Inspired by Calculus Thursday math circle, week 3

These are notes about our activities at the third meeting of the Math Circle. The main goals of activities were to contrast linear and non-linear growth, and to approximate curves with straight lines. Children’s activities brought my attention to three other ideas: consistency of patterns; differences; and measurement.

Here are all photos from the Circle. Yelena took notes (here they are) and Sally took photos this day – big thanks, ladies!

Fractal lichen. Mark found fractals and infinity in gray lichen. Every meeting, we find math in interesting objects. Do try it at home.

New words in what we did the last time. We talked about the activities from the last time. Gray suggested the word “smallz” to describe objects that grow smaller and smaller and smaller. I brought up the old long word “infinitesimal.” I think kids like “smallz” better. It has a nice ring to it! I asked kids to come up with examples of infinitesimal things. They got stuck, I sent them to ask parents. It’s a good lifetime practice to go ask your elders. Examples: squares (from the last time), Hotel Infinity rooms, bacteria… At home: search for infinitesimals; invite your kids to name their ideas.

Guess my pattern. I should have invited kids to make growth patterns of any sort, but I first invited them to guess what comes next in my pattern (1, 3, _). Guessing particular patterns before building your own causes two issues. First, it does not allow kids to work out the idea of consistency, which is easier if they are free to build any pattern they want. So we had to work on consistency too much.

Second, kids always want and need free play before they work systematically. Guessing someone else’s pattern is systematic. So children built their own patterns anyway, in the background, as we worked on my rule… At home: take turns guessing patterns you build, but let the kids build theirs first, any way they want!

Building patterns. I asked children to build a growing pattern “all in one line.” Note Hayden building complex shapes that formally follow this rule! We then checked if tops of each column formed straight lines or not. At home: come up with stories about things that look like histograms or growing columns. I am thinking of supporting columns for roller coasters. We need more motivation to look at this type of graphs. Kids assumed “linear is good” – Jeremy was somewhat concerned his pattern that grew by 1, 2, 1, 2… wasn’t linear. Again, we need story motivations for when we want linear and non-linear patterns to happen. Roller coasters? Skylines? Meanwhile, kids focused on differences from one column of LEGO to the next. Difference equations and the attention to differences (“deltas”) are key ideas in calculus. We will work on these ideas some more. Slopes (derivatives) have to do with differences!

Seeking straight lines. We went outside to look at straight and curvy lines – that was our “move-around” activity for the day. My hypothesis was that most straight lines were human-made. Kids found many straight lines in nature: blades of grass, leaf veins, and the flat ground (“technically not human-made” – Jeremy).

The ground example surprised me, because it’s not a line but a surface (a plane). Jeremy extrapolated the idea of lines to the next dimension. At home: look for human-made and naturally occurring straight lines, curves, planes, and curved surfaces.

The doubling pattern. This activity illustrates the simple principle of learning math: don’t show, don’t tell, but help to do. I invited kids to make a pattern where every next column is twice as high as the previous one. Telling was confusing. I started to build the example, with four columns. Showing my example was confusing too. Extra explanations did not work… Then grown-ups helped each kid to make two columns as high as the last one in the sequence, and stack them to make the next column in the sequence. Bingo!

Kids were very involved in this exponential growth, and making their towers higher and higher. They even took apart their past freeform creations to have enough blocks for tall towers. After a good while of playing with this amazing pattern, I asked kids to go through it backwards. Smaller and smaller and smaller, 8, 4, 2, 1… Then what? Jeremy: “If you had another type of brick…” I asked kids to imagine bricks getting thinner and thinner and thinner. Infinitesimal bricks. They were quiet for a while, imagining, thinking.

Runaway measurement. While measurements or counting are not the goals of these meetings, kids keep bringing them up. “This column is almost as long as my leg!” – this expresses the surprise at the speed of exponential growth. The length is more tangible if you compare it to familiar objects, as journalists often do. We used rulers to see if lines were straight or curved, but kids also wanted to measure the lengths of their LEGO columns. “My column is literally as tall as yours” (Gray) – with some hint of wonder. The fact that the same rule (doubling) produces the same outputs every time is not quite obvious, at least not in its applications. We probably need to do more measurement activities, since it’s a big interest. At home: check out the Universcale interactive for relative measurements in our universe; look for examples of exponential growth.

From polygons to circles. Adding more and more sides to a regular polygon makes it look closer and closer to a circle. Jeremy: “Interesting that you started doing this as soon as I mentioned Pi!” Me: “That’s because you keep mentioning Pi, and I prepared a Pi-related activity.” At home: look for more examples of how straight lines can form curves.

Video of the day: Sierpinski Dream.

Notes from home: Lynna sent a couple of observations. It’s always interesting to see what discussions happen at home! Do share what you do, and what kids say.

Maria – we stopped on the way to math tonight and our car wont restart. We are awaiting help and are sad to miss our wonderful class this evening. We are having an interesting discussion about zero and negative integers after he said that numbers have a beginning at 1 and go on forever to infinity in one direction.

 I had to share the saying on my tea bag this morning with you ~ Act selfless, you will be infinite.  Once we start thinking of these ideas we start to notice them everywhere!