## Everyday Mathematics 6th Grade Answer Key Unit 4 Expressions and Equations

Using Order of Operations
Question 1.
Insert parentheses to make the expression equivalent to the target number.
Numerical Expression
8 – 2 + 5
15 – 3 ∗ 4 + 2
3 ∗ 5 + 4 ∗ 6

Target Number
1
50
162

Question 2.
Simplify each expression.
a. (3 + 9)2 ________
b. 24 ∗ 22 ________
c. 20 – (6 – 4) ________
d. ($$\frac{1}{2}$$ ÷ $$\frac{1}{4}$$) ∗ 6 ________

Question 3.
Complete the table

Question 4.
Use the given calculator keys to find an expression equivalent to the target number. You may use the keys more than once or not at all.

Practice
Write the opposite of each number.
Question 5.
12 ________

Question 6.
-2 ________

Question 7.
-3.5 ________

Question 8.
$$\frac{3}{5}$$ ________

Practicing Order of Operations
In Problems 1–3, tell whether the number sentence is true or false. If it is false, rewrite it with parentheses to make it true.

Question 4.
Evaluate.
a. 45 – (1 + 4)2 + 3

b. (2 + 4)2 ∗ (1 + 2)4

Question 5.
Write an expression for AT LEAST three of the following numbers using six 7s. All values can be found using only addition, subtraction, multiplication, and division.
1 = ____________
2 = ____________
3 = ____________
4 = ____________
5 = ____________
6 = ____________

Practice
Find the greatest common factor.
Question 6.
GCF (10, 50) = _______

Question 7.
GCF (80, 24) = _______

Question 8.
GCF (90, 54) = __________

Using Expressions
Question 1.
Write a numerical expression for calculating the number of shaded border tiles for the pictured 12-by-12 tiled floor.

b. Circle the expressions below that also represent the number of shaded tiles in the 12-by-12 tiled floor.
11 + 11 + 11 + 11
4 ∗ 12 + 4
(12 – 2) + (12 – 2) + 12 + 12
4 ∗ 12 – 2

c. Choose one of the expressions you circled in Part b and explain how it represents the number of shaded tiles

Question 2.
A rectangular tiled floor is shown at the right. Write an expression that models how you can find the number of shaded tiles in the 3-by-10 rectangular floor.

Question 3.
Write an expression that models how you can find the number of shaded tiles in the 3-by-13 rectangular floor shown at the right.

Try This
Question 4.
Write an algebraic expression for the number of shaded tiles in a 3-by-n rectangular floor. Use your expression to find the number of shaded tiles in a 3-by-125 tiled floor.

Practice
Find the least common multiple.
Question 5.
LCM (3, 5) = ________

Question 6.
LCM (10, 12) = ________

Question 7.
LCM (6, 12) = ________

Algebraic Expressions
Write an algebraic expression. Use your expression to solve the problem.
Question 1.
Kayla has x hats. Miriam has 6 fewer hats than Kayla. _______
If Kayla has 22 hats, how many hats does Miriam have? _______

Question 2.
The width of Rectangle A is half of its height. Write an algebraic expression for the width of Rectangle A.
a. Define your variable. Let ____ represent ________.
b. Algebraic expression: _______________
c. Using the variable you defined in Part a, write an algebraic expression for the perimeter of Rectangle A __________.

Question 3.
Larry ran 2.5 miles more than Jusef.
Write an algebraic expression for how far Larry ran.
a. Define your variable. Let _______ represent __________.
b. Algebraic expression: __________
c. If Jusef ran 5 miles, how many miles did Larry run? __________

Question 4.
For each situation, choose an expression from the box that matches the situation, and write it in the matching blank. You may use an expression more than once.

a. With 4 bags of n potatoes, the total number of potatoes is __________.
b. If you exchange n quarters for dollars, you get __________ dollars.
c. There are n pens in a box. Denise has 4 pens more than 2 boxes of pens. The total number of pens Denise has is __________.

Practice
Use <, >, or = to make the number sentence true.
Question 5.
$$\frac{3}{4}$$ ________ $$\frac{3}{7}$$

Question 6.
0.4 _______ 0.400

Question 7.
0.8 ________ 0.67

Question 1.
Look for a pattern in the set of numerical equations. Describe the pattern in words. Use a variable and write an equation that represents the pattern.
36 = 32 ∗ 34
586 = 582 ∗ 584
(0.25)6 = (0.25)2 ∗ (0.25)4
a. Description: ____________________
b. Equation that generalizes the pattern: ____________________
c. Write two more examples of the pattern: ____________________

Question 2.
For each equation, circle the number of solutions you could find.

Question 3.
Circle the answer that best describes each equation.

Question 4.

Try This
Question 5.
The numbers 4, 5, and 6 are called consecutive numbers because they follow each other in order. The sum of 4, 5, and 6 is 15—that is, 4 + 5 + 6 = 15. Circle all equations that generalize finding a sum of 170 for three consecutive numbers.
a. x + 2x + 3x = 170
b. 170 = x + (x + 1) + (x + 2)
c. 3x + 3 = 170

Practice
Estimate whether each sum is closest to 0, $$\frac{1}{2}$$, 1, or 1 $$\frac{1}{2}$$.
Question 6.
$$\frac{8}{9}$$ + $$\frac{5}{8}$$ _________

Question 7.
$$\frac{1}{10}$$ + $$\frac{1}{11}$$ __________

Question 8.
$$\frac{5}{6}$$ + $$\frac{2}{16}$$ _________

The Distributive Property
Question 1.
Each of the expressions describes the area of the shaded part of one of the rectangles. Write the letter of the correct rectangle next to each expression.

a. 4 ∗ (11 – 6) _________
b. 44 – 20 _________
c. 30 _________
d. (6 ∗ 9) – (6 ∗ 4) _________
e. (4 ∗ 11) – (4 ∗ 6) _________
f. (11 – 5) ∗ 4 _________
g. (11 ∗ 4) – (5 ∗ 4) _________
h. 6 ∗ (9 – 4) _________

Question 2.
Circle the equations that are examples of the Distributive Property.
a. (80 ∗ 5) + (120 ∗ 5) = (80 + 120) ∗ 5
b. 6 ∗ (3 – 0.5) = (6 ∗ 3) – 0.5
c. (9 ∗ $$\frac{3}{8}$$) – ($$\frac{2}{3}$$ ∗ $$\frac{3}{8}$$) = (9 – $$\frac{2}{3}$$) ∗ $$\frac{3}{8}$$
d. (16 ∗ 4) + 12 = (16 + 12) ∗ (4 + 12)

Write an equation to show how the Distributive Property can help you solve each problem.
Question 3.
Kelly signed copies of her new book at a local bookstore. In the morning she signed 36 books, and in the afternoon she signed 51 books. It took her 5 minutes to sign a book. How much time did she spend signing books?
Equation: __________________
Solution: __________________

Question 4.
Mr. Katz gave a party because all the students scored 100% on their math tests. He had budgeted $1.15 per student. It turned out that he spent$0.25 less per student. How much money did he spend for 30 students?
Equation: __________________
Solution: __________________

Practice
Write the reciprocal.
Question 5.
5 ________

Question 6.
$$\frac{2}{9}$$ ________

Question 7.
3 $$\frac{1}{3}$$ _______

Applying the Distributive Property

Question 1.
Match each property with a generalized form of the property

Question 2.
For each equation below, use general equations for properties to determine whether it is true or false. For each true number sentence, list the property or properties that apply. For false number sentences, write “None.”
a. (9 – 4) ∗ 3 = (9 – 3) ∗ (4 – 3) _______ Property: ____________
b. (8 + 5) ∗ 2 = (8 + 2) ∗ (5 + 2) _______ Property: ____________
c. (8 + 5) ∗ 2 = 2 ∗ (8 + 5) _______ Property: ____________

Use the Distributive Property to solve Problems 3–4.
Question 3.
Show how to solve the problems mentally.
a. 85 ∗ 101 = ____________
b. 156 ∗ 9 = ____________
c. 48 ∗ 24 = ____________

Question 4.
Rewrite each expression as a product by taking out a common factor
a. 48 + 24 = _______ ∗ (_______ + _______) = _______ ∗ _______
b. 72 – 56 = _______ ∗ (_______ – _______) = _______ ∗ _______
c. (2y) + (3 ∗ y) = (_______ + _______) ∗ _______ = _______ ∗ _______

Practice
Use <, >, or = to make the sentence true.
Question 5.
$$\frac{2}{3}$$ _________ $$\frac{2}{5}$$

Question 6.
0.7 _______ $$\frac{4}{5}$$

Question 7.
0.3 ______ 0.23

Question 8.
1 $$\frac{1}{4}$$ _______ 1.25

Building with Toothpicks
Yaneli is building a pattern with toothpicks. The pattern grows in the following way:

Question 1.
How many toothpicks are needed for Design 5? _________

Question 2.
How many toothpicks are needed for Design 10? __________

Question 3.
Describe in words how you see the toothpick design growing. What stays the same from one figure to the next? What changes?

Question 4.
Write an expression to represent how many toothpicks are needed for Design n?

Question 5.
What toothpick design number could you build with exactly 82 toothpicks? ________

Question 6.
Describe how you can figure out the number of toothpicks you need for any design number.

Practice
Evaluate each expression.
Question 7.
72 = ________

Question 8.
_________ = 24

Question 9.
15 = __________

Question 10.
43 = ____________

Inequalities
Question 1.
Amelia’s cell phone plan lets her send a maximum of 500 text messages per month.
Define a variable.
Write an inequality to represent Amelia’s situation.

Question 2.
The temperature in the freezer should be no higher than -18°C.
Define a variable.
Write an inequality to represent the situation.

Question 3.
Sam scored 68 in miniature golf. What score would beat Sam’s score?
Define a variable:
Write an inequality to represent the situation.

Question 4.
Choose the number sentence that represents each statement

A number is less than 42. __________
b. A number is greater than 42. __________
c. A number is at least 42. __________
d. A number is no greater than 42. __________

Practice
Question 5.
______ = 5.6 + 11.7

Question 6.
9.2 + _______ = 12.1

Question 7.
19.37 – 9.29 = _______

Question 8.
______ = 0.834 – 0.75

Solving and Graphing Inequalities
Describe the solution set for each inequality. Graph the solutions for each inequality.
Question 1.
a. 5 < n _____________

b. q < 5 ______________

c. w > -3 __________________

Question 2.
Write the inequality represented by each graph below.
a.

b.

c. List three numbersthat are part of the solution set for Part a.

Question 3.
a. Write an inequality with a solution set that is all numbers less than 0.
b. Find three numbers that are not in the solution set for Part a.
c. Write an inequality with a solution set that does not have any numbers in common with the solution set in Part a or the numbers you wrote in Part

Practice
Solve.
Question 4.
3.45 ∗ 2 = ________

Question 5.
3.2 ∗ 4.5 = _________

Question 6.
________ = 1.53 ∗ 3.3

Graphing Alligator Facts
Question 1.
If the temperature of an alligator nest is below 86°F, the female alligators hatch.
Define a variable: _______________
Represent the statement with inequalities: _______________
Graph the solution set that makes both inequalities true.

Describe how your graph represents the situation.

Question 2.
If the temperature of an alligator nest is above 93°F, the male alligators hatch. Use the same variable you used in Problem 1.
Represent the statement with inequalities: _______________
Graph the solution set that makes both inequalities true.

Question 3.
Adult alligators are at least 6 feet long. The longest one on record was 19 feet.
Define a variable: _______________
Represent the statement with inequalities: _______________
Graph the solution set that makes both inequalities true.

Question 4.
Alligators lay 20–50 eggs in a clutch. Variable: _______________
Represent the statement with inequalities: _______________
Graph the solution set that makes both inequalities true.

Describe how your graph represents the situation.

Practice
Evaluate.
Question 5.
15% of 60 ______

Question 6.
25% of 300 ______

Question 7.
250% of 18 ________

Absolute Value
Question 1.
a. On the number line, plot points at two numbers whose absolute values are 8.

b. Explain why you get a positive number when you take the absolute value of a negative number.

Question 2.
Complete.
a. |20| = ________
b. |8.25| = ________
c. |-79| = ________
d. |-0.004| = ________
e. |-10 $$\frac{1}{2}$$ | = ________
f. |0| = ________

Question 3.
Find at least three numbers that answer each riddle.
a. A number with an absolute value that is equal to itself ___________
b. A number with an absolute value that is its opposite ____________

Question 4.
Make up your own absolute value riddle

Try This
Question 5.
Find at least three numbers that make each statement true
a. |x| = – x ___________
b. |x| > – x ___________

Practice
Divide.
Express your remainder as a fraction.
Question 6.

Question 7.

Using Absolute Value
For Problems 1–2, do the following:

• Plot the numbers on the number line.

Question 1.
The freezing point of water is 0°C. In Chicago, it is -7°C. In Montreal, it is -9°C.

Which city’s temperature is farther from 0? ________
-7 > -9 or |-9| > |-7|

Question 2.
Rita has a debt of $14, and Jamal has a debt of$18.

Whose balance is farther from 0? ________
|-18| > |-14| or -18 < -14

Question 3.
Explain how you know whether you need to use absolute value to answer the question. What do you have to consider?

Question 4.
Find the distance between the ordered pairs.
a. (-2, -1) and (-2, 3) Distance: ________
b. (-2, 3) and (3, 3) Distance: ________
c. (3, -1) and (3, -4.5) Distance: ________
d. (-11, 9) and (-11, -32) Distance: ________

Practice
Solve
Question 5.
2 $$\frac{1}{2}$$ ÷ $$\frac{3}{4}$$ = __________

Question 6.
1 $$\frac{2}{3}$$ ÷ $$\frac{1}{3}$$ = ________

Question 7.
3 $$\frac{3}{4}$$ ÷ $$\frac{1}{3}$$ = _________

Temperatures in Seattle
The city of Seattle is located in the state of Washington. It is located 113 miles south of the U.S.–Canadian border at a latitude of 47°37′ N. The city is located at sea level on Puget Sound, near the Pacific Ocean.
Question 1.
Use the information above to predict whether Seattle’s monthly average temperature data will have a large or small mean absolute deviation. Explain your answer.

Question 2.
The average monthly temperatures for Seattle are given below. Find the listed data landmarks and measures of spread. Round your answers to the nearest tenth.

a. Minimum: _________
b. Maximum: _________
c. Median: _________
d. Mean: _________
e. Range: _________
f. Mean absolute deviation: _________

Question 3.
Use the data landmarks and measures of spread you found in Problem 2 to draw some conclusions about Seattle’s average monthly temperatures.

Bring in one 3-dimensional shape with faces made up of polygons. It will go in the class Shapes Museum. Find a shape that has at least one face that is not a rectangle. See pages 246–248 in your Student Reference Book for examples of the kinds of shapes to bring.
Practice
Solve.
Question 4.
_______ = 0.09 ÷ 0.03

Question 5.
0.75 ÷ 0.3 = _______