Partition Shapes into Equal Areas

Students will partition circles, rectangles, and other polygons into equal parts using halves, thirds, fourths, sixths, and eighths. They will verify that each part has equal area by reasoning about the size and symmetry of the parts.

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Grade 3 · Partitioning Shapes and Unit Fractions · question 1 of 3

How it is taught

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A whole can be cut into equal pieces.

8 equal pieces are eighths.

Read it: Equal parts, one part named

A rectangle is cut into 4 equal parts. One part is shaded. It is 1/4 of the whole.

Check that the parts are all the same size.
Count the equal parts: 4.
Name one part: 1 of 4 equal parts is 1/4.

Equal parts cover the same amount of the shape.

The bottom number tells how many equal parts make the whole.

4 parts of 1/4 make the whole rectangle.

When a whole is cut into 4 equal parts, each part is 1/4 of the whole.

Calling a part 1/4 when the 4 parts are not the same size.

Only equal parts are fourths. Check the sizes first.

A garden is cut into 2 equal parts. What fraction of the garden is one part? — 1/2. 2 equal parts, so one part is 1/2 of the whole.

Partition a rectangle into 6 equal parts (two ways)

Here is one rectangle, and we want six equal parts. Is there only one way to cut it?

Grade 3 · Cutting one rectangle into sixths two different ways

Start with the whole rectangle, uncut.

Here's the first way most people think of: five evenly spaced vertical cuts, making six tall columns.

Check every column. Same width, same height, so every column covers the same amount of the rectangle.

Now a second way. Two horizontal cuts and one vertical cut give three rows of two — six cells.

Check these too. All six cells cover the same amount.

Look at the two side by side. A tall skinny column does not look like a small wide cell — but each one is exactly one sixth of the rectangle. Equal parts have to cover the same area. They do not have to be the same shape.

Six equal parts means each part is one sixth, whichever way you cut it.

Partitioning means equal areas, not identical-looking pieces. Cut a shape into six equal parts any way you like, and each part is one sixth.

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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