ARE GLASSES TALLER THAN ROUND?: Diameter, Circumference, and Height of Cylinder

I hope you all are enjoying having your children find every cylinder in your home to see which is greater: circumference or height (is it taller than it is round?). They should record the results on pages 6-8 of the pdf.

Ask someone to guess if a glass or cup is taller than it is round. They will almost always guess that it is taller than round unless they are suspicious. If they guess that the cup is taller than it is round, explain why they are wrong. If they guess the opposite, ask them why they think it is more round than tall. Then teach them what you know.

HINT: YOU MAY WANT TO DO A SUBTLE (without arousing suspicion) CHECK BEFORE YOU DO THIS. TRY TO WRAP YOUR HAND ALL THE WAY AROUND YOUR DRINKING GLASS. YOUR FINGERS AND THUMB WILL LIKELY NOT REACH AROUND THE OTHER SIDE. NOW WITH YOUR THUMB AND INDEX FINGER, TRY TO SPAN THE HEIGHT OF THE GLASS. YOU WILL EITHER SUCCEED OR COME VERY CLOSE. 

I am not supporting child gambling, but ….

If someone tells you that of course it is taller than it is round, tell them that you will bet them (a piece of candy) that the cup is more round than it is tall. Then, prove it to them. Of course, this is not a fair bet (don't take the candy, share it) because you know that 99 out of 100 drinking cups are more round than they are tall (that is, the distance around the circumference of the glass opening is going to be greater than the height of the cylinder).

 The differentiation in this lesson is how they estimate the circumference from the diameter. The K students will just measure the circumference with the centimeter ruler I laminated for them. Just reading a centimeter ruler is challenging for them as they are learning how adding ten and then 1-9 more will give you numbers in the teens; adding 20 and then 1-9 more will give you numbers in the 20s; and so on. 

All other students were challenged to first find the diameter in centimeters. Then they will be multiplying that diameter by 3 (or adding the diameter to itself three times); and remember to add one seventh of the diameter to the circumference estimate (if the diameter is close to 7, then add one; if the diameter is close to 14, add two; if the diameter is close to 21, add three and so on). We talked briefly about PI and how it is close to 3 and 1/7. I am not a big fan of having students memorize multiple digits of PI as engineers rarely use more than 5-9 digits (3.141592654…). Understanding how to take one seventh of the diameter and add it to three times the diameter is a very close estimate of circumference. 

For students will a little advanced knowledge (4-7th graders), they can do all of the calculations in millimeters. Of course, this complicates the multiplication by 3 and the division by 7. They will further develop their estimating skills and multiples of 7. Pages 10-12 of the pdf have these worksheets for millimeters.

The key for everyone is that they check their estimate of circumference by using the ruler to check actual measurement. Then, they measure the height and compare the circumference to height. They will now get to use the greater than , less than and equals symbols to indicate which is greater. 

The attached pdf explains this phenomenon graphically with color and a focus on vocabulary. The fourth page even shows that a test tube is only slightly taller than it is round. In your house, only a champagne flute should be taller than it is round as a drinking glass. Of course, they should find many cylinders that are taller than round such as a pencil, pen, certain cleaning solutions, etc.

The children should continue to practice use of compass.