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PowerTeaching Math 3rd Edition Level 6 Unit 7 Cycle 1 Lesson 1 © 2014 Success for All Foundation Homework Problems 1

Homework Problems

Name

Team Name Team Complete?

Team Did Not Agree On

Questions…

#’s

Quick Look

Write the vocabulary introduced in this cycle:

Today we learned about exponents. Exponents are a way to rewrite expressions that multiply the same

number more than once. For example, we can rewrite 4 × 4 × 4 × 4 × 4 as 45 because we are using 4 as a

factor 5 times. We also see exponents in some formulas.

We find the area of a square by multiplying length times width.

On a square, the length and the width are the same size. So our formula is A = s × s. That’s

the same as A = s2. What is the area of this square?

A = s2

A = (5 in.)2 = 5 in. × 5 in. = 25 in.

2

____________________________________________________________________________________

Directions for questions 1 and 2: Rewrite using an exponent.

1) 12 × 12 × 12 × 12

2) 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7

PowerTeaching Math 3rd Edition Level 6 Unit 7 Cycle 1 Lesson 1 2 ©2014 Success for All Foundation Homework Problems

Directions for questions 3 and 4: Write the prime factorization using exponents.

3) 64

4) 600

Directions for questions 5–8: Show the math in the story using multiplication and an exponent.

5) The zoo has 4 lions. Each lion eats 4 meals a week. Each meal has 4 slabs of meat. How many slabs are eaten in all in one week?

6) There are 8 artists in the class.

Each artist drew 8 pictures.

There are 8 aliens in each picture.

Each alien has 8 arms.

Each arm has 8 fingers.

How many fingers in all?

7) There are 7 people playing a game.

Each person gets 7 clues.

Each clue has 7 facts.

Each fact is 7 words.

How many words in all?

8) 10 teams were competing in a contest.

Each team had 10 people on it.

Each person had to attempt 10 events.

Each event had 10 obstacles.

How many obstacles in all?

Directions for questions 9–12: Find the area of the square using the formula A = s2

9)

10)

11)

12)


Mixed Practice

13) Solve.

18 – 9 ÷ 3 × 2

14) There are 129 balloons at a party and 43

children in attendance. How many balloons

does each child receive if the balloons are

shared equally?

15) Charlene is paid $23.75 for every 3 hours

she babysits. How much would Charlene

earn if she babysat for 5.5 hours?

16) Write the integer to represent this phrase:

343 dollars in debt.

Word Problem

17) Giana’s garden is shaped like a square. Each side of the garden measures 12 feet. If Giana wants to

cover the garden with a layer of new potting soil, what is the area that she needs to cover? Explain

your thinking.

PowerTeaching Math 3rd Edition Level 6 Unit 7 Cycle 1 Lesson 1 4 ©2014 Success for All Foundation Homework Problems

For the Guide on the Side

Today your students learned about exponents. We use exponents as a shorter way to write multiplication

sentences that include the same variable or number. We would rewrite 3 • 3 • 3 • 3 • 3 • 3 as 36.

Exponents help us with measurements as well. We find the area of a square using the formula A = s2,

and we find the volume of a cube using the formula V = s3. Additionally, the units for area are always

units2, like cm

2 or in.

2. And the units for volume are always units

3, like mm

3 or ft

3.

Your student should be able to answer these questions about exponents.

1) Explain how you used an exponent to rewrite this problem.

2) What is the formula to find the area of a square; how is it different from the formula to find the area of a rectangle?

3) What is the exponent in this number sentence, what does it mean?

4) If a number is being squared, what exponent does that mean? What about cubed?

Here are some ideas to work with exponents with your student.

1) Write your own exponent stories, like the first problem of the homework assignment, about

your family and friends.

2) Find examples of squares in your house and measure an edge. Then find the area using the

formula A = s2.

3) Use Khan Academy to review exponents by watching helpful videos:

http://www.khanacademy.org/math/arithmetic/basic-exponents/v/understanding-exponents

4) Or to practice exponent problems:

http://www.khanacademy.org/math/algebra/exponents-radicals/e/exponents_1


Homework Answers

1) 124

2) 78

3) 26 4) 2

3 × 5

2 × 3

5) The lions eat 4 × 4 × 4 or 43 slabs of meat

each week. 6) There are 8 × 8 × 8 × 8 × 8 or 8

5 fingers in

all.

7) There are 7 × 7 × 7 × 7or 74 words in all. 8) There are 10 × 10 × 10 × 10 or 10

4

obstacles in all.

9) 784 ft2

10) 153.76 cm2

11) 219.04 mm2

12) 324 in.2

Mixed Practice

13) 12 14) Each child will get 3 balloons.

15) Charlene would make $43.54 for babysitting for 5.5 hours.

16) –343

Word Problem

17) Giana needs to cover 144 ft2. I multiplied, using the formula for area:

A = s2 = (12 ft)

2 = 12 ft × 12 ft = 144 ft

2. I built a math model to help me solve this problem (TLM

practice #4) because I used the formula for finding the area of a square.


Homework Problems

Name



Questions…

#’s

Quick Look

Vocabulary words introduced in this cycle:

exponent, squared, cubed, evaluate, numeric expression, Order of Operations

Today we used the Order of Operations to find the value of expressions. The Order of Operations is the

set of rules that mathematicians have agreed upon for which operations to do first, second, and so on.

Here’s an example; let’s use PEMDAS to help us find the value of this expression.

5 × (27 – 6) + 43 ÷ 2

Parentheses � 5 × 21 + 43 ÷ 2

Exponents � 5 × 21 + 64 ÷ 2

Multiplication & � 105 + 64 ÷ 2

Division 105 + 32

Addition & � 137

Subtraction

Directions for questions 1–8: Evaluate the expression.

1) 34 – 4 × (4 + 3)

Explain your thinking.

2) 17 + 82 × 0.25


Remember, we do Multiplication and

Division in order from left to right.

Then we do Addition and

Subtraction in order from left to right.

PowerTeaching Math 3rd Edition Level 6 Unit 7 Cycle 1 Lesson 2 2 © 2014 Success for All Foundation Homework Problems

3) 2 + (42

× 2

1 ) – 7

4) 10 + (2 × 6 ÷ 4)

5) 4

3 × (8 + 4) – 10

6) (3 + 24) ÷

2

1

7) 12 + (72 – 15 ÷

3

1 )

8) 100 ÷ 22 – 3

Mixed Practice

9) Rewrite using exponents.

7 × 4 × 4 × 3 × 7 × 4

10) Find the area using the formula A = s2.

Word Problem

11) Sara and Jarreth both evaluated the expression 54 + (2

2 • 8) – 4 ÷

5

2 . Here is their work.

Sara Jarreth

54 + (2

2 • 8) – 4 ÷

5

2 54 + (2

2 • 8) – 4 ÷

5

2

625 + (4 • 8) – 4 ÷ 5

2 625 + (4 • 8) – 4 ÷ 5

2

625 + 32 – 4 ÷ 5

2 625 + 32 – 4 ÷ 5

2

625 + 32 – 10 657 – 4 ÷ 5

2

657 – 10 = 647 653 ÷ 5

2 = 1,6322

1

Who got the correct value? Describe the mistakes the other made.



Today your student used the Order of Operations to find the value of numeric expressions. The

expressions contained parentheses, exponents, and the four basic operations: addition, subtraction,

multiplication, and division. Your student learned about two mnemonic devices to help remember the

correct order. The expressions include whole numbers, fractions, and decimals.

1) PEMDAS

Parentheses

Exponents

Multiplication and Division from left to right

Addition and Subtraction from left to right

2) Please Excuse My Dear Aunt Sally � The first letter of each word corresponds to an operation in

the order they should be completed.

We will continue to use the Order of Operations to help us simplify and evaluate algebraic expressions.

We will also use it to help us solve equations.

Your student should be able to answer the following questions about the Order of Operations:

1) Which operation will you do first to evaluate this expression?

2) If you just did the operations in this problem from left to right, would you get a different

answer than if you followed the Order of Operations?

3) Why do you think we do the multiplication and division and the subtraction and addition from

left to right?

Here are some ideas to work with the Order of Operations with your student:

1) Watch a video to review the Order of Operations.

Introduction to the Order of Operations:

khanacademy.org/math/arithmetic/order-of-operations/v/introduction-to-order-of-operations

Order of Operations:

khanacademy.org/math/arithmetic/order-of-operations/v/order-of-operations


Homework Answers

1) The answer is 53. Possible explanation: First, I did the operation in the parentheses, 4 plus 3, which equals 7. Next, I evaluated the exponent. 3

4 = 81. Then, I multiplied 4 times 7, which equals 28 and subtracted that

from 81, which equals 53. I used accuracy and precision (TLM practice #6) to carefully do each operation in the problem, according to the Order of Operations.

2) The answer is 33. Possible explanation: First, I found the value of the expressions with exponents, so 8

2 = 64. Next, I

did the multiplication and division from left to right, so 64 × 0.25 = 16. Finally, I did the addition and subtraction from left to right, so 17 + 16 = 33. I used the pattern and structure of the problem (TLM practice #7) and the Order of Operations to help me solve this problem one step at a time.

3) 3

4) 13

5) –1

6) 38

7) 16

8) 22

Mixed Practice

9) 72 × 4

3 × 3

10) 225 cm2

Word Problem

11) Possible answer: Sara found the correct value. Jarreth did not use the Order of Operations correctly. In the 4th line of his work, it shows that he added 625 and 32. The Order of Operations tells you that multiplication and division should come before addition and subtraction. Then the 5th line of his work again shows that he subtracted before dividing. I used what I know about the Order of Operations (TLM practice #3) to think about Jarreth’s work and to determine that he didn’t follow the correct order of PEMDAS.


Homework Problems

Name



Questions…

#’s

Quick Look


exponent, squared, cubed, evaluate, numeric expression, Order of Operations

Today we used our knowledge of the Order of Operations and problem solving to answer real-world math

problems. First, we found the key words and numbers in the math stories and situations. Then, we turned

those words and numbers into numeric expressions. Finally, we used the Order of Operations to evaluate

the numeric expressions and answer the questions.

Here’s an example:

Gerald’s Toy Factory makes 3 colors of teddy bears: brown, tan, and yellow. Each type of teddy

bear uses 3 yards of fabric. The factory can create 50 teddy bears each hour. They make brown

teddy bears for 3 hours each day, tan teddy bears for 4 hours each day, and yellow teddy bears

for 2 hours each day. So how much fabric does Gerald’s Toy Factory use in a day to create

teddy bears?

The important information in the problem is highlighted in yellow. The question we have to answer is

highlighted in green. Let’s write a numeric expression to help us.

Teddy bears created each day: 50 hour

bearsteddy × (3 hours + 4 hours + 2 hours) × 3

bearteddy

fabric of yards

50 hour

bearsteddy × 9 hours × 3

bearteddy

fabric of yards

450 teddy bears × 3 bearteddy

fabric of yards = 1,350 yards of fabric each day.

No matter what color teddy bear is being made, Gerald’s Toy Factory makes 50 teddy bears each hour.

So we found the total hours that the factory makes teddy bears, multiplied that by the rate of teddy bears

created each hour. Then we multiplied that by the ratio of fabric needed to create each bear. We used the

Order of Operations to do the work in the parentheses first, then to multiply from left to right.


1) Chen is creating a pattern for the floor of his patio using wooden tiles in two types: pine wood and

cedar wood. Each tile is a square that is 6 inches on each side. Chen needs 44 pine wood squares

and 20 cedar wood squares to create the pattern for his patio.

Chen is purchasing the wood from the local hardware store as longer planks then cutting the planks

into the 6-inch squares.

Type of Wood Length of Plank Cost of Plank

Cedar 72 inches $21.34

Chestnut 60 inches $36.50

Pine 4.5 feet $15.11

Birch 7 feet $29.99

a) If the hardware store only allows customers to buy whole planks of wood, then how many planks

of each type of wood does Chen need to purchase?

b) If the tiles completely cover Chen’s patio, what is the total area of the patio?

c) What will Chen spend on wood to create the pattern for his patio?

Mixed Practice

2) Show the math in the story using

multiplication and using an exponent.

2 friends each have 2 cats.

Both cats have 2 types of collars.

Each color has 2 I.D. tags on it.

How many I.D. tags to the friends have

in all?

3) Evaluate the expression.

102 × (16 – 6 + 25) ÷ 5


4) What is the value of –213 ?

5) Ms. Connelly purchased 23 fluid ounces of

mercury for the chemistry lab for $28.98.

How much did she spend per ounce?

Word Problem

6) Jeremiah buys 3 cans of white paint and 4 cans of beige paint. Each can of paint costs $8.99.

Gheta buys 4 rolls of wall paper with flowers and 2 rolls of wall paper with stripes. Each roll of

wall paper costs $5. Write and evaluate a numeric expression to find how much more Jeremiah

spent on decorating supplies than Gheta. Explain your thinking.



Today your student used the Order of Operations with exponents to solve real-world math problems.

The problems included whole numbers, fractions, decimals, and money amounts that were being added,

subtracted, multiplied, and divided. Your student sorted the information in the problems by identifying the

key words and numbers and formed a plan to answer the questions. The information in the problem was

sometimes given as words, in charts, on figures, or in graphs. Your student also identified information that

was included in the problem but was not needed to answer the question or questions.

Your student should be able to answer the following questions when solving multiple-step math problems:

1) What is the key information in this problem?

2) What question or questions do you need to answer to solve this problem?

3) Is there any information in the problem that you will NOT use to answer the question?

4) What numeric expression did you write to help you solve this problem?

5) Explain how your numeric expression represents what is happening in the problem.

Here are some ideas to work with numeric expressions and problem solving with your student: 1) Have your student write and evaluate numeric expressions to help plan a meal: the total ingredients that must be purchased, the total cost to purchase the ingredients, the total cook time, etc.

2) Have your student plan a party or event for their whole school. Have them determine what items they need to collect or purchase for the event. Then write and evaluate numeric expressions to find the total number of items for their whole school, or to compare the total for their class versus the total for another class.


Homework Answers

1) a. Cedar 20 ÷ (72 ÷ 6) = 1.667, so 2 planks of cedar wood.

Pine 44 ÷ (4.5 × 12 ÷ 6) = 4.8, so 5 planks of pine wood.

b. (20 + 44) × (6 in.)2 = 2,304 in.

2, so Chen’s patio has an area of 2,304 in.

2.

c. (2 × $21.34) + (5 × $15.11) = $118.23, so Chen would spend $118.23 on the wood for his patio.

Mixed Practice

2) 2 × 2 × 2 × 2 or 24

3) 700

4) 213

5) Ms. Connelly spent $1.26 per ounce.

Word Problem

6) (3 + 4) × $8.99 – (4 + 2) × $5 = $32.93; Jeremiah spent $32.93 more than Gheta. First, I thought about what was happening in the story, and then turned the story into a numeric expression with addition, multiplication, and subtraction. Since I had to determine how much Jeremiah and Gheta each spent, I put that operation in parentheses because according to the Order of Operations, parentheses need to be done first. I used what I know (TLM practice #3) to help me determine what operations to put in parentheses and how to set up the problem to accurately reflect the math in the story problem.


Homework Problems

Name



Questions…

#’s


Quick Look

Today we identified the unknown quantity in situations. Then, we assigned a variable to it.

Here’s an example!

It snowed 10.5 inches more this year than it did last year.

known quantity unknown quantity

If we know how many inches it snowed last year, we can determine how many inches it snowed this year.

Let s represent the amount of snow last year.

Also, we realized that because the variable x looks so similar to a multiplication sign, we’ll switch to using

the multiplication dot or parentheses instead. Now multiplication looks like this: 6 • 5 = 30, or 6(5) = 30.

Directions for questions 1 and 2: Assign a variable to an unknown quantity in the situation.

1) Omar slept 30 minutes longer on Tuesday

night than on Monday night.

2) Trip and Anne-Marie both have coin

collections. Trip has twice as many

coins as Anne-Marie.


3) What is an unknown in this situation? What variable could you assign to that unknown value?

Directions for questions 4–8: Assign a variable to an unknown quantity in the situation.

4) Grant drove an average speed of 54 miles per hour on his trip to Nevada.

5) Sasha babysat for her neighbor last week. She earns $12 for each hour she babysits.


6) Logan and Trevor competed in an apple bobbing contest. Trevor picked up 3 more apples than Logan.

7) Dimitri finished his homework 7 minutes faster tonight than on Wednesday night. Explain your thinking.

8) Frank donated 0.25 of his lottery winnings to charity.


9) What is an unknown in this situation? What variable could you assign to that unknown value?

Directions for questions 10–12: Assign a variable to an unknown quantity in the situation.

10) Simon and Camilla both ride bicycles to stay in shape. On Monday, Camilla biked 1.2 miles farther

than Simon.

11) Nina was baking a cake and accidently put 4 times the amount of flour she was supposed to!

12) Kelvin and Rachel held the record at their school for the longest egg toss. This year, they completed

a toss 2 yards shorter than their previous record. Explain your thinking.

Mixed Practice

13) Evaluate the expression.

30 – 4 • 3 ÷ 6

14) Write the prime factorization of 100

using exponents.


15) Divide.

40.3 ÷ 0.25

16) Estimate the sum.

3558

9 + 5

35

2

Word Problem

17) Max compared sports drinks at two different grocery stores. The first grocery store sold the drinks for

2 cents an ounce less than the second grocery store. Assign a variable to an unknown quantity in the

situation. Explain your thinking.



Today your student was introduced to variables. A variable is an unknown value, or a value that can

change. Variables are usually represented with lowercase letters. For example, since the number of

cars in a parking lot on any day might not be known, or the number of cars can change, it is a variable.

We could use c to represent the number of cars in the parking lot.

Your student determined the unknown quantities in situations and assigned variables to them.

This prepares students for writing expressions with variables, finding the value of the expression when a

quantity is assigned to the variable, and solving for variables in equations.

Your student should be able to answer the following questions about variables:

1) What information is known in the problem?

2) If you were solving this problem, what information would you need? How would you solve it?

3) Explain what a variable is.

Here are some ideas to work with variables:

1) Can you give an example when you may use a variable in your own life? To get started, think of a situation where a quantity may vary (the hours you work per week, the pounds of apples you purchase, etc.).

2) Watch a video from Khan Academy to review what a variable is:

http://www.khanacademy.org/math/algebra/solving-linear-equations/v/variables-expressions-

and-equations


Homework Answers

All answers are possible answers.

1) Let s represent the minutes Omar slept on Monday night.

2) Let c represent the number of coins Anne-Marie has.

3) An unknown value is the number of jelly beans in all. Let t stand for the number of jelly beans in all.

4) Let h represent how many hours Grant drove for.

5) Let r represent the hours Sasha babysits in one week. Possible explanation: I know Sasha’s rate of pay. In order to determine her earnings in a week, I need to know how many hours she worked. Since that is an unknown quantity, I assigned a variable to it. I used what I know (TLM practice #3) about unknown quantities and assigning variables to help me determine what was unknown in the problem and then represented that information with a letter.

6) Let a represent the number of apples that Logan successfully picked up.

7) Let h represent the amount of time, in minutes, it took Dimitri to complete his homework on Wednesday night. Possible explanation: The sentence tells me that Dimitri finished 7 minutes faster tonight than on Wednesday, but it does not tell me how long it took him to complete his homework on Wednesday night. That is an unknown quantity, so I assigned the variable h to represent it. I used what I know (TLM practice #3) about unknown quantities and assigning variables to help me determine what was unknown in the problem and then represented that information with a letter.

8) Let w represent Frank’s total winnings.

9) An unknown value is the capacity at that particular stadium. Let c stand for the designated capacity.

10) Let m represent the number of miles Simon biked on Monday.

11) Let f represent the amount of flour Nina was supposed to use.

12) Let y represent their record distance, in yards, in the egg toss. Possible explanation: Let y represent their record distance, in yards, in the egg toss. The sentence tells me that the team’s egg toss this year was 2 yards short of their record; however it does not tell me how many yards the record toss was. This is an unknown quantity, so I assigned the variable y to represent it. I used what I know (TLM practice #3) about unknown quantities and assigning variables to help me determine what was unknown in the problem and then represented that information with a letter.

Mixed Practice

13) 28 14) 22 • 5

2

15) 161.2 16) Possible estimate: 35 + 5 = 40

Word Problem

17) Possible answer: Let s represent the price per ounce of the sports drink at the second grocery store. Possible explanation: I know that the first grocery store sold the drinks for 2 cents less per ounce. I do not know how much the second grocery store sells them for. Since that is an unknown quantity, I assigned a variable to it. Using what I know about unknown quantities and variables (TLM practice #3) helped me to determine what amount was missing in the problem and what letter I should use as a variable to represent that amount.


Homework Problems

Name

Team Name Team Complete? Team Did Not Agree On

Questions…

#’s

Quick Look


variable, constant, coefficient, term, algebraic expression

Today we identified terms associated with algebraic expressions: Constant– A constant is a quantity that always stays the same. Variable– A variable is an unknown value or a value that can change. Coefficient– A coefficient is a number multiplied by a variable. Term– A term is a single constant, variable, or variable with a coefficient. Algebraic Expression– An algebraic expression is a collection of terms connected by operations that includes one or more variables.

Here’s an example.

We can also represent the expression, x + 7, with algebra tiles:

Directions for questions 1–6: Identify the variables, constants, number of terms, and coefficients.

1) 11w + 0.9 + 202

1

a. variable(s)

b. constant(s)

c. number of terms

d. coefficient(s)


2) 103y

a. variable(s)

b. constant(s)

c. number of terms

d. coefficient(s)

3) 3 + 3m + 4.5 + r

a. variable(s)

b. constant(s)

c. number of terms

d. coefficient(s)

4) 5p + 3p + 82

a. variable(s)

b. constant(s)

c. number of terms

d. coefficient(s)

5) x + 24

a. variable(s)

b. constant(s)

c. number of terms

d. coefficient(s)

6) 72 + 4k

a. variable(s)

b. constant(s)

c. number of terms

d. coefficient(s)

Directions for questions 7 and 8: Write an algebraic expression that represents the given algebra tiles.

7)

8)

Directions for questions 9 and 10: Sketch algebra tiles to represent the algebraic expression.

9) 4u + 2

10) b + 10


Mixed Practice

11) Assign a variable to an unknown quantity in the situation.

Henry slept for 45 minutes longer on Saturday night than he did on Thursday night.

12) Convert 3.5% to a decimal.

13) What is 17% of 200?

14) Evaluate.

3 • (45 – 20 ÷ 4)

Word Problem

15) Sue added y and 3. How many terms are in her expression? What is the constant in her expression? What is the coefficient in her expression? Explain your thinking.



Today your student identified vocabulary associated with algebraic expressions. Algebraic expressions are expressions that include one or more variables. A variable is an unknown value or value that can change. It is often written as a lower case letter such as x or y.

Your student also learned that a constant is a quantity that always stays the same. In z + 34, 34 is the constant and z is the variable. A coefficient is a number multiplied by a variable. In 5a, 5 is the coefficient because 5a = 5 • a. Finally, a term is a single constant, variable, or variable with a coefficient. For example, in 35 + 7b, there are two terms: 35 and 7b.

Your student should be able to answer the following questions about algebraic expressions and vocabulary:

1) Explain what the coefficient of n is in the expression n + 3.

2) How many terms are in this expression? How do you know?

3) Is this expression a numeric or algebraic expression? How do you know?

4) Why do you think it is important to know these vocabulary words?

Here are some ideas to work with algebraic expressions and vocabulary:

1) Play 21 questions! Have someone write an algebraic expression for your student. Your student does not see the expression and asks questions (they do not need to be yes or no questions) to try to guess the expression (e.g. How many terms? Is the constant greater than 5?) You may want to set guidelines for the expressions (e.g. all numbers in the expression are less than 10) so that it is not too difficult to guess.

2) Represent expressions with household items! Write an algebraic expression for your student. Have your student represent the expression with household items (perhaps pencils could represent variables and erasers could represent units).


Homework Answers

1) a. variable(s): w b. constant(s): 0.9, 202

1

c. number of terms: three d. coefficient(s): 11

2) a. variable(s): y b. constant(s): none c. number of terms: one d. coefficient(s): 103

3) a. variable(s): m, r b. constant(s): 3, 4.5 c. number of terms: four d. coefficient(s): 3, 1

4) a. variable(s): p b. constant(s): 82 c. number of terms: three d. coefficient(s): 5, 3

5) a. variable(s): x b. constant(s): 24 c. number of terms: two d. coefficient(s): 1

6) a. variable(s): k b. constant(s): 72 c. number of terms: two d. coefficient(s): 4

7) Possible answer: 8a + 5 8) Possible answer: c + 9

9)

10)

Mixed Practice

11) Possible answer: Let s represent the time Henry slept on Thursday night.

12) 0.035

13) 34 14) 120

Word Problem

15) There are two terms in her expression. The constant is 3 and the coefficient is 1. I used what I know (TLM practice #3) about the parts of an algebraic expression to help me identify what the terms, constant, and coefficient are in this expression.


Homework Problems

Name



Questions…

#’s

Quick Look



Today we wrote algebraic expressions for math statements. Here’s what that looks like!

For the phrase, 6 more than k, if something is more than, it means it’s greater, so we need to add.

We can write that as 6 + k or k + 6.

We also wrote math statements for algebraic expressions.

For the expression, h ÷ 7, we can write that as:

h is divided into 7 equal parts or the quotient of h and 7

Directions for questions 1–5: Write an algebraic expression for the phrase.

1) twice x

2) 5 less than p

3) y more than 28.9

4) k squared

5) 10 divided into b parts

Directions for questions 6–10: Write a math statement to describe the algebraic expression.

6) 50 + w

7) 16 – h

8) 3t

9) g3


10) 7

q

Mixed Practice

11) Estimate.

5 • 0.45

12) Find the greatest common factor.

50 and 25

13) Write a variable for the unknown quantity in

the situation.

Mrs. Jane’s college fund was split equally

between her 5 children.

14) Identify the variables, constants, number of

terms, and coefficient.

12 + 10y + 4 + n

a. variable(s)

b. constant(s)

c. number of terms

d. coefficient(s)

Word Problem

15) Amelia lengthened her jump rope of x inches by 5 more inches. Write an algebraic expression to

describe the new length. Explain your thinking.



Today your student wrote one-step expressions to represent math statements with variables and wrote

math statements for given algebraic expressions. For example, the math phrase, h in addition to 34,

means h + 34 or 34 + h. The algebraic expression, 4 ÷ x, can be written as a math statement: 4 divided

into x equal parts. Practicing writing expressions both ways will help students communicate about math

more clearly as well as interpret algebraic expressions.

Your student should be able to answer the following questions about writing expressions.

1) What is the variable?

2) What numbers are being used?

3) What operation will you use? How do you know?

4) Explain what this algebraic expression means.

Here are some ideas to work with writing expressions:

1) Create your own algebraic expression! What math statements can you use to describe it?

2) Have someone give your student a math statement verbally or written down. Can he or she write

it as an expression?


Homework Answers

1) 2x

2) p – 5

3) y + 28.9 or 28.9 + y

4) k2

5) 10 ÷ b or b

10

6) Possible answer: w more than 50

7) Possible answer: h deducted from 16

8) Possible answer: 3 times t

9) Possible answer: g to the third power

10) Possible answer: the quotient of q and 7

Mixed Practice

11) Possible estimate: 5 • 0.5 = 2.5

12) 25

13) Possible answer: Let x represent the total amount of money in the college savings fund originally.

14) a. variable(s): y, n b. constant(s): 12, 4 c. number of terms: four d. coefficient(s): 10, 1

Word Problem

15) Possible explanation: x + 5 or 5 + x If something is lengthened, it means length is added, so you need to add x and 5. I used what I know about determining an unknown quantity (TLM practice #3) to help me determine an algebraic expression for this problem.


Homework Problems

Name



Questions…

#’s

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Today we wrote algebraic expressions for multi-step math statements. Here’s what that looks like!

20 divided by the product of r and 30

20 is divided by a product, so it’s written as r30

20 or 20 ÷ (30r).

We also wrote math statements for multi-step algebraic expressions. For example:

(y + 3) – 20.5 can be represented in words: the sum of y and 32 minus 20.5.

Directions for questions 1–5: Write an algebraic expression for each phrase.

1) the difference between 4 times x and 18

2) 25 times the difference of 2

1 and f

3) the product of 10 and the sum of g and 2

4) the square of the quotient of 5 and y

5) t divided by the sum of 9.7 and 14


Directions for questions 6–10: Write a math statement to describe each algebraic expression.

6) 5b ÷ 3

7) 2

–5 x

8) 40 • (p – 0.4)

9) 9 – 18w

10) k • (6 + 3

2)

Mixed Practice

11) How many terms are in the following

expression?

y

2 + x

12) Order from least to greatest.

8

18, 2

4

3,5

9

13) How many liters are there in 84 ounces?

(33.8 ounces in 1 liter)

14) Is 306

158 closest to 0,

2

1, or 1?

Word Problem

15) Write the statement as an algebraic expression. Then explain your thinking.

Cindy divided the sum of 56 and w by 2.



Today your student wrote multi-step expressions to represent math statements with variables and wrote

math statements for given multi-step algebraic expressions. Practicing writing expression both ways will

help students communicate about math more clearly as well as interpret algebraic expressions.

This builds on our experience with single-step expressions and will help us as we begin writing and

solving equations.

Your student should be able to answer the following questions about writing expressions:

1) What is the variable?

2) What numbers are being used?

3) What operation will you use? How do you know?

4) Explain what this algebraic expression means. Here are some ideas to work with writing expressions:

1) Create your own multi-step algebraic expression! What math statements can you use to describe it?

2) Have someone give your student a multi-step math statement with a variable verbally or written

down. Can he or she write it as an expression?


Homework Answers

1) 4x – 18

2) 25 • (2

1 – f)

3) 10 • (g + 2)

4) (y

5)2

or (5 ÷ y)2

5) t ÷ (9.7 + 14) or 14+7.9

t

6) Possible answer: the quotient of 5 times b and 3

7) Possible answer: the quotient of 5 minus x and 2.

8) Possible answer: 40 times the difference of p and 0.4

9) Possible answer: 9 minus the product of 18 and w

10) Possible answer: k times the sum of 6 and 3

2

Mixed Practice

11) There are two terms.

12) 5

9,

8

18, 2

4

3

13) 2.49 liters in 84 ounces

14) It is closest to 2

1.

Word Problem

15) Possible explanation: 2

+56 w or (56 + w) ÷ 2. I saw that a sum was being divided by 2. The sum of

56 and w is 56 + w. Then, I put it in the numerator of the fraction and put 2 in the denominator so that the sum is divided by 2. I translated the algebraic expression into math (TLM practice #2) because I put the expression into mathematical vocabulary to reflect the mathematical operations in the algebraic expression.


Homework Problems

Name



Questions…

#’s

Quick Look


Today we evaluated algebraic expressions for a given value of a variable. Here’s an example!

Evaluate 5 • (42 + h) when h = 2.5.

5 • (42 + 2.5) Substitute 2.5 for h.

5 • (16 + 2.5) By the Order of Operations, evaluate exponents first.

5 • 18.5 Add within parentheses.

92.5 Multiply.

Directions for questions 1–10: Evaluate the expression for the given variable.

1) 4

6c when c = 8.

2) 24g if g = 2.

3) (y

60 + 6.78) – (12 – 2.32) when y = 12.

4) 131.5 – 2p if p = 47.2.

5) 20 – 4v for v = 24

3.

6) 8w2 when w = 10.


7) v4 for v = 3

8) (3y + 14 ÷ 7) – 15 for y = 7.

9) 9n + (15.7 • 5) when n = 2.5.

10) h ÷ (3

2 •

10

21) • 2 if h = 14.

Mixed Practice

11) Find the area of the square.

12) Find the volume.

13) Patricia has x pairs of shoes. Kendra had 3 times the number of shoes that Patricia has, but lost

4 pairs. Write an expression to describe the number of pairs of shoes Kendra has.

14) The point (–8, 10) is in which quadrant of the coordinate plane?

Word Problem

15) Tina has 4x days of school left. If x = 14, how many days of school does Tina have left?




Today your student learned to evaluate algebraic expressions for a given value of a variable. Evaluate means to substitute a number for a variable and then find the numerical value of the expression. Your student used the Order of Operations to find the numerical value of the expression.

In the next lesson, your student will apply this concept to what he or she already knows about geometry and explore how the value of the expression changes when the value of the variable changes. For example, if you enter an expression for the volume of an object in a spreadsheet, you could graph how the volume increases or decreases when one of the measurements of the object increases or decreases.

Your student should be able to answer the following questions about evaluating expressions:

1) How did you evaluate the expression? 2) How did you know what operation to do first? 3) Will you get the same result if the variable equals a different number? 4) Once you substitute a value for a variable in an algebraic expression, what kind of expression results.

Here are some ideas to work with evaluating expressions:

1) Make your own expressions! Then, evaluate the expressions with different numerical values. Explain what happens when you evaluate the expression with different numbers.

2) Watch a video from Khan Academy to review evaluating expressions: http://www.khanacademy.org/math/algebra/solving-linear-equations/v/variables-and-expressions-1?exid=evaluating_expressions_1


Homework Answers

1) 12

2) 48

3) 2.1

4) 37.1

5) 9

6) 800

7) 81

8) 8

9) 101

10) 20

Mixed Practice

11) 20.25 ft2

12) 5,832 cm3

13) 3x – 4

14) Quadrant II

Word Problem

15) Tina has 56 days of school left. First, I estimated 15 times 4, which is 60. Next, I substituted 14 for x, then multiplied to get 56. I used the information in the problem to estimate the problem first (TLM practice #1), then solved it and checked to see that the answer made sense with my estimate.


Homework Problems

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evaluate

Today we wrote and evaluated expressions to describe perimeter and area. Here’s an example!

We can write an expression to describe the area of the rectangle: 7m.

Let’s find the area when m = 18

7m = 7 • 18 = 126, so the area is 126 units.

1)

a. Write an algebraic expression that

represents the perimeter of the figure.

b. Evaluate the algebric expression to find

the perimeter when p = 6, and then

when p = 2.


2)




the perimeter when y = 20.

3)

a. Write an algebraic expression that represents the area of the rectangle.

b. Evaluate the algebric expression to find the area when k = 3, and then when k = 7

4.

4)


represents the area of the square.


the area when g = 2, and then when g = 8.


5) Which shape has a greater area? Explain your thinking.

6)

a. Write an algebraic expression that represents the area of the rectangle.

b. Evaluate the algebric expression to find the area when c = 10, and then when c = 6.

7)



b. Evaluate the algebraic expression to find the

perimeter when d = 54.


8) Charlie designed two diagrams for his school project. Which diagram has a greater perimeter?


Mixed Practice

9) Write an algebraic expression to represent the sum of 6.5 and the difference of x and 14.

10) What is the coefficient of m? What is the coefficient of 3m?

11) Simplify.

2 + (29 – 42)

12) Jane bought a pack of cards for $3.12. If 52 cards come in the deck, what is the unit price for

each card?

Word Problem

13) Write an expression for the area of a rectangle that has a length of t and height of 4. What will the

area be if the length of the rectangle is 21?



Today your student learned to write and evaluate algebraic expressions that describe area and perimeter.

Given a diagram, your student used what he or she already knows about geometry to write an

expression. Then, he or she evaluated the expression. Sometimes, your student evaluated the

expression for more than one value. For example, each side of a square is x. The area of the square is

x • x or x2. If x = 5, the area is 5

2 = 25 square units, and if x = 7, the area if 7

2 = 49 square units.

Your student should be able to answer the following questions about evaluating expression:

1) How does your expression describe area/perimeter?

2) How did you know all the lengths of the sides of the figure?

3) How did you evaluate the expression?

4) Will you get the same result if the variable equals a different number?

5) What would happen to the value of the expression if the length of this side was shorter/longer?

Here are some ideas to work with evaluating expressions:

1) Write an algebraic expression to find the area or perimeter of an object in your home. Take a

guess at the length or width of the object so you can evaluate the expression. Then measure the

object and find the perimeter or area. How close was your guess?

2) Watch a video from Khan Academy to review evaluating expressions:

http://www.khanacademy.org/math/algebra/solving-linear-equations/v/variables-and-expressions-

1?exid=evaluating_expressions_1


Homework Answers

1) a. 3p + 3p +3p + 3p + 2.5p + 2.5p b. The perimeter is 102 units when p = 6 and 34 units when p = 2.

2) a. 0.5y + y + 17 b. The perimeter is 47 units if y = 20.

3) a. 4 • 7k

b. The area is 84 square units when k = 3, and 16 square units for k = 7

4.

4) a. 4.5g • 4.5g

b. The area is 81 square units for g = 2 and 1,296 square units if g = 8.

5) Shape 2 has a greater area. Possible explanation: Each shape has a width that is 6 units. However, the length of Shape 2 is 13t + 2, whereas Shape 1 has a length of only 13t. If t = 1, Shape 1 would have an area of 19 square units, and the area of Shape 2 would be 21 square units. I used the pattern and structure of the problem (TLM practice #7) to help me compare the two figures’ lengths and widths.

6) a. 8 • 22

1c

b. The area is 200 square units when c = 10 and 120 square units if c = 6.

7) a. d + d + d b. The perimeter is 162 units when d = 54.

8) Diagram A has the greater perimeter. Possible explanation: Diagram A and diagram B have all of the same measurements except diagram A has 2 sides that are longer than diagram B’s sides. If b = 2, Diagram A’s perimeter is 68 units and Diagram B’s perimeter is 56 units. I used what I know about area, perimeter, length and width (TLM practice #3) to help me determine which diagram has the greater perimeter. I recognized that Diagram A’s 2 sides are longer than Diagram B, therefore, the perimeter would be greater.

Mixed Practice

9) (x – 14) + 6.5

10) 1 and 3

11) 15

12) The unit price is $0.06 per card.

Word Problem

13) The expression that represents the area of the rectangle is 4t. If the length of the rectangle is 21 units, then the area is 84 square units.


Homework Problems

Name



Questions…

#’s

Quick Look


evaluate

Today you wrote an algebraic expression to describe a pattern. Here’s an example!

Complete the chart by writing an expression for the pattern.

The pattern is 3t.

3 • 14 = 32 3 • 21 = 63 3 • 37 = 111

We know 3t is the correct expression for the pattern because it applies to all the values in the table.

Directions for questions 1–5: Complete the chart by writing an expression for the pattern.

1)

2)

3)

4)


5)

Directions for questions 6–10: Read and solve each problem.

6) Jafar was working on his golf game, hitting

more balls each day. On day y, how many

balls will he hit?

7) When Jake’s garden produces n tomatoes,

how many tomatoes will Amy’s garden

produce?

8) Joana has red and yellow marbles. If the

pattern continues, how many yellow marbles

will she have if she has b red marbles?

9) This chart shows how much Chris was paid

for the number of hours he worked. How

much will he be paid for working x hours?

10) Coach Franks was tracking how many basketball players they would have in all in the league if more

students signed up. How many players would there be in all if w more students join?


Mixed Practice

11) Write an expression to describe the

perimeter for a rectangle with a length

of p and a width of p + 12.

12) Evaluate 4 + (2p + 22) when p = 5.

13) Order from least to greatest.

68%,3

2, 65%, 0.67

14) Find the value.

| –4.5 |

Word Problem

15) If Wanda makes a $23.00 purchase, she gets $27.00 back in change. If she makes a $25.00

purchase, she gets $25.00 back in change. If she makes a $40.00 purchase, she gets $10.00

back in change. If she makes a purchase of h dollars, how much change will she get back?



Today your student used tables of values to determine how changing the value affects the value of the

expression. Your student predicted what expression described the pattern and then tested it to make sure

it applied to the entire table. This process helps your student summarize calculations and interpret

expressions, which will prepare them for solving equations.

Your student should be able to answer the following questions about expressions and patterns:

1) What is the pattern? How do you know?

2) What if the number in the top row is 100? What number would be below it?

3) How can tables of patterns help you predict future events?

4) How are the numbers in the table related? Are there numbers that aren’t related? Explain.

Here are some ideas to work with expressions and patterns:

1) Find a pattern in your everyday life! Can you write an expression to describe it? How can a table of patterns and expressions help you predict a future event?

2) Watch a video from Khan Academy to review patterns and expressions (first 2 minutes only).

http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/patterns-and-equations


Homework Answers

1) 5z

2) c – 8

3) v + 1.5

4) g ÷ 5 or 5

g

5) a – 20

6) He will hit 50y golf balls.

7) Amy’s garden will produce n ÷ 3 or 3

n tomatoes.

8) Joana will have b + 24 yellow marbles.

9) Chris will make 10.5x dollars.

10) There would be 85 + w players in all in the league.

Mixed Practice

11) p + p + (p + 12) + (p + 12)

12) 18

13) 65%, 3

2, 0.67, 68%

14) 4.5

Word Problem

15) Wanda will get 50 – h dollars back in change. I found what all the amounts had in common ($50) by finding the difference between the price paid and the change given for each purchase. I used the pattern and structure in the problem (TLM practice #7) to help me determine an expression to represent the change Wanda would get back from a purchase of h dollars.


Homework Problems

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Questions…

#’s

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evaluate

Today we wrote an algebraic expression to describe a real-world situation. Then, we evaluated the

expression at a specific value. Here’s an example!

Sim made 78 scarves. He sold t scarves on Saturday and 30 scarves on Sunday.

Write an expression to describe how many scarves Sim has left.

78 – t – 30 or 78 – (t + 30)

Sim had 78 scarves. If he sold 20 on Saturday and t on Sunday, he sold 20 + t scarves in all. So to find

how many he had left, subtract the scarves he sold in all from the total amount of scarves he made.

Evaluate the expression when t = 45

If t = 45, that means he sold 45 scarves on Saturday. Replace t with 45 and evaluate using the

Order of Operations.

78 – (t + 30)

78 – (45 + 30)

78 – 75 = 3; Sim has 3 scarves left.

1) At an amusement park, x people are on a roller coaster. 40% of the people are kids under

the age of 13.

a. Write an algebraic expression to describe how many people on the roller coaster

are 13 and younger.


b. Evaluate the expression for x = 30.

2) The local car wash charges $8.00 to wash the outside of the car, and an additional $0.50 per minute

to clean the inside.

a. Write an algebraic expression to describe how much the car wash makes if a person gets both

the inside and outside of their car cleaned.

b. How much does it cost a customer that has the outside of his car washed and it takes 20 minutes

to clean the inside?

3) At a town hall meeting, w people came to vote on a new town ordinance. 3

2 of the people voted ‘yes’.

5 voters abstained from the vote.

a. Write an algebraic expression to describe the number of people who voted ‘no’.

b. Evaluate the expression if w = 45.

4) Janet donated pencils to four different disadvantaged schools and distributed them evenly.

a. Write an algebraic expression to describe how many pencils each school received.


b. How many pencils did each school get if Janet donated 8,000 pencils?


5) At the golf tournament, g golfers scored well enough to advance to the next round. 66 players

did not advance.

a. Write an algebraic expression to describe how many golfers there were in total.

b. Evaluate the expression for g = 34. Explain your thinking.

6) Ms. Sager donated 128 ounces of glue to the 6th grade art class. Mr. McEvoy donated 256 ounces of

glue. The 6th grade art class has z students.

a. Write an algebraic expression that describes the amount of glue each student could have if the

glue is divided equally.

b. Evaluate the expression for z = 15.

7) A restaurant paid their employees $150,000 from their total revenue.

a. Write an algebraic expression that describes how much money the restaurant has left after they

pay their employees.

b. How much money do they have left if their total revenue income was $400,000?

8) To dock a boat at Jerry’s Waterside Dock, it costs $250.00 in advance, and a $150 each month the

boat is docked in the spot.

a. Write an algebraic expression that describes how much it will cost to dock a boat.

b. How much will it cost if you want to dock a boat for 12 months? Explain your thinking.


Mixed Practice

9) Write an algebraic expression for the

math phrase.

2 less than the product of 4 and g

10) Write an expression to complete the pattern for

the table.

11) Add.

7

24 + 0.782 =

12) Write one unit rate to describe the

conversion between Teaspoons

and Tablespoons.

Word Problem

13) Each class has g reading groups. Each reading group has 5 students. Each student needs 5 binders.

Each binder has 5 tabs. Write an expression to describe the amount of tabs each class needs.

How many tabs does a class with 4 reading groups in it need? Explain your thinking.



Today your student wrote algebraic expressions to represent real-life situations. He or she built on writing

algebraic expressions from mathematical statements to apply expressions to real-life problems. Your

student defined an unknown quantity with a variable and then wrote an expression paying close attention

to Order of Operations. Your student then evaluated the expression at a specific value.

Your student should be able to answer the following questions about writing and evaluating expressions:

1) What’s going on in the problem?

2) Explain why you wrote the expression this way. Why are the numbers, variables, and operations

in this particular order?

3) Explain how the value of the expression will change when the value of the variable changes.

Here are some ideas to work with writing and evaluating expressions:

1) Write an algebraic expression to describe a real-life situation. It could be about your life, school

community, or world! Then, evaluate your expression at different values. What happens to the

value of the expression as you change the value of the variable?

2) Watch a video from Khan Academy to review evaluating expressions:

http://www.khanacademy.org/math/algebra/solving-linear-equations/v/variables-and-

expressions-1


Homework Answers

1) a. Possible answer: The number of kids under the age of 13 is 0.4x, where x represents the total number of people on the roller coaster. b. If there are 30 people on the roller coaster, then there are 12 children under the age of 13.

2) a. Possible answer: The cost to get the inside and outside of your car cleaned is 8 + 0.50k, where k is equal to the number of minutes it takes to clean the inside.

b. It will cost $18.00 if the car takes 20 minutes to clean inside.

3) a. Possible answer: The number of people who vote ‘no’ is 3

1w – 5, where w represents the total

number of people at the meeting. b. If there are 45 people at the meeting in total, there are 10 people that voted ‘no’.

4) a. The number of pencils each schools receives is s ÷ 4 or 4

s where s is equal to

the total number of pencils Janet donated. The value that is missing is how many pencils Janet donated in total, so I represented that with the variable s. Then I divided s by 4 (because there are four schools splitting the donation) to find out how many pencils each school received. I made sense of the problem (TLM practice #1) and made sure I understood the situation, then made a plan to solve it.

b. If Janet donated 8,000 pencils, each of the four schools received 2,000.

5) a. Possible answer: The total number of golfers in the tournament is g + 66, where g represents the number of golfers who advanced to the next round.

b. If 34 golfers advanced to the next round, there were 100 golfers in all to start the tournament. Possible explanation: To find how many golfers there were in the tournament in total,

I substituted 34 for g and added 66 to get 100. I made sense of the problem (TLM practice #1) and made sure I understood the situation, then made a plan to solve it.

6) a. Possible answer: The amount of glue each student receives is (128 + 256) ÷ z, where z represents the total number of students.

b. There would be 25.6 ounces of glue for each student if there are 15 students in the class.

7) a. Possible answer: The restaurant has p – $150,000 after they pay their employees, where p is equal to the total revenue.

b. If their total revenue was $400,000, the restaurant would have $250,000 remaining after paying their employees.

8) a. Possible answer: The total cost to dock the boat is 250 + 150y, where y is equal to the number of months the boat is docked.

b. It will cost $2,050 to dock the boat for 12 months. Possible explanation: To find the total cost of docking the boat for 12 months, I substituted

12 for y and followed the Order of Operations by first multiplying 150 • 12 = 1,800, then adding 250 to get $2,050. That’s TLM practice #5.

Mixed Practice

9) 4g – 2 10) y ÷ 2 or 2

y 11) 4.211 12) 3 teaspoons per

1 tablespoon.

Word Problem

13) 53 • g or 125 • g. If the class has 4 reading groups, they need 500 tabs.

Possible explanation: 5 students who need 5 binders each with 5 tabs each means multiplication, that’s 5 • 5 • 5 or 5

3. Then I multiplied that by the number of reading groups in each class, g. If g = 4

that means that each class has 4 reading groups. I substituted 4 for g and used the Order of Operations to evaluate. I got 500 which means that each class needs 500 tabs.


Homework Problems

Name



Questions…

#’s

Quick Look


evaluate

Today we used the Work Backwards strategy to solve word problems. Here’s an example.

Larry has a cup of paper clips. He took out 10 and gave them to his friend. Then he put half of what was

left in his desk. After that, 7 were left in the cup. How many paper clips were in the cup when he started?

First, organize the data:

Then work backwards:

So there were 24 paper clips to begin with.

Don’t forget to check your work! Just use your answer and work forward through the steps of the problem:


Directions for questions 1–5: Solve each problem.

1) Diane won a lottery game. She spent $120 of the winnings on clothes. She spent twice that amount

on groceries. Then, outside the grocery store, she found $20! With that $20 plus her remaining

winnings, Diane had $1,200 left. How much did Diane win?

2) Jimmy is a huge sports fan! He has 5 more football jerseys than baseball jerseys. He has twice as

many soccer jerseys as baseball jerseys. He has 3 more basketball jerseys than soccer jerseys.

Jimmy has 5 basketball jerseys. How many jerseys does Jimmy have in all?

3) At the start of a concert, the venue was not filled up. Five minutes later, 350 more people showed up.

Half the people at the end stayed around to get an autograph. If 2,000 people stayed around trying to

get an autograph, then how many people were there at the start of the concert?

4) Nicole has a jar of rubber bands. She used half of the rubber bands to make a necklace. Then she

took the rest and divided them into 4 equal groups. Each pile had 20 rubber bands. How many rubber

bands were in the jar when Nicole started?

5) Lindsay walked from her house to Rico’s house in a half of an hour. They played whiffle ball for 2

hours and then had dinner. Dinner took twice as long as the walk to Rico’s. Dinner was over at 8:00

p.m. What time did Lindsay leave for Rico’s house?

Mixed Practice

6) Is 7a • 5 • a equivalent to 40a?

7) Use the Distributive Property to write the

expression a different way.

48x + 36

8) Use the Properties of Addition to simplify the

expression. Be sure to combine like terms.

(5k + 20) + 14k + 0.4


9) It took Angie 24 minutes to read 2 chapters.

a. What is Angie’s reading rate? Write her rate in words and as a ratio.

b. Write a unit rate to describe how much Angie reads in 1 hour.

Word Problem

10) Bill put his pennies in two piles. The first pile had 150 pennies, and he put that pile in his coin bank.

He put the pennies from the second pile into paper rolls. It took 5 rolls to hold all those pennies.

Each roll holds 50 pennies. How many pennies did Bill have in all? Explain your thinking,



Today your student learned to use the Work Backwards strategy to solve problems. He or she started

with the last piece of data and worked back through the problem using the opposite operations to solve.

He or she then worked forward through the problem with the answer to check his or her work.

As we move into solving equations and inequalities, we use inverse operations to help us. Working

Backwards is a strategy that will help your student be prepared to use inverse operations in the

upcoming units.

Your student should be able to answer the following questions about problem solving:

1) What is the situation? What is the mathematical data? What is the question?

2) What is your plan? Where will you start?

3) How can you check your work?

4) How did working backwards help you solve this problem?

Here are some ideas to work with problem solving:

1) Create your own problems where you need to work backwards to solve. Explain how you created the problem. Did you work forwards or backwards to create it?

2) Model a simple addition, subtraction, division, or multiplication problem with manipulatives at home (pennies, paper clips). Can you model the problem backwards and forwards?


Homework Answers

1) Diane won $1,540.

2) Jimmy has 14 jerseys.

3) There were 3,650 people at the start of the concert.

4) Nicole started with 160 rubber bands.

5) Lindsay left at 4:30 p.m.

Mixed Practice

6) No, they are not equivalent.

7) 12(4x + 3)

8) 19k + 20.4

9) a. minutes 24

chapters 2, 2 chapters per 24 minutes

b. hour 1

chapters 5

, 5 chapters per 1 hour

Word Problem

10) Bill had 400 pennies in all. Possible explanation: I used the Work Backwards strategy to solve. Each paper roll holds 50 pennies and there are 5 rolls. That’s 50 • 5 = 250 pennies in the second pile. There were 150 pennies in the first pile and 250 pennies in the second pile. 150 + 250 = 400 pennies in all. I rearranged the problem and used the pattern and structure (TLM practice #7) to work backwards and solve the problem. Starting at the end and working back helped me to determine how many pennies Bill had in all.

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