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What Is Pi?

03 Mar

What do your students think about pi before they get to your geometry class? Mine know that it has something to do with a circle…formulas, in particular. And many think that π is both 22/7 and irrational.

I used to have students measure the circumference and diameter of circles in class and submit those via a Lists Quick Poll using TI-Nspire Navigator. I would aggregate the data to send back to them for creating a scatterplot of Circumference vs. diameter. Eventually, I will have students who have already made sense of pi in CCSS-M grade 7. 7.G.B.4: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Last year I started collecting data from students that they measured outside of class using a Google form. I am still able to aggregate the data into a TI-Nspire document to share with the students, and we have more class time for our exploration.

We also use the automated data capture feature of TI-Nspire to collect data (diameter and circumference) from a circle on a Graphs page into two lists so that we can graph that data as well as the data from our own measurements. (See Exploring Diameter and Circumference from Geometry Nspired.)

03-02-2014 Image002 03-02-2014 Image001

Now we are ready to graph our data.

What do you notice?

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They go up.

Okay, so as the diameter increases, the circumference increases. Does that make sense?

Why are the data so different?

Students recognize pretty quickly that their measured data are not as accurate as the data collected using the technology.

How would you describe the basic shape of the data?

A line.

Yes. Linear. How do we model a line?

y=mx+b

Okay – so let’s start with the identity line: y=x.

03-02-2014 Image004

Does this line model the data?

No.

What do we need to do to it?

It needs to be steeper.

How do we change the steepness?

We use the movable line feature of TI-Nspire to change the slope of the line. When does it fit?

03-02-2014 Image005

Oh – so y=πx.

03-02-2014 Image006

What does that mean?

What did we plot on the y-axis? Circumference.

What did we plot on the x-axis? Diameter.

So Circumference=π*diameter.

Have we seen that before?

Screen Shot 2014-02-07 at 9.52.40 AM

And so we continue, making sense of π as the rate of change of circumference to diameter for all circles.

 

What would we expect a graph to look like if we were to plot the Area vs. radius?

Increasing.

Okay, so as the radius increases, the area increases.

What about the basic shape?

We collect data using TI-Nspire, and then we create a scatterplot.

03-02-2014 Image00803-02-2014 Image007

What about the basic shape?

03-02-2014 Image009

It looks exponential.

So while these students know the shape of exponential, they didn’t have CCSS-M Algebra I, and they don’t know the model of an exponential graph.

I plot y=ex, and we try to move it to fit the data.

02-07-2014 Image028

Hmmm. That doesn’t look as close as I thought.

Is it a parabola?

Do you remember an equation of a parabola?

y=x2?

So we try it.

02-07-2014 Image030

Is it better?

It needs to be steeper.

So we use the grab and drag transformation feature of TI-Nspire to fit the quadratic curve to our data.

y=3.14x2, or even better, y=πx2.

02-07-2014 Image031

What did we plot on the x-axis? Radius

What did we plot on the y-axis? Area

So Area=π*radius2.

Screen Shot 2014-02-07 at 9.52.47 AM

π is also the ratio of the area of a circle to the square of its radius.

And so the journey continues, trying to make sense of the irrational …

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2 responses to “What Is Pi?

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