• Making the Shifts

  How does this task exemplify the instructional Shifts required by CCSSM?

  Focus	Belongs to the Widely Applicable Prerequisites for College and Careers
  Coherence	Builds on work with algebraic expressions begun in grades 6–8, with added complexity appropriate to high school
  Rigor	Conceptual Understanding: primary in this task Procedural Skill and Fluency: not targeted in this task Application: secondary in this task

• 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:

  $$\begin{align} \\P(x)=-2x^2+24x-54 \\P(x)=-2(x-3)(x-9) \\P(x)=-2(x-6)^2+18\,\end{align}$$

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  1. Which of the equivalent expressions for $P(x)$ reveals the price which gives a profit of zero without changing the form of the expression?

  $P(x)=-2x^2+24x-54$ $P(x)-2(x-3)(x-9)$ $P(x)=-2(x-6)^2+18$

  1. Find a price which gives a profit of zero.
     
     

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  2. Which of the equivalent expressions for $P(x)$ reveals the profit when the price is zero without changing the form of the expression?
     $P(x)=-2x^2+24x-54$ $P(x)-2(x-3)(x-9)$ $P(x)=-2(x-6)^2+18$
  3. Find the profit when the pice is zero.
     
     

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  4. Which of the equivalent expressions for $P(x)$ reveals the price which produces the highest possible profit without changing the form of the expression?
     $P(x)=-2x^2+24x-54$ $P(x)-2(x-3)(x-9)$ $P(x)=-2(x-6)^2+18$
  5. Find the price which gives the highest possible profit.
     
     

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  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.

• Illustrative Mathematics Commentary and Solution

  Commentary:

  This task is part of a joint project between Student Achievement Partners and Illustrative Mathematics to develop prototype machine-scorable assessment items that test a range of mathematical knowledge and skills described in the CCSSM and begin to signal the focus and coherence of the standards.

  Task Purpose:

  The primary purpose of this task is to assess students' knowledge of certain aspects of the mathematics described in the High School domain A-SSE: Seeing Structure in Expressions. 

  This tasks emphasizes the choice aspect of the full standard, giving students the option of several forms of a quadratic expression to choose from when answering questions about a real-world scenario governed by that expression. The task avoids having the students do the algebraic manipulations themselves, focusing instead on using the structure of the expressions.

  Mathematical Content:

  Students will likely have encountered several "complete the square" problems which focus on the algebraic manipulation of quadratic expressions. By stripping this procedural aspect of the content away, students are assessed on a more conceptual understanding of quadratic expressions.

  Mathematical Practices:

  The task, and the standard A-SSE.B.3 in general, ties closely to MP.7, Look for and make use of structure.

  At the high school level, structure of the type mentioned in MP.7 manifests itself in the form of algebraic expressions, and the current task concerns itself quite explicitly with exploiting the structure to solve problems. For example, students need to recognize that the structure of the factored form (along with the zero-product property for real numbers) allows them to much more easily find a price that leads to a profit of zero than would, say, the standard form of the quadratic.

  Solution:

  For each of the three questions, choosing the correct form of the expression allows the answering of the question without doing any computations. Specifically:

  1. $P(x)=-2(x-3)(x-9)$ shows the values of $x$ that make the profit zero without any computation because a quadratic function in factored form $P(x)=a(x-r_1)(x-r_2)$ has roots $r_1$ and $r_2$ and the profit is zero when $P(x)=0$.
  2. The profit is zero when $-2(x-3)(x-9)=0$ which occurs when $x=3$ or $x=9$. The company breaks even if the price charged for the product is $\$3$ or $\$9$.
  3. $P(x)=-2x^2+24x-54$ shows the profit when the price is zero without any computation because the value of a quadratic function in standard form $P(x)=ax^2+bx+c$ when $x=0$ is $c$.
  4. In this case $c=-54$, so the profit is $-54$ (in thousands of dollars) when the price is zero. If the company gives the product away for free, it loses $\$54,000$.
  5. $P(x)=-2(x-6)^2+18$ gives us the price which maximizes profit without any computation because the maximum value of a quadratic function in vertex form $P(x)=a(x-h)^2+k$  is $k$ and occurs at $x=h$.
  6. From $P(x)=-2(x-6)^2+18$, we see that the maximum profit is 18 thousand dollars, and it occurs when $x=6$. The company should charge a price of $6 for this product in order to maximize its profit.

  As part of an assessment, this would be a 3-point task.

• Additional Thoughts

  The Standards avoid the term "simplify" as there isn't always an obvious simplest form. The emphasis instead is on, "purposeful transformation of expressions into equivalent forms that are suitable for the purpose at hand." This task shows three equivalent expressions and requires that students understand the structure and purpose of each one. This is a critical aspect of Seeing Structure in Expressions. For more information on the expectations of this domain, read pages 4–6 of the progression document, High School, Algebra.

  For further analysis on this task from an assessment perspective, read the Cognitive Complexity section on the Illustrative Mathematics site.

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