What is the Question?

I'm trying to foster problem solving early with my year 7 class... here's what we've been trying for decimals.

Inspired by the Rio Olympic 100m swimming finals, they had about 6 standard questions to give some scaffolding first, such as "how much faster did Pieter swim 100m that Simone?"

Then we moved on to the outstanding questions below (questions are normally split up into good, excellent, outstanding/mastery).

Cubic Tangents at Root Midpoints

In our geogebra inset session I was challenged to investigate cubic graphs. Specifically, for any cubic graph that has three distinct real roots....

(1) sketch the graph

(2) draw a tangent to the curve at all the midpoints of the roots

(3) what do you notice

(4) can you prove it?

Spoiler alert... Don't look below!

Proving it algebraically was a challenge initially, I'd recommend looking initially at the case where the midpoint of two of the roots is 0, then extend it. Have fun!