Equal shares that look different
Two people each get one half of the same cracker, but the pieces are different shapes. Did they get the same amount?
Where you are 4 in A fraction as part of a whole
Why this exists
A child who has met equal shares knows that a half is one of two pieces of the same size, and that a square cut down the middle or from corner to corner gives two halves. That lesson showed one such pair. This one asks how far the idea goes.
Hand a child a square cracker scored into small squares and ask for a fair share for two people. Most will snap it straight down the middle. But a staircase break, a hook, or a slanted cut can be just as fair, and the child who sees that has understood something deeper than “cut down the middle”: a share is fair because of how much it holds, not because of how it looks.
That raises two questions a child cannot answer by looking. How do you show that a staircase piece and a triangle piece hold the same amount? And if two halves of one cracker are always the same size, why is one half of a big cracker not the same as one half of a small one? This lesson answers both, over three days.
Definition
Recall from Equal shares and naming the whole and Equal shares and naming the whole: a whole is cut into equal shares when the pieces are all the same size and together make up the whole, with no gaps and no overlaps. Two equal shares are halves, three are thirds, four are fourths.
Two pieces are the same size when they hold the same amount of the whole. They do not need to be the same shape. On a cracker or a paper square marked in small squares that are all alike, there are two ways to show two pieces are the same size:
- count: they cover the same number of small squares, and
- move: one piece can be cut into parts and the parts slid or turned, with no gaps and no overlaps, until they lie exactly on the other piece.
A small square cut from corner to corner is cut into two half squares, and two half squares together cover one small square.
Intuition
The lesson runs over three days. Part 1 cuts one whole into equal shares in many ways and counts squares to check them. Part 2 shows the same thing by moving and turning pieces, which works even where counting gets hard. Part 3 asks why all the shares of one whole are the same size, and why shares of different wholes need not be.
Part 1: many ways to cut one whole
Each cracker in Figure 1 has 16 small squares, 4 rows of 4. A is cut straight down the middle. B is cut in a staircase. C is cut so that one piece is a hook. Count the shaded piece in each: A has 4 rows of 2, which is 8; B has 3 + 2 + 2 + 1 = 8; C has 4 + 1 + 1 + 2 = 8. Each shaded piece covers 8 squares, so the other piece covers 16 - 8 = 8 too, and since 8 + 8 = 16 every cracker is cut into two pieces of the same size. All three are cut into halves, though no two of the shaded pieces look alike.
Children enjoy finding their own. Give each child a square of grid paper 4 squares on a side and ask for a way to cut it into halves along the lines that nobody else has found. The test is always the same: does each piece cover 8 squares?
Part 2: moving a piece to show its size
Counting answers the question, but moving pieces shows it. In Figure 2 the staircase half has one square sticking out at the top right and a gap at the bottom. Cut off the dotted square and drop it into the gap: the piece becomes the left two columns, the plain half from Figure 1 A. Moving a part of a piece does not make the piece bigger or smaller, so the staircase piece was a half all along.
Moving also works where the cut runs through the small squares, so that some of them are cut into parts. Then counting means putting the parts back together, as in Example 3.
Part 3: one whole, or two?
Here is why two halves of one cracker must be the same size even when they look different. Each half, together with its partner, makes the whole cracker. If one person’s half were bigger than another person’s half of the same cracker, then two of the bigger halves would make more than two of the smaller ones, which is more than the whole cracker. That cannot happen, because each pair makes exactly the cracker. So all halves of one whole are the same size, and the same goes for thirds and fourths.
That argument used one whole. Two crackers that are different sizes have halves of different sizes, whatever their shapes (Example 7).
Results
Part 1: many ways to cut one whole
One whole can be cut into equal shares in many ways. The shares from one way of cutting may have different shapes from the shares from another way, and even the shares from one way of cutting may differ in shape from each other (Figure 3, D).
On a grid, counting decides. When all the small squares are the same size, two pieces that cover the same number of small squares are the same size. A whole of 16 squares has halves of 8 squares and fourths of 4; a whole of 12 squares has thirds of 4.
Part 2: moving a piece to show its size
Moving does not change size. A piece cut into parts that are slid or turned, with no gaps and no overlaps, holds the same amount as before.
A half turn can prove halves. If a straight cut goes through the center of a square or a rectangle, turning either piece half way round the center lays it exactly on the other, so the two pieces are halves.
Part 3: one whole, or two?
All equal shares of one whole are the same size, whatever their shapes. Every half of a whole is the same size as every other half of it, however it was cut. The same is true of thirds and of fourths, and of shares of two wholes that are the same size.
Shares of different wholes need not be the same size. A half of a bigger whole is bigger than a half of a smaller one.
Why it is true
Counting decides. Each small square holds the same amount. A piece of 8 squares is 8 of those amounts put together, so any two pieces of 8 squares hold the same amount. Half squares work the same way, because the two halves of a small square cut corner to corner are the same size (turn one half way round the middle of the small square and it lands on the other), so two of them hold one small square.
Halves of 16 squares cover 8. Two halves are the same size and together cover all 16 squares, so the number of squares in a half added to itself makes 16. The only number that does that is 8: 7 + 7 = 14 is too few and 9 + 9 = 18 too many. In the same way a fourth of 16 squares covers 4, because 4 + 4 + 4 + 4 = 16, and a third of 12 squares covers 4, because 4 + 4 + 4 = 12.
Moving does not change size. Sliding or turning a part does not stretch it or shrink it, so it covers the same number of squares wherever it goes. Putting the parts down with no gaps and no overlaps means nothing is added and nothing is counted twice.
The half turn. A half turn about the center of a square or a rectangle carries the whole shape onto itself: each corner goes to the opposite corner. A straight cut through the center is carried onto itself as well, with its two ends swapped. So the piece on one side of the cut is carried exactly onto the piece on the other side. The two pieces are the same size, and together they make the whole, so they are halves.
All halves of one whole are the same size. Call one half P and another half of the same whole Q, cut any way. Two pieces the size of P make the whole, and two pieces the size of Q make the whole. If Q were bigger than P, two pieces the size of Q would be bigger than two the size of P, so the whole would be bigger than itself. If Q were smaller, the whole would be smaller than itself. Neither can be, so P and Q are the same size. The argument for thirds uses three pieces in place of two, and for fourths four.
Different wholes. The argument above needed both pairs of halves to make the same whole. If two halves of a bigger cracker make the bigger cracker and two halves of a smaller cracker make the smaller one, the halves of the bigger one must be bigger, or the bigger cracker would be no bigger.
Worked examples
Have grid paper, scissors and a few real crackers or paper squares on the table. Cutting and moving the pieces is the best proof a child can give.
Part 1: fourths four ways
Marama has four square crackers just alike, each scored into 4 rows of 4 small squares. She breaks each one into four pieces along the lines, a different way each time (Figure 3). Which crackers are broken into fourths?
Fourths are four pieces of the same size. Each cracker has 16 small squares, and four equal shares of 16 squares have 4 squares each, since
So count the squares in each piece.
A: four columns, each 4 squares tall, so each covers 4 squares. B: four 2 by 2 squares, each 2 + 2 = 4 squares. C: four hooks, each a row of 3 with 1 more square, 3 + 1 = 4. D: the top row is 4 squares, the second row is 4 squares, and each 2 by 2 square below is 4 squares.
Every piece of every cracker covers 4 squares, so all four crackers are broken into fourths. Look at D: its four shares are not even the same shape as each other, two long rows and two squares, and they are still fourths, because each holds 4 of the 16 squares.
Part 1: thirds of a long cracker
Ugonna’s cracker is 6 small squares across and 2 down. Figure 4 shows three ways to break it into three pieces along the lines. Are all three ways thirds?
The cracker has 6 + 6 = 12 squares. Three equal shares of 12 squares have 4 squares each, because
A: three 2 by 2 squares, 4 squares each. B: a top row of 4, a bottom row of 4, and a 2 by 2 square of 4 at the right end. C: an L at the left end, 3 on top and 1 below, so 3 + 1 = 4; the same L at the right end, 4 squares; and a row of 4 along the bottom middle, shaded.
Every piece covers 4 squares, so all three ways are thirds. In B and C the three shares are not all the same shape, and in C two of them are L shapes, neither a square nor a straight row.
Part 2: corner to corner on a bigger cracker
Selorm breaks a 4 by 4 cracker from one corner to the opposite corner (Figure 5). The cut runs through some of the small squares. Is the shaded triangle a half?
A half of 16 squares covers 8, so count the squares in the triangle. Some are whole and some are cut.
Whole squares: under the cut there are 3 in the bottom row, 2 in the row above it and 1 in the row above that, so 3 + 2 + 1 = 6 whole squares.
Cut squares: the cut runs corner to corner through 4 small squares, one in each row, and the triangle holds one half of each. Two half squares make one small square, so the 4 half squares make 2 small squares: 1 + 1 = 2.
So the triangle covers 6 + 2 = 8 squares, and the triangle is a half of the cracker.
Moving shows it too. Pair up the half squares: turn one half way round and slide it next to another, and the two together fill one small square. A simpler check is a half turn: turn the shaded triangle half way round the center of the cracker and it lands exactly on the other triangle, so the two are the same size.
Part 2: straightening a hook
Figure 1 C shows a half shaped like a hook. Show that it is a half by moving squares, without counting.
The plain half, the left two columns, is a half: a straight cut down the middle of a square makes two pieces that lie on each other when one is flipped over. So it is enough to turn the hook into the left two columns without adding or losing any squares.
The hook has the whole top row, the whole left column, and one more square in the bottom row. The left two columns hold 2 squares that the hook does not, the two just inside the hook in the second column. And the hook has 2 squares the left two columns do not, the right end of its top row. Cut off those two dotted squares and set them into the two empty places, as in Figure 6. The hook becomes exactly the left two columns.
Nothing was added and nothing was lost, only moved, so the hook holds as much as the plain half. It is a half.
Part 2: a slanted cut through the center
Marama cuts a paper square with no grid on it, using one straight slanted cut. The cut passes through the center of the square (Figure 7). Are the two pieces halves?
There are no small squares to count, so use a half turn. Put a pencil point on the center and turn the shaded piece half way round it.
Each corner of the square goes to the opposite corner, so the square lands on itself. The top end of the cut is as far from the top right corner as the bottom end is from the bottom left corner. The turn carries the top right corner to the bottom left corner, so it carries the top end of the cut to the bottom end. So the cut lands on itself, with its ends swapped, and the shaded piece lands exactly where the other piece was.
The two pieces lie exactly on each other, so they are the same size, and together they make the square. They are halves. The same turn works for any straight cut through the center of a square or a rectangle, and for the staircase in Figure 1 B. It does not work for the hook in Figure 1 C, whose two pieces are different shapes; there counting or moving squares is needed.
Part 3: two halves of one cracker
Marama and Ugonna have crackers just alike, 4 by 4. Marama breaks hers along the staircase of Figure 1 B; Ugonna breaks his corner to corner, as in Figure 5. Each eats one half. Did they eat the same amount?
SolutionCount: Marama’s staircase half covers 3 + 2 + 2 + 1 = 8 squares, and Ugonna’s triangle half covers 6 whole squares and 4 half squares, 6 + 2 = 8 squares. Both covered 8, so they ate the same amount.
The answer could be known before counting. Any half of a 16 square cracker covers 8 squares: the two halves are the same size and together make 16, and 8 + 8 = 16, while 7 + 7 = 14 and 9 + 9 = 18. The crackers are just alike, so they are the same whole, and every half of the same whole is the same size, whatever its shape.
Part 3: halves of different crackers
Ugonna’s cracker is 4 squares across and 4 down. Selorm’s is 4 across and 2 down. Each breaks his cracker straight down the middle and eats one half (Figure 8). Did they eat the same amount?
Ugonna’s cracker has 4 rows of 4, 16 squares, so his half covers 8, since 8 + 8 = 16. Selorm’s cracker has 2 rows of 4, 8 squares, so his half covers 4, since 4 + 4 = 8.
Both ate one half, but Ugonna ate 8 squares and Selorm ate 4, so Ugonna ate more. Both pieces are halves of their own crackers. The rule that halves are the same size holds for halves of one whole, or of wholes just alike. These are different wholes, so their halves are different sizes.
Boundaries
Counting needs squares that are all alike. On a cracker scored into squares of different sizes, two pieces with the same number of squares need not hold the same amount.
Half squares need a cut through the middle of the small square. A small square cut corner to corner gives two half squares. A straight cut that crosses a small square without passing through its middle gives two parts of different sizes, and counting by halves no longer works; moving or a half turn may still settle it.
The half turn needs the right whole. A square, a rectangle and a circle each land on themselves under a half turn about their center, so a straight cut through the center halves them. A triangle does not land on itself under a half turn, and a cut through its middle need not halve it.
Same number of pieces is not enough. Four pieces are fourths only when each holds a fourth of the whole (Exercise 6).
One whole. Shares are compared as the same size only when they come from the same whole, or from wholes that are the same size (Example 7).
Where this goes next
Seeing that a share’s size does not depend on its shape is the start of area, where counting the squares a shape covers becomes a way to measure any flat shape, and cutting and moving pieces becomes the tool for finding areas of shapes that are not rectangles. In the study of fractions as shaded parts of a shape it means the shaded part can be any shape at all, as long as it holds the right amount. Later, two shares of different shapes that each hold one half grow into unit fractions and into the idea that one half can be built from two fourths in many different-looking ways.
Exercises
Part 1: equal shares or not?
Use Figure 9. For each cracker, say whether it is cut into equal shares, and if it is, name the shares.
- Cracker A.
- Cracker B.
- Cracker C.
- Cracker D.
- Cracker E.
- Cracker F.
Part 1: cut your own
- On grid paper, draw a square 4 squares on a side. Cut it into halves along the lines, in a way that is not in Figure 1.
- A cracker is 4 squares across and 3 down. It is broken into fourths along the lines. How many squares are in each fourth?
- A cracker is 5 squares across and 2 down. How many squares are in each half? Show a way to break it into halves along the lines that is not a straight line.
Part 2: show the size
Use Figure 10.
- How many small squares does the shaded triangle in A cover? Is it one fourth of the square?
- How many small squares does the shaded slanted piece in B cover? Is it one half of the rectangle? Show it by moving a part as well.
- In Figure 2, the unshaded piece is the other half of the staircase cracker. Which one square can be moved to turn it into the right two columns?
- A cracker 4 squares across and 2 down is cut from one corner to the opposite corner. Without counting, explain why the two triangles are halves.
Part 3: one whole, or two?
- Marama and Ugonna have crackers just alike, 4 squares across and 3 down. Marama breaks hers into three rows. Ugonna breaks his into an L at each side and a 2 by 2 square in the middle of the bottom two rows. Each eats one third. How many squares does each eat? Did they eat the same amount?
- One cracker is 6 squares across and 2 down, and another is 3 across and 2 down. Each is broken into thirds. How many squares are in a third of each?
- Two paper squares are just alike, 4 by 4. Selorm cuts one into halves corner to corner and takes one half. Marama cuts the other into fourths and takes two of them. How many squares does each take? Did they take the same amount?
- A piece covers 6 small squares. Could it be a half of a cracker of 16 squares? Could it be a half of a cracker of 12 squares?
Draw it
- On a 4 by 4 grid, draw fourths that are not all the same shape, different from Figure 3 D.
- On a grid 6 squares across and 2 down, draw thirds that are not all rectangles, different from Figure 4 C.
Explain
- A cracker is 6 squares across and 3 down. Explain why every half of it covers 9 small squares, however it is cut.
- Two pieces of paper look different. Explain how you could show they are the same size without a grid.
- Are two halves always the same size? Explain.
- Two pieces cover the same number of small squares on the same grid. Must they be the same shape? Explain with an example.
Answers
- Equal shares: halves. Each piece covers 4 squares, 3 + 1 = 4, and 4 + 4 = 8.
- Not equal shares. The top piece covers 9 squares, 4 + 4 + 1 = 9, and the bottom piece 7, so the two pieces are not halves.
- Not equal shares. The three pieces cover 3, 4 and 5 squares, so they are not thirds.
- Equal shares: thirds. Each piece covers 4 squares, and 4 + 4 + 4 = 12.
- Equal shares: fourths. The four T shapes each cover 4 squares, and 4 + 4 + 4 + 4 = 16.
- Not equal shares. The pieces cover 4, 4, 3 and 5 squares, so they are not fourths.
- Any cut along the lines that leaves 8 squares on each side is right. For example, the top two rows as one piece and the bottom two rows as the other, or a staircase going the other way.
- 3 squares. The cracker has 4 + 4 + 4 = 12 squares, and 3 + 3 + 3 + 3 = 12.
- 5 squares, since 5 + 5 = 10. For example, one piece is the first 4 squares of the top row and the first square of the bottom row, 4 + 1 = 5, and the other piece is the rest.
- 4 squares: 2 whole squares and 4 half squares, and the 4 half squares make 2 squares, so 2 + 2 = 4. It is one fourth, since 4 + 4 + 4 + 4 = 16.
- 4 squares: 2 whole squares and 4 half squares, 2 + 2 = 4. It is one half, since 4 + 4 = 8. To show it by moving, cut the slanted piece straight down through its middle and slide the left triangle to the right end: the piece becomes a 2 by 2 square, the right half of the rectangle.
- The square at the bottom row, second column. Move it up to the empty place in the top row, third column, and the piece becomes the right two columns, 8 squares.
- Turn one triangle half way round the center of the rectangle. The rectangle lands on itself and the cut lands on itself, so the triangle lands exactly on the other one. The two are the same size and make the whole, so they are halves.
- 4 squares each. The cracker has 4 + 4 + 4 = 12 squares, Marama’s row is 4, and each of Ugonna’s pieces is 4: an L of 2 + 1 + 1 = 4 or a 2 by 2 square of 2 + 2 = 4. They ate the same amount: thirds of crackers just alike are the same size, whatever their shape.
- 4 squares and 2 squares, since 4 + 4 + 4 = 12 and 2 + 2 + 2 = 6.
- Selorm takes 8 squares and Marama takes 4 + 4 = 8 squares, so they took the same amount.
- Not of 16 squares, since a half of 16 squares covers 8. It could be a half of 12 squares, since 6 + 6 = 12.
- For example, the top row as one share, the bottom row as another, and the middle two rows split into a 2 by 2 square on the left and a 2 by 2 square on the right. Each share covers 4 squares. Any drawing whose four shares each cover 4 squares is right.
- For example, rows AABBBC on top and AABCCC below: a 2 by 2 square of 4, an L of 3 + 1 = 4, and another L of 1 + 3 = 4. Any drawing whose three shares each cover 4 squares is right.
- The cracker has 6 + 6 + 6 = 18 squares. The two halves are the same size and together cover all 18, so a half added to itself makes 18. Only 9 does that, since 9 + 9 = 18, while 8 + 8 = 16 and 10 + 10 = 20.
- Cut one piece into parts and slide or turn the parts, with no gaps and no overlaps, until they lie exactly on the other piece. Or, if both are halves of the same square cut through its center, turn one half way round the center and it lands on the other.
- Only halves of the same whole, or of wholes the same size. Then every half is the same size, whatever its shape. A half of a big cracker is bigger than a half of a small cracker.
- No. In Figure 1, the staircase piece and the hook piece each cover 8 squares, but they are different shapes.