A few weeks ago, we watched Nathan Kraft’s Soccer Ball Inflation video on 101 questions.

What mathematical question could we explore?

how many more pumps did it take to fill the bigger balls 1

how many pumps did it take to inflate the largest soccer ball 1

How many pumps did it take to fill the other 2 soccer balls? 1

Did each ball take the same amount of pumps to fill it up? 1

how are the balls proportional? 1

are all three balls proportional? 1

are the amount of pumps proportional to the little ball 1

are the 3 balls similar? 1

how many did it take to pump the other ones 1

how many pumps does the 2 soccer ball take? 1

how many pumps did it take to inflate the other balls 1

How much capital does this person contribute to the economy? 1

how many pumps of air did each soccer ball need 1

is the amount of air used to fill the soccer balls proportionate 1

how does the size of the ball affect how many pumps it takes to fill it up, what is the proportional size related to each ball 1

how many pumps of air does it take for the big ball 1

do the bigger soccer balls take more pumps to pump up 1

Are the circumferences proportional? 1

how many more pumps did it take every time there was a bigger ball 1

how many pumps did take for the biggest ball lo inflate? 1

how many pumps taken to fill the second largest ball and the largest ball. 1

Will it take twice the amount to pump the medium ball? 1

are the no. of air pumps porportional for the smallest and largest soccer balls 1

did it take the same amount of pumps for the bigger soccer balls? 1

if how many pumps it takes to inflate the soccer ball is proportional to its size 1

are the balls pproportional? 1

how many pumps more on the big ball than the medium and small ball 1

do the bigger soccer balls take more pumps to fill up 1

Is the number of pumps reqiured to inflate the small ball multi 1

how many more pumps did it take to fill the bigger balls 1

number of pumps reqiured to inflate the small ball multi 1

Our whole class conversation started with the question: Are the balls similar?

Yes.

How do you know?

Students think about similarity in terms of dilation because of our definition of similarity (CCSS-M 8.G.: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. Because of the learning experiences that we have created, students think about dilations and similarity synonymously.

So the spheres are dilations of each other. We can translate the centers to coincide and then dilate to map one figure onto the other.

Are the circumferences proportional?

Why?

So how many pumps will it take to fill the medium ball?

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

Most students wanted to know the radii of the small and medium balls.

I gave them the circumferences (because that’s what Nathan included in Act 2, and I didn’t want to do any calculating).

More than one team asked for the volume of a sphere while they were working, which surprised me, as it didn’t occur to me to actually calculate volume to determine the number of pumps in the medium ball. But I gave them permission to look up the formula on their electronic device or from a textbook in the back of the room, and I also asked them to find a second way to solve when they finished using the formula.

You should know that students have generalized the results for the relationships between the similarity ratio, ratio of perimeters, ratio of areas, and ratio of volumes for similar figures. In fact, since my students are coming to me now knowing the relationship, we didn’t spend as much time as we have in the past developing those relationships. We did look at the two solids (1:3 similarity ratio) and review the ratio of surface areas (1:9) and volume (how many of the small will it take to fill the large? 27).

Why is this concept so easy to “get” but so difficult to “apply”? All of my students can tell me to square the similarity ratio to get the ratio of the areas and to cube the similarity ratio to get the ratio of the volumes. But very few can use that information to solve a problem.

Maybe the student questions at the beginning of the lesson provide some insight into the difficulty:

“is the amount of air used to fill the soccer balls proportionate”

“are the no. of air pumps porportional for the smallest and largest soccer balls”

“if how many pumps it takes to inflate the soccer ball is proportional to its size”

Only one of these questions recognizes that we talk about proportionality between two quantities.

It turns out that only those who used the volume formula calculated the number of pumps correctly (33ish). They had used the given circumferences to calculate the radii, used the radii to calculate the volumes, and then set the ratio of the volumes equal to the ratio of the number of pumps.

Why doesn’t it work to set the ratio of the circumferences equal to the number of pumps?

How can we determine the number of pumps without having to calculate the volume?

Then more practice with ratios of perimeters, areas, and volumes.

And more evidence of misconceptions.

And then students **asked** to do more. So we did. And they provided evidence of understanding.

And so the journey continues … figuring out the questions to ask to uncover misconceptions and learn from them.