Recognizing that Equal Shares Need Not Have the Same Shape - Second Grade Mathematics
When learners first discover fractions, they often assume that for two shares to be equal, they must look like identical twin shapes. If one piece is a triangle, they think the other piece must also be a triangle! In advanced geometry, you discover an astonishing and liberating truth: two shares of the exact same whole shape can have completely different geometric shapes, yet still possess the exact same amount of surface area! A long thin rectangle can have the exact same area as a short squat square or a triangle. In this chapter, you will explore how identical wholes can be partitioned into equal shares of different shapes, and prove that equal shares need not have the same shape.
Partitioning the Same Whole in Different Ways
Consider two identical square sheets of paper, Square A and Square B.
Partitioning Identical Squares into Fourths:
Square A (Four Long Strips): Square B (Four Small Squares):
+-----+-----+-----+-----+ +-----------+-----------+
| | | | | | | |
| 1/4 | 1/4 | 1/4 | 1/4 | | 1/4 | 1/4 |
| | | | | +-----------+-----------+
| | | | | | 1/4 | 1/4 |
+-----+-----+-----+-----+ | | |
+-----------+-----------+
The Shape Comparison:
In Square A, each fourth is a tall, narrow vertical rectangle. In Square B, each fourth is a short, wide square. Do they look alike? No! One is a long skinny rectangle, and the other is a square. Do they have the exact same area? YES! Both Square A and Square B were identical whole squares. In both cases, the whole was partitioned into 4 equal shares. Therefore, one long thin strip from Square A has the EXACT same amount of paper as one small square from Square B!
Diagonals: Rectangles and Triangles from Halves
Let us look at two identical rectangles cut in half.
Partitioning Identical Rectangles into Halves:
Rectangle 1 (Vertical Cut): Rectangle 2 (Diagonal Cut):
+-------------+-------------+ +-------------+
| | | |\ |
| RECTANGLE | RECTANGLE | | \ TRIANGLE|
| HALF | HALF | | \ |
| | | | \ |
+-------------+-------------+ | \ |
|TRI- \ |
|ANGLE \ |
+-------------+
Rectangle Half vs. Triangle Half
In Rectangle 1, a straight vertical cut down the center creates two identical smaller rectangles. Each piece is one-half of the whole. In Rectangle 2, a diagonal cut from the top-left corner to the bottom-right corner creates two identical triangles! Each triangle is one-half of the whole. Because both cuts divide the identical starting rectangle into two equal shares, the rectangle half from Rectangle 1 and the triangle half from Rectangle 2 contain the EXACT same amount of surface area!
The Geoboard and Unit Square Proof
How can we prove with 100 percent certainty that two completely different shapes have the exact same area? We can count unit squares on a grid!
Proving Equal Area on a Grid:
Shape 1 (A 1 by 4 Strip):
[#][#][#][#]
Area = 4 unit squares.
Shape 2 (A 2 by 2 Square):
[#][#]
[#][#]
Area = 4 unit squares.
Shape 3 (An L-Shape):
[#]
[#]
[#][#]
Area = 4 unit squares.
Three Different Shapes, One Identical Area!
Look at the three figures above: Shape 1 is a long stick. Shape 2 is a box. Shape 3 is an L-shape. All three shapes look completely different to your eyes, but every single one of them contains exactly 4 unit squares! If a whole shape made of 8 unit squares is partitioned into two 4-square pieces, it does not matter what shape those pieces take: both are genuine, legitimate halves.
Practice Exercises
Exercise 1: Describe two different ways to partition an identical square into four equal shares: one way that creates smaller squares, and one way that creates triangles.
Exercise 2: A square is cut into two rectangles using a vertical line. An identical square is cut into two triangles using a diagonal line. Does one of the rectangle halves have more paper, less paper, or the same amount of paper as one of the triangle halves? Explain your reasoning.
Exercise 3: On a grid of unit squares, Shape A is a 2 by 3 rectangle, and Shape B is a 1 by 6 strip. Calculate the area of each shape in unit squares, and state whether they represent equal shares of a 12-square whole.
Exercise 4: Explain why someone who thinks "equal shares must always look identical" is mistaken, using a real-world example like sharing a sandwich.
Solutions and Explanations
Solution to Exercise 1: One way to partition a square into fourths is drawing one vertical line down the middle and one horizontal line across the middle, creating 4 smaller squares. A second way is drawing two diagonal lines from corner to corner (forming an X), creating 4 equal triangles.
Solution to Exercise 2: The rectangle half has the exact same amount of paper as the triangle half. Because both starting squares were identical in size, and both cuts divided the whole square into two equal shares, each piece represents exactly one-half of the starting area. The shape of the piece does not alter the total amount of paper it contains.
Solution to Exercise 3: Shape A (a 2 by 3 rectangle) has an area of 2 x 3 = 6 unit squares. Shape B (a 1 by 6 strip) has an area of 1 x 6 = 6 unit squares. Because both shapes contain exactly 6 unit squares, each represents exactly half (an equal share) of a 12-square whole.
Solution to Exercise 4: Someone who believes equal shares must look identical is confusing physical shape with physical quantity (area). For example, if you have two identical square sandwiches, you can cut sandwich A into two rectangles and sandwich B into two triangles. Both sandwiches were the exact same size, and both were cut in half. A person eating a triangle half receives the exact same amount of bread, cheese, and nutrition as a person eating a rectangle half, proving that equal shares do not need to share the same shape.