Partitioning Shapes into Four Equal Shares (Fourths/Quarters) - First Grade Mathematics

What happens when you have a delicious treat that you want to share equally among four friends? You must partition the whole into four equal shares! In mathematics, each of those four equal shares is called a fourth, or a quarter. Just like four quarter coins join together to make one whole dollar, four fourths of a shape join together to make one whole figure! In this chapter, you will master partitioning rectangles, squares, and circles into four equal shares, explore multiple slicing strategies, and understand the relationship between fourths and halves.

What Is a Fourth (or Quarter)?

When a whole shape is partitioned into four equal shares, each share is called one fourth, or one quarter.

Partitioning a Square into Fourths (The Window Cut):
       +---------------+---------------+
       |               |               |
       |   1 Fourth    |   1 Fourth    |
       |  (1 Quarter)  |  (1 Quarter)  |
       +---------------+---------------+
       |               |               |
       |   1 Fourth    |   1 Fourth    |
       |  (1 Quarter)  |  (1 Quarter)  |
       +---------------+---------------+
           4 Equal Shares = 4 Fourths!

Two Names for the Same Share:

Name 1: A FOURTH (because the whole was cut into 4 equal parts). Name 2: A QUARTER (from the Latin word meaning four, like 4 quarters in a dollar or 4 quarters in a football game). Both names mean the exact same fraction: one of four equal shares.

Halving the Halves: The Secret to Making Fourths

How do you partition a shape into four equal shares without guessing? The easiest method is to CUT IN HALF, THEN CUT IN HALF AGAIN!

The Halve-the-Halves Strategy:
Step 1: Cut the whole in half! (2 Halves)
        +-------|-------+
        |  1/2  |  1/2  |
        +-------|-------+

Step 2: Cut each half in half! (4 Fourths!)
        +---|---|---|---+
        |1/4|1/4|1/4|1/4|
        +---|---|---|---+

Fourths Are Smaller than Halves!

Notice something critical: when you cut a shape into four pieces instead of two, each individual piece becomes SMALLER! A fourth of a pizza is smaller than a half of a pizza, because four people are sharing instead of two.

Partitioning Circles into Fourths

To partition a circle into fourths, draw two crossing lines that meet at the center like a plus sign (+).

Fourths of a Circle:
                 12
             .---+---.
            /  1 | 2  \
      9  --+-----+-----+--  3   <- Two perpendicular lines through center!
            \  3 | 4  /
             '---+---'
                 6
         4 Equal Pizza Slices!

Equal Pie Slices

Each slice has the exact same angle at the center and covers the exact same area. All 4 slices are equal fourths.

Practice Exercises

Exercise 1: How many equal shares are in a whole that has been partitioned into fourths? What is another word for a fourth?

Exercise 2: Which is bigger: one half of a cookie, or one fourth of the exact same cookie? Explain why.

Exercise 3: Describe the two-step strategy called "halving the halves" used to partition a shape into fourths.

Exercise 4: Draw or describe two different ways to partition a square into four equal shares.

Solutions and Explanations

Solution to Exercise 1: There are 4 equal shares in a whole partitioned into fourths. Another common word for a fourth is a quarter.

Solution to Exercise 2: One half of the cookie is bigger. When you divide a cookie into halves, it is shared between only 2 parts. When you divide it into fourths, it must be shared among 4 parts, making each individual piece smaller.

Solution to Exercise 3: First, partition the entire shape into 2 equal halves. Second, partition each of those halves in half again. This creates 4 equal shares (fourths).

Solution to Exercise 4: Way 1: The cross/window cut, drawing one vertical line down the middle and one horizontal line across the middle to make 4 small equal squares. Way 2: The strip cut, drawing three vertical lines to divide the square into 4 tall, equal vertical rectangular columns.