End-of-Unit Assessment

1.

Select all expressions whose value is negative.

A.

$\frac{\text-15}{4}$

B.

$\frac{15}{\text-4}$

C.

$\frac{\text-15}{\text-4}$

D.

$(\text- 9)\boldcdot\frac23$

E.

$(\text- 9)\boldcdot(\text- \frac23)$

Answer:

A, B, D

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 the sign of the number. 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 make computations, and ask them what shortcuts they might take.

2.

A sunken ship is resting at 1,000 meters below sea level. Above the ship, a whale is swimming 600 meters below sea level. Above the ship and the whale, a plane is flying at 1,400 meters above sea level.

<p>A vertical number line labeled from bottom to top, negative 1000, negative 500, 0, 500, 1000, 1500, 2000.</p>

Which statement is true?

A.

The difference in elevation between the ship and the plane is 400 meters.

B.

The difference in elevation between the whale and the plane is -2,000 meters.

C.

The difference in elevation between the whale and the ship is -400 meters.

D.

The difference in elevation between the plane and the whale is 800 meters.

Answer: The difference in elevation between the whale and the plane is -2,000 meters.

Teaching Notes

Students who select choice C may be finding the change in elevation from $a$ to $b$ instead of the difference between $a$ to $b$ , which is defined as $a-b$ . Students who select choice D may not recognize that “below sea level” means the whale’s elevation is -600 meters.

3.

Let

x=\text- \frac56

and

y=\frac43

.

Select all the expressions that have a positive value.

A.

$x \boldcdot y$

B.

$\frac{y}{x}$

C.

$\frac{x}{y}$

D.

$x+y$

E.

$x-y$

F.

$y−x$

Answer:

D, F

Teaching Notes

Students who select an incorrect option may have a misunderstanding about the meaning of that operation 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.

$60 − (\text- 80)$
$2.3 + (\text- 4.9)$
$4.5 + (\text- 4.5)$
$\text-6+\frac13$

Answer:

140
-2.6
0
$\text-5\frac23$

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.

5.

Solve each equation.

$\text-3a = \text-27$
$b + 14.5 = 10$
$2 = \text- \frac{1}{5}c$

Answer:

$a = 9$
$b = \text-4.5$
$c = \text-10$

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	57	57
transaction 2	-25
transaction 3	40
transaction 4		10
transaction 5		-6

Explain what the number -6 tells you about transaction 5.
Explain what the number -25 tells you about transaction 3.

Answer:

	transaction amount	account balance
transaction 1	57	57
transaction 2	-25	32
transaction 3	40	72
transaction 4	-62	10
transaction 5	-16	-6

Sample response: Someone owes the bank $6. They need to deposit $6 into their account to bring their account balance back to $0.
Sample response: They took $25 out of the account.

Minimal Tier 1 response:

Work is complete and correct.
Sample response:

See table.
The person is $6 in debt.
They withdrew $25.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: One incorrect entry in the table; response for part b 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 negative signs; response for part b 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 scuba diver is diving at a constant rate when her team on the surface requests a status update. She looks at her watch which says her elevation is 18 feet below sea level. 8 seconds later, her elevation is 20 feet below sea level.

At what rate is her elevation changing? Use a signed number, and include the unit of measurement in your answer.
How many more seconds until she reaches her goal depth of 50 feet? Explain or show your reasoning.
How many seconds before her team requested an update was the diver at the surface of the water? Explain or show your reasoning.

Answer:

$\text-\frac14$ feet per second or equivalent. Her elevation decreases 2 feet in 8 seconds, which as a unit rate is $\frac28$ feet in 1 second. The value is negative because her elevation is decreasing.
120 more seconds (or equivalent). Sample reasoning: 30 feet of elevation must be lost at a rate of $\text-\frac14$ feet per second. It will take 120 seconds since $\text-30 \div (\text-\frac14)=120$ .
72 seconds before, or -72 seconds (or equivalent). Sample reasoning: She had descended 18 feet at a rate of $\text-\frac14$ feet per second, and $\text-18 \div (\text-\frac14)=72$ .

Minimal Tier 1 response:

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

$\text-\frac28$ feet per second.
She needs to descend 30 more feet, which is 15 times the 2 feet we know about, so it takes 120 seconds because $15 \boldcdot 8 = 120$ .
72 seconds before, or -72 seconds, because it takes 4 seconds to descend 1 foot, and $18 \boldcdot 4=72$ .

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; 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 $\frac{8}{20}$ feet every second; 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 elevation. 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.