End-of-Unit Assessment

1.

Lin uses a $50 gift card to buy a game on her phone for $9.99. She also uses the gift card to buy upgrades for her characters in the game. Each upgrade costs $1.29.

Which of these inequalities describes this situation, where $n$ is the number of upgrades Lin can buy?

A.

$9.99 + 1.29n \ge 50$

B.

$9.99 + 1.29n \le 50$

C.

$9.99 - 1.29n \ge 50$

D.

$9.99 - 1.29n \le 50$

Answer:

$9.99 + 1.29n \le 50$

Teaching Notes

Students selecting C or D may be thinking that, because each upgrade costs $1.29, the amount should be subtracted. However, the $1.29 per upgrade is added to the amount being spent on the gift card. Students selecting A 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 $x$ that make the inequality $\text-3x + 1 < 7$ true?

A.

A number line labeled A with the numbers negative 5 through 5 indicated. There is a closed circle at negative 2 and an arrow is drawn from the closed circle extending to the left.

B.

A number line labeled B with the numbers negative 5 through 5 indicated. There is an open circle at negative 2 and an arrow is drawn from the open circle extending to the left.

C.

A number line labeled C with the numbers negative 5 through 5 indicated. There is a closed circle at negative 2 and an arrow is drawn from the closed circle extending to the right.

D.

A number line labeled D with the numbers negative 5 through 5 indicated. There is an open circle at negative 2 and an arrow is drawn from the open circle extending to the right.

Answer:

Teaching Notes

Students selecting A or B instead of D should be reminded of the recent work on checking numbers to determine the solution of an inequality. For example, $x = 0$ makes the inequality true, so it should be part of the graph. Students selecting A or C need to review the meaning of open and closed circles for graphed inequalities.

3.

Select all expressions that are equivalent to $6x + 1 - (3x - 1)$ .

A.

$6x + 1 - 3x - 1$

B.

$6x + \text-3x + 1 + 1$

C.

$3x+2$

D.

$6x - 3x +1 -1$

E.

$6x + 1 +\text - 3x -\text - 1$

Answer: B, C, E

Teaching Notes

Students selecting A have not distributed the negative sign to the subtracted 1. Students failing to select B might be thrown by the different ordering of the terms, or they might not be distributing the negative sign correctly. Students failing to select C could have made a variety of algebra mistakes on their way to rewriting the expression with fewer terms. Students selecting D have also incorrectly distributed the negative sign. Choice E shows the process for distributing the negative sign explicitly. A student failing to select E may need some extra coaching with the distributive property.

4.

At midnight, the temperature in a city was 5 degrees Celsius. The temperature was dropping at a steady rate of 2 degrees Celsius per hour.

Write an inequality that represents $t$ , the number of hours past midnight, when the temperature was colder than -4 degrees Celsius. Explain or show your reasoning.
On the number line, show all the values of $t$ that make your inequality true.

Answer:

$5 - 2t < \text-4$ or $t > \frac 9 2$ (or equivalent). Sample reasoning:
- $5-2t$ shows the temperature starting at 5 degrees Celsius and decreasing by 2 degrees every hour after midnight, $t$ . Because we want this quantity to be less than -4 degrees Celsius, write $5-2t < \text-4$ .
- The temperature will reach -4 degrees Celsius after $\frac 9 2$ hours. Since the temperature needs to be colder, $t$ must be greater than $\frac 9 2$ .
The graph shows an open circle at $4\frac12$ with shading to the right.

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$5 - 2t < \text-4$ . The temperature starts at 5 degrees, then it goes down 2 degrees every hour. The temperature has to be less than -4 degrees.
The graph shows an open circle at $4\frac12$ with shading to the right.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Acceptable errors: Bases graph in part b on an incorrect inequality in part a.
Sample errors: Gives no explanation for a correct inequality; writes an inequality with only one mistake, like $5+2t<\text-4$ ; does not use “open circle” notation for graph; uses incorrect direction of inequality in parts a, b, or both.

Tier 3 response:

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

Teaching Notes

Students are not required to solve the inequality in part a, but many will anyway, perhaps by first finding $t = \frac 9 2$ as the time when the temperature was exactly -4 degrees Celsius.

5.

Expand to write an equivalent expression:

$\text{-} \frac{1}{4}(\text-8x+12y)$
Factor to write an equivalent expression:

$36a-16$

Answer:

$2x-3y$ (or equivalent)
$4(9a-4)$ or $2(18a-8)$ (or equivalent)

Teaching Notes

Look for sign errors in part a. There are multiple ways to factor the expression in part b.

6.

Tyler is rewriting the expression $6 - 2x + 5 + 4x$ with fewer terms. Here is his work:

$\displaystyle 6 - 2x + 5 + 4x$

$\displaystyle (6-2)x + (5+4)x$

$\displaystyle 4x+9x$

$\displaystyle 13x$

Tyler’s work is incorrect. Circle the step where he made a mistake. Explain or show why this expression is not equivalent to $6 - 2x + 5 + 4x$ .
Write an expression equivalent to $6 - 2x + 5 + 4x$ that only has two terms.

Answer:

Students circle $(6-2)x + (5+4)x$ . Sample reasoning:
- $(6-2)x$ is not equivalent to $6-2x$ . I can show this because when $x$ is 0, one expression equals 0 but the other equals 6. The same is true for $(5-4)x$ and $5+4x$ .
- Tyler did not use the distributive property correctly. $(6-2)x = 6x-2x$ , not $6-2x$ . The same is true for $5−4x$ and $5+4x$ .
Sample response:

$\displaystyle 6 - 2x + 5 + 4x$

$\displaystyle 6 + \text-2x + 5 + 4x$

$\displaystyle \text-2x + 4x + 6 + 5$

$\displaystyle (\text-2+4)x+11$

$\displaystyle 2x+11$

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$6-2x$ does not equal $(6-2)x$ .
$2x+11$

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Identifies Tyler’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; has an algebra mistake in part b 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 Tyler’s reasoning; gives incorrect answer to part b with no work shown.

Teaching Notes

Students who are very comfortable with algebra can jump straight to the answer in part b without showing reasoning. However, an incorrect answer with no reasoning earns an automatic Tier 3 rating.

7.

A teacher only uses his car to drive to and from work each day, so the car only uses 0.6 gallon of gas each day. The car holds 14 gallons of gas. A warning light comes on when the remaining gas is 1.5 gallons or less.

If $d$ represents the number of days of driving, what does $14−0.6d$ represent?
Write and solve an equation to determine the number of days the teacher can drive the car without the warning light coming on.
Write and solve an inequality that represents this situation. Explain clearly what the solution to the inequality means in the context of this situation.

Answer: Sample responses:

$14 - 0.6d$ is the amount of gas in the tank, in gallons, after $d$ days of driving.
$14 - 0.6d = 1.5$ . Subtract 14 from each side to get $\text-0.6d = \text-12.5$ . Divide each side by $\text-0.6$ to get $d \approx 20.83$ . The car can drive for about 20 days, and the warning light will come on near the end of the 21st day.
The inequality $14 - 0.6d > 1.5$ represents the times when the warning light is off. The solution to this inequality is $d < 20.83$ , so the warning light is off for all times until near the end of the 21st day.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Acceptable errors: Says the teacher can drive for 21 days if it is specified that the light will come on during the 21st day.
Sample:

The expression is how much gas is left. $t$ is the number of days.
$14 - 0.6d = 1.5$ , $\text-0.6d = \text-12.5$ , $d \approx 20.83$ . He can drive for 20 days.
$14 - 0.6d > 1.5$ . $d<20.83$ because 20.83 is when the light comes on and before that the light would be off. This means that Diego’s father can drive for 20 full days before the warning light comes on.

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 any arithmetic errors with work shown; gives poor explanations (especially in part c) with otherwise correct work; asserts that since $d \approx 20.83$ is a solution to the equation, the teacher can drive for 21 full days; fails to justify the direction of the inequality in the solution to part c (where justification can involve algebra, testing points, or referring to the context).

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Misinterprets of the situation, leading to incorrect answers for part a; writes incorrect equation in part b; reverses the inequality sign in either the original inequality or the solution to part c; omits real-world interpretation in parts b and 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 $0.6d = 12.5$ and $14 - 0.dt \le 1.5$ .