Centroid of a Triangle

Yesterday, my grade 10 classes learned about triangle centres. They each had a triangle awaiting them on their desk as they walked in (I copy them onto thicker-than-normal paper). 

I gave instructions along these lines:
- cut out the triangle
- figure out how to balance it on your pencil
- make note of the balance point
- figure out how the balance point relates to the coordinates of the vertices of the triangle
- when you think you know, create your own coordinates for a triangle ABC, write them on the board along with their balance point

That last part was new. Its inspiration is Joel Bezaire's Variable Analysis Game which you can learn more about here. It helped some students see the pattern (especially the last line that I added to make it a little more obvious) and kept those who had found it early engaged as they were checking that other student's coordinates and balance points worked.

It was a fun way to start the class so I thought I would share... Thanks, Joel!

MPM2D - Day 37: Triangle Centres (day 2)

Today we actually found a triangle centre algebraically. I started by correcting what I had written on Friday - we needed to find the intersection of the perpendicular bisectors, not the medians. Oops! (my diagram was even correct...sigh)

 

They did pretty well and someone even said that it none of the parts was difficult. I think this example helped them understand why having a plan was important.

When I checked homework over the weekend I noticed that very few students knew what to do when a slope gave them a number over 0. So we looked at this example next, which was #12a) from Thursday's work.

There was quite a lot of "side work" done to explain what these line segments looked like and how these equations make sense.

We spent the last 10 minutes playing a round of Headbanz! It was great to listen to them asking each other questions and to see the excitement when they figured out their equation.

Here is today's homework.

MPM2D - Day 36: Triangle Centres

Now that my students have had the chance to practice finding equations of medians, altitudes and perpendicular bisectors, it was time to look at triangle centres. I started by giving them a piece of bright orange paper (it is the day before Halloween, after all) with a triangle on it, which I asked them to cut out.

Then I asked them to balance the triangle on the eraser end of a pencil and take note of the coordinates of the balance point.

Then I asked them how they could calculate the coordinates of the balance point and had them discuss ideas in their groups.

It was really interesting to see the groups that had thought to find the intersection of the medians react to the alternate, and much simpler, approach of finding the average of the x-coordinates and the average of the y-coordinates.

I told them that this point is called the centroid and is one of the triangle centres. Then I had them go to GeoGebra on the Chromebooks and we started constructing and investigating.

I asked them for conjectures on why the orthocentre is sometimes outside the triangle and they did a good job of figuring it out. Some needed a little hint so I added angle measures to my triangle.

There seemed to be the most confusion around perpendicular bisectors, but I think we cleared that up. 

I showed them that I could construct a circle through all three vertices whose centre was the circumcentre of the triangle. Then I constructed line segments from the centre to each vertex and asked what was special about those segments.

I think they saw that this could be useful when they heard some examples.

Then we started the one example for the day, but only got this far.

So I asked if anyone would be upset if I held onto the homework until Monday after we finish this example. No one objected so they can work on their parabolic art over the weekend. Here is today's handout.