Lesson 24
Polynomial Identities (Part 2)
24.1: Revisiting an Old Theorem (5 minutes)
Warm-up
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, such as whether or not the instructions will always work (MP1). 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.
Student Facing
Instructions 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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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 \(A^2+B^2=C^2\) for a triangle with sides \(A\), \(B\), and \(C\), 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 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 students do not notice that these instructions do not work when the two integers chosen are the same, do not bring it up since students will continue to work with these instructions in the following activity.
24.2: Theorems and Identities (10 minutes)
Activity
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 2 chosen integers, the identity can be used to generate Pythagorean Triples.
Monitor for students who answer the last question by proving that \((a^2+b^2)^2\) is equivalent to \((a^2-b^2)^2 + (2ab)^2\) in different ways. For example, some students may use the distributive property on each expression to rewrite them without parentheses while others may only rewrite one of the expressions in order to prove its equivalence to the other.
Launch
Design Principle(s): Optimize output (for explanation)
Student Facing
Here are the instructions to make a right triangle from earlier:
- 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.
- Using \(a\) and \(b\) for the two integers, write expressions for the three side lengths.
- Why do these instructions make a right triangle?
Student Response
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Anticipated Misconceptions
Some students may not be sure which of the three expressions is the hypotenuse of the right triangle. Remind these students that the hypotenuse is always the longest side of a right triangle and that they could test some values of \(a\) and \(b\) to see which of the three expressions results in the longest side.
Activity Synthesis
Select students to share the expressions they wrote for the three sides and record these for all to see, making sure to note which of the three expressions is the hypotenuse. Once students are in agreement on the three expressions, select previously identified students to share why the instructions lead to a right triangle, recording different strategies for all to see.
If not brought up during the warm-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 zero. They also must be positive since side length is always positive.)
If not brought up by students earlier, conclude the discussion by asking students if the equation \((a^2+b^2)^2 = (a^2-b^2)^2 + (2ab)^2\) is an identity. After some quiet think time, invite 2–3 students for each side of the argument to share their reasoning. Make sure students keep their reasoning focused on the equation itself and not on how it can be used to create right triangles. In particular, when using it to make right triangles there are restrictions, such as \(a \neq b\), but those restrictions do not affect whether or not the equation as written is an identity. If the two expressions in an equation can be shown to be equivalent expressions, the equation is an identity.
24.3: Identifying Identities (15 minutes)
Optional activity
This activity is optional because it includes additional practice manipulating expressions. Use this activity if students need more practice rewriting expressions.
Launch
Supports accessibility for: Organization; Attention; Social-emotional skills
Student Facing
Here is a list of equations. Circle all the equations that are identities. Be prepared to explain your reasoning.
- \(a = \text-a\)
- \(a^2+2ab+b^2=(a+b)^2\)
- \(a^2-2ab+b^2=(a-b)^2\)
- \(a^2-b^2 = (a-b)(a-b)\)
- \((a+b)(a^2-ab+b^2)=a^3-b^3\)
- \((a-b)^3=a^3-b^3-3ab(a+b)\)
- \(a^2(a-b)^4-b^2(a-b)^4 = (a-b)^5(a+b)\)
Student Response
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Activity Synthesis
For each equation, select students to explain why the equation is or is not an identity.
If time allows, ask students to come up with an equation that is never true. For example, \(x^2=x^2+1\) or \(x+1=x+2\).
24.4: Egyptian Fractions (20 minutes)
Activity
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 \(\frac{2}{15}\) 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.
Launch
Arrange students in groups of 2. Tell students to complete each problem individually, and then compare their work with their partner. Partners should discuss any similarities and differences in their methods, and decide which method they prefer.
If students need some practice adding fractions with unlike denominators, begin the activity by displaying the equation \(\frac78 = \frac12 + \frac13 + \frac{1}{24}\) for all to see and ask students how they can figure out without using a calculator that the equation is true. After some quiet work time, select 1–2 students to share their thinking about why the equation is true. The purpose of looking at this equation before the main activity is to make sure students recall that adding fractions requires a common denominator.
Design Principle(s): Support sense-making
Student Facing
In Ancient Egypt, all non-unit fractions were represented as a sum of distinct unit fractions. For example, \(\frac49\) would have been written as \(\frac13 + \frac19\) (and not as \(\frac19+\frac19+\frac19+\frac19\) or any other form with the same unit fraction used more than once). Let’s look at some different ways we can rewrite \(\frac{2}{15}\) as the sum of distinct unit fractions.
- Use the formula \(\frac{2}{d}=\frac{1}{d}+\frac{1}{2d}+\frac{1}{3d}+\frac{1}{6d}\) to rewrite the fraction \(\frac{2}{15}\), then show that this formula is an identity.
- Another way to rewrite fractions of the form \(\frac{2}{d}\) is given by the identity \(\frac{2}{d} = \frac{1}{d} +\frac{1}{d+1} + \frac{1}{d(d+1)}\). Use it to re-write the fraction \(\frac{2}{15}\), then show that it is an identity.
Student Response
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Student Facing
Are you ready for more?
For fractions of the form \(\frac{2}{pq}\), that is, fractions with a denominator that is the product of two positive integers, the following formula can also be used: \(\frac{2}{pq}=\frac{1}{pr} + \frac{1}{qr}\), where \(r = \frac{p+q}{2}\). Use it to re-write the fraction \(\frac{2}{45}\), then show that it is an identity.
Student Response
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Anticipated Misconceptions
Some students may not be sure how to find a common denominator when the denominators all involve variables. Remind them that a common denominator is a multiple of all of the denominators. If needed, students can guess possible common denominators, and then test them by checking whether they're divisible by each of the denominators.
Activity Synthesis
The purpose of this discussion is for students to share how they proved the given formulas are identities. Ask 2–4 previously identified groups who used different methods to share how they each showed that the formula is an identity, recording their strategies for all to see.
Lesson Synthesis
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.
24.5: Cool-down - A Cubic Identification (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
Sometimes we can think something is an identity when it actually isn’t. Consider the following equations that are sometimes mistaken as identities:
\(\displaystyle (a+b)^2=a^2+b^2\)
\(\displaystyle (a-b)^2=a^2-b^2\)
Both of these are true for some very specific values of \(a\) and \(b\), for example when either \(a\) or \(b\) is 0, but they are not true for most values of a and b, for example \(a = 2\) and \(b = 1\) (try it!). The actual identities associated with the expressions on the left side are \((a+b)^2=a^2+2ab+b^2\) and \((a-b)^2=a^2-2ab+b^2\).
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:
\(\displaystyle \frac{1}{x}=\frac{1}{x+1} + \frac{1}{x(x+1)}\)
We can show that this identity is true by adding the terms in the expression on the right using a common denominator:
\(\displaystyle \begin{array} \\ \frac{1}{x+1} + \frac{1}{x(x+1)} &= \frac{1}{x+1} \boldcdot \frac{x}{x} + \frac{1}{x(x+1)} \\ &= \frac{x}{x(x+1)} + \frac{1}{x(x+1)} \\ &= \frac{x + 1}{x(x+1)} \\ &= \frac{1}{x} \\ \end{array}\)
An important difference from polynomial identities is that identities involving rational expressions could have a few exceptional values of \(x\) where 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 \(x\) except \(x=0\) and \(x =\text-1\).