Cutting into two equal parts
Here is one bar. The dotted line runs down its middle. Cut along it, and the bar is in 2 parts that are exactly the same size: lay one part on the other, and they match.
Parts that are exactly the same size are called equal parts.
One bar, with a dotted line down its middle.
One of two equal parts is a half
Color in 1 of those 2 equal parts. The colored part is one half of the bar.
The part that is not colored is a half too. Two halves make the whole bar again.
The bar cut into 2 equal parts, with 1 part colored in: one half.
Not equal, not halves
Now move the cut away from the middle. The bar is still in 2 parts, but one part is bigger than the other. Neither part is a half, because a half is one of 2 equal parts, and these parts are not equal.
Counting the parts is not enough. The parts must also be the same size. Each part is one half only when the cut is exactly in the middle.
one piece > the other, so neither piece is a half
Move the cut until every piece is the same size
The cut starts far from the middle, so the two parts are not the same size. Move it until they match: only then is each part a half.
Count the parts, not the lines
A bar can be cut into more than 2 equal parts. This bar is cut into 5 parts, all the same size. To say how many parts there are, count the parts themselves: 1, 2, 3, 4, 5.
Count carefully, because the lines can trick you. There are 4 cuts inside the bar, and 4 cuts make 5 parts. Counting the cuts gives one too few. Counting every line, the two ends of the bar as well, gives 6, which is one too many.
A bar cut into 5 equal parts. There are 4 cuts inside it.
When no parts are equal
This bar is cut into 3 parts, and no two of them are the same size. So how many equal parts does it have? None. There are 3 parts, but none of them is an equal part.
A small part, a middle-sized part and a big part: 3 parts, and none of them equal.
Worked example: A Chocolate Bar Broken into Parts That Are Not Equal
Question A chocolate bar has 12 small squares in a row. Dad breaks it into 4 parts. The parts have 6, 2, 2 and 2 squares. Ben takes a part with 2 squares. (a) Is Ben's part 14 of the bar? (b) Dad breaks a new bar of 12 squares into 4 equal parts. How many squares are in each part?
1.Draw the bar as 12 squares in a row. Mark the 4 parts. They have 6, 2, 2 and 2 squares.
The bar has 12 squares. It is broken into 4 parts with 6, 2, 2 and 2 squares. 2.Look at the sizes. One part has 6 squares and the other parts have 2 squares. The parts are not equal.
One part has 6 squares and the other parts have 2 squares. The parts are not equal. 3.(a) A quarter is 1 of 4 equal parts. These 4 parts are not equal, so Ben's part is not 14 of the bar.
(a) A quarter is 1 of 4 equal parts. These parts are not equal, so Ben's part is not 14 of the bar. 4.Now share the 12 squares of the new bar into 4 equal parts. Give 1 square to each part in turn until all the squares are used.
Share the 12 squares of the new bar into 4 equal parts, 1 square to each part in turn. 5.(b) Each part gets 3 squares. Check: 3 + 3 + 3 + 3 = 12. Now each part is 14 of the bar.
(b) Each part gets 3 squares, because 3 + 3 + 3 + 3 = 12. Now each part is 14 of the bar.
Answer: (a) No, because the 4 parts are not equal; (b) 3 squares
Common mistakes
- Saying that Ben's part is 14 because the bar has 4 parts. Counting the parts is not enough. The parts must also be equal, and a part with 6 squares is not equal to a part with 2 squares.
- Writing 4 squares for part (b) because there are 4 parts. Four parts of 4 squares make 4 + 4 + 4 + 4 = 16 squares, and the bar has only 12.
Worked example: A Sandwich Cut in Half in Two Different Ways
Question Mei and Tom each have a square sandwich of the same size. On square paper, each sandwich covers 4 small squares. Mei cuts her sandwich in half from side to side. Tom cuts his sandwich in half from corner to corner. Each child eats 1 part. (a) What fraction of her sandwich does Mei eat? (b) Who eats more, Mei or Tom?
1.Draw each sandwich as 4 small squares. Draw Mei's cut from side to side. Draw Tom's cut from corner to corner.
Each sandwich is 4 small squares. Mei cuts from side to side. Tom cuts from corner to corner. 2.(a) Mei's cut makes 2 equal parts. One of 2 equal parts is a half, so Mei eats 12 of her sandwich.
(a) Mei's cut makes 2 equal parts, so Mei eats 12 of her sandwich. 3.Count the squares in Mei's part. It has 2 whole squares.
Mei's part has 2 whole squares. 4.Count the squares in Tom's part. It has 1 whole square and 2 half squares. Two half squares make 1 whole square, so his part has 1 + 1 = 2 squares.
Tom's part has 1 whole square and 2 half squares. Two half squares make 1 whole square: 1 + 1 = 2 squares. 5.(b) Both parts have 2 squares, so they are the same size. Tom also eats 12 of his sandwich. Mei and Tom eat the same amount.
(b) Both parts have 2 squares. Each is 12 of a sandwich, so Mei and Tom eat the same amount.
Answer: (a) 12; (b) They eat the same amount. Each part has 2 small squares
Common mistakes
- Saying that Tom eats more because his cut is longer. The length of the cut does not tell the size of the part. Count the squares: both parts have 2 squares.
- Saying that Tom's part is not a half because it is a triangle. A half does not need a special shape. It is 1 of 2 equal parts of the whole.