Saturday, October 11, 2014

Implement The Core: f(3)=5,f(5)=5 and much more

Twitter Problem 10-11-14
f is a linear function with f(3)=5 and f(5)=3. f(0)=?
(1) The title has an error and omits the critical linear condition. Note that f(5)=3 not f(5)=5.
(2) The Mathematics Practice Standards ask us to extend student thinking, make connections and go beyond the superficial qualities of a problem.
My hope is that you will see the Twitter problem as a  door marked ENTER not EXIT...
How do we do this?
One possibility is to ask our students to generalize. Note that the responsibility is shifted from us to them. We can guide this by prompting with: "Suppose f(a)=b..."
Here's one possible generalization:
f is linear, f(a)=b,f(b)=a. Show that f(0)=a+b
Since the answer to the Twitter problem is 8, do you believe some of your students will make the connection from 8=3+5?
Yes, this is time-consuming. Some of the best food requires slow cooking! (Sorry for all the metaphors...)
(3) Would you also want your students to relate f(a)=b and f(b)=a to the reflection relationship between the points (a,b) and (b,a)?

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