Pick’s Theorem

Yesterday my older son struggled a little bit with this problem from the 1996 AJHME:

Problem 22 from the 1996 AJHME

The problem involves calculating the area of this triangle (as displayed in the link above):

Talking through that problem with my older son yesterday motivated me to try to talk about Pick’s theorem with the boys today. Although this project didn’t go quite as well as I was hoping, we had an interesting discussion about geometry and making conjectures in math. I will probably revisit this project in the near future to hopefully do better with a 2nd try.

We began by having my younger son talk about the geometry problem from yesterday. Right way he has the idea to find the area of the big rectangle and then subtract away other bits of area that we know. He struggles to find the shapes to subtract away – I think that it would have been better if I’d used peg board with larger squares.

 

Because he was struggling to see some of the shapes to subtract away on the peg board, we re-drew the problem on our white board. With a little help from his brother he was able to find some triangles and squares to subtract away:

 

Having found that the area of our triangle was 1/2, we went back to the peg board to find other triangles that had area of 1/2. There was a lot of nice talk about the area of triangle and also symmetry here. We ended up finding lots of triangles with area equal to 1/2.

The next thing we asked was – what did we notice that was the same about about all of the triangles that had an area of 1/2? It took a while to find the right math language to use to answer this question.

 

At the end of the last video my older son noticed something about a triangle that had an area of 1 – it seemed to have one extra dot. We explored that idea in this part of the project.

One thing that I didn’t do a good job with here is helping the boys with the confusion surrounding the dots that were the verticies of the triangle and the other dots that were on the perimeter.

Despite that little bit of confusion, I love the way the boys were exploring different triangles in this part of the project.

 

During the last project the boys speculated that adding an extra dot to the perimeter increased the area by 1/2. In this part of the project we looked more carefully at the relationship between the number of dots on the perimeter and the area for triangles with no dots in the middle of the triangle.

I was really happy with the back and forth between the boys here that eventually ended up finding a nice little formula hiding in our data.

 

Next we moved on to studying areas of triangles with dots in the middle. It was neat to hear them talk about the difference that the dots in the middle made with these triangles. Eventually we arrive at the formula in Pick’s Theorem 🙂

 

For the final part of the project we looked to see if the formula held with quadrilaterals. Here the small grid came back to cause a little confusion – so one lesson for me is USE A BIGGER GRID!!! – but we did find that our the formula we found for triangles held for quadrilaterals.

 

So, a fun project, though I left feeling that it could have been better. Will definitely be revisiting this one soon.