Aren’t infinite geometric series cool? If you just shouted “yes”, then you are potentially as geeky as I am. A “proof without words” from MathFail kicked off today’s discussion:

I wasn’t quite sure what sort of observations I would receive from my class. But just enough ideas were generated to get us going:

There are an infinite number of triangles down the right side.

All those triangles on the right add up to the half-triangle on the left.

Both are great starts for what I hope my students will learn today. A video I made in my driveway continued the ideas of geometric series and their infinite terms.

A few students wanted to argue that the sequence in the video was arithmetic, but some meaningful debate yielded agreement that geometric made more sense. Groups then worked through a similar problem involving a Superball being dropped, leading to terms and total distance traveled.

Many groups employed a “brute force” method to find their answers. Using the Desmos calculator (many students chose to use the iPhone app), we found value in developing the equation and using tables and summation symbols to find solutions. This was my first time usign Desmos with this particular lesson, and it was an awesome addition, which added value to the need for writing a clear function to define your situation.

I was messing about with triangles the other day and came up with the following:
Take any triangle. Join the mid points of the sides to get 4 half height triangles. Repeat this operation with the middle triangle, and again, and again…Now look at the top triangle of the first 4. Its median is half way to the bottom. Take the matching triangle in the middle triangle – its median is 1/8 of the original median. keep going and get the series
1/2 + 1/8 + 1/32 + 1/128 + …
which is a nice geometric series.
Do the same from below and get 1/4 + 1/16 + 1/64 + …
These two add up to 1 by a geometrical limit argument, and the first is twice the size of the second by a simple algebraic argument. So the medians meet at a point 2/3 of the way along each one.
No, it’s not as simple as yours, but it’s fun !

As i was hunting for different photos to use as openers, I came across one similar to what you describe. Lots of ways to adjust the fraction up the sides we will cut off at an explore.

I was messing about with triangles the other day and came up with the following:

Take any triangle. Join the mid points of the sides to get 4 half height triangles. Repeat this operation with the middle triangle, and again, and again…Now look at the top triangle of the first 4. Its median is half way to the bottom. Take the matching triangle in the middle triangle – its median is 1/8 of the original median. keep going and get the series

1/2 + 1/8 + 1/32 + 1/128 + …

which is a nice geometric series.

Do the same from below and get 1/4 + 1/16 + 1/64 + …

These two add up to 1 by a geometrical limit argument, and the first is twice the size of the second by a simple algebraic argument. So the medians meet at a point 2/3 of the way along each one.

No, it’s not as simple as yours, but it’s fun !

As i was hunting for different photos to use as openers, I came across one similar to what you describe. Lots of ways to adjust the fraction up the sides we will cut off at an explore.

I just did a picture for you:

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