# Day 6 – Income Tax and Local Linearity

Almost done reviewing in AP Calc. Today we took income tax systems from different countries and tried to make a piecewise graph and piecewise equation out of them. Students worked on whiteboards in groups of 3. This is a bit of a random activity, but I wanted to challenge them with something interesting about piecewise functions (and it was actually the perfect level of challenge for most) and I wanted something that hit the four representations of math that the AP curriculum emphasizes – graphical, numerical, analytical and verbal. I think that this activity does this well. Here are the boards halfway done (we will finish tomorrow).

In non AP, we are transitioning from algebra to calculus with a nice activity I got from college board somewhere. Students are given three linear looking functions (left) and are asked to write equations for them. Then I reveal that the equations are actually really zoomed in curvy equations, so which graph is which curvy function? We had an awesome discussion about local linearity, and the idea of the “slope” on a curvy line. I had them write “tweets” at the end with their big ideas (thanks @Fouss) and most were SPOT ON. One of my favorite classes to date!