Unit 5 Plan - Grade 7

Lesson 1

Interpreting Negative Numbers

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Signed Numbers (1 problem)

Here is a set of signed numbers: 7, -3, $\frac12$ , -0.8, 0.8, - $\frac{1}{10}$ , -2

Order the numbers from least to greatest.
If these numbers represent temperatures in degrees Celsius, which is the coldest?
If these numbers represent elevations in meters, which is the farthest away from sea level?

Show Solution

-3, -2, -0.8, - $\frac{1}{10}$ , $\frac12$ , 0.8, 7
-3
7

Lesson 2

Changing Temperatures

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Stories about Temperature (1 problem)

Write a story about temperatures that the following expression could represent: $27 + (\text-11)$
Draw a number line diagram and write an expression to represent this situation: “On Tuesday at lunchtime, it was $29 ^\circ \text{C}$ . By sunset, the temperature had dropped to $16 ^\circ \text{C}$ .”

Show Solution

Sample response:

It was 27 degrees at lunch time, and by the evening the temperature had dropped 11 degrees.
$29 + (\text-13)$

Lesson 3

Changing Elevation

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Add 'Em Up (1 problem)

Find each sum.

$56 + (\text- 56)$
$\text- 240 + 370$
$\text- 5.7 + (\text- 4.2)$

Show Solution

0
130
-9.9

Lesson 4

Money and Debts

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Buying a Bike (1 problem)

Clare has $150 in her bank account. She buys a bike for $200. What is Clare's account balance now?
If Clare earns $75 the next week from delivering newspapers and deposits it in her account, what will her account balance be then?

Show Solution

-$50
$25

Lesson 5

Representing Subtraction

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Same Value (1 problem)

Which other expression has the same value as $(\text-14) - 8$ ? Explain your reasoning.
1. $(\text-14) + 8$
2. $14 - (\text-8)$
3. $14 + (\text-8)$
4. $(\text-14) + (\text-8)$
Which other expression has the same value as $(\text-14) - (\text-8)$ ? Explain your reasoning.
1. $(\text-14) + 8$
2. $14 - (\text-8)$
3. $14 + (\text-8)$
4. $(\text-14) + (\text-8)$

Show Solution

$(\text-14) + (\text-8)$ . Sample reasoning: Adding -8 results in the same value as subtracting 8.
$(\text-14) + 8$ . Sample reasoning: Subtracting -8 results in the same value as adding 8.

Lesson 6

Finding Differences

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A Subtraction Expression (1 problem)

Select all of the choices that are equal to $(\text-5)-(\text-12)$ .

-7
7
The difference between -5 and -12
The difference between -12 and -5
$(\text-5)+12$
$(\text-5)+(\text-12)$

Show Solution

B, C, E

Lesson 7

Adding and Subtracting to Solve Problems

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Coffee Shop Cups (1 problem)

Here is some record keeping from a coffee shop about their paper cups. Cups are delivered 2,000 at a time.

day	change
Monday	+2,000
Tuesday	-125
Wednesday	-127
Thursday	+1,719
Friday	-356
Saturday	-782
Sunday	0

Explain what a positive and negative number means in this situation.
Assume the starting amount of coffee cups is 0. How many paper cups are left at the end of the week?
How many cups do you think were used on Thursday? Explain how you know.

Show Solution

Sample response: Positive might mean the number of cups delivered or delivered minus used. Negative might mean the number of cups used.
2,329 cups
281. Sample reasoning: It looks like some were delivered and some were used. Since they are delivered 2,000 at a time, $2,000-1,719$ would be the number used.

Section A Check

Section A Checkpoint

Problem 1

Find each sum or difference. If you get stuck, consider using a number line.

$8 + \text-11$
$\text-1 + 10$
$12 - (\text-6)$
$\text-4.5 - 2.3$
$\text-\frac79 + \frac29$

Show Solution

-3
9
18
-6.8
$\text-\frac59$

Problem 2

A librarian keeps track of how many books are checked in or out of the library during the day.

3 books checked out
6 books checked in
2 books checked in
5 books checked out
1 book checked in
4 books checked out

How could the librarian use signed numbers to represent this information?
How does the total number of books in the library at the end of the day compare to the beginning of the day?

Show Solution

Sample response: The librarian could use positive numbers to represent books that get checked in and negative numbers to represent books that get checked out.
There are 3 fewer books in the library at the end of the day, because $\text-3 + 6 + 2 + \text-5 + 1 + \text-4 = \text-3$ .

Lesson 8

Multiplying Rational Numbers (Part 1)

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Multiplication Equations (1 problem)

Two runners start at the same point. For each runner, write a multiplication equation that describes their journey.

Lin runs for 25 seconds at 8 meters per second. What is her finish point?
Diego runs for 30 seconds at -9 meters per second. What is his finish point?

Show Solution

Sample response:

$8 \boldcdot 25 = 200$
$\text-9 \boldcdot 30 = \text-270$

Lesson 9

Multiplying Rational Numbers (Part 2)

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True Statements (1 problem)

Decide if each equation is true or false.

$7 \boldcdot 8 = 56$
$\text-7 \boldcdot 8 = 56$
$\text-7 \boldcdot \text-8 = \text-56$
$\text-7 \boldcdot \text-8 = 56$
$(3.5) \boldcdot 12 = 42$
$(\text-3.5) \boldcdot \text-12 =\text -42$
$(\text-3.5) \boldcdot \text-12 = 42$
$\text-12 \boldcdot \frac{7}{2} = 42$

Show Solution

true
false
false
true
true
false
true
false

Lesson 10

Multiply!

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Making Mistakes (1 problem)

Noah was doing some homework and answered the following questions. Do you agree with his answers? If you disagree, explain your reasoning.

$\text-5 \boldcdot 8 = \underline{\phantom{00}40\phantom{00} }$
$(2.7) \boldcdot (\text-2.5) = \underline{\phantom{00}\text-6.75\phantom{00}}$
$\text-\frac34 \boldcdot \text- \frac57 = \underline{\phantom{00}\text-\frac{15}{28}\phantom{00}}$

Show Solution

Disagree. Sample reasoning: A negative number times a positive number is a negative number.
Agree.
Disagree. Sample reasoning: A negative number times a negative number is a positive number.

Lesson 11

Dividing Rational Numbers

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Matching Division Expressions (1 problem)

Match each expression with its value.

$15 \div 12$
$12 \div (\text-15)$
$12 \div 15$
$15 \div (\text-12)$

-0.8
0.8
-1.25
1.25

Show Solution

$15 \div 12= 1.25$
$12 \div (\text-15) = \text-0.8$
$12 \div 15 = 0.8$
$15 \div (\text-12) = \text-1.25$

Lesson 12

Negative Rates

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Submarines (1 problem)

A submarine is descending to examine the seafloor 2,100 feet below the surface. It takes the submarine 2 hours to make this descent. Write an equation to represent the relationship between the submarine's elevation and time.
Another submarine’s descent can be represented as $y=\text-240x$ , where $y$ is the elevation in feet and $x$ is time in hours. How long will it take this submarine to make the descent?

Show Solution

$y=\text-1050 x$ , where $y$ is the elevation in feet and $x$ is the time in hours
8.75 hours

Section B Check

Section B Checkpoint

Problem 1

Find each product or quotient.

$\text-4 \boldcdot 10$
$\text-5 \boldcdot \text-11$
$(3.6) \boldcdot (\text-0.2)$
$\text-20 \div \text-4$
$\frac34 \div \left( \text-\frac13 \right)$

Show Solution

-40
55
-0.72
5
$\text-\frac94$ (or equivalent)

Problem 2

One train is traveling north at 10 meters per second. On a parallel track, another train is traveling south at 7 meters per second.

How could we use signed numbers to represent this situation?
Where was each train engine 5 seconds before they passed each other?

Show Solution

Sample response: We could use positive numbers to represent movement to the north and negative numbers to represent movement to the south.
The first train was 50 meters south of the crossing point, because $10 \boldcdot \text-5 = \text-50$ . The second train was 35 meters north of the crossing point, because $\text-7 \boldcdot \text-5 = 35$ .

Lesson 13

Expressions with Rational Numbers

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Make Them True (1 problem)

Complete each equation with an operation to make it true.

$24\ \_\_\_\ \frac34 =18$
$24\ \_\_\_ \ \text- \frac34 =\text-32$
$12\ \_\_\_\ 15=\text-3$
$12\ \_\_\_\ \text-15=27$
$\text-18\ \_\_\_\ \text-\frac34 =24$

Show Solution

$24 \boldcdot \frac34 =18$
$24 \div \text- \frac34 =\text-32$
$12 - 15=\text-3$
$12 - \text-15=27$
$\text-18\div \text-\frac34 =24$

Lesson 14

Solving Problems with Rational Numbers

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Charges and Checks (1 problem)

Lin's sister has a checking account. If the account balance ever falls below $0, the bank charges her a fee of $5.95 per day. Today, the balance in Lin's sister's account is -$2.67.

If she does not make any deposits or withdrawals, what will be the balance in her account after 2 days?
In 14 days, Lin's sister will be paid $430 and will deposit it into her checking account. If there are no other transactions besides this deposit and the daily fee, will Lin continue to be charged $5.95 each day after this deposit is made? Explain or show your reasoning.

Show Solution

-$14.57
No. Sample reasoning: Even if the fee was $10 per day, that would total $140, which is much less than what she will deposit.

Lesson 15

Solving Equations with Rational Numbers

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Hiking Trip (1 problem)

The Hiking Club is taking another trip. The hike leader has a watch that shows that they have gained 296 feet in altitude from their starting position. Their altitude is now 285 feet. The equation $x + 296 = 285$ can be used to represent the situation.

Solve for $x$ .
What does $x$ mean in the situation?

Show Solution

$x = \text- 11$
Sample response: Since $x$ represents starting elevation, the Hiking Club started at an altitude of -11 feet, or 11 feet below 0.

Lesson 16

Representing Contexts with Equations

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Floating above a Sunken Canoe (1 problem)

A balloon is floating above a lake, and a sunken canoe is below the surface of the lake. The balloon’s vertical position is 12 meters, and the canoe’s is -4.8 meters. The equation $12 + d = \text-4.8$ represents this situation.

What does the variable $d$ represent?
What value of $d$ makes the equation true? Explain your reasoning.

Show Solution

The difference in elevation. (“Change in elevation” should also be accepted.)
-16.8. Sample reasoning: The equation $12+d=\text-4.8$ can be rewritten as $d=\text-4.8-12$ , and $\text-4.8-12=\text-16.8$ .

Section C Check

Section C Checkpoint

Problem 1

Solve each equation.

$a + 6.5 = \text-3.2$
$\text-4b = 30$

Show Solution

$a = \text-9.7$
$b = \text-7.5$ (or equivalent)

Problem 2

Choose the equation that best represents the situation:

An elevator is descending at a rate of 2.5 feet per second. How long will it take the elevator to go down 20 feet?

A.$20 - x = 2.5$

B.$x + 2.5 = \text-20$

C.$20 \div \text-2.5 = x$

D.$\text-2.5x = \text-20$✓

Show Solution

D

Problem 3

A scientist is using a drone to explore a cave. Over a period of 6 seconds, the drone goes from 25 feet below the surface to 16 feet below the surface.

How fast is the drone moving upward?
At this rate, when will the drone be 5 feet above the ground?

Show Solution

$\frac32$ feet per second (or equivalent), because $\text-16 - \text-25 = 9$ and $9 \div 6 = \frac32$
in 14 seconds, because $5 - \text-16 = 21$ and $21 \div \frac32 = 14$

Lesson 17

The Stock Market

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No cool-down

Unit 5 Rational Number Arithmetic — Unit Plan