Lesson 5

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.

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.

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.

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.

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.

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?”