5.1: Notice and Wonder: Factored Form (5 minutes)
This prompt gives students opportunities to see and make use of structure while making connections between graphs and mathematical expressions (MP7). The specific structure they might notice is the relationship between the horizontal intercepts of a graph and the factors of the expression being graphed. This warm-up also provides an opportunity to remind students what a zero of a function is as they begin to make connections between zeros of functions and features of graphs of functions.
Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the graphs for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.
What do you notice? What do you wonder?
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, point out contradicting information, etc.
If not brought up during the conversation, ask students why it makes sense that, for example, \(f(x)=(x+5)(x+1)(x-3)\) has a graph that crosses the horizontal axis at -5.
5.2: What Values of $x$ Make These Equations True? (10 minutes)
Continuing the thinking started in the warm-up, in this activity, students focus on what values of \(x\) will make a factored expression equal zero. The focus of this task is for students to use the structure of the equations to reason about the solutions (MP7), so graphing technology should not be employed at this time.
Identify students using different strategies to find values of \(x\). For example, some students may write down short equations to solve, such as \(x+4=0\), while others may skip writing that step and go directly to writing down the solution.
Supports accessibility for: Organization; Attention; Social-emotional skills
Find all values of \(x\) that make the equation true.
- \((x+4)^2 (x+2)^2=0\)
Are you ready for more?
- Write an equation that is true when \(x\) is equal to -5, 4, or 0 and for no other values of \(x\).
- Write an equation that is true when \(x\) is equal to -5, 4, or 0 and for no other values of \(x\), and where one side of the equation is a 4th degree polynomial.
Students may mistakenly go from \(x+a = 0\) to \(x = a\), from \(2x+8 = 0\) to \(x = 8\), or \(x = \text-8\). Encourage them to check their answer by substituting it for \(x\) in the original equation.
Display the polynomial \(f(x)=x^3 + 5x^2 + 2x - 8\) for all to see and ask, “How could you figure out what values of \(x\) make \(f(x)=0\)?” (We could graph \(y=x^3 + 5x^2 + 2x - 8\) and see where the curve crosses the \(x\)-axis. If the equation was written in factored form, we could use the factors to identify the zeros of the function that way.) Tell students that another way to write the function is \(f(x)=(x+4)(x+2)(x-1)\), which is the expression from the first question, and display a graph of \(y=f(x)\) for all to see and check against their answer. The visual provided by the graph is helpful for understanding the relationship between zeros and factors that will be used throughout the unit and beyond.
An important takeaway from this activity for students is that it takes only one of the factors equaling zero at a specific input for the entire expression to equal zero. If needed, have a brief discussion about why this is true of zero but not other numbers (for example, if \(m\boldcdot n = 10\), that doesn't guarantee that either \(m\) or \(n\) is 10).
Select 2–3 previously identified students to share how they identified values of \(x\) starting with those who wrote short equations and continuing with those who worked out the solutions mentally. If not discussed, explicitly call out the common error that a factor of \(x+a\) means that \(x=a\) is a solution.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
5.3: Factors, Intercepts, and Graphs (20 minutes)
In this partner activity, students take turns using the structure of equations to match them to either a graph or a description of a graph, building their fluency identifying the horizontal intercepts of a graph of a polynomial from the equation of the polynomial written in factored form. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). All the polynomials in the activity are 3rd degree with similar factors, so students must pay close attention to signs as they match cards. The focus of this task is for students to use the structure of the equations to reason about the matches (MP7), so graphing technology should not be employed at this time.
Arrange students in groups of 2. Give each group a set of cut-up slips. Ask students to take turns: the first partner identifies a match and explains why they think it is a match, while the other listens and works to understand. When both partners agree on the match, they switch roles.
Design Principle(s): Support sense-making; Maximize meta-awareness
Supports accessibility for: Visual-spatial processing
Your teacher will give you a set of cards. Match each equation to either a graph or a description.
Take turns with your partner to match an equation with a graph or a description of a graph.
- For each match that you find, explain to your partner how you know it’s a match.
- For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
Students may match the graphs incorrectly if they do not check enough points. For example, cards 7 and 8 both have intercepts at -5 and 4. Encourage students to double-check their matches by making sure that the properties they’ve found eliminate all possible matches but one. If more than one match is possible, they should look for other important properties.
Once all groups have completed the matching, discuss the following:
- “Which matches were tricky? Explain why.”
- “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”
Display the equation \(V(x)=(11-2x)(8.5-2x)(x)\) for all to see. Students may remember this equation from an earlier lesson in which \(V(x)\) is the volume, in cubic inches, of a box made from a single sheet of paper with squares of side length \(x\) inches cut from each corner. Ask students to write a short explanation to another student who is not in the class about how to identify what values of \(x\) make \(V(x)=0\). (0, 4.25, and 5.5.) If time allows, invite students to share their explanations.
5.4: Cool-down - Polynomial Graphing Error (5 minutes)
Cool-downs for this lesson are available at one of our IM Certified Partners
Student Lesson Summary
When a polynomial is written as a product of linear factors, we can identify several facts about it.
For example, the factored form of the polynomial shown in the graph is \(P(x)=0.5(x-3)(x-2)(x+1)\).
Looking back at the equation, can you see why the graph has \(x\)-intercepts at \(x=\) 3, 2, and -1? Each of these \(x\)-values makes one of the factors in the expression \(0.5(x-3)(x-2)(x+1)\) equal to zero, and so makes the equation \(P(x)=0\) true. The numbers 3, 2, and -1 are known as the zeros of the function. When a polynomial is not written as the product of linear factors, identifying the zeros from the expression for the polynomial can be more challenging. We’ll learn how to do that in future lessons.