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?”

The radius.

 

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.

 

I asked, ”Is it easy to measure the radius?”

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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Systems of Equations – Take 3

What do you do with Systems of Equations in a high school Algebra 1 class?

Our students have some experience with systems from grade 8, but not as much as they eventually will with full implementation of our new standards.

NCTM’s Principles to Actions lists Establish mathematics goals to focus learning as one of the Mathematics Teaching Practices. We can tell that having a common language to talk about what we are doing is helping our students communicate to us about what they can and can’t (yet) do.

We started our unit on Creating Equations & Inequalities with the following leveled learning progression and questions:

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

Level 2: I can graph a linear equation y=mx+b in the x-y coordinate plane.

Level 3: I can solve a system of linear equations.

Level 4: I can create a system of equations to solve a problem.

 

NCTM’s Principles to Actions lists Elicit and use evidence of student thinking as one of the Mathematics Teaching Practices. We need to know what students are thinking so that we can move their thinking forward. We created a leveled learning formative assessment so that we could see where students are.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

 

 

We have this y= Question configured to generate a graph preview. The equations students enter graph as they are typed, so the students are able to check their thinking as they go. (Note that we don’t have to configure the question to graph the equation as it is entered; we are choosing to do so while students are learning.)

The following are the results from one of our Algebra 1 classes. This is my favorite question.

How can we use the equation to know whether it contains the point (2,3)?

How can we use the graph to know whether it contains the point (2,3)?

What happens when we give students a point & ask them to create two equations that contain the point?

Do they know that they are creating a system?

Does it help them to know that eventually we will give them the equations and ask them for the point?

 

Level 2: I can graph a linear equation y=mx+b in the x-y coordinate plane.

And results from the second question:

Level 3: I can solve a system of linear equations.

What’s significant about the green point?

What does it have to do with the given equations?

 

At least half of our students don’t yet understand what the solution is.

 

 

And only a very few are able to solve the system from the question on 3.3, which is okay. This is the first day of the unit. We have the information we need to know how to proceed with the lesson.

 

 

I wrote about some of the tasks we used on the first two days of the unit here.

On the third day, we tried Dueling Discounts from Dan Meyer’s 101 Questions, which went better than Dan’s Internet Plans Makeover.

And more:

 

Maybe it’s time to generalize our results?

On the last day we tried Candy & Chips from 101 Questions, which also went well.

NCTM’s Principles to Actions lists Build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. We know we aren’t there yet, but we are definitely making progress as the journey continues …