Lesson 6

Different Forms

6.1: Which One Doesn’t Belong: Small Differences (5 minutes)

Warm-up

This warm-up prompts students to compare four equations. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. In particular, encourage students to be clear about any connections they make between the equation and what the corresponding graph must look like. 

Monitor for any students who use the distributive property to rewrite the first equation in standard form.

Launch

Arrange students in groups of 2–4. Display the equations for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular item does not belong, and together, find at least one reason each item doesn’t belong.

Student Facing

Which one doesn’t belong?

A: \(y=(x+4)(x-6)\)

B: \(y=2x^2-8x-24\)

C: \(y=x^2+5x-25\)

D: \(y=x^3+3x^2-10x-24\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as constant term or coefficient. Also, press students on unsubstantiated claims.

If not brought up during the discussion, select previously identified students to share how they rewrote the first equation, including any organizing techniques they used, such as a diagram.

6.2: The Return of the Box (15 minutes)

Activity

In an earlier activity, students graphed the function describing the relationship between the volume of a box made from a single piece of paper and the side length of the squares cut from the corners of the piece of paper. In this activity, students identify features of a similar polynomial function from the context and expression alone without a graph, so technology is not an appropriate tool during the activity. During the discussion, students use two equivalent forms of the expression to identify features of the graph, including intercepts.

Monitor for students using different strategies to rewrite the equation in standard form to share during the discussion.

Launch

Action and Expression: Develop Expression and Communication. Maintain a display of important terms and vocabulary. During the launch, take time to review the following terms from previous lessons that students will need to access for this activity: horizontal and vertical intercepts, degree, leading term, factored form, standard form, constant, coefficient, equivalent, and volume.
Supports accessibility for: Memory; Language

Student Facing

Earlier, we learned we can make a box from a piece of paper by cutting squares of side length \(x\) from each corner and then folding up the sides. Let’s say we now have a piece of paper that is 8.5 inches by 14 inches. The volume \(V\), in cubic inches, of the box is a function of the side length \(x\) where \(V(x)=(14-2x)(8.5-2x)(x)\).

  1. Identify the degree and leading term of the polynomial. Explain or show your reasoning.
  2. Without graphing, what can you say about the horizontal and vertical intercepts of the graph of \(V\)? Do these points make sense in this situation?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may not be sure how to identify the degree or leading term of the polynomial written in factored form. Encourage these students to try writing the expression in a different form, referring back to the discussion of the previous activity.

Activity Synthesis

The goal of this discussion is for students to see some of the advantages and disadvantages of the two main forms polynomials are written in: factored form and standard form.

Begin the discussion by selecting 2–3 previously identified students to share their strategies for rewriting the equation in standard form. Depending on student familiarity with the distributive property, discussion of how to find the degree and leading term may need additional time and practice for students to explore different strategies and find one that works for them, such as using a diagram to stay organized while multiplying.

Display \(V(x)=(14-2x)(8.5-2x)(x)\) and \(V(x)=4x^3 - 45x^2 + 119x\) for all to see. Invite students to choose one and then use it to identify a feature of the graph of the polynomial until all major features (such as intercepts) have been identified. If possible, add these features to a set of scaled axes throughout the discussion and then sketch in the cubic curve of \(y=V(x)\) at the conclusion as a small preview of the work students will do in future lessons investigating the general shape of polynomials with different degrees.

6.3: Using Diagrams to Multiply (15 minutes)

Optional activity

This activity is optional practice that not all classes may need. If students struggle with multiplying in the previous activity, this activity may be useful.

This activity prompts students to notice the structure that relates expressions in factored form to their equivalent counterparts in standard form. When translating between these structures, it’s very helpful to have a way of staying organized. In this activity, students see one such way of staying organized: diagrams. 

When working with negative numbers, such as linear factors of the form \((x-n)\), it is helpful to think of subtracting \(n\) as adding \(\text-n\), and labeling the diagram accordingly. For example, for the last expression, \((x-1)(x-7)\), one side of the diagram should be labeled with \(x\) and -1, and the other labeled with \(x\) and -7.

Launch

Depending on students’ needs, not all of the following examples may be necessary. The first example is a completed diagram and the multiplication it represents. Ask students where they see multiplying in the diagram and how they could use it to find the result of the multiplication. The next diagram has not been filled in. Students should say which expressions are being multiplied (\((x-6)(x-2)\)) before filling in the diagram. The third diagram is the completed version.

Arrange students in groups of 2. Ask them to work on the first problem independently and then check answers with their partner.

\((x+2)(x+3)\)

  \(x\) 2
\(x\) \(x^2\) \(2x\)
3 \(3x\) 6
  \(x\) \(\text-6\)
\(x\)    
\(\text-2\)    
  \(x\) \(\text-6\)
\(x\) \(x^2\) \(\text-6x\)
\(\text-2\) \(\text-2x\) 12
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, invite students to color code like terms in their diagrams to reveal the terms they will combine before putting their answer in standard form. Some students may also benefit from adding in an exponent of 1 for any first-degree variables. This may help some students to more efficiently see structure while analyzing a sum and its corresponding addends, or a pair of factors and its corresponding product within the box diagram.
Supports accessibility for: Visual-spatial processing

Student Facing

  1. Use the distributive property to show that each pair of expressions is equivalent.
    1. \((x+2)(x+4)\) and \(x^2 + 6x + 8\)
    2. \((x+6)(x+\text-5)\) and \(x^2 + x -30\)
    3. \((x^2+10x+7)(2x-1)\) and \(2x^3+19x^2+4x-7\)
    4. \((4x^3-8)(x^2+3)\) and \(4x^5+12x^3-8x^2-24\)
  2. Write a pair of expressions that each have 2 or 3 terms, and trade them with your partner. Multiply the expressions they gave you.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to share their diagrams. If some students used methods other than diagrams to multiply their partner’s expression, invite them to share and explain their method.

6.4: Spot the Differences (15 minutes)

Activity

In this activity, students graph two related cubic polynomials to investigate the effect on input-output pairs of a function when the function is multiplied by a constant. Students are also invited to use mathematical language such as relative minimums and maximums to describe with precision (MP6) as they say how the features of the graphs are similar and different. They use two equivalent forms, standard and factored, to find properties of the functions.

This activity also asks students to multiply three factors together to rewrite polynomials in standard form, increasing the level of difficulty from the previous activity. Monitor for students who complete the multiplication in different ways. For example, given three factors \(A\), \(B\), and \(C\), some students may multiply \((A \boldcdot B)(C)\), while others choose \((A)(B \boldcdot C)\). Students should understand that order of multiplication with polynomials, like that of integers, does not change the product.

This activity works best when each student has access to devices that can run the Desmos applet, because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, display the applet from the digital version of the activity for all to see during the synthesis. 

Launch

Arrange students in groups of 2. Ask students to predict the differences between the functions first. Assign each partner either \(f(x)\) or \(g(x)\) to rewrite for the second problem. If time allows, have students rewrite both expressions individually and then compare their result and method with a partner. 

Representation: Internalize Comprehension. Begin by providing students with questions that support them in investigating generalizations and help them frame their conclusions. Display the questions and ask students to read and make a prediction before the launch, and also to revisit the questions in preparation for the synthesis. Possible questions include: “Do all functions with the same factors have the same horizontal intercepts?”, “If functions have the same intercepts, do they have the same graph?”, and “What happens to a function’s graph when we multiply the function by a constant?”
Supports accessibility for: Conceptual processing

Student Facing

Let \(f(x)=(x-2)(x+3)(x-7)\) and \(g(x)=\frac12 (x-2)(x+3)(x-7)\).

  1. Use the applet to explore both functions in the same window of \(\text-10 \leq x \leq 10\) and \(\text-100 \leq y \leq 100\). Describe how the two graphs are the same and how they are different.
  2. What degree do these polynomials have? Rewrite each expression in standard form to check.
  3. Let \(h(x)= (3x-6)(x+3)(x-7)\). What do you think the graph of \(y=h(x)\) will look like compared to \(y=f(x)\)? Use the applet to check your prediction.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Here are the graphs of two polynomial functions, \(f\) and \(g\). We know that \(g(x)=k\boldcdot f(x)\).

  1. Why do the two graphs have different vertical intercepts but the same horizontal intercepts?
  2. What is the value of \(k\)?
Two curves on a coordinate plane.  X-intercepts are the same for both functions. F of x has more extreme maximums and minimum than G of x. F of 0 is negative 5, G of 0 is negative 1.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Arrange students in groups of 2. Ask students to predict the differences between the functions first. For the question about rewriting the expressions in standard form, assign each partner either \(f(x)\) or \(g(x)\) to rewrite. If time allows, have students rewrite both expressions individually and then compare their result and method with a partner. Provide access to devices that can run Desmos or other graphing technology.

Representation: Internalize Comprehension. Begin by providing students with questions that support them in investigating generalizations and help them frame their conclusions. Display the questions and ask students to read and make a prediction before the launch, and also to revisit the questions in preparation for the synthesis. Possible questions include: “Do all functions with the same factors have the same horizontal intercepts?”, “If functions have the same intercepts, do they have the same graph?”, and “What happens to a function’s graph when we multiply the function by a constant?”
Supports accessibility for: Conceptual processing

Student Facing

Let \(f(x)=(x-2)(x+3)(x-7)\) and \(g(x)=\frac12 (x-2)(x+3)(x-7)\).

  1. Use graphing technology to graph both functions in the same window of \(\text-10 \leq x \leq 10\) and \(\text-100 \leq y \leq 100\). Describe how the two graphs are the same and how they are different.
  2. What degree do these polynomials have? Rewrite each expression in standard form to check.
  3. Let \(h(x)= (3x-6)(x+3)(x-7)\). What do you think the graph of \(y=h(x)\) will look like compared to \(y=f(x)\)? Use graphing technology to check your prediction.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Here are the graphs of two polynomial functions, \(f\) and \(g\). We know that \(g(x)=k\boldcdot f(x)\).

  1. Why do the two graphs have different vertical intercepts but the same horizontal intercepts?
  2. What is the value of \(k\)?
Two curves on a coordinate plane.  X-intercepts are the same for both functions. F of x has more extreme maximums and minimum than G of x. F of 0 is negative 5, G of 0 is negative 1.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may be unsure of the notation used to describe the graphing windows in the first problem. It may be helpful for them to make a sketch of the axes and mark off points before setting a window size on their graphing technology.

Activity Synthesis

The purpose of this discussion is for students to understand that multiplying by a constant \(k\) causes all output values of a function to be \(k\) times farther from the horizontal axis, which in turn means \(k\) has no effect on the zeros of a function.

Begin the discussion by selecting 2–3 previously identified students to share how they rewrote \(f(x)\) and \(g(x)\) in standard form. Display different methods used for all to see and compare. If all students multiplied the factors in the same order, demonstrate how multiplying in a different order results in the same product.

Ask students, “How can you identify the constant term without multiplying out the entire expression?” (With an expression like \(\frac12 (x-2)(x+3)(x-7)\), the constant term is 21 and comes from \((\frac12)(\text-2)(3)(\text-7)\).)

If not brought up during the discussion, it is important to note that all three polynomials have the same degree and the same zeros, yet the three have distinct outputs for all other inputs. The idea that knowing the degree and zeros of a polynomial is insufficient for identifying a specific polynomial will be revisited in future lessons. If time allows, ask students to write an equation for a fourth function with the same degree and zeros as the first three.

Conversing: MLR2 Collect and Display. During the discussion, listen for and collect the language students use to describe the effect on output values when a function is multiplied by a constant. Call students’ attention to language such as “same horizontal intercepts,” “closer to/farther from horizontal axis,” or comparisons between the coordinates of the same input values on different graphs. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students analyze the effect of a constant on cubic polynomials by paying attention to specific features on graphs.
Design Principle(s): Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Display both equation forms and the graph of \(y=f(x)\) for all to see.

\(\displaystyle f(x) = 2x^3 - 18x^2 - 44x + 240\)

\(\displaystyle f(x) = (x+4)(2x-6)(x-10)\)

Graph of a polynomial function.

Here are some questions for discussion.

  • “Where can you see the 240 in the factored form of the equation? In the graph?” (240 is the constant term, which is the result of multiplying the 4, -6, and -10 values in the factored form. The graph intercepts the vertical axis at 240.)
  • “Which form do you think is better?” (It depends on what information you want. The factors help you identify the zeros, but with factored form, you need to do a bit of work to figure out the vertical intercept and the degree.)
  • “Let’s say \(g(x)\) is a polynomial with the same zeros and degree as \(f(x)\), but with a vertical intercept at \((0,120)\) instead of \((0,240)\). What could the equation for \(g(x)\) be?” (\(g(x)=0.5 (x+4)(2x-6)(x-10)\))

6.5: Cool-down - Identifying Polynomial Features (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We can express polynomials in different, equivalent, algebraic forms. These forms can give us different information about features of the polynomial and its graph. Earlier, we learned about expressing a polynomial function in factored form to identify zeros. The standard form of a polynomial, that is, the expanded version of factored form, makes it easier to identify different information about a polynomial.

For example, here are the expressions for a polynomial \(P(x)\) written in factored form and standard form:

\(\displaystyle P(x) = 0.25(x-1)^2(x+2)(x-3)(x+3)\)

\(\displaystyle P(x) = 0.25x^5 - 3x^3 + 0.5x^2 + 6.75x - 4.5\)

In standard form, two key features of the polynomial function can be identified: the constant term and the degree.

The constant term, shown as -4.5 in the example, tells us the value of the function when \(x=0\). In a graph of the function, this point is known as the vertical intercept.

The degree, shown as 5 in the example, tells us about the general shape of the graph of the polynomial, which is something we’ll learn more about in future lessons.