Unit 3, Lesson 12
Polynomial Identities (Part 2)
Integrated Math 3
12.1
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.
Instructional Routines
NoneMaterials
NoneActivity Narrative
The goal of this activity is for students to follow directions to build several right triangles in order to become familiar with the process and form initial opinions (MP1), such as whether or not the instructions will always work. The following activity builds on the work done here, asking students to write and then reason about expressions representing the different sides built from the two chosen integers.
Monitor for students making labeled sketches of their triangles and using the converse of the Pythagorean Theorem to check that the triangle is an actual right triangle.
Launch
NoneActivity
NoneInstructions to make a right triangle:
- Choose two integers.
- Make one side length equal to the sum of the squares of the two integers.
- Make one side length equal to the difference of the squares of the two integers.
- Make one side length equal to twice the product of the two integers.
Follow these instructions to make a few different triangles. Do you think the instructions always produce a right triangle? Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to record a variety of triangles that work using the directions and at least one that does not work. Additionally, make sure the converse of the Pythagorean Theorem (if for a triangle with sides , , and , then the triangle must be a right triangle) is stated during the discussion since students will need to use it in the following activity.
Select 2–3 previously identified students to share their reasoning about whether or not the instructions will always work. Record for all to see the integers students pick and the triangles they do, or do not, create.
If not brought up, ask, “Are there any restrictions on the two integers chosen at the start of the instructions?” (They cannot be the same value since that makes one of the sides of the triangle equal to 0. They also must be positive since side length is always positive.)
12.2
Activity
Instructional Routines
MLR7: Compare and ConnectMaterials
NoneActivity Narrative
Building on the work done in the Warm-up, in this activity students write expressions for the different parts of the instructions and then use the Pythagorean Theorem to show why the instructions work for making right triangles. The goal of this activity is for students to recognize that the equation they create is an identity and that, for certain restrictions on the two chosen integers, the identity can be used to generate Pythagorean Triples.
Monitor for students who answer the last question by proving that is equivalent to in different ways, such as:
- Use the distributive property on each expression to rewrite them without parentheses.
- Rewrite one of the expressions in order to prove its equivalence to the other.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
12.3
Activity
Optional
Instructional Routines
NoneMaterials
NoneActivity Narrative
Use this activity if students need more practice rewriting expressions.
Launch
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 3–4 of the equations on the list and circle the ones that are identities and then explain their reasoning.
Supports accessibility for: Organization, Social-Emotional Functioning
Supports accessibility for: Organization, Social-Emotional Functioning
Activity
None12.4
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
The purpose of this activity is to bring together identities and the work students did earlier with rational equations. For each question, students first rewrite the fraction using the given formula and then prove that the formula is an identity.
The whole-class discussion at the end of the activity should focus on how students showed that the formula is an identity. Monitor for groups who use different methods to share during the discussion. For example, they may identify different common denominators when adding together the fractions in the expression on the right side of the formula.
Lesson Synthesis
Tell students to look back at the different identities they have investigated so far. If possible, display a list for all to see. Ask students to pick one that they found most interesting and write a few sentences explaining why. For example, students may like an identity because:
- They never thought of relating fractions that way before.
- The algebra used to prove the expressions in the identity are equivalent was particularly satisfying.
- It is an identity they have used a lot.
After quiet work time, invite 2–4 students to share their identity and why they chose it with the class. If time allows, consider arranging students in groups of 3 to make displays for different identities that includes the arithmetic proving that it is an identity and why the students find it interesting.
Student Lesson Summary
Sometimes we can think something is an identity when it actually isn’t. Consider the following equations that are sometimes mistaken as identities:
Both of these are true for some very specific values of and (for example, when either or is 0), but they are not true for most values of and , for example and (try it!). The actual identities associated with the expressions on the left side are and .
Are polynomials the only types of expressions you can find in identities? Not at all! Here is an identity that shows a relationship between rational expressions:
We can show that this identity is true by adding the terms in the expression on the right using a common denominator:
An important difference from polynomial identities is that identities involving rational expressions could have a few exceptional values of for which they are not true because the rational expressions on one side or the other are not defined. For example, the identity above is true for all values of except and .