
• one solution (intersecting lines)
sphere • infinite solutions (same line)

a three-dimensional figure
where every point on the surface
is the same distance from the
center; a round ball

square root a value that, when multiplied 16 = 4
by itself, gives the number
4, 9, 16, 25
square number the result after multiplying an
integer by itself; also called a
perfect square

standard form an equation where x and y are
on the same side of the equal
sign

straight angle an angle of 180° 10
8
substitution solves for one variable in the
method 424Grade 8 Cumulative Glossary TE
equation and then substitutes
that variable in the other

equation

Copyright © Swun Math

Grade 8 • Glossary

Term Definition Example Unit

supplementary two angles that add up to 180°; 2x — y = 2 10
2x + y = 6 8
angles both are adjacent angles 4

system of a set of two or more equations 4
equations with the same variables
5
tape diagram a visual model representing
9
parts of a whole; also called a 10
bar model

term a constant, variable, or a number 52x2 — 9x + 36 = 7m + 82
theorem and variable, multiplied in an
translation expression and separated by + or
transversal ─

a rule in math; a statement that 9 is a perfect square: 32= 9
can be proven Pythagorean Theorem:
a2 + b2 = c2

a slide; moving every point on a

shape the same distance and in
the same direction

two lines crossed by another
line

two-way table shows data from one sample 12

group as it relates to two
different categories

425 Copyright © Swun Math Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

unit rate a ratio showing a comparison 7m = 21; m is the variable 6
to one; a rate with a
unobserved denominator of 1 12
data
data that has not been 4
variable collected, but can be predicted
by using a line of best fit 10

a letter or symbol that
represents an unknown value

vertical angles angles opposite each other angles 2 and 4
angles 1 and 3
when two lines cross; are
congruent

vertical line uses a vertical line to check that 7
test points on the graph intersect at 11
exactly one point; visual way to
determine if a graph is a function

volume the amount of space a solid V= B · h
figure occupies; capacity Cylinder: V=
measured in cubic units Cone: V=

Sphere:

Copyright © Swun Math 426Grade 8 Cumulative Glossary TE

Grade 8 • Glossary

Term Definition Example Unit

whole number the natural numbers, including 0, 1, 2, 3, 4, ... 4
zero 12

x-intercept the point where a straight line 6, 12

crosses the x-axis; located at

(x, 0)

y-intercept (b) the point where a straight line
crosses the y-axis; located at

(0, y)

427 Copyright © Swun Math Grade 8 Cumulative Glossary TE

Resource Pages
Volume 1

428

Resource Page
Q1 Coordinate Grid

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Resource Page
Q1 Coordinate Grid

Copyright © Swun Math 430Grade 8 Volume 1 Resource Page TE

Resource Page
Four Coordinate Grid

10

9

8

7

6

5

4

3

2

1
0

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

431 Copyright © Swun Math Grade 8 Volume 1 Resource Page TE

Resource Page
Four Coordinate Grid

10 10

9 9

8 8

7 7

6 6

5 5

4 4

3 3

2 2

1 1
0 0

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10
-1

-2 -2

-3 -3

-4 -4

-5 -5

-6 -6

-7 -7

-8 -8

-9 -9

-10 -10

10 10
9
9 8
7
8 6
5
7 4
3
6 2
1
5
0
4 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1 1 2 3 4 5 6 7 8 9 10

3 -2
-3
2 -4
-5
1 -6
0 -7
-8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1 1 2 3 4 5 6 7 8 9 10 -9
-10
-2

-3

-4

-5

-6

-7

-8

-9

-10

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Resource Page
Graph Paper

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Resource Page
Base Ten Blocks

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Resource Page
Number Line (horizontal)

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Resource Page
Number Line (vertical)

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Resource Page
Colored Chips

Colored Chips front - copy back-to-back

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Resource Page
Colored Chips

Colored Chips back - copy back-to-back

Copyright © Swun Math 438Grade 8 Volume 1 Resource Page TE

Resource Page
Algebra Tiles

─1 ─1 1 1
─1 ─1 1 1

11

─x ─x 1 1

11
11
11

─x ─x 1 1

11

x
x
x2
─x2

─1 x
─1 x
x x2
x ─x2

439 Copyright © Swun Math Grade 8 Volume 1 Resource Page TE