options: Analyze patterns and relationships.

Understand properties of multiplication and the relationship between multiplication and division.

Understand the relationship between zeros and factors of polynomials.

Reason about and solve one-variable equations and inequalities.

Represent and solve problems involving addition and subtraction.

Rewrite rational expressions.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Represent and analyze quantitative relationships between dependent and independent variables.

Work with radicals and integer exponents.

Write expressions in equivalent forms to solve problems.

Generate and analyze patterns.

Use the four operations with whole numbers to solve problems.

Perform arithmetic operations on polynomials.

Add and subtract within 20.

Work with addition and subtraction equations.

Gain familiarity with factors and multiples.

Understand and apply properties of operations and the relationship between addition and subtraction.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand the connections between proportional relationships, lines, and linear equations.

Solve equations and inequalities in one variable.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Use properties of operations to generate equivalent expressions.

Solve systems of equations.

Work with equal groups of objects to gain foundations for multiplication.

Interpret the structure of expressions.

Create equations that describe numbers or relationships

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Write and interpret numerical expressions.

Represent and solve equations and inequalities graphically.

Understand solving equations as a process of reasoning and explain the reasoning.

Use polynomial identities to solve problems.

Represent and solve problems involving multiplication and division.

Multiply and divide within 100.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
problem_activity: Task
The profit, $P$ (in thousands of dollars), that a company makes selling an item is a quadratic function of the price, $x$ (in dollars), that they charge for the item. The following expressions for $P(x)$ are equivalent:
$$\begin{align}P(x)=&-2x^2+24x-54\\
P(x)=&-2(x-3)(x-9)\\
P(x)=&-2(x-6)^2+18
\end{align}$$
Which of the equivalent expressions for $P(x)$ reveals the price which gives a profit of zero without changing the form of the expression?
P(x)= -2x$^2$+24x-54
P(x)= -2(x-3)(x-9)
P(x)= -2(x-6)$^2$+18
Find a price which gives a profit of zero.
Which of the equivalent expressions for $P(x)$ reveals the profit when the price is zero without changing the form of the expression?
P(x)= -2x$^2$+24x-54
P(x)= -2(x-3)(x-9)
P(x)= -2(x-6)$^2$+18
Find the profit when the price is zero.
Which of the equivalent expressions for $P(x)$ reveals the price which produces the highest possible profit without changing the form of the expression?
P(x)= -2x$^2$+24x-54
P(x)= -2(x-3)(x-9)
P(x)= -2(x-6)$^2$+18
Find the price which gives the highest possible profit.
For each of parts (a), (c), and (e), students choose which form of the function is displayed, and then use that to answer the corresponding questions in parts (b), (d), and (f) respectively.  Students are assessed both on selecting the correct form of the function and on their numerical answer for each part.


correct_option_index: 9

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