# Equal Means Equal > **Do the parts of a fraction have to be equal?** Yes. A fraction like 1/4 means one of 4 equal parts of a whole, so 4 pieces of different sizes are not fourths. The parts must be the same size, but they don’t have to be the same shape. - Big idea: Fractions need equal shares of one whole. Four parts are fourths only when all four are the same size, even if their shapes differ. - Common confusion: Any four pieces are fourths, even when the pieces are different sizes. - It often sounds like this: “I cut it into four pieces, so each piece is a fourth.” - Family: Parts & Wholes (slice 1 of 20) - Try it yourself (about 3 minutes, free): https://slices.org/slices/equal-means-equal ## In plain words A fraction describes equal shares of a whole. Fourths are what you get when one whole is split into 4 parts that are all the same size. Counting pieces is not enough. A rectangle cut into 4 strips of different widths has 4 parts, but they are not fourths, because some strips are bigger than others. To check, compare the sizes. When all 4 parts are equal, each one is 1/4 of the whole: exactly 4 of them fill it, with nothing left over. Equal parts do not have to be the same shape. Cut the left half of a square into two small squares and the right half into two triangles: all four parts are fourths, because each half was cut into 2 equal parts. The same rule holds for every fraction. Thirds are 3 equal parts of a whole, fifths are 5 equal parts, and so on. ## A worked example Three friends share a pizza cut into 3 slices of different sizes. Everyone gets one slice, but not everyone gets the same amount, so the slices are not thirds. Cut the pizza into 3 equal slices and each friend gets exactly 1/3. ## Why this confusion is common Children usually meet fractions by counting: shade 1 of 4 pieces and call it one fourth. Because most pictures in books are already cut into equal pieces, the word “equal” quietly drops out of view. Everyday language doesn’t help either: people talk about “the bigger half”. Asking “are the parts the same size?” every time brings the missing condition back. ## Try this away from the screen Take two identical paper squares. Fold one in half and in half again the other way to make 4 small squares. Fold the other along both diagonals to make 4 triangles. Cut out one small square and one triangle. They look different, but cut the small square along its diagonal and the two halves fit exactly onto the triangle: the same amount of paper, so both are fourths. ## Helpful ways to talk about it - Instead of “There are 4 pieces, so each one is a fourth.”, try: “Are all 4 parts the same size? How could we check?” - Instead of “Fourths always look like this.”, try: “These parts are different shapes. Do they cover the same amount of the whole?” ## Practice questions 1. Which shape is cut into fourths? **Answer:** Shape B. The two diagonals cut the square into 4 matching triangles, so each one is 1/4. The other shapes have 4 parts, but not 4 equal parts. 2. Three friends share this pizza, one slice each. Does each friend get 1/3 of the pizza? **Answer:** No. Thirds are 3 equal parts of one whole. These slices are different sizes, so the friends don’t all get the same amount, and they don’t all get 1/3. 3. These 4 parts are not all the same shape. Is each part 1/4 of the square? **Answer:** Yes. Each half of the square is cut into 2 equal parts: two small squares on one side, two triangles on the other. All 4 parts are the same size, so each is 1/4. 4. Which strip of paper is cut into thirds? **Answer:** Strip B. Thirds are 3 equal parts. Only one strip has 3 pieces of the same width. ## Questions people ask ### Do equal parts have to be the same shape? No. Equal parts must be the same size (the same area), not the same shape. A square cut into two small squares and two triangles is still cut into fourths, as long as each part covers exactly one fourth of the square. ### How can I tell whether parts are equal? Compare them directly: fold one part onto another, cut them out and stack them, or count the squares on grid paper. Slices of a circle that meet in the middle are equal when the angles where they meet are equal. Looking alone can mislead, so it is worth checking. ### Can one piece be 1/4 if the other pieces are different sizes? Yes. A piece is one fourth of the whole when exactly 4 pieces of its size would fill the whole, however the rest is cut. But the pieces only count as fourths together when every one of them is that size. ### When do children learn about equal parts? In the US Common Core standards, children split circles and rectangles into equal shares in grade 2, and in grade 3 they describe each equal part as a unit fraction such as 1/4. Every later fraction idea builds on it. ## Curriculum connections (US Common Core) - 3.G.A.2: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. - 2.G.A.3: Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. - 3.NF.A.1: Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. ## Related slices - [The Bottom Number](https://slices.org/slices/the-bottom-number): What the bottom number of a fraction counts: all the equal parts that make one whole. - [What’s the Whole?](https://slices.org/slices/whats-the-whole): Why half of a big pizza is more than half of a small one: a fraction depends on its whole. - [The Top Number](https://slices.org/slices/the-top-number): What the top number of a fraction counts: how many of the equal parts you have.