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Hi Tracy – If x does not equal y then I think any values for x and y will produce an a^2 + b^2 = c^2 situation where (a,b,c) is a Pythagorean triple. So, if x = 2 and y = 5, then (4+25)^2 = (4-25)^2 + (2*10)^2. [Note (-21)^2 = 21^2]
Then 29^2 = 21^2 + 20^2 and (20,21,29) is the triple.
If x = y then you still get a Pythagorean triple like (0,2,2) but I guess it would not represent the sides of a right triangle since one side has zero length. Somebody much smarter than me will have to weigh in on the technicalities there 🙂
I’m having trouble with figuring out the meaning of this standard (“Prove polynomial identities and use them to describe numerical relationships.”) It seems there are two general aspects to the standard: proving identities, and using them. The standard, the progression doc, and the Illustrative Math example all put much more focus on the applied aspect of the standard. However, while there are infinite polynomial identities to prove, we have only two specific examples of polynomial identities we can use, and these uses are both pretty hard to replicate with other polynomial identities.
The example suggested by the standard itself (let’s call it the “Pythagorean” identity) implies that the use of polynomial identities is such that there are some special identities we can use by plugging in values and obtaining meaningful sets of numbers. I would really appreciate more examples of polynomial identities we can use this way.
Being unable to find even one more example of a such an identity, I came up with a different interpretation of the standard. According to this interpretation, the standard is aiming for students to arrive at different polynomial identities by themselves, in order to prove theorems that regard numerical relationships. The “Pythagorean” identity doesn’t fit this interpretation, since I don’t think we can expect students to derive it by themselves. It also isn’t used as a part of a grander proof. The case of (n+1)^2-n^2=2n+1 seems more to the point here. With some guidance, students should be able not only to prove this identity, but to actually derive it as a part of a modeling effort to explain why the difference between consecutive perfect squares is always odd.
Under this interpretation, I thought it could be possible to use polynomial identities to prove some divisibility issues. For instance, the identity n^2+n=n(n+1) can explain why for any value of n, the result of n^2+n is an even number. Similarly (but more elaborately), the identity n^3+3n^2+3n=n(n+1)(n+2) can explain why for any value of n, n^3+3n^2+3n is divisible by 6.
To sum it up, I would appreciate:
a. More examples of polynomial identities that can be useful in a similar manner to the “Pythagorean” identity.
b. Your opinion of the two different interpretations and the examples that follow.I’m not clear on the distinction between your two interpretations, but the examples you came up with are really great, so I guess the second one is best! I’ll get the Illustrative Mathematics team working on more. Can we use yours?
First of all, of course you can use my examples. I should also probably mention that I’m writing questions for Khan Academy.
Second, I think we’re in agreement that the intention in “use polynomial identities” is that students should be able to use identities as a tool in reasoning about numbers.
I was mainly confused by the example of the “Pythagorean” identity. I think that under my interpretation, the “use” of this identity is to explain why for any integers x and y, the three expressions (x^2+y^2)^2, (x^2-y^2)^2, and (2xy)^2 form a Pythagorean triple. This is different from the “use” suggested by the example, which is the act of finding triples by substituting specific integers for x and y. Finding Pythagorean triples is a very specific use, which I wasn’t able to generalize to a broader category of application.
Hope that was clearer.
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