Gr 06 Introduction to Pi foldable

The day before this activity, my students measured circular objects and their diameters to try and answer the question "how many diameters will fit along the outside of a circular object." We were looking for some kind of relationship between the two things. Kind of like with radius and diameter. We know that r= d÷2 and d= 2×r. So it seems like there should be some kind of relationship between diameter and circumference, too!

What we discovered was that circumference was about 3 times larger than the diameter. Or the diameter was about circumference divided by 3.

So today we examined that relationship a little more by making these foldables. We did the inside first. A copy of the cut-outs is available here: [link]

1. Students had to color and then cut out the circles and the strip. The circles' circumferences and the strip had to be the same color; and the diameters had to all be the same color, but one different from the circumference.
2. Students then drew a "starting line" on the left of their page. They glued the strip down touching the line. I explained that this line was the same length as a circle's circumference "unrolled" into a straight line.
3. Then we glued down the circles to see how many diameters could fit along that circumference. We drew an ending line where the diameters stopped and could tell the circumference was longer. We added a second line to show where the circumference stopped and looked at that distance in-between them. I asked students to estimate about how much of a diameter that extra bit was.
   
4. I told students that mathematicians have been looking at the extra little bit for thousands of years because it turns out, it's not such a simple number to get. I shared this YouTube video about the history of pi [link]. I prefer to use ViewPure to show less clutter on the screen (it strips out all the adds and the sometimes questionable video links that follow a YouTube video).
5. Then we go back and add the information on the above the circles, that pi is a symbole used to represent the circumference divided by diameter (the idea of how many diameters fit on a circumference); and that two estimates of pi (mentioned in the video) are 3.14 and 22/7. Any time we use those numbers (3.14 or 22/7), we are estimating the number of diameters on a cirucmference... just like we did when we estimated that there were only three. But 3.14 and 22/7 are better estimates -- they are not precise measurements.
6. We next add that + ~ 0.14 to the diagram and we go to the cover next to learn a few ways that people write the pi symbol (I have learned that students need to be taught some method of writing pi, or they end up with REALLY sloppy approximations). So they can choose any of the representations on the cover to write (and maybe there are other variations, but these are pretty common).
   

Gr 06 Parts of a circle Foldable

This was heavily adapted from some Waldorf materials I found online. I don't have the time to let the kids explore "circle" like Waldorf does. I'm sure they'd benefit from it. Nonetheless, my students appeared to like the process of creating and learning parts of circles this way.

Here's how I used it:

1. First we folded the paper (fold a large + sign first, then fold the corners in). This is a picture of the finished product, but at this point, there shouldn't be any writing on it.
   
2. Draw a center point on the fold underneath each flap. Each point should be a different color and should get a different label (A, B, C, D are fine, but kids could use whatever letters or colors they wanted).
3. Point A: Ask kids to use a ruler to mark all the points they can find that are exactly 2 cm from point A. When they think they know what shape it's making, they can use a colored pencil to sketch it in. Ask them what it is that defines this shape (all the points that are the same distance from the center point). Then record the definition with them. I spiraled it in to show it was revolving around the center, but that's optional and it was kind of hard for the kids. Close the flap, place the title on it (Definition of a circle), and move on to the flap for point B.
   
4. Point B: Write the title "Radius and Diameter" and ask students to construct another r= 2 cm circle around point B just like they did for point A before. Draw a line segment from the center to a point on the circle and ask them to describe what you have drawn. Ask them it if stays the same no matter what point on the circle you draw a line segment to. Formalize this distance with them as the definition of the radius. Then do a similar process with the diameter.
   
5. Point C: Write the title "Circumference (C) and Arc" and ask students to construct another r= 2 cm circle around point C. Draw a point on the circle and a series of arrows that goes all around the circle and ends at the same point again. Ask students to describe what you have drawn using their geometry vocabulary, then formalize the definition of circumference with them. Now draw two new points on the circle and shade in the distance on the circle between them. Ask students to describe what you've drawn using their geometry vocabulary, then formalize the definition of Arc with them.
   
6. Point D: Write the title "Chord and Tangent" and ask students to construct another r= 2 cm circle around point D. Draw two points on the circle and connect them with a straight line segment. Ask students to describe what you've drawn with their geometry vocabulary, then formalize the definition of chord with them. Do the same for tangent. Now what's cool is that students may recognize that the diameter of a circle is a special case of a chord. And if a tangent dips into the circle, it will no longer be a tangent because it will connect two points on the circle.
   

Gr 06 Introduction to Geometry Vocabulary Foldable

Just starting in on exploring geometry with my students this year and trying out foldables to help them organize and illustrate their vocabulary and concepts. These are the teacher versions made in my own notebook; my students made theirs in their own notebooks. It's really just note-taking, but they find it more entertaining to use colors and foldables and it makes a great reference.

This foldable was created on a single 8.5" x 11" piece of paper, folded in the middle. It was taped into the notebooks no the left side only so the tape acted as a hinge and all 4 pages can be accessed.

PAGE 1

PAGES 2 and 3

PAGE 4

If π is good, is 2π better? How about 2πr?

Okay, quick math quiz: what does this formula mean?
2πr= C

If you answered "two times pi times diameter equals the Circumference," you'd be kinda' right... and you'd probably also be kinda' wrong. 

You'd be kinda' right because that would calculate the correct answer for the circumference of a circle. But you'd be kinda' wrong because the relationship (as taught to kids -- and most likely to 80% of adults) is not "two quantities of pi times the radius."

Instead, kids are taught that the circumference of a circle is πd (or about 3.14 diameters). So when you change diameter into multiples of the radius, it should stay together as a unit (d= 2r). And so the formula should read 2r•π= C.

Ahh, but that's not what we write in textbooks.

Instead of letting the kids get accustomed to 2r as a unit to represent d, we rewrite the formula to better match algebraic convention (or to match trigonometric understandings): none of which makes sense to a middle schooler. So instead of 2r•π= C (or even π•2r= C), they get 2πr= C which totally blows their chance at following along with what has been measured and summarized into a formula format.

Now, if we used τ (tau) instead, it might be easier yet. With that little replacement (where τ is the ratio C/r... or the number of radii that can fit along the outside of a circle), the formula becomes even easier!

τ•r= C or r•τ= C

And that would be GREAT! Except that the official state test requires them to be able to calculate -- BY HAND -- the area and circumference of a circle using "3.14" and "22/7" as estimates of pi.

*head thwack*