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

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

Write expressions in equivalent forms to solve problems.

Perform arithmetic operations on polynomials.

Represent and solve problems involving multiplication and division.

Interpret the structure of expressions.

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

Work with radicals and integer exponents.

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

Rewrite rational expressions.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Understand the relationship between zeros and factors of polynomials.

Represent and solve equations and inequalities graphically.

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

Work with addition and subtraction equations.

Use properties of operations to generate equivalent expressions.

Reason about and solve one-variable equations and inequalities.

Use the four operations with whole numbers to solve problems.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Use polynomial identities to solve problems.

Gain familiarity with factors and multiples.

Write and interpret numerical expressions.

Analyze patterns and relationships.

Solve equations and inequalities in one variable.

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

Represent and analyze quantitative relationships between dependent and independent variables.

Create equations that describe numbers or relationships

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

Add and subtract within 20.

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

Represent and solve problems involving addition and subtraction.

Multiply and divide within 100.

Generate and analyze patterns.

Solve systems of equations.
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: 2

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