My favorite elementary geometry proof

My favorite elementary geometry proof is that medians divide a triangle into 6 triangles having equal area.

I’m a tiny bit disappointed that our Geometry book covers this theorem before it covers Ceva’s theorem because having to assume that the medians intersect in a single point is a bit of a shame. No matter, though, I’ve always found it amazing that you can prove this nice fact just from the area formula for a triangle. It is a really nice example of mathematical ideas for kids.

I broke the proof into three parts of about 3 minutes each. Not much commentary is necessary as the proof really stands on its own. This sequence shows another example of why I love talking about math with my kids.

Desmos and Circumscribed Circles

Last week I saw a request for additional solutions to this problem:

I wasn’t in much of a writing mood that night, so I made a short video showing two different solutions involving circumscribed circles:


The second solution – noting that the formula for the area of a triangle is ABC / 4R – really requires trigonometry, and is likely to be too advanced for an intro geometry course.

Today, by interesting coincidence, I saw this neat post on Patrick Honner’s blog:

Patrick Honner on working with Circumcircles in Desmos

That post links to a Desmos program about Circumcircles that is hosted here:

Patrick Honner’s Circumcircle program in Desmos

You can see the solution to the original circle / square problem by playing around with the three points of the triangles in Mr. Honner’s program. In fact if you set (a_1,b_1) = (10,10), (a_2,b_2) = (0,-10), and (a_3,b_3) = (-10,10) you get this picture:

Desmos Pic

If you know how to use Desmos, you can probably even add in the original square. Even without the square, though, from the picture above you can see that the diameter of the circle is indeed 25. Yay!

Can’t say that I know how to use Desmos at all (same with Geogebra), but examples like the one Patrick Honner published today show me that these tools are really powerful.

Always happy to see these neat little math coincidences with people sharing math online!

A neat “last digit” problem from James Tanton

My younger son and I finished up the “last digit” chapter in our Introduction to Number Theory book today, and I was looking for a fun last exercise. Luckily James Tanton had my back this weekend:

I decided to use this problem as tonight’s exercise and even asked my older son to join in since it looked like we’d have a nice discussion.

We started by just talking through the problem a little bit. I asked each kid to rank the four problems from what they thought would be the easiest to what they thought would be the hardest. It was interesting that their ranking was not the same.

My younger son started off on what he thought was the easiest problem – the pattern in the last digit of the powers of 3:


Next up was the powers of 5. My older son recognized that this would be an easy problem. We talked for a little bit about why the last digit was always 5 and managed to squeeze out a pretty good (if short) discussion. After that we moved on to talking about the harder problems. Neither kid had any ideas about how to proceed with the larger numbers. Eventually my younger son suggested that it would be a good idea to move to the computer for the more difficult problems, so that’s what we did.


We ended the last movie trying to guess how long it would take the powers of 13 to repeat the last two digits. Both kids thought the repetition would happen pretty quickly – about every 2 or every 4 times. We’ve mostly been talking about last digits, so I guess that conditions you to believe that repetitions will happen pretty frequently.

We saw immediately that the repetition didn’t occur quickly. My younger son realized that we should be looking for a last two digits of 01 to singal the repetition – I was happy to hear that since that showed he was starting to understand what to look for.

At the end of this movie we were talking about moving on to look at 113. My older son thought there might be a pattern going from 3 to 13 to 113. Love when they have ideas like that.


The last thing we did was look at the last three digits of powers of 113. It turns out that my son’s guess for the length of the repeating pattern was right. He was pretty excited about that! To see if his patterned continued we looked to see if the last four digits of 1113 cycled every 500 powers.

In all honesty, I don’t know why this pattern is happening, but the kids really liked it. Even if it was just a guess, it was neat to see how happy my son was that he’d guessed the pattern correctly.


So, another really fun math conversation with the kids from a James Tanton tweet. I’ve really enjoyed the section in our number theory book on last digits – it seems to have really helped my younger son build on his existing number sense. The problems with the last two, or three, or more digits are fun, too. They also are great, and relatively straightforward, ways to introduce a little computer math to kids. Definitely a fun little conversation tonight.