standard_description: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
problem_activity: Activity
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.
Launch
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.
Conversing: MLR8 Discussion Supports.
In their groups of 2, students should take turns explaining their reasoning for how they matched the equation to the graph or the description of the graph. Display the following sentence frames for all to see: “The equation _____ matches _____ because . . . .”, and “I noticed _____ , so I matched . . . .” Encourage students to challenge each other when they disagree. When monitoring the discussions, point out key words and phrases, such as “factored form,” “zeros,” or “horizontal intercepts.” This will help students explain how the factored form of a polynomial equation connects to the horizontal intercepts of the graph.
Design Principle(s): Support sense-making; Maximize meta-awareness
Representation: Internalize Comprehension.
Activate or supply background knowledge. Ask students to brainstorm important features of a graph (making clear that this list is not exhaustive). Display the list for class reference. Provide students with highlighters or colored pencils and encourage them to mark defining features of their graphs and their equations (horizontal intercepts, relative minimums and maximums, and so on).
Supports accessibility for: Visual-spatial processing
Student Facing
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.
Student Response
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Anticipated Misconceptions
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.
Activity Synthesis
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?”


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