Inspired by Calculus Math Circle – Week 5

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Young calculus - bodies of revolution

If we don’t teach kids to do actual calculations, what do we teach them in our calculus circle? And can it even be called calculus? Wouldn’t it be better to wait a few years until kids get older?  These are the questions we are focusing on as our “Inspired by Calculus” local math circles for 7-11 year olds come to an end. Do try these activities at home by yourself (good), with your child (better), or invite some friends over (the best). If you do, please share your experience with us. As always, we welcome your questions and comments.

Week 1 activities | Week 2 activities | Week 3 activities | Week 4 activities

Week 5 – The Tale of Two Worlds

We started the first meeting of the series with free play. We also wanted to end our last meeting with it. Only this time we wanted the kids to move beyond noticing existing patterns and toward making their own by solving a series of open-ended puzzles. We also wanted them to think about and experiment with pattern-making in the abstract calculus world, not just in the concrete, physical, world.

The kids appeared eager to follow the White Rabbit. As soon as Maria explained that calculus world is the world of imagination, several of them exclaimed: “I want to go to the calculus world!”

Our first puzzle became our starting point on this journey.

Puzzle 1 – Draw a point

Inspired by Calculus - Point and Line

Draw a point. Does a point you draw exist in our world (physical world) or in the calculus world?

The points kids drew, however small, had some length and width. But a point in the calculus world does not have either. It has no size.

Imagining something with no size is very hard! Most kids, when asked, said they could not do it. Did they get upset or discouraged? Not at all!

Mathematical Lesson Learned – Difficult Can be Enjoyable

Deep math is difficult, but that does not make or break motivation. When we enjoy doing something, whether it’s yoga, crafting or math, we do so despite the difficulties or even because of them. Children’s joyful reaction to the challenges we presented is a good indicator of their enjoyment of playful, deep mathematics.

Puzzle 2 – Draw a line

Next, we asked the kids to draw a line made out of points: to integrate a line. We got straight lines and curves, lines with points very close together and those with points farther apart. After a brief discussion of the lines, one of the kids said: “Everything has a point in it!” They seem to get the point (pun intended).

Since the individual points that make up a line have no size, the line itself has no width. Can you imagine something infinitely thin? Once again, it’s hard, although perhaps a bit easier because a line does have something – its length.

Puzzle 3 – Draw a shape

Once all the kids drew their lines, they faced something new – the idea of dimensions. Although all them are familiar with the words “2D” and “3D” and use these terms often, they were confused when the concept of dimensions was brought up.

Mathematical Lesson Learned – the Importance of Asking Questions

It seems to raise more questions than answers. Will a line always be one-dimensional? What if it loops? What about a disk? How many dimensions does it have? A quadrilateral has four sides, so how come it only has two dimensions? The confusion happens between the spatial dimensions and abstract dimensions. Seems like another rabbit hole, promising another exciting adventure, just opened up!

Puzzle 4 – Integrate a Circle

The new challenge was to integrate a circle out of lines. It seemed we got as many solutions as there were kids, and then some! This is another great sign of children’s enjoyment of and interest in the topics. Although they worked at different speeds and could easily see each other’s solutions, every child in the circle continued working on the problem. With closed-ended problems that only have one answer, many kids stop working once someone in the group finds that answer.

Mathematical Lesson Learned – True Understanding Happens When You Do Things Your Way

But it is also a good check of children’s understanding. We have talked about integrating a circle, but kids tried it first with the triangles. In exploring and imagining so many other ideas, the kids showed they understood what it means to “integrate” a shape.

Puzzle 5 – Integrate a shape, any shape

Integrating a triangle

Next came an even more challenging task. We asked the kids to draw a shape, any shape, and integrate it.  As usual, math activities must be full of meaningful choices. While some kids got started, others said they were confused.

Mathematical Lesson Learned – It’s OK to Get Emotional About Math

Expressing one’s feelings about a problem, reacting to it emotionally, is a necessary step toward a solution. Confusion, irritation, annoyance – it’s okay to feel that way about life’s problems, including math problems. If you do something about it, the negative feeling just might go away. The children worked hard on this task, composing (integrating) shapes out of lines, but also out of small 2D shapes, and out of points. You can build bridges from this activity to different types of formal integration, such as the shell method or double integrals. It was time to kick it up a notch and move to the third dimension.

Puzzle 6 – Turn a Surface into a Body of Revolution

The kids were handed thin wooden disks. What can we integrate out of them? The kids explored different solutions which depended on the sizes of disks they used for stacking as well as on the order in which disks of different sizes were stacked: cylinders, cones, and more complex bodies of rotation.

Calculus Math Circle - Integration

Sometimes when you put surfaces together, you just get another surface. But sometimes you don’t. The kids played with pieces of craft foam trying to solve this puzzle. What if we roll a piece into a spiral? What if the piece is infinitely long and infinitely thin? Have we just crossed from the physical world to the calculus world? Looks like it!

Puzzle 7 – For the Road

We finished the final math circle with a brief foray into integrating into 4th and 5th dimension. And the kids got to watch the trailer for Flatland. There were no challenges other than our suggestion to “think of it on your own” and “try it at home”.


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