Unit 3, Lesson 8
Polynomial Identities (Part 1)
Algebra 2
8.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSA-APR.C.4
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity can be used to generate Pythagorean triples.
Materials
NoneActivity Narrative
The purpose of this Math Talk is to invite students to consider what is happening when we look at the difference between the squares of two consecutive integers. It encourages students to think about the relationship between consecutive squares and to rely on patterns to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students show why this relationship is always true for any two consecutive integers.
To efficiently evaluate each expression, especially the final expression, students need to look for and make use of structure (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneFind the value of each expression mentally.
Activity Synthesis
The goal of this discussion is to review students’ strategies for finding the difference of consecutive squares.
To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone use the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
If the fact that the differences are the sum of the two numbers being squared does not come up during the conversation, ask students to discuss this idea. Also ask students if they think this relationship is true for all consecutive integers. Invite a student from each side to explain why they think the relationship will or will not remain true.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:
Advances: Speaking, Representing
- “First, I because .”
- “I noticed , so I .”
Advances: Speaking, Representing
8.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students take a second look at the differences from the Warm-up and generalize what is happening using variables. Students learn that this is an example of an identity, which is the focus of this and the following lessons.
Monitor for students that reason about the problems by using variables for the integers, such as and for Clare's and and for Andre's.
8.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
In this activity, students continue to work with identities while also seeing a type of structure they will recognize in a future lesson and use to derive the formula for the sum of a geometric sequence. In the first part of the activity, students are asked to look for regularity in repeated reasoning as they find the product of and an increasingly more complicated polynomial (MP8).
Lesson Synthesis
The purpose of this discussion is for students to think about identities from a graphing perspective. That is, if two expressions form an identity, then the graphs of the expressions are identical. If possible, display a graph of the relevant equations for all to see to help illustrate the discussion.
Here are some possible questions for discussion:
- “What is another equation in the form that has the same graph as ? Explain how you know.” ( will have the same graph because we can use the distributive property to rewrite as .)
- “How would you explain why and have the same graph?” (Since the expressions on the right are equivalent expressions, they have the same outputs for any input of , which means they share the same input-output coordinates on a graph.)
Student Lesson Summary
In earlier grades we learned how to do things like apply the distributive property and combine like terms to rewrite expressions in different ways. For example, . The new algebraic expression on the right comes from writing the original on the left in a different way. More precisely, the expression on the left has the same value for all possible inputs as the expression on the right, making them equivalent expressions. This is an example of an identity.
Two examples of identities seen in earlier grades are:
For all possible values of and , the left and right sides of these equations are equal. In fact, the first of these identities can be extended to show that for any positive integer value of the expression
is equivalent to