End-of-Unit Assessment

Answer: A, D

Answer: B, D, E

Answer:

$\begin{cases} x+2y=10\\ \text-4x-y=2 \end{cases}$

Answer: $x=12$. Sample reasoning: Use the distributive property to rewrite the equation as $\frac{1}{3}x + 2 = x - 6$. Then, subtract $\frac{1}{3}x$ from each side to get $2 = \frac{2}{3}x - 6$. Add 6 to each side to get $8 = \frac{2}{3}x$. Then $12 = x$.

Minimal Tier 1 response:

• Work is complete and correct.

• Sample:
  \begin{align} \frac{1}{3}x + 2 &= x-6\\ 2 &= \frac{2}{3}x - 6 \\ 8 &= \frac{2}{3}x \\ 12 &= x\end{align} 

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 $\frac{1}{2}$ initially, but dividing only $\frac{1}{3}x$  or 2 by $\frac{1}{2}$; incorrect use of the distributive property; failure to use inverse operations.

Answer:

$x=\text-12$, $y=3$

Lin and Han have money in their school lunch accounts. Han starts with $100 in his account. He spends $15 each week on lunches. 

The amount of money in Lin’s lunch account is shown by the line in the graph.

1. Write an equation representing the amount of money in Han’s account after $x$ weeks.
2. After how many weeks will Han and Lin have the same amount of money in their lunch accounts? Explain or show your reasoning.

Answer:

1. $H = 100 - 15x$ (or equivalent)
2. 3 weeks. Sample reasonings:
   • Graphing Han’s account balance on the same graph as Lin’s, I see that the lines intersect at $(3, 55)$, so they each have $55 in their accounts after 3 weeks.
   • An equation for Lin’s account balance is $y = 10x + 25$. Solve $100 - 15x = 10x + 25$ to get $x = 3$.

Minimal Tier 1 response:

• Work is complete and correct.
• Sample:
  1. $y = \text{-}15x + 100$
  2. 3 weeks. Correct graph of Han’s account balance and intersection marked.

Tier 2 response:

• Work shows general conceptual understanding and mastery, with some errors.
• Sample errors: Han’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 Han’s equation; no reasonable equations, tables, or graphs to represent amount of money in the account earned after $x$ weeks; no reasonable method for finding the number of weeks after which Han and Lin will have the same amount of money.

Answer:

1. number of 6-player games	number of 2-player games
   0	25
   8	1
   1	22
   7	4
   4	13

2. 6 six-player games and 7 two-player games

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:

1. See table.

2. Let $x$ represent number of 6-player games and $y$ represent number of 2-player games. Solve the system $6x+2y=50$ and $x+y=13$

$6x+2(-x+13)=50$

$6x-2x+26=50$

$4x+26=50$

$4x=24$

$x=6$

$6+y=13$

$y=7$

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 such as $6x+2y=13$ or $x+y=50$.

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 "3 on 3" and "1 on 1" division games constraint means that the table is filled out completely incorrectly.