Lesson 6.2.3
HW: Day 1: Problems 6-81 to 6-86
Day 2: Problems 6-87 to 6-92
Learning Target: Scholars will use formal notation while simplifying expressions and solving equations.
They will compare arithmetic and algebraic methods for solving problems.
In this lesson, you will continue to improve your skills of simplifying and solving more complex
equations. You will develop ways to record your solving strategies so that another student can understand
your steps without seeing your Equation Mat. Consider these questions as you work today:
 How can I record the steps I use to solve?
 How can I record what is on the Equation Mat after each step?
6-71. Gene and Aidan were using algebra tiles to solve equations. Aidan was called away.
 Help Gene finish by completing the table shown below and on the Lesson 6.2.3 Resource Page.
6-72. Aidan was frustrated that he needed to write so much when solving an equation. He decided to
come up with a shortcut for recording his work to solve a new equation.
As you look at Aidan’s recording at right of how he solved 2x + 4 = −12 below,
visualize an Equation Mat with algebra tiles. Then answer the questions for each step below.
1. What legal move does writing –4 twice represent?
2. What legal move does circling the +4 and the –4 represent?
3. What does the box around the
represent?
4. Why did Aidan divide both sides by 2?
5. Check Aidan’s solution in the original equation. Is his solution correct?
6-73. The method of recording the steps in the solution of an equation is
useful only if you understand what operations are being used and how they relate to the legal moves on
your Equation Mat.
Find the work shown at right on your resource page for this lesson.
1. For each step in the solution, add the missing work below each line that shows what legal moves
were used. You may want to build the equation on an Equation Mat.
2. Check that the solution is correct.
6-74. For each equation below, solve for x. You may want to build the equation on your Equation
Mat. Record your work in symbols using Aidan’s method from problem 6-72. Remember to check your
solution.
1. −2x + 5 + 2x − 5 = −1 + (−1) + 6x + 2
2. 3(4 + x) = x + 6
6-75. Oliver is building a train depot for his model railroad. As his final step, he needs to apply rain
gutters around the roof of the rectangular building. He has 52 cm of rain gutters. The length of the
depot is 19 cm. Explore how Oliver can find the width of the depot by answering the questions below.
1. Find the width of the depot using arithmetic (that is, solve the problem without using any
variables). Record your steps.
2. Use w to represent the width of the depot. Write an algebraic equation that shows the
perimeter is 52 cm, and solve your equation. Record your steps.
3. Which method, the arithmetic or algebraic, did you prefer? Why?
6-78. Maggie’s mom agrees to let Maggie buy small gifts for some of her friends. Each gift costs
$4. Maggie’s mom gave her a budget of $19. When Maggie went online to order the gifts, she
discovered there was a $7 shipping fee no matter how many gifts she bought.
1. Use arithmetic (without variables) to determine how many gifts Maggie can buy. Record your
steps.
2. Write an algebraic equation to determine how many gifts Maggie can buy with $19, and solve
your equation. Record your steps.
3. Compare and contrast the two methods of solving the problem.
6-79. Your teacher will explain the way you will be working on solving the equations below for x. You
may want to build the equations on your Equation Mat. Record your work and check your solution.
1.
2.
3.
4.
2(x + 1) + 3 = 3(x − 1)
−2x − 2 = 3(−x + 2)
3 + 4(2 − x) = 3x + (−x) − 7
6(3 − x) + (−20) = 10 + 3(4x + 2)
6-81. Solve each equation below for x. Check your final answer.
1. 4x = 6x − 14
2. 3x + 5 = 50
6-82. Forty percent of the students at Pinecrest Middle School have a school sweatshirt. There are 560
students at the school. Draw a diagram to help you solve each problem below.
3. How many students have a school sweatshirt?
4. If 280 students have school t-shirts instead of sweatshirts, what percentage of the
school has a t-shirt?
5. What percentage of the school does not have a t-shirt or a sweatshirt?
6-83. Latisha wants to get at least a B+ in her history class. To do so, she needs to have an overall
average of at least 86%. So far, she has taken three tests and has gotten scores of 90%, 82%, and 81%.
6. Use the 5-D Process to help Latisha determine what percent score she needs on the
fourth test to get the overall grade that she wants. The fourth test is the last test of the
grading period.
7. The teacher decided to make the last test worth twice as much a regular test. How does
this change the score that Latisha needs on the last test to get an overall average of
86%? Support your answer with mathematical work. You may choose to use the 5-D
Process again.
6-84. Factor each expression. That is, write an equivalent expression that is a product instead of a sum.
8. 20y − 84
9. 24b2 + 96b
6-85. Copy and complete each of the Diamond Problems below. The pattern used in the Diamond
Problems is shown at right.
a.
b.
c.
d.
6-86. A cattle rancher gave
of his land to his son and kept the remaining
for himself. He kept
34 acres of land. How much land did he have to begin with? 6-87. Solve each equation. Record your
work and check your solution.
10. 5(x − 2) + (−9) = −7(1 − x)
11. −6x − 7 = −1(−9 + 2x)
6-88. Simplify each expression.
12.
13. 0.4 · 0.3
14. −
15.
16.
17.
6-89. Sketch the parallelogram shown at right, and then redraw it
with sides that are half as long. 6-89 HW eTool (Desmos) Homework Help ✎
18. Find the perimeters of both the original and smaller parallelograms.
19. If the height of the original parallelogram (drawn to the side that is 6 units) is 2 units,
find the areas of both parallelograms.
6-90. Evaluate the expression 10 – 2x for the x-values given below.
20. x = 2
21. x =
22. x = –2
6-91. Set up a four quadrant graph and graph the points below to make the four-sided shape PQRS. 6-91
HW eTool (Desmos).
 P(–2, 4) Q(–2, –3) R(2, –2) S(2, 3)
1. What shape is PQRS?
2. Find the area of the shape
6-92. Write and solve an inequality for the following situation.

Robert is painting a house. He has 35 cans of paint. He has used 30 cans of paint on the
walls. Now he needs to paint the trim. If each section of trim takes
can of paint, how many
sections of trim can he paint? Show your answer as an inequality with symbols, in words, and
with a number line. Make sure that your solution makes sense for this situation.
Lesson 6.2.3

6-71. See answers in bold in the table below:

6-72. See below:
1. To remove four ones from each side or to add four negative unit tiles to both
sides.
2. Forming a zero pair.
3. A Giant One.
4. To have only one positive x-tile remaining so that the remaining unit tiles on the
other side will be the solution.
5. 2(−8) + 4 = −12. Yes.
6-73. See below:
1. Students should show the terms that were combined in the first step, adding four
to each side, subtracting 3x from each side, and dividing each side by 4.
2. 2 + (−4) + 6(2) = 3(2) − 1 + 5; 10 = 10
6-74. See below:
1. x = 0
2. x = −3
6-75. See below:
1. 52 − 19 − 19 = 14 ÷ 2 = 7 cm, width = 7 cm
2. 19 + 19 + 2w = 52; w = 7 cm
3. Answers vary.



 6-78. See below:
1. 19 − 7 = 12 ÷ 4 = 3 gifts
2. 4g + 7 = 19, g = 3 gifts
3. Possible answer: when using arithmetic you have to think of the end and work
backwards; it is easy to mix up the order when working backwards, and harder to explain
your steps; with algebra you first have to write a variable for the unknown, but then
simplifying the equation and recording steps is a little easier.
 6-79. See below:
1.
2.
3.
4.
x=8
x=8
x=3
x = −1

6-81. See below:
1. x = 7
2. x = 15

6-82. See below:
1. 224 students
2. 50%
3. 10%

6-83. See below:
1. 91%
2. 88.5%

6-84. See below:
1. 4(5y − 21)
2. 24b(1b + 4)


6-85. See answers in bold in the diamonds below:



6-86. 51 acres
6-87. See below:
1. x = −6
2. x = −4
6-88. See below:
1.
2. 0.12
or
3.
4. 13
5.


or –1
6. −1.8 or −1
6-89. See below:
1. P = 17 units, P = 8.5 units
2. A = 12 square units, A = 3 square units
6-90. See below:
1. 6
2. 9
3. 14

6-91. See below:
1. Trapezoid
2. 24 square units

6-92.
p + 30 ≤ 35, 0 ≤ p ≤ 10, less than or equal to 10 sections but greater than or
equal to 0 sections, since there cannot be a negative number of trim pieces. See diagram
below.