End-of-Unit Assessment

1.

Select all of the equations that are equivalent to

2x + 6 = x - 4

.

A.

x + 6 = \text{-}4

B.

2x = x + 2

C.

2x + 8 = x - 2

D.

2(x+3) = x - 4

E.

2x + 3 = x - 2

Answer: A, C, D

2.

Select all the systems of equations that have exactly 1 solution.

A.

$\begin{cases} y = 3x + 1 \\ y = \text-3x - 7 \end{cases}$

B.

$\begin{cases} y = 3x + 1 \\ y = x + 1 \end{cases}$

C.

$\begin{cases} y =3x + 1 \\ y = 3x + 7 \end{cases}$

D.

$\begin{cases} x+y = 10 \\ 2x + 2y = 20 \end{cases}$

E.

$\begin{cases} x + y = 10 \\ x + y = 12 \end{cases}$

Answer: A, B

3.

Which system of equations has a solution of $(3,\text{-}4)$ ?

A.

\begin{cases} y = 2x -10 \\ y = \text{-}x +1 \end{cases}

B.

\begin{cases} x + y = \text{-}1 \\ y = \text{-}4x + 8 \end{cases}

C.

\begin{cases} y = 4 - 2x \\ y + 3x = 5 \end{cases}

D.

\begin{cases} y - 5x = \text{-}17 \\ x - y = \text{-}7 \end{cases}

Answer: $\begin{cases} x + y = \text{-}1 \\ y = \text{-}4x + 8 \end{cases}$

4.

Solve this equation. Explain or show your reasoning.

$\displaystyle \frac 1 2 x - 7 = \frac 1 3 \left(x - 12\right)$

Answer:

$x = 18$ . Sample reasoning: Use the distributive property to rewrite the equation as $\frac 1 2 x - 7 = \frac 1 3 x - 4$ . Then, subtract $\frac 1 3 x$ from each side: $\frac 1 6 x - 7 = \text{-}4$ . Add 7 to each side: $\frac 1 6 x = 3$ . Then $x = 3 \div \frac 1 6 = 3 \boldcdot 6 = 18$ .

Minimal Tier 1 response:

Work is complete and correct.
Sample:
$\frac 1 2 x - 7 = \frac 1 3 x - 4$
$\frac 1 6 x - 7 = \text{-}4$
$\frac 1 6 x = 3$

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Algebra mistakes not directly related to the work of this unit: incorrectly subtracting or dividing fractions; incorrectly adding or subtracting integers.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Solution given with no work shown; algebra mistakes that are pertinent to the work of this unit: dividing both sides of the equation by $\frac13$ initially, but dividing only $\frac12x$ or -7 by $\frac13$ ; incorrect use of the distributive property; failure to use inverse operations.

Teaching Notes

Watch for students having difficulty with the step of combining terms. Students may be having difficulty understanding that the same steps that work for integers also work for fractions and decimals.

5.

Solve this system of equations.

$\begin{cases} 3x + 4y = 36\\ y=\text-\frac{1}{2} x + 8 \end{cases}$

Answer:

$x = 4$ , $y = 6$

Teaching Notes

This system is most likely solved by substitution, but can also be solved by inspection after graphing.

6.

Andre and Elena are each saving money. Andre starts with $100 in his savings account and adds $5 per week.

The amount of money Elena has saved is shown by the line in this graph

Write an equation representing Andre’s savings after $x$ weeks.
After how many weeks will Andre and Elena have the same amount of money in their savings accounts? Explain or show your reasoning.

Answer:

$y = 5x+100$ (or equivalent)
6 weeks. Sample reasonings:
- Graphing Andre’s savings on the same graph as Elena, I see that the lines intersect at $(6,130)$ , so they each have $130 after 6 weeks.
- An equation for Elena’s savings is $y = 20x+10$ . Solve $5x+100 = 20x+10$ to get $x = 6$

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$y = 5x + 100$
6 weeks. Correct graph of Andre’s equation and intersection point marked (or correctly set up and solved one-variable equation).

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Andre’s equation is written as an expression; algebra or graphing mistakes in Part B work contains a correct solution to the equation in Part B, but the explanation does not answer the number of weeks.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: No sensible approach to writing Andre's equation; no reasonable equations, tables, or graphs to represent amount of money saved after $x$ weeks; no reasonable method for finding the number of weeks after which Andre and Elena will have the same amount of money.

Teaching Notes

While the most likely technique is to set up and solve a one-variable equation, it is also possible for students to solve this problem through graphing, or through making a table of values for each person.

7.

At a game night, people can choose to play chess, a 2-player game, or to play hearts, a 4-player card game.

60 people are playing the games with $x$ representing the number of chess games being played and $y$ representing the number of hearts games.

Complete this table showing some possible combinations of the number of each type of game being played.

chess games ( $x$ ) hearts games ( $y$ )

30 0

8

10

15

2
There are 3 more games of hearts being played than games of chess being played. How many of each game are being played? Explain or show your reasoning.

chess games ( $x$ )	hearts games ( $y$ )
30	0
	8
10
	15
2

Answer:

chess games ( $x$ ) hearts games ( $y$ )

30 0

14 8

10 10

0 15

2 14
8 chess games and 11 hearts games. Sample reasoning: The solution solves the system of equations $2x + 4y = 60$ and $y = x + 3$ . Solve by substitution: $2x + 4(x+3) = 60$ . Then $x = 8$ , and $y = 11$ because $y = x + 3$ .

chess games ( $x$ )	hearts games ( $y$ )
30	0
14	8
10	10
0	15
2	14

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Solutions that simply involve filling out more rows of the table are acceptable.
Sample:

See table.
Solve the system $2x + 4y = 60$ and $y = x + 3$ .
$2x + 4(x+3) = 60$
$2x+4x+12=60$
$6x=48$
$x=8$
$y=11$ .
8 chess games, 11 hearts.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: One row of the table is incorrect; system of equations is present but work to solve those equations contains algebra errors; equation to represent “3 more games of hearts than chess” actually represents “3 more games of chess than hearts.”

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Acceptable errors: One of the equations in Part B is incorrect because of an error filling out the table in Part A.
Sample errors: Several rows of the table are incorrect; a system of equations is present and their solution is correct, but the equations do not come close to representing the situation; a correct system of equations is present but the work is incorrect and does not involve the substitution method.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Incorrect or missing answer to Part B with no system of equations written; misinterpretation of the 4-player game/2-player game constraint means that the table is filled out completely incorrectly.

Teaching Notes

Most likely, students will solve this system of equations by substitution, but the system can also be solved by graphing or by making a detailed table.