M5M1 - Developing Area Formula (8)

M5M1. Students will extend their understanding of area of geometric plane figures.
g. Derive the formula for the area of a circle.

Deriving the area formula for circles is a bit more complex than deriving the formulas for polygons. The fundamental idea, "When you are given an unfamiliar shape, you may be able to calculate its area if you can make a familiar shape (or a collection of familiar shapes)," is still applicable. However, we can't really make a polygons out of a circle. So, the process typically requires an additional step.

However, before we get to the derivation of the area formula, let's think about another idea first. Here is a little puzzle for you. I'm sure many of you have seen racquetball balls sold in a canister like this one:

So, which do you think is loner, the height of the canister or the distance around the canister, or are they about the same? I encourage you to actually find a canister and test it yourself.

Sometimes, they come in a package of 3 balls in a canister like this one:

Again, which is longer, the height or the distance around it, or are they about the same?

Tennis balls usually sold in canisters, sometimes with 3 balls and others may have 4.

Again, which do you think is longer, the height or the distance around it, or are they about the same?

I have once seen a set of 3 softballs being sold in a canister like these, too. Again, which do you think is longer, the height or the distance around it, or are they about the same?

It turns out that whenever there are 3 balls in a canister, the height and the distance around the canister are about the same, no matter what kind of balls they are - the distance around is actually a bit longer but they are pretty close. So, what is happening? A simplified picture of the question will be like this:

In other words, the question was asking you to compare the length around a circle (we call it circumference) and the sum of three diameters. What we observed in the canisters of balls is that the circumference of a circle is about three times of its diameter no matter how big a circle is. In other words, the ratio of the circumference to the diameter is constant, and the value of this ratio is a little more than 3. This ratio is called the ratio of circumference, and we often use the Greek letter, π (pi). This number is an example of irrational number, that is, a number that cannot be expressed exactly as a fraction - you may have seen people use 22/7 as an approximation, but it is not exactly equal to pi. Another, and perhaps more common, approximation of the ratio of circumference is 3.14, and this is mentioned in M5M1h.

What we just observed can be summarized in the following equations:

Because the diameter is twice the radius, we can also express the relationship between the circumference and the radius this way:

It turns out the circumference plays an important role when we are trying to calculate the area of a circle. I will discuss the area of the circle in the next post.

M5M1 d - Developing Area Formulas (7)

M5M1. Students will extend their understanding of area of geometric plane figures.
d. Find the areas of triangles and parallelograms using formulae.

The most common difficulty I observe when children (or adults) are asked to find the area of triangles or parallelograms isn't that they can't remember the formulas, but rather they don't seem to have a clear understanding of what "b" (base) and "h" (height) are supposed to be. This is particularly telling when students are asked to find the area of parallelogram like the one shown below:

Although there are sputtering of errors such as adding all measurements, the most common error is multiplying the lengths of two adjacent sides, which is an overgeneralization of the area formula for rectangles. Some students may even write, "a = bh," on the paper, yet still multiply two adjacent sides. Even if we help students derive the formulas, it will be a good idea to provide students to deepen their understanding of the formulas. When you give problems, you may want to include extra information like the one above. Another potentially useful activity is to have students find all triangles with the area of 2 square units on a regular 5 by 5 geoboard.

Many students will find the following triangles:

Note that the one on the bottom right is really the same as the one in the top middle, just in different orientation.

If you ask them what strategy they used, many will say that the product of the base and the height must be 4. So, the possible combination is either 2 and 2 or 1 and 4 (or 4 and 1). If students are having difficult time going beyond those found above, you may want to focus on the two triangles on the left and ask if there is any other triangle you can make with the base of 2 units and the height of 2 units. You may even want to draw both triangles on the same base:

Eventually, someone will notice that you can "slant" the triangle a bit further:

It is important to discuss here if the height is still 2 units for this triangle. Once they agree that the height of this new triangle is 2 units, others may find another one.

If you put all of these triangles (and a mirror image of one of them) on the same base, you can see a picture like this:

Students may notice that the "top" vertex is moving along a line, and if the geoboard was larger, we could slant the triangle even further by moving the vertex further and further to the right along the same line. If you ask them to describe the relationship between the line and the base of the triangle, students will notice that they are parallel. From here, we can generalize that the area of triangle does not change if we move a vertex along the line that is parallel to the opposite side. We can then re-emphasize that this is indeed the case because all of these triangles have the same height, and the height is really the distance between the base and the parallel line containing the opposite (from the base) vertex. The distance between the two parallel lines is measure by the length of perpendicular segments from one to the other, so the height of a triangle can be measured anywhere although some are more commonly used in textbook drawings than others:

You can do a similar activity with parallelograms, as well. Through these questions, you can deepen students understanding of the formulas - in particular the understanding of the height in the triangle (and parallelogram) formula.