End-of-Unit Assessment

1.

Select all the true statements.

A.

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

B.

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

C.

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

D.

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

E.

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

Answer: C, E

Teaching Notes

This problem targets students’ understanding of sums and differences of two rational numbers. Some students may wish to draw number lines to help.

Students who select choice A may have thought it was the same as $72 - 100$ . Students who select choice B may have misread the problem as $(\text{-}3.7) - (\text{-}4.1)$ . Students who do not select choice C may have a major misconception about opposites. Students who select choice D may have misunderstood the problem as $\text-\frac52 - \frac72$ . Students who do not select choice E may have misread the additional negative sign inside the parentheses or made an error calculating the value of $\text{-}\frac{12}4$ .

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.

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

Teaching Notes

Students who select choice A may be finding the change in elevation from $a$ to $b$ instead of the difference between $a$ and $b$ , which is defined as $a−b$ . Students who select choice B or C may not recognize that “below sea level” means the trout’s elevation is -23 feet.

3.

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

Which expression has a negative value?

A.

$x + y$

B.

$x - y$

C.

$x \boldcdot y$

D.

$\frac x y$

Answer:

$x + y$

Teaching Notes

Students who select an incorrect option may have a misunderstanding about the meaning of the operation in the expression or may have misinterpreted addition as moving to the right on a number line.

The numbers are small enough that the answers can be computed, but it is simpler to use the sign and size of the numbers. Watch for students who compute, and ask them what shortcuts they might take.

4.

Calculate the value of each expression.

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

Answer:

6
-6
6
-6
-6
6

Teaching Notes

Students should see the relationships between the first three expressions and between the last three.

5.

Solve each equation.

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

Answer:

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

Teaching Notes

Students solve equations of the form $x + p = q$ and $px=q$ when the solution or coefficients are negative.

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.

Answer:

	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.

Teaching Notes

Students work with signed numbers in a money context and interpret the meaning of negative numbers in that context.

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.

Answer:

$\text{-}\frac32$ 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.

Teaching Notes

As mentioned in the solution rubric, a student finding an incorrect rate should be given credit for correct calculations based on that rate.

If discussing this problem, look for multiple solution methods, especially those that involve working with signed numbers for both the rate and number of gallons. While it is not necessary to use signed numbers to solve these problems, it is an excellent way to work here. Do not penalize students who solve the problem without using signed number arithmetic, although they must use a signed number for the rate.