End-of-Unit Assessment

1.

Tyler has run 15 miles this month. For the rest of the month, he plans to run the same number of miles each day. There are 12 days left in the month. Before the end of the month, Tyler needs to run at least 35 miles to meet his goal.

Which of these inequalities describes this situation, where $m$ is the number of miles Tyler runs each day?

A.

$12m-15\geq35$

B.

$12m-15\leq35$

C.

$35\geq15+12m$

D.

$35 \leq 15+12m$

Answer:

D

Teaching Notes

Students selecting A or B may be thinking that, because Tyler has already run 15 miles this month, that amount should be subtracted. However, the miles he runs each of the 12 days left in the month is added to those 15 miles he already ran. Students selecting B or C have made a direction error in the inequality and may need a reminder about how to check the correct direction of a linear inequality.

2.

Which number line shows all the values of $y$ that make the inequality $\text-6y-2 \geq 10$ true?

A.

B.

C.

D.

Answer:

A

Teaching Notes

Students selecting C or D instead of A should be reminded of the recent work on checking numbers to determine the solution of an inequality. For example, $y=0$ makes the inequality false, so it should not be part of the graph. Students selecting B or D need to review the meaning of open and closed circles for graphed inequalities.

3.

Select all expressions that are equivalent to $\text-4(x+2)-2x+4$ .

A.

$\text-6x-4$

B.

$\text-4x+2-2x+4$

C.

$\text-10x$

D.

$\text-4x + \text-8 - 2x + 4$

E.

$\text-4x-2x-8+4$

Answer:

A, D, E

Teaching Notes

Students selecting B have not distributed the negative sign to both terms within the parentheses. Students failing to select E might be thrown by the different ordering of the terms, or they might not be distributing the negative sign correctly. Students failing to select A could have made a variety of algebra mistakes on their way to rewriting the expression with fewer terms. Choice D shows the process for distributing the negative sign explicitly. A student failing to select D may need some extra coaching with the distributive property.

4.

A hiker starts at an elevation of -50 feet (where 0 represents sea level). The hiker climbs at a rate of 15 feet per minute.

Write an inequality that represents $t$ , the number of minutes after the start of the hike, when the hiker’s elevation was higher than 5 feet above sea level. Explain or show your reasoning.
On the number line, show all the values of $t$ that make your inequality true.

Answer:

$\text-50+15t>5$ or $t>\frac{11}{3}$ (or equivalent). Sample reasoning:
- $\text-50+50t$ shows the elevation of the hiker after $t$ minutes of hiking. Because we want this quantity to be greater than 5, write $\text-50+15t>5$ .
- After $3\frac{2}{3}$ minutes, the hiker will be at an elevation higher than 5 feet above sea level.
The graph shows an open circle at approximately $3\frac{2}{3}$ with shading to the right.

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$\text-50+15t>5$ . The hiker begins at an elevation of -50 and climbs 15 feet per $t$ minutes to an elevation greater than 5 feet above sea level.
The graph shows an open circle at approximately $3\frac23$ with shading to the right.

Tier 2 response:

Work shows general conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Acceptable errors: Creates a correct graph based on an incorrect inequality.
Sample errors: Gives no explanation for a correct inequality; Writes an inequality that has one mistake, like $\text-50-15t>5$ ; does not use “open circle” notation in graph; uses incorrect direction of inequality in either or both parts; gives an answer of 3 minutes.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Writes an inequality that is not close to correct; omits graph or it is simply incorrect.

Teaching Notes

Students will need to denote the initial elevation by showing -50. Students are not required to solve the inequality, but many will anyway, perhaps by first finding $t>\frac{11}{3}$ as the number of minutes it would take to be higher than 5 feet above sea level.

5.

Factor to write an equivalent expression:

$\text-9x+15y$
Expand to write an equivalent expression:

$\text-5(7m-2)$

Answer:

$3(\text-3x+5y)$ or $\text-3(3x-5y)$ (or equivalent)
$\text-35m+10$ (or equivalent)

Teaching Notes

Look for sign errors.

6.

Priya is rewriting the expression $\text-2(4-2x)-3-2x$ with fewer terms. Here is her work:

$\text-2(4-2x)-3-2x$

$\text-8-2x-3-2x$

$\text-11-2x-2x$

$\text-11-4x$

Priya's work is incorrect. Circle the step where she made a mistake. Explain or show why this expression is not equivalent to $\text-2(4-2x)-3-2x$ .
Write an expression equivalent to $\text-4(x+3)-12x+5$ that has only two terms.

Answer:

Students circle $\text-8-2x-3-2x$ . Sample reasoning:
- $\text-8-2x$ is not equivalent to $\text-2(4-2x)$ . I can show this because when $x$ is 1, one expression equals -1 and the other equals -4.
- Priya didn’t use the distributive property correctly. $\text-2(4-2x)=\text-8+4x$ , not $\text-8-2x$ .
Sample response:

$\text-4(x+3)-12x+5$

$\text-4x-12-12x+5$

$\text-16x-7$

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$\text{-}8-2x$ does not equal $\text{-}2(4-2x)$ .
$\text{-}16x-7$ .

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Identifies Priya’s error correctly, but the explanation is either incorrect or vague; identifies the problem step by substituting a value like $x=0$ at each step, but no algebra mistake is identified; makes an algebra mistake when writing the equivalent expression with otherwise correct work shown.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Fails to identify the error in Priya’s reasoning; gives incorrect answer when writing the equivalent expression.

Teaching Notes

Students who are very comfortable with algebra can jump straight to the equivalent expression that has only two terms. However, an incorrect answer with no reasoning earns an automatic Tier 3 rating.

7.

A community center rents their hall for special events. They charge a fixed fee of $200 plus an hourly fee of $22.50. Lin has $300 to spend on renting the hall for a fundraiser.

If $h$ represents the number of hours, what does $22.50h+200$ represent?
Write and solve an equation to determine the number of hours Lin can rent the community center.
Write and solve an inequality that represents this situation. Explain what the solution to the inequality means in this situation.

Answer: Sample responses:

$22.50h+200$ represents the total amount Lin would pay to rent the community center for $h$ hours.
The equation is $22.50h+200=300$ . Subtract 200 from each side to get $22.50h=100$ . Divide each side by 22.50 to get 4.4. Lin can rent the community center for 4.4 hours before she runs out of money.
The inequality $22.50h+200 \leq 300$ represents the amount of money Lin would pay to rent the community center for a given number of hours. The solution to this inequality is $h \leq 4.4$ (rounded to the nearest tenth), so Lin can rent the community center for no more than 4.4 hours (or 4 hours if they only rent increments of whole hours).

Minimal Tier 1 response:

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

The expression represents the amount of money Lin would spend to rent the community center for $h$ hours
$22.50h+200=300$ . $h\approx4.4$ . Lin can rent the community center for 4 hours (assuming they only rent increments of whole hours).
$22.50h+200\leq300$ . $h \leq 4.4$ (rounded to the nearest tenth). Lin can rent the community center for no more than 4 hours (assuming they only rent increments of whole hours).

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Has arithmetic errors with work shown; gives poor explanations (especially the meaning of $22.50h+200$ ) with otherwise correct work.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Misinterprets the situation, leading to incorrect answer to part a; writes incorrect equation; reverses the inequality sign in either the original inequality or the solution; omits real-world interpretation in parts b or c.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Shows little progress on most problem parts; has three or more error types under Tier 3 response.

Teaching Notes

The first part helps students interpret the situation and represent it algebraically. Writing and solving the equation in the second part should help students identify the correct direction of the inequality for the last part. There are multiple other possible equations and inequalities, including $22.5h=135$ and $300 \geq 22.50h+200$ .