In honor of 2pi day (June 28th)...
From the awesome Dinosaur Comics, of course.
Why celebrate half the holiday? Get the whole circle! That old pi day is for squares. Er, semicircles, at least.
Seriously, we did this activity in the preservice elementary class and I thought I'd share. It culminates in the usual measure a bunch of circles activity, but tries to motivate it. Sure it's amazing that all those ratios are close, but why would you do it in the first place? And why are they all the same anyway? For me the answer is similarity. This activity was motivated by my students wanting to know more about pi, as it came up when doing volume of solids (someone remembered a formula), and that got people wondering. In addition, our geometry class comes before our number class, and lots of students had said that fractions were what was most confusing. And pi practically lives in ratio city.
Two objects are similar in geometry if one is an enlargement of the other. Mathematicians often use ratios to investigate them.
Consider a 2x3 rectangle. A 4x6 rectangle is an enlargement. But a 4x5 rectangle is not. Can you tell by looking? Describe what you see.
People discussed how the 4x5 should be similar because its 2 wider and 2 longer, but in the 4x6, which looks the same, you can see 4 or the original rectangle.
What ratios can you make with the two similar rectangles that are the same?
Students made both 2/3 to 4/6 and 2/4 to 3/6.
If you wanted to make a rectangle that was similar and 5 squares wide, how long should it be? Can you prove your answer is right?
Some students set up a proportion and solved with cross multiplying... although they confessed that they didn't know why cross multiplication worked. One student saw 5 as 2 + 2 + half a 2, and calculated 3 times 2.5. We talked about how that was excellent proportional thinking because instead of 2+2+1, she relatedit back to the original as half of 2. I showed how in the picture you could see the 7 as 2.5 of the 3's, or 3+3+half of 3.
A 24x36 poster is supposed to be an enlargement of an 18 x 27 poster. Is that possible or not? Explain?
Saw quickly with ratios.
Find the lengths of the diagonal of each poster using the Pythagorean Theorem.
Refresh, connect question.
Sum up what you see about ratios and enlargements/similarity:
Jotted down and discussed at their tables. Enlargement and proportional were heard in a lot of the conversations.
Common photograph sizes are 3x5, 4x6, 5x7, 8x10, 11x14 and 16x20 and 20x30. Sketch, outline or shade in the ones you can on graph paper. Which are similar? Which are closest to similar?
The sketched them separately, so I did a nested version on the board to show another way. (Often a student will have done it that way.)
Most cameras record in a ratio of 3:2, movie camcorders in 4:3 and hi-def in 16:9. How do these compare to common print sizes? What sizes would you use for your regular prints and for an enlargement?
General agreement on the 4x6.
Looking at these circles, they look pretty similar in the usual English sense. Are they similar in the mathematical sense? How do you know?
This was an unexpectedly interesting discussion. Most people thought yes, they are all similar, and then one student made them hesitate. That led to proportional talk and someone brought ovals into it. I asked if people knoew what made a circle a circle and they didn't. They discussed how the only things you could measure are the radius/diameter or the circumference.
Give three different pairs of similar rectangles. Compute the ratios of their perimeter to their longest side. What do you notice?
Students saw the constant ratios quickly when they shared examples. One student made the jump to the idea of scale. Pretty cool.
What might that mean about circles? Give your reasons. (This is basically asking for an educated guess.)
Time was tight, so I was a bit leading here. What ratio would the rectangle example be like for circles? "Circumference to radius." So if circles are all similar... "Then the ratios should be the same."
Divide and Conquer
• Data collection: in your group measure the circumference and diameter of at least 10 different circles. (details to follow)
• Add your data to the class stem and leaf.
• Which is the best choice for typical (mean, median, mode) and why?
We discussed how to measure diameter and circumference, and agreed on precision. (Two decimal places for the ratio.) We got over 50 data points, and it made a pretty nice bell curve between 2.8 and 3.8. There were two measurements of 6.00, but she realized she had divided by the radius.
The mode was 3.20 (5) followed by 3.18 (4). The median was 3.19. The mean about 3.22. Curiously, the majority of students felt that the spread was too big to be explained by measurement error, so circles must not all be similar! I did let them know that they were, but it would take a more theoretical argument to prove it to them.
All in all it felt much more contextual to start with similarity, and students made a lot of sense about why pi might have been discovered.
The activity sheets are below. But also be sure to check the comments, where Alexander Bogomolny has a couple of very relevant links about the idea of measuring one circle as accurately as possible.
Similar to Circles