I hope that you find this resource helpful in your classroom. Please feel free to contact me with any questions as you implement this in your class.
Maneuvering the Middle is an education blog with valuable tips for lesson planning, teacher organization, and math concepts in the middle school classroom.
Print and copy this resource for your personal classroom use only.
Save this to both home and school computers.
Only post this only for students on a password protected class website.
Reproduce or distribute this resource to other colleagues.
Post this on the internet in any form, including classroom/personal websites, network drives, or other sharing websites (i.e. Amazon Inspire, etc).
Products by Maneuvering the Middle, LLC may be used by the purchaser for their classroom use only. This is a single classroom license only. All rights reserved. If you wish to share this product with your team or colleagues, you may purchase
Note: This is a strategy for moving information from short-term to long-term memory. I usually ask students to write 2-3 sentences or bullet points summarizing what they’ve learned.
Unit: Linear EquationsHomework 1
Name _____________________________________Date _____________________________Pd______
SIMPLIFYING EXPRESSIONSMatch each expression with the correct simplified expression. Use the corresponding letter with each solution to help you solve the riddle.
Solve the following two-step equations, and show all work.
Write an equation that could be used to represent each situation. Then, solve the equation.
14. On a beach trip, Lucy rents a bike from “Wheels by the Waves”, where they rent bikes for $12 plus $3 per hour. If Lucy spent $30, how many hours (h) did she rent a bike?
Equation:__________________________
Solution:__________________________
15. Paul is scuba diving and is 3.5 feet below sea level. He is descending at a rate of 0.5 feet per minute. If Paul is now at 12 feet below sea level, how many minutes has he been diving?
2. Which of the following is a solution to the equation 3 – 4y = 19?
525
a. y = -19b. y = 20c. y = 4d. y = -4
3. Solve.
7m – 17 = 60
_______________
4. Solve.
c
−9+ 6 = 14
5. Solve.
18 = 5m + 3
_______________
6. Solve.
4
3y = 16
_______________
7. Solve.
w
−2.5= 8
8. Solve.
1
5x − 2 = 4
_______________
9. The water temperature at the beach started at 82 degrees, and it is rising 0.6 degrees each hour. If the water temperature is now 85 degrees, write and solve an equation to find h, the number of hours that have passed.
Equation: _________________________
Solution: _________________________
10. While at the beach, Daniel buys lunch for his family from a food stand. He purchases one hot dog for $2.50 and 3 hamburgers. If he spent $13 total, write and solve an equation to find h, the amount each hamburger cost.
Equation: _________________________
Solution: _________________________
Unit: Linear EquationsHomework 2
ONE AND TWO-STEP EQUATIONS
Name _____________________________________Date _____________________________Pd______
1. Calvin goes shopping over the weekend and buys a new pair of shoes, a t-shirt, a pair of sunglasses, and a hat that costs $9. The shoes are twice as much as the shirt, and the sunglasses are half as much as the shirt. The total cost of his purchase is $65.
9. The perimeter of the rectangle below is 65 inches.
a. What is the value of x? ___________
b. What are the dimensions of the rectangle? ________________
10. James bought a cheeseburger, fries and a drink for dinner. The cheeseburger was three times the price of the fries, and the drink and the fries were the same price. If the entire meal cost $12.50, what was the price of each item?
Name _____________________________________Date _____________________________Pd______
Each of the equation cards on the left has the same solution as one of the equation cards on the right. Find the matching equations to complete the sentences below.
1. Card A and Card _______ have the same solution of x = ________.
2. Card B and Card _______ have the same solution of x = ________.
3. Card C and Card _______ have the same solution of x = ________.
4. Card D and Card _______ have the same solution of x = ________.
5. Card E and Card _______ have the same solution of x = ________.
Name _____________________________________Date _____________________________Pd______
1.8x – 7 = 4x + 21
_______________
2.-11x + 9 = -2x + 45
_______________
3.13x + 9 = 6x – 12
_______________
4.14x + 1 = 8x + 13
_______________
5. Jane is going to solve the equation shown below:
8x – 19 = -12x + 61
Jane thinks the first step should be to subtract 12x from both sides of the equation, but Allie thinks she should add 12x to both sides of the equation. Who is correct, and how do you know?
Solve the equation:
_______________
Combine
variables
constants
Solve
x = -3 x = 2
Answer Key
*Note: I often tell students to keep the variables on the side that has “more” variables to try and keep the variable positive. Some teachers prefer to always have students collect variables on the left. My advice is to pick what works for you and stay consistent for your students!
Allie is correct. In order to remove -12x from the right, you would need to add 12x to create a zero pair, or cancel it out.
x = 4
x = 7 x = -4
Solve the following equations with variables on both sides.
11. The perimeter of the rectangle and the triangle shown are the same. Write an equation to represent the situation, and solve for x. Then, find the perimeter of each shape.
Solve the problems below, and be sure to show all work.
Unit: Linear EquationsHomework 4
EQUATIONS WITH VARIABLES ON BOTH SIDES
Name _____________________________________Date _____________________________Pd______
1.22 + 54x = -20 + 60x
2.-5x – 16 = 8x – 3
______________
3.-2x – 10 = -14x – 7
______________
4.
2x – 1 = 34x + 9
5.-8.75x + 14 = -5.25x – 3.5
______________
6.8 + 9.8x + 9 = 3.8x – 16 + 3.8x
______________
7. The steps below show Cam’s work in solving an equation. Circle the line where he makes a mistake. Then, find the correct solution to the equation.
______________
8. The perimeter of the square and the equilateral triangle shown are the same. Write an equation to represent the situation, and solve for x. Then, find the perimeter of each shape.
Name _____________________________________Date _____________________________Pd______
Let’s start by reviewing different ways we can state mathematical operations. List as many synonyms that you can think of in each section below. Then, use the keywords to help you solve the following real-world problems.
KEY
WO
RDS
ADDITION: SUBTRACTION:
MULTIPLICATION: DIVISION: EQUALS:
I KNOW: I NEED TO KNOW:
PLAN AND WORK: MY SOLUTION:
1. Hudson and Knox are in a race. Hudson is running at a speed of 8.8 feet per second. Knox got a 30-foot head start and is running at a speed of 6.3 feet per second. How many seconds will it take until Hudson and Knox have run the same number of feet?
Husdon’s expression is 8.8x and Knox’s is 6.3x + 30.
The value for x that will make the expressions equal to one another.
8.8x = 6.3x + 30x = 12 seconds
Same
More thanIncreased bySumGain
AltogetherTotalIn allCombined
Less thanDecreased byDifferenceLoss
TimesEachPer
ProductTwice
HalfSplit
QuotientShared equally
Is
Answer Key
Fewer
Use the keywords to help you set up and solve an equation for each of the situations below.
SITUATION EQUATION & SOLUTION
2. Celebrity A has 400 followers on social media and is gaining 75 followers each day. Celebrity B has 1,000 followers on social media, but is losing 25 followers a day. How many days will it take them to have the same number of followers?
Equation: ____________________________
x = _____________
3. Ben and his friends are going to play putt-putt. At one location, they can pay $2.00 for rentals plus $7.50 per game. At another location, it would cost $8.00 for rentals plus $6.00 per game. How many games would they have to play in order for either location to cost the same amount?
Equation: ____________________________
x = _____________
4. Hayley has $155 in savings, and her brother Hank has $230. Hayley is saving $10 each week, and her brother is spending $15 each week. After how many weeks will their savings accounts have the same amount of money in them?
Equation: ____________________________
x = _____________
5. Isabelle takes the bus to work. The bus ride to work costs her $2.50 each time. She could buy a bus pass for a one-time fee of $15.00 and would then only pay $1.00 each time to ride the bus. How many times would she have to take the bus in order for the cost to be the same?
Equation: ____________________________
x = _____________
6. Thirty-five more than .8 times a number is the same as 43 less than the product of -7 and the number.
1. Graham is hiking at an altitude of 14,040 feet and is descending 50 feet each minute. Max is hiking at an altitude of 12,500 feet and is ascending 20 feet each minute. How many minutes will it take Graham and Max to meet at the same altitude?
a. Equation: ________________________________
b. Solution: _________________________________
c. At what altitude will Graham and Max meet?
________________
2. Nick and Adam workout together. Nick weighs 160 pounds and is gaining about 3 pounds per week. Adam weighs 195 pounds and is losing about 2 pounds each week. Write an equation that can be used to find the number of weeks that it will take them to weigh the same amount.
a. Equation: ________________________________
b. Solution: _________________________________
c. How much will Nick and Adam weigh?
________________
3. Jaycie has a VIP membership to a movie theater, so she pays $27 a year and $6.00 for each movie she sees. Claire doesn’t have a membership, so she pays $8.25 each movie she sees. How many movies would they have to see in a year in order to pay the same amount that year?
a. Equation: ________________________________
b. Solution: _________________________________
c. How much will Jaycie and Claire pay for the year?
Name _____________________________________Date _____________________________Pd______
14,040 – 50x = 12,500 + 20x
x = 22
12,940 feet
27 + 6x = 8.25x
x = 12
$99
160 + 3x = 195 – 2x
x = 7
181 pounds
x + 2 = 78
x – 15
x = -136
Answer Key
5. Use the equation shown to answer a-b.
0.75x + 3 = 0.25x + 5.25
a. Can the equation above be used to solve the following problem? If not, explain why.
“At an ice cream shop, customers pay $3.00 for a medium ice cream plus $0.75 per topping. At a second ice cream shop, customers pay $5.25 for a medium ice cream plus $0.25 per topping. How many toppings would a customer have to order for the ice cream to cost the same at either shop?”
b. Can the equation above be used to solve the following problem? If not, explain why.
“Joseph has $3.00 in his piggy bank and he adds $0.75 each day. Kelsey has $5.25 in her piggy bank and she spends $0.25 each day. When will Joseph and Kelsey have the same amount in their
piggy banks?”
Yes.
No; because Kelsey is spending money each day, the equation should be 0.75x + 3 = -0.25x + 5.25.
Note: 8.8A does not specify that students solve the inequality, however I included it in the notes as it is helpful to reinforce the skills needed to solve equations with variables on both sides.
x ≥ 123 x ≤ 8x < 7
x ≤ 250 x > 56.75 x ≥ 22
8. To have a birthday party at an ice skating rink, Lizzy would pay $40 plus $18.25 per person. To have a party at a trampoline park she would pay $124 plus $7.75 per person. How many people would Lizzy need to have at her party in order for the cost of a party at the ice skating rink to be more than a party at the trampoline park?
Inequality: ________________________________
Solution: ________________________________
9. In Neil’s city, the temperature is 72.6 degrees and is rising 1.3 degrees each hour. In Sheppard’s city, the temperature is 80.4 degrees and is dropping 1.82 degrees each hour. In how many hours will the temperature in Neil’s city be no less than the temperature in Sheppard’s city?
Inequality: ________________________________
Solution: ________________________________
10. Choose the inequality that could be used to solve the problem below:
Sarah plans to sell decorative signs at an upcoming craft market. It costs her $3.50 to make each sign, and she paid a one-time fee of $185 to rent a booth at the market. If Sarah sells each sign for
$15.75, how many signs will she need to sell in order for her expenses to be no more than her earnings?
11. In a-c, determine if the description matches the inequality below. If not, explain what should be corrected.
213x + 8 ≥ 11
6x + 1
a. “8 more than the quotient of 213
and a number is greater than or equal to the quotient of 116
and
the same number increased by 1.”
b. “213
times a number plus 8 is at most 1 more than 116
times the same number.”
c. “8 more than the product of 213
and a number is no less than 1 more than the product of 116
1. Clint has to spend a minimum of $75 in order to use his coupon at the store.
_________________
2. Molly went shopping and her bank account got below $7.50.
_________________
3. Sharon’s organization hopes to raise more than $10,000 at their annual 5k race.
_________________
4. Mrs. Johnson informed her students that their exam will have at most 25 questions.
_________________
In 5-6, choose the inequality that could be used to solve each problem.
5. Caroline’s hair is 7 inches long and grows about 15
inch each month. Trinity’s hair is 10 inches
long and grows about 110
inch each month. How many months will it take for Caroline’s hair to be at
least as long as Trinity’s hair?
6. A puppy weighs 2.8 pounds at birth and gains about .5 pounds per week. A kitten weighs 1.4 pounds at birth and gains about 1.2 pounds per week. In how many weeks will the weight of the kitten be no less than the weight of the puppy?
In 7-8, write an inequality that could be used to solve each problem.
7. Tess has 42 words of an essay typed and is typing about 1.3 words per second. Margo has 122 words of an essay typed and is erasing about 1.2 words per second. In how many seconds will the number of words in Tess’s essay exceed the number of words in Margo’s essay?
_______________________________
8. Hank has picked 1.3 pounds of berries and picks about .75 pounds per minute. Tom has picked 2.5 pounds of berries and picks about .5 pounds per minute. In how many minutes will Hank have picked at least as many pounds of berries as Tom?
Name _____________________________________Date _____________________________Pd______
In 1-4, write an inequality to represent each situation.
A. 7x + 15
≥ 10x + 110
C. 7 + 15x > 10 +
110
x
B. 7 + 15x ≥ 10 +
110
x D. 7 + 15x ≤ 10 +
110
x
A. 1.4 + 1.2x > 2.8 + .5x C. 1.4 + 1.2x ≤ 2.8 + .5x
B. 1.4 + 1.2x < 2.8 + .5x D. 1.4 + 1.2x ≥ 2.8 + .5x
Answer Key
42 + 1.3x > 122 – 1.2x
1.3 + .75x ≥ 2.5 + .5x
x ≥ 75 x < 7.50
x > 10,000 x ≤ 25
Name _____________________________________Date ____________________________ Pd ______
Unit: Linear EquationsReview
LINEAR EQUATIONS STUDY GUIDESolve each of the problems below. These represent the types of questions on your test. Be sure to ask questions if you need more help with a topic.
I CAN WRITE AND SOLVE LINEAR EQUATIONS. 8.8C
1. Solve the equation:
x−3.5
+ 33 = 30
2. Solve the equation:
17 – 19x = 10
3. A sprout in Jessie’s garden is 515
cm tall and
is growing 1310
cm each week. Write and solve
an equation in order to determine how many weeks it will take for the plant to be 13 cm tall.
Equation: ___________________
Solution: __________________
4. An acorn is falling from 13 feet in the air. If it descends 4.8 feet every second, write and solve an equation to determine how long it will take for the acorn to be 1 foot above the ground.
Equation: ___________________
Solution: __________________
I CAN WRITE AND SOLVE MULTI-STEP EQUATIONS. 8.8C
5. Solve the equation:
-4 + w + 17 – 23w = 16
6. Solve the equation:
-12c – 4 + 7.5c – 11 = -6
7. The perimeter of the rectangle shown is 36 inches. Write and solve an equation to find the value of x.
I CAN SOLVE EQUATIONS WITH VARIABLES ON BOTH SIDES. 8.8C
8.8x – 4 = 12x + 6 + x
9.-2 – 0.5b + 12 = 0.3b + 6.2
10.23x – 24 =
16x – 6
11.5.9 – m = -4.5 – 5m
12. Write and solve the equation modeled below:
Equation: ____________________
Solution: ____________________
13. Hector is solving an equation with variables on both sides, and his work is shown below. Explain Hector’s mistake in his work, and find the correct solution to the equation.
-x-1
=
x
x
x
x
-1
-1
-1
-1
-1
-x
-x
-x
-124 + 478x = -9 – 1
8x
-124 + 468x = -9
468x = 115
x = 24.2…
x = -2 b = 4.75
x = 36 m = -2.6
4x – 4 = -4x – 2
x = 1/4
Hector did not combine the variables correctly. He should add 1/8x to both sides, which would give him 5x
I CAN WRITE INEQUALITIES THAT REPRESENT REAL-WORLD PROBLEMS. 8.8A
17. Vera is currently using 16,000 MB of storage on her phone, and she is deleting 2.2 MB of music each minute. Keira is currently using 11,275 MB of storage on her phone, and she is adding 16.7 MB of music each minute.
Write an inequality that could be used to find the number of minutes it will take for Vera to be using no more than the amount of storage that Keira is using.
__________________________________
18. Xander’s water bottle has 15.4 oz of water in it and Xander is drinking 0.5 oz of water each second. Leon’s water bottle has 5.4 oz of water in it and Leon is adding 0.75 oz of water each second.
Write an inequality that could be used to find the number of seconds it will take for Xander’s water bottle to have a fewer number of ounces than Leon’s water bottle.
__________________________________
I CAN WRITE AND SOLVE EQUATIONS THAT REPRESENT REAL-WORLD PROBLEMS. 8.8A
14. Marshall earns a salary of $36,000, and each year he receives a $4,000 raise. Jim earns a salary of $51,000, and each year he receives a $1,500 raise. How many years will it take before Marshall and Jim earn the same salary?
Equation: _________________________________
Solution: ____________________________
15. The square and the equilateral triangle below have the same perimeter. Find the value of x. Then, find the perimeter of each shape.
Equation: ____________________________
Value of x: ____________________________
Perimeter: ____________________________
16. Peter’s suitcase weighs 1425
pounds, and he adds 35
pounds each minute. William’s suitcase
weighs 48 pounds, and he removes 45
pounds each minute. After how many minutes will the
I CAN WRITE REAL-WORLD PROBLEMS GIVEN AN EQUATION. 8.8B
19. For a-b, determine whether or not the equation below correctly represents each situation. If so, write “yes”. If not, write “no” and write the correct equation for the situation described.
12 – 110
x = 2 + 110
x
a. Jill’s bowl of pudding contains 12 ounces, and she eats 110
of an ounce each second. Barb’s
bowl of pudding contains 110
of an ounce, and she adds 2 ounces each second. After how
many seconds will the two bowls contain the same amount of pudding?
b. Raven runs 12 miles at the beginning of the week and decreases the length of her run by 110
of a mile each day after that. Josh runs 2 miles at the beginning of the week and increases
the length of his run by 110
of a mile each day after that. After how many days will Raven and
Josh run the same amount?
I CAN WRITE REAL-WORLD PROBLEMS GIVEN AN INEQUALITY. 8.8B
20. For a-b, determine whether or not the inequality below correctly represents each situation. If so, write “yes”. If not, write “no” and write the correct inequality for the situation described.
100 + 23.75x ≤ 10 + 43.75x
a. Gym A charges members $100 to sign up plus $23.75 each month. Gym B charges members $10 to sign up plus $43.75 each month. How many months would a member need to work out for Gym A to be less expensive than Gym B?
b. Moving Company A charges a fee of $23.75 plus $100 per hour. Moving Company B charges a fee of $43.75 plus $10 per hour. How many hours would it take for Moving Company A to be no more expensive than Moving Company B?
No; 12 – 110
x = 110
+ 2x
Yes; the equation correctly represents the situation.
2. The perimeter of the equilateral triangle shown is 67.5 inches. Write and solve an equation to find the value of x.
1
5
_______________
3. Austin solved the equation as shown below. Which describes the error that Austin made in his work?
4. Erin weighs 135 pounds and is gaining 12
pound each week. Miranda weighs 143 pounds and is
losing 14
pound each week. Which of the following equations could be used to determine x, the
number of weeks that it will take until the two girls weigh the same amount?
5. A video game store allows customers to rent games for $4.75 each. Customers can also buy a membership for $54 annually, and video games would only cost $2.50 each. Write and solve an equation to find the number of video games a customer would have to rent in a year in order for the two options to be equal.
Equation: ____________________________
Solution: ____________________________
Name _____________________________________Date ____________________________ Pd ______
INSTRUCTIONS: Solve the problems below. Be sure to show your thinking.
8. Identify the equation modeled below, and then find its solution.
A. Equation: -4x + 6 = -2x – 2; Solution: x = 4
B. Equation: 6x + 4 = -2x + 2; Solution: x = -14
C. Equation: 6x – 4 = -2x – 2; Solution: x = -32
D. Equation: 6x – 4 = -2x – 2; Solution: x = 14
9. Joe’s Catering Company charges a $45 set-up fee plus $12.50 per person. Jimmy’s catering company charges a $75 set up fee plus $7.50 per person. Which inequality could be used to determine x, the number of people for which Joe’s Catering Company would be less expensive than Jimmy’s catering company?
A. 45 + 12.5x > 75 + 7.5x
B. 45 + 12.5x ≥ 75 + 7.5x
C. 45 + 12.5x < 75 + 7.5x
D. 45 + 12.5x ≤ 75 + 7.5x
10. Keith is in a hot air balloon that is 200 feet
above the ground and is ascending 112
feet per
second. Erica is in a hot air balloon that is 548
feet above the ground and is descending 125
feet per second. After how many seconds will the hot air balloons be the same number of feet above the ground?
A. 52 seconds
B. 120 seconds
C. 75 seconds
D. 3,480 seconds
11. The rectangles shown below have the same perimeter. Which of the following is the value of x?
A. x = 15
B. x = 12
C. x = -2.64
D. x = 3.25
Solve the problems below. Be sure to show your thinking.
12. Which of the following is the value of x needed to make the equation true?
-18 + 3x + x = -4x + 50
A. x = 11.3
B. x = 4.5
C. x = 8.5
D. x = -4.5
13. Solve the equation below.
-314x + 25 = 2
14x – 13
12
x = _______________
14. Mike is driving a boat at 12 mph and is increasing his speed by 0.2 mph each second. Hank is driving a boat at 25 mph and is decreasing his speed by 0.7 mph each second. Which of the following equations could be used to determine x, the number of seconds it will take for the boats to be traveling the same speed?
15. Norah has never lost a tooth, but she will receive $0.75 from the tooth fairy each time she loses a tooth. Kate has $1.25 from the tooth fairy, and she gets $0.25 each time she loses a tooth. Which inequality could be used to find x, the number of teeth that each girl would have to lose for Norah to have at least as much money from the tooth fairy as Kate?
A. 0.75x ≥ 1.25 + 0.25x
B. 0.75x ≥ 1.25 – 0.25x
C. 0.75x ≤ 1.25 + 0.25x
D. 0.75x > 1.25 + 0.25x
16. Which of the following equations does not have a solution of x = 0.5?
A. 10x – 5 = -2x + 1
B. -2x + 6 = -6x + 8
C. -5x + 10 = x + 7
D. 3x + 3 = -2x + 13
17. Edith’s bird feeder contains 412
pounds of bird seed and decreases by 34
pound each day.
June’s bird feeder contains 3 pounds of bird seed and decreases by 14
pound each day. If
neither person adds or removes seed from the bird feeder, how many days will it take for Edith and June’s bird feeders to contain the same amount of bird seed?
_______________
Solve the problems below. Be sure to show your thinking.
18. Find the value of x needed to make the equation true.
-7.2x + 3.5 = -8.7x – 10
x = _______________
19. A tree in Micah’s yard has 120 leaves and loses an average of 4.5 leaves each day. A tree in Heather’s yard has 45 leaves and gains an average of 0.5 leaves each day. Write and solve an equation to find the number of days it will take for both trees to have an equal number of leaves.
Equation: ______________________________
Solution: ______________________________
20. Which of the following represents a problem that could be solved by using the inequality below?
3 + 13x > 8 – 2
3x
A. The ground near Brad’s house is covered with 3 inches of snow and gains 13
of an inch every
hour. The ground near Wendy’s house is covered with 8 inches of snow and loses 23
of an
inch every hour. After how many hours will the ground near Brad’s house be covered with more snow than the ground near Wendy’s house?
B. A snail is 3 inches up a tree and climbs 13
inch higher every minute. A beetle is 8 inches up a
tree and climbs 23
inch lower every minute. After how many minutes will the snail be above the
beetle?
C. Both A and B can be solved using the inequality.
D. Neither A or B can be solved using the inequality.
Solve the problems below. Be sure to show your thinking.
