M5M1 & M5A1 - Developing Area Formulas (6)

M5M1. Students will extend their understanding of area of geometric plane figures.
M5A1. Students will represent and interpret the relationships between quantities algebraically.

The topic of deriving the area formulas lie in the intersection of the four of the five content strands in the GPS: Number & Operations, Geometry, Measurement, and Algebra. Obviously, since we are determining the area, we are measuring the size of various shapes. Although we can determine the area by covering the given shape with unit squares or drawing it on a grid paper, we are going beyond measurement based on counting. Rather, we are calculating the area. Therefore, students will have to utilize their understanding of number and operations - and students can practice their skills as they study this topic as well. As we try to calculate the area of new shapes, students will have to somehow make familiar shapes for which they know how to calculate the area. In that process, they will have to utilize their knowledge and skills of geometry. Finally, we are not just interested in calculating the area. We want to generalize the process and derive formulas that can be applied to other shapes of the same type. Thus, students are utilizing and developing their understanding and skills of algebra as they generalize and express the relationship among length measurements of different parts of the given shape and its area.

Although the GPS does not expect students to derive area formulas for other polygons, deriving formulas for other polygons may be helpful for students to deepen not only their understanding of area formulas but also advancing their understanding of algebraic thinking and representations. Two possible polygons to consider are trapezoids and kites. The formula for calculating the area of trapezoids may be familiar to many people, so I will focus on the formula for the area of kites today.

Students can derive the formulas for these shapes using what they already know. The general steps are:
(1) Find the area of various trapezoids (or kites) by making a familiar shape (or a collection of familiar shapes) using the strategies discussed before.
(2) Determine what measurements (or the given trapezoid/kite) will be needed to calculate the area.
(3) Determine which strategy of calculating area might be useful for generalization.
(4) Derive the formula - i.e., summarize the process of calculation in an equation using variables (words or letters).

Note that a kite is a quadrilateral with two distinct pairs of equal adjacent sides. Squares and rhombuses are special types of kites because they have four equal sides. One of the properties of rhombuses is that their diagonals are perpendicular to each other and they intersect at their mid-points. The diagonals of kites are also perpendicular, but they intersect at the mid-point of one of the diagonals but not necessarily the other. In the figure below, the diagonals are all perpendicular. However, in (a), they are intersecting at the mid-point of both diagonals while in (b) ~ (e), they are intersecting at a mid-point of one of the diagonals but not the other. However, since they all have two distinct pairs of equal adjacent sides, they are all kites. Since all four sides are equal, (a) is also a rhombus.

So, how can children find the area of a kite? One of the big idea from earlier is that, when we are given an unfamiliar shape, we may be able to find the area by making a familiar shape (or a collection of familiar shapes). We have also seen that there are three general strategies that can be used to make a familiar shape: (1) sub-dividing, (2) making-it-bigger, and (3) cutting-and-moving. For example, here are three different ways the area of a kite can be calculated:

In (1), we are using the sub-dividing method. Since we already know how to calculate the area of triangles, we can divide the kite into two triangles. The area of the kite is the sum of the area of the two triangles. In (2), we surround the kite by a rectangle. We can see that the length and the width of this rectangle are the length of the two diagonals of the kite. In (3), we cut and moved the parts of the kite to form a rectangle. One of the dimensions of the rectangle is the same as the length of one of the diagonals while the other dimension is a half of the other diagonal.

With each of these methods, we need to know the lengths of the diagonals. Thus, in order to calculate the area of kites, the length of diagonals are needed. However, in method (1), if we use the diagonal whose mid-point is the point of intersection of diagonals (as this example above shows), we will need to know exactly how the other diagonal is split to calculate the area of the two triangles.

For method (3), we can make a new rectangle differently, yet still with the dimension of one diagonal of the kite by a half of the other diagonal of kite as shown below.

However, it may be difficult for some children to understand what the vertical dimension (in this example) should be a half of the diagonal.

Therefore, although each of these methods may be generalized to derive a formula, method (2) may be simpler one to derive a formula. With method (2), you simply enclose the kite with a large rectangle, and the new rectangle you create has the area that is twice the area of the given kite:

Therefore, the area of the kite can be calculated by dividing the area of the rectangle. Thus, the formula for the area of kites might look like this:
Area of Kites = Diagonal 1 x Diagonal 2 ÷ 2.

What's important is not the actual formula but understanding the general process of generalizing and deriving the formula. This can be said of the formulas for triangles and parallelograms, too. We should emphasize the reasoning involved in the process of deriving the formulas. If students understand the reasoning, they will be able to re-derive the necessary formula even if they can't simply recall it.

M5M1 c - Developing Area Formulas (5)

M5M1. Students will extend their understanding of area of geometric plane figures.
c. Derive the formula for the area of a triangle.
In some textbooks, the area of triangles is studied before the area of parallelograms. In those cases, students typically examine how to calculate the area of right triangles. They notice that the area of a right triangle is a half of the rectangle you can make by using two copies of the right triangle.

They will then investigate how the area of more general triangles may be calculated, for example a triangle like this one:

Typically, students will split the triangle into two right triangles and apply the same method as before:

From here, those textbooks often generalize the way to calculate the area of triangles as:

However, sometimes students develop a misconception that the base must be the side of triangle that can be split to form 2 right triangles. In particular, they may not be able to calculate the area of the following triangle:

They want to split the triangle into two right triangles as shown, but they cannot determine the length of "base" and "height" in this case:

What they could do is to use the make-it-bigger approach and form a larger right triangle by adding on a smaller right triangle as shown below:

You can not only calculate the area of the given triangle by using this method, this method can be generalized to support the derivation of the formula. However, the challenge for students to derive (or verify the previously developed) formula is that they have to apply the distributive property as shown below:

Students may be used to applying the distributive property over addition, but they may not have had much experiences with the distributive property of multiplication over subtraction. Moreover, they have to treat the expression 4 ÷ 2 as a quantity.

Although it is possible to discuss the area of triangles before the area of parallelograms, my preference is to discuss parallelograms first. If parallelograms are "familiar" shapes, students can use a variety of methods to find the area of triangles. Here are just a couple students have come up on their own:

From the method on the left, we can see that the area of the triangle is the half of the area of the parallelogram, and the area of the parallelogram may be calculated by multiplying the "base," which is one side of the triangle, and the "height," which is the distance between the base and the parallel line containing the third vertex. From the method on the right, we can see that the area of the triangle is equal to the area of the new parallelogram. The area of the new parallelogram may be calculated by multiplying the "base," which is a side of the given triangle, and a half of the "height" of the triangle.

Either way, we can generate the formula, Area = Base x Height ÷ 2. Just as was the case with parallelograms, it is important that students understand that any of the three sides of the triangle may serve as the base and for each base, there is a corresponding height. The height is the distance between the base and the third vertex, which is the same thing as the length of perpendicular segment from the third vertex to the base.

By the way, it is probably important to write the formula with "÷ 2" since students only learn to model fraction multiplication in Grade 5 according to the GPS.