Partitioning Shapes into Equal Unit Areas - Third Grade Mathematics

Geometry and fractions merge in a beautiful way when we partition two-dimensional shapes into equal areas. Partitioning means dividing a whole shape into parts that have the exact same size. When a shape is partitioned into equal sections, each section represents a unit fraction of the whole shape's total area. In this chapter, you will discover how to partition rectangles, squares, and circles into equal unit areas and express those areas as fractions.

Partitioning and Unit Fraction Areas

When you divide a geometric shape into equal parts, you are partitioning it. If a shape is partitioned into b parts of equal area:
- Each individual part has an area equal to 1/b of the entire shape!


A Rectangle Partitioned into 4 Equal Areas:
+---------------+---------------+---------------+---------------+
|               |               |               |               |
|      1/4      |      1/4      |      1/4      |      1/4      |
|               |               |               |               |
+---------------+---------------+---------------+---------------+
The whole rectangle is divided into 4 equal parts.
Each part has an area equal to 1/4 of the whole rectangle's area!

If you shade 3 of those parts, the shaded region represents 3/4 of the total area of the shape.

Different Ways to Partition the Same Shape into Equal Areas

One of the most fascinating discoveries in geometry is that a shape can be partitioned into equal areas in multiple different ways!

Look at four different ways to divide the exact same square into fourths (each piece having 1/4 area):

Way 1: Vertical Slices          Way 2: Horizontal Slices
+---+---+---+---+               +---------------+
|   |   |   |   |               |      1/4      |
|1/4|1/4|1/4|1/4|               +---------------+
|   |   |   |   |               |      1/4      |
+---+---+---+---+               +---------------+
                                |      1/4      |
                                +---------------+
                                |      1/4      |
                                +---------------+

Way 3: Four Small Squares       Way 4: Diagonal Slices (Triangles)
+-------+-------+               +---------------+
|  1/4  |  1/4  |               | \     1/4   / |
+-------+-------+               |   \       /   |
|  1/4  |  1/4  |               | 1/4 \   / 1/4 |
+-------+-------+               |       X       |
                                |     /   \     |
                                |   /       \   |
                                | /     1/4   \ |
                                +---------------+

Notice something extraordinary about Way 3 and Way 4: In Way 3, each part is a small square. In Way 4, each part is a triangle. Even though a small square and a triangle look completely different in shape, their areas are completely identical! Each one covers exactly 1/4 of the original square's total area.

Non-Equal Partitions Do NOT Form Unit Fractions

Be on high alert: if the partitioned sections do not have equal areas, they cannot be described using standard unit fractions!

Equal Halves (1/2 each):          NOT Equal Halves:
+---------------+---------------+  +----------+--------------------+
|               |               |  |          |                    |
|      1/2      |      1/2      |  |  Small   |       Large        |
|               |               |  |          |                    |
+---------------+---------------+  +----------+--------------------+
Valid fraction partition!          INVALID! Cannot name as 1/2!

Always verify that every partitioned piece has the exact same amount of surface area before writing a fraction.

Chapter Practice Exercises

  1. A rectangle is partitioned into 6 equal parts. a. What unit fraction represents the area of each part? b. If you shade 5 of those parts, what fraction of the total area is shaded?
  2. Draw or describe three different ways to partition a rectangle into 2 equal areas.
  3. Draw or describe two different ways to partition a square into 4 equal areas.
  4. A square is partitioned into 4 equal triangles. True or False: Each triangle represents 1/4 of the total area of the square. Explain your answer.
  5. A circle is divided into 3 pieces by drawing straight lines, but one piece is much bigger than the other two. Can you say that each piece has an area of 1/3 of the circle? Explain why or why not.

Solutions and Step-by-Step Answers

  1. Rectangle partitioned into 6 equal parts: a. Each part represents the unit fraction 1/6 of the total area. b. Shading 5 parts covers 5/6 of the total area.
  2. Three ways to partition a rectangle into 2 equal areas:
  3. Way 1: Draw a vertical line down the exact middle (creating 2 tall rectangles).
  4. Way 2: Draw a horizontal line across the exact middle (creating 2 wide rectangles).
  5. Way 3: Draw a diagonal line connecting opposite corners (creating 2 congruent triangles).
  6. Two ways to partition a square into 4 equal areas:
  7. Way 1: Draw 3 vertical lines to create 4 equal vertical columns.
  8. Way 2: Draw 1 vertical line and 1 horizontal line through the center to create 4 smaller squares.
  9. True. Since the square was partitioned into 4 parts of equal area, each triangle represents exactly 1/4 of the total area of the square, regardless of its triangular shape.
  10. No. To name each part as 1/3, all three pieces must have the exact same equal area. Since one piece is noticeably larger than the other two, the pieces are unequal and cannot be called 1/3.