Triangles in planes and spheres

Today our fun Family Math project was about geometry.  We did a little playing around with triangles in the plane and triangles on the sphere.   A more advanced version of this discussion would probably include some mention of Euclid’s 5th postulate.

Our first topic of discussion was parallel lines on a plane.  What does it mean to be parallel?  My youngest son sees parallel lines as lines that do not intersect and my oldest wants to define parallel in terms of the slope of the line.

After talking about parallel lines for a bit, we went on to talk about parallel lines and angles:

Next we go on to talk about triangles.  The point of this discussion is to see that the angles in a triangle can be rearranged to make the same angle as a straight line.   The main idea here is just the idea that we discussed in the last video:

 

Now we move on to some fun ideas about triangles.   Just using some of the  basic facts about angles that we talked about in the last movie + the Pythagorean theorem, we find the area of a equilateral triangle, and also some simple properties of an equilateral right triangle:

Finally the punch line – what happens if we try to extend some of these geometric ideas beyond the plane?  The easiest example to show is a sphere, and I illustrate a triangle with three right angles by drawing the picture on a softball.  I love that my youngest son’s reaction was that this triangle was impossible.  Ha, not impossible, you are looking at it right now!!

 

Feels like there are a lot of different directions to go introducing basic geometric ideas to young kids.  One unexplored idea here is to show a surface where a triangle’s angles add up to less than 180 degrees.   Maybe there’s a 3D printing / basic geometry project in the near future!

 

Using snap cubes to talk about the 4th dimension

Had a friend from college visiting for Memorial Day and thought it would be fun to do a video explaining the 4th dimension to all of the kids in the house this weekend.  This project didn’t go quite as well as I was hoping, but I think the idea here is fun.  Will probably try it again in a few months.

In the first video we walk through the concept of a zero dimensional object sliding in time.  Our model for a zero dimensional object is a snap cube.  We talk through how a zero dimensional object sliding in time can create a one dimensional object.  The concept may seem a little strange when you talk (or read) about it, but seeing the trail of the snap cube as it moves helps the idea make sense (I hope!).

One other thing that we’ll be keeping track of in each of the videos is the number of cubes we have at every stage of the sliding.  With a single sliding snap cube, counting the cubes is easy – we just get 1,2,3,4,5, . . .

Next we try to make a two dimensional object by paying careful attention the sliding zero dimensional object from the previous video.   We build a two dimensional object – sort of a triangle – out of the pieces that the sliding snap cube created in the last video.  In this section the number of cubes we need to build our object at each stage is 1,3,6,10,15, and etc:

 

Now we take the idea from the last video and apply again to make a three dimensional object.   This time we have to keep track of the shapes at every stage of the “sliding” in the last film and combine those shapes together as they slide in time.  The object we create this time around is a 3-dimensional pyramid.  The number of blocks at each stage is 1,4,10,20,35, and etc . . .

Now for the 4-D challenge.  We want to apply the same idea as in the previous two videos, but there’s a little snag.  We don’t have any dimensions left in our kitchen, so how are we going to put the 3D object together?  Unfortunately the 4D shape we are creating here is pretty hard to visualize, but we can at least understand what the slices look like – they are exactly the shapes from the prior video!  One neat thing is that even though it is difficult to understand the picture of the full shape, we actually can count the number of cubes at each stage – 1, 5,15,35, and etc.

Finally, having build and sort of understood a 4 dimensional object, I wanted to show a neat connection this project has to Pascal’s triangle.  In every video we found an interesting sequence of numbers by counting the number of blocks needed to build our object.  Each of those sequences comes from a diagonal in Pascal’s  Triangle!  Pretty amazing that Pascal’s triangle tells us how to count blocks in 4 dimensional pyramids.   The kids even speculated that other diagonals count blocks in higher dimensions.  Pretty fun:

So, although this one didn’t go as well as I’d hoped, it was still really fun.  At least it was nice to end on a really cool note with the connection to Pascal’s triangle.  Will definitely try to improve on this one later.