Unit 5 Rational Number Arithmetic — Unit Plan

Title

Assessment

Lesson 1

Interpreting Negative Numbers

Signed Numbers

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 4

Money and Debts

Buying a Bike

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

Same Value

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

A Subtraction Expression

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

Coffee Shop Cups

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)

Multiplication Equations

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)

True Statements

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 12

Negative Rates

Submarines

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

Make Them True

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

Charges and Checks

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.

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

$\text-2.5x = \text-20$

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

No cool-down

Unit 5 Assessment

End-of-Unit Assessment

Problem 1

Select all the true statements.

$72 - (\text{-}100)$ is negative.

$(\text{-}3.7) + (\text{-}4.1)$ is positive.

$2.3 + (\text{-}2.3)$ is equal to zero.

$\text-\frac 5 2 + \frac72$ is negative.

$\text{-}2.6 - (\text{-} \frac{12} 4)$ is positive.

Show Solution

C, E

Problem 2

A heron is flying 50 feet above sea level. Below the heron, a pelican is flying 17 feet above sea level. Below both birds, a trout is swimming 23 feet below sea level.

A vertical number line labeled from bottom to top, negative 40, negative 20, 0, 20, 40, 60.

Which statement is true?

A.The difference in elevation between the heron and the pelican is -33 feet.

B.The difference in elevation between the heron and the trout is 27 feet.

C.The difference in elevation between the pelican and the trout is 6 feet.

D.The difference in elevation between the trout and the heron is -73 feet.

Show Solution

The difference in elevation between the trout and the heron is -73 feet.

Problem 3

Let $x = \text{-}\frac {11} 8$ and $y = \text{-}\frac {11} 4$ .

Which expression has a negative value?

$x + y$

$x - y$

$x \boldcdot y$

$\frac x y$

Show Solution

$x + y$

Problem 4

Calculate the value of each expression.

$8 \boldcdot \frac34$
$\text{-}8 \boldcdot \frac34$
$\text{-}8 \boldcdot (\text{-}\frac34)$
$\text{-}\frac{18} 3$
$\frac{18}{\text{-}3}$
$\frac{\text{-}18}{\text{-}3}$

Show Solution

6
-6
6
-6
-6
6

Problem 5

Solve each equation.

$6a = \text-30$
$12-b = 15$
$\text-4 = c + \frac16$

Show Solution

$a = \text-5$
$b = \text-3$
$c= \text-4\frac16$ (or equivalent)

Problem 6

When the table here is complete, it shows five transactions and the resulting account balance in a bank account. Fill in the missing numbers.

	transaction amount	account balance
transaction 1	70	70
transaction 2	25
transaction 3	-32
transaction 4		40
transaction 5		-10

Explain what the number -32 tells you about transaction 3.
Explain what the number -10 tells you about transaction 5.

Show Solution

	transaction amount	account balance
transaction 1	70	70
transaction 2	25	95
transaction 3	-32	63
transaction 4	-23	40
transaction 5	-50	-10

Sample response: They took $32 out of the account.
Sample response: They owe the bank $10. They need to deposit $10 into their account to bring their account balance back to $0.

Minimal Tier 1 response:

Work is complete and correct.
Sample:

See table.
They withdrew $32.
The person is $10 in debt.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: One incorrect entry in the table; response for part c acknowledges debt but is flawed in other ways.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Multiple incorrect entries in the table, including systematic errors like always making the two columns equal or omitting the negative signs; response for part c does not involve debt; multiple error types under Tier 2 response.

Problem 7

A water tank can hold 30 gallons when completely full. A drain is emptying water from the tank at a constant rate. When Jada first sees the tank, it contains 21 gallons of water. After draining for 4 more minutes, the tank contains 15 gallons of water.

At what rate is the amount of water in the tank changing? Use a signed number, and include the unit of measurement in your answer.
How many more minutes will it take for the tank to drain completely? Explain or show your reasoning.
How many minutes before Jada arrived was the water tank completely full? Explain or show your reasoning.

Show Solution

$\text{-}\frac32$ gallons per minute. 6 gallons leave in 4 minutes, which as a unit rate is $\frac32$ gallons in 1 minute. The value is negative because the amount of water is decreasing.
10 minutes. Sample reasoning: 15 gallons must be drained at a rate of $\text{-}\frac 32$ gallons per minute. It will take 10 minutes since $(\text{-}15) \div (\text{-}\frac 32) = 10$ .
6 minutes before, or -6 minutes. Sample reasoning: 9 gallons have already drained from the tank when Jada arrives. It took 6 minutes since $9 \div (\text{-}\frac32) = \text{-}6$ .

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample response:

$\text{-}\frac32$ gallons per minute
10, because 1.5 gallons drain every minute. $15 \div 1.5 =10$ .
-6, because $\text-9 \div 1.5 = \text-6$ .

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Minor visible calculation errors in multiplying, dividing, or determining the rate; incorrect or omitted units used on rate; rate of $\frac32$ instead of $\text{-}\frac32$ ; correct answers without justification.
Acceptable errors: An incorrect rate coming from a calculation error is used correctly through the rest of the problem.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Incorrect rate from conceptual misunderstanding, such as $\frac23$ gallon per minute; omitted rate; invalid methods or omissions on either of the time questions.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Incorrect or omitted rate and invalid methods or omissions on at least one of the time questions.