Area of a Circle

After finding the formula for the circumference of a circle, my 6th graders were ready to work on finding the area of a circle.

I asked them to draw a circle on notebook paper, any size, but not too small. Then I gave each a centimeter cube to trace one face onto their paper to remind them how much the area of one square centimeter covered. (You can see it drawn on top right pic below.)

Question 1: Give me a guess — only by looking — what is the area of your circle, in square centimeters? Please write that number on your paper, label it "guess."

Question 2: Now use whatever tools you need, give me a better answer. I know some of you already know the formula for the area of a circle, but you may not use it unless you can tell me where it came from. (No one even tried.)

I can't tell you how happy I was to see all the different ways the the kids had tried to approximate area. Their perseverance humbled me. A few students drew triangles, knew to use "base times height divided by two," but erroneously used radius as height.

Last week I was reading Mimi's post about estimating area of circles, and Sue VanHattum reminded me of the rectangle model in her comment.

I had the kids fold their circles like this.

They cut out the pieces — turning every other piece 180 degrees — and glued them together. Cristian said, "First we had a circle, then triangles, then a rectangle. That's crazy!"

Never once did I answer any of the questions. I just asked them. I began with, "What is the area of this rectangle or parallelogram?" Step by step, as a class, the kids walked this equation all the way to Area = pi x radius^2.

This morning, two days after the activity, I did a My Favorite No to see if they remembered: 31 out of 33 students got the correct circumference formula; 24 of 33 got the correct area formula. Bonus if they showed me how the rectangle model helped explain the area of a circle.

Follow Up on Friday Bubbles

My 6th graders had a blast last Friday blowing bubbles.

Upon their return on Monday, I asked for ten groups to volunteer just one set of their circumference and diameter numbers. Then I asked, "What is the relationship between these numbers?" Some quiet mumbling, then just quiet. I asked again, "I have these 10 sets of numbers. What should I do with each set of numbers to see if there's a relationship between them? I want you to use your calculator to do something with these two numbers and find out for me."

They were busy punching in numbers. After a few minutes...

Me: What are you punching into your calculator?

Matt: I'm dividing the numbers.

Me: Why divide? Why not add or subtract? Or multiply the numbers?

Matt: Adding and subtracting don't tell you anything. Just doesn't make sense.

Sophia: The circumference is always bigger than the diameter. So I think I want to know how many times it's bigger.

Me: Class, do you agree that we should divide circumference by diameter? (Lots of nods around the room.)

While they continued with their own bubble numbers, I filled in the "relationship" column for the 10 volunteered sets from earlier.

They were skeptical — didn't know what to make of the "relationship" column. I said, "I think measuring a bubble imprint with a string is not easy. It was fun to blow bubbles, for sure. But what about it that may have affected how your numbers came out?"

Their answers:

The bubbles were sometimes oval shaped.

Hard to keep the string perfect.

It can get messy!

Maybe some people didn't measure the widest part.

Maybe they measured in centimeters for one part and inches for another.

It was so much fun! (I can always count on a random answer.)

I said, "Because our bubble numbers could use some help. Let's do a clean run through this again, this time using paper and pencil."

I asked them to draw two to three different sized circles on notebook paper — measure the diameter and circumference of each one. I showed them how to "walk" the string around the circle for better accuracy — press and let go, press and let go... They knew that if they started from one end of a diameter, walked the string around the circle, to the other end of the diameter, then they just needed to multiply this by two to get the circumference.

I asked them to find the "relationship" number again for each circle.

The paper in top row, left, reads: All of them are around 3, which is close to pi. Other papers didn't include observations or conjectures, and a few wrote they weren't so sure. But when we shared ALL the "relationship" numbers that the class had — over 60 of them as each student did at least 2 circles — and they were pretty well convinced that the diameter was "about 3 times longer" than the diameter!

I showed them this very short video to reflect the simple work that we did.

Next up, we are going to find the AREA of a circle!!!!!!!

Friday Bubbles

This activity is a lot of fun for 6th graders to discover for themselves the relationship between diameter and circumference.

Given a small container of bubble solution, the student pours a little of it onto his/her desk and use a straw to blow a bubble. When a bubble pops, it leaves enough of an imprint on the desk for the student to measure its diameter and circumference with a string. Students continue to blow different-sized bubbles and record their measurements.

Mix these three ingredients together well for a perfect bubble solution. I doubled this recipe to make enough for 34 students working in pairs.

• 1 cup distilled water

• 2 tbsp Karo light corn syrup

• 4 tbsp Dawn dishwashing liquid

I put the bubble solution in small plastic cups. Paper cups do not work! I give students embroidery threads to use as "string" to measure diameter and circumference. Of course they then have to measure the string against a ruler to get a reading in centimeters.

They work in pairs — one person blows a bubble and measures, the other person records; they wipe off desk, string, and ruler after each bubble. Switch roles. I ask for at least five different-sized bubbles.

Not wanting to waste the leftover bubble solution, I let the kids go outside to blow bubbles in the Friday afternoon sun. One kid said, "Today is such a good day." Other kids quickly chimed in to agree.

I really think they will remember that the circumference is about three times the diameter.