# Tag Archives: proportional reasoning

## Soccer Ball Inflation

We tried Soccer Ball Inflation again this year.

I haven’t found many opportunities during our first semester of geometry for students to engage in multiple steps of the modeling cycle. So I’m glad for the few problems that at least let students define the problem, decide what information is useful to know, and begin to formulate a model to describe relationships between what is important.

We watched Nathan Kraft’s Soccer Ball Inflation video on 101 questions.

Most students wanted to know how many pumps it would take to fill the other balls.

What information do you need to know to figure it out?

This was the end of November. It’s not the last time I’ll ask my students what information they need to know to figure out the answer to a question, but it was the first. It takes practice figuring out what information is useful, especially when it has been given for so long. Most of what they wanted to know (except for the answer) isn’t very useful or even possible without complicated measurement tools.

So I asked, “What’s easy for us to know? What’s easy to measure?”

Last year, I noted in my blog post that I gave them the circumferences (because that’s what Nathan included in Act 2, and I didn’t want to do any calculating). Dan called me out on this:

I’m not the only one who’s been living inside the “ideal” math world for too long. So have my students.

Oh. I guess not. The circumference.

Okay – so the circumference. I gave them the circumferences of all three balls. They knew from the video that it had taken 9 pumps for the smaller ball.

And finally … What assumptions are we making here?

They worked. I watched.

I’ve learned not to be surprised at the faulty proportional reasoning that happens every single year.

Most students said 14 pumps would fill the medium ball.

Why doesn’t that work?

The few who had gotten it correct actually calculated the radii from circumferences, and then calculated volumes from the radii.

No one recognized that the cube of the ratio of the circumferences would equal the ratio of the volumes.

And so the journey continues … trying to escape the “ideal” math world, one lesson at a time.

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Posted by on December 13, 2015 in Dilations, Geometry

## The Similarity Ratio

The Similarity Ratio

How would your students solve the following problem? (After they discuss their love or hatred of clowns, that is.) A clown’s face on a balloon is 4 in. high when the balloon holds 108 in.3 of air. How much air must the balloon hold for the face to be 8 in. high?

My students try to take the given measurements and make a proportion out of them. For several years now, I have tried to figure out a way to help students not just understand that the areas and volumes of two similar figures are not proportional to their side lengths (or perimeters) but know how to apply that concept to solve problems.

This year we started with the Soccer Ball Inflation video on 101 Questions.

Then we explored what happened with similar rectangles – their similarity ratios, perimeters, and areas.

We summarized the results:

And then moved to right triangles.

And then we worked problems similar to the clown problem, uncovering misconceptions, figuring out the incorrect thinking that occurred for incorrect responses.

And the results on the summative assessment were better than usual. Last year, on a problem like the clown problem, less than 50% of students answered it correctly. This year, 72% answered it correctly.

And so, the journey continues, where some years the results are better than others …