Grade 1 · 1.G.A.3 · 2.G.A.3

Equal shares and naming the whole

Two children share one sandwich, and one cut makes two pieces. When is each piece really a half?

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A fraction of a setGrade 3 Area and length models of a fractionGrade 3 Equal shares that look differentGrade 2 Markup and marginMath 7 +1 more

Where you are 4 in A fraction as part of a whole

Fractions, ratio and percent3

Fractions15

A fraction as part of a whole4

A fraction of a set Area and length models of a fraction

Equal shares and naming the whole

Equal shares that look different

Why this exists

Two children share one sandwich. The grown-up makes one cut and hands each child a piece. If the cut is down the middle, nobody minds. If the cut is near one end, one child gets far more, and any child will say that is not fair.

The words half, third and fourth carry exactly that idea of fair. In everyday talk “half” can mean any piece. In mathematics it cannot: a half is one of two pieces that are the same size. Without that rule, “I ate a half” would not tell you whether it was a crumb or nearly all.

Even with that rule, “I ate a half” leaves one question open: a half of what? Half of a cupcake and half of a birthday cake are both halves, and nobody would call them the same amount. A fraction always belongs to a whole, and saying which whole is the second idea of this lesson, naming the whole.

Every fraction a child meets later is counted in equal shares of a named whole, so these two ideas are the first ideas of the whole fractions story.

Definition

Equal shares

A whole, such as a sandwich, a pizza or a strip of paper, is cut into equal shares when

every piece is the same size, and
the pieces make up the whole exactly: no gaps, no overlaps, nothing left over.

When a whole is cut into equal shares:

with 2 equal shares, each share is one half of the whole, and the shares are halves;
with 3 equal shares, each share is one third, and the shares are thirds;
with 4 equal shares, each share is one fourth, also called one quarter, and the shares are fourths or quarters.

Figure 1Both sandwiches are cut into two pieces. Only A is cut into halves.

In Figure 1 both sandwiches are cut into two pieces, but only A is cut into halves. The two pieces of B are not the same size, so neither of them is a half, even though there are two of them.

Same size means the pieces hold the same amount. For a flat shape that is the same amount of surface, its area: if one piece could be laid on the other, cut up and moved if need be, it would cover it exactly. It does not mean the same shape (Example 3). For a piece of string, or a strip cut straight across, the same size means the same length.

When the child starts writing fractions, one half is written \displaystyle \frac{1}{2}, one third \displaystyle \frac{1}{3} and one fourth \displaystyle \frac{1}{4}. The bottom number says how many equal shares the whole is cut into. The top number says we mean one of them. That way of writing is the subject of Unit fractions.

Naming the whole

Every fraction is a fraction of something, and that something is its whole: one sandwich, one pizza, one strip of paper, one bar of 12 squares. Naming the whole means saying which whole a fraction is of.

The name of a fraction says how the whole was shared and how many shares are meant. “One half” says: the whole was cut into 2 equal shares, and we mean one of them. It does not say how big the whole was, so on its own it does not say how much that is: one half of a small pizza and one half of a large pizza are both halves, but not the same amount of pizza (Figure 3). The fraction and its whole together say how much.

Two wholes count as the same for this purpose when they are the same size: half of one sandwich and half of another sandwich just like it are the same amount.

Intuition

Equal shares

Fold a strip of paper so its two ends meet, and crease it. Open it out. The crease cuts it into two halves, and the fold proved they are the same size: one lay exactly on top of the other. Fold it again, end to end, and open it: now there are four fourths. Folding is the best first tool, because it shows “the same size” instead of asking the child to judge it by eye.

Figure 2The same bar cut into more equal shares gives smaller shares.

Figure 2 shows one bar three times, cut into halves, thirds and fourths. The whole did not change, but it was shared among more pieces, so each piece got less: four children sharing one pizza each get less than two children sharing it.

The name of a share also says how many of them make the whole. Children can point as they count: “one fourth, two fourths, three fourths, four fourths, that’s the whole pizza”.

Naming the whole

Whenever someone says “a half”, ask “a half of what?”. The answer is the whole.

Figure 3One half of a small pizza and one half of a large pizza. Both are halves, but they are not the same amount.

In Figure 3 each pizza is cut through its center into two equal shares, so each shaded piece is one half of its own pizza. The large pizza’s half is bigger, because its whole is bigger: two of its halves make the large pizza, and two of the small halves make only the small pizza.

Folding shows it too. Fold a short strip and a long strip each in half and lay the two halves side by side. Both creases made halves, but the long strip’s half is longer. The word “half” was the same both times; the whole was not.

Results

Equal shares

Equal shares make up the whole. Two halves make one whole, three thirds make one whole and four fourths make one whole.

More equal shares of the same whole means smaller shares. A half of a sandwich is bigger than a third of the same sandwich, and a third is bigger than a fourth.

Equal shares of the same whole need not be the same shape. Two halves of one square may be a rectangle and a triangle (Example 3). What must match is the size.

A piece’s name depends only on its size. A piece is one third of a whole exactly when three pieces of that size would make up the whole. So a whole cut into three pieces of different sizes is not cut into thirds. And one piece can be a fourth of the whole even when the pieces are not all fourths (Example 6).

Naming the whole

How much a fraction is depends on its whole. The same fraction of a bigger whole is a bigger amount. The same fraction of two wholes of the same size is the same amount.

A piece’s name depends on the whole it is measured against. A piece is one half of a whole two pieces long and one fourth of a whole four pieces long (Example 8).

One share gives back the whole. If a piece is one third of a whole, three pieces of that size make the whole; if it is one fourth, four pieces of that size make the whole (Example 9).

Fractions are compared by their names only within one whole. “One half is more than one fourth” is true for the halves and fourths of one whole, or of wholes of the same size. Of different wholes, one fourth can be more than one half (Example 10).

Why it is true

Equal shares

Equal shares make up the whole. That is part of what “equal shares” means: the cut used up the whole, with nothing left over and nothing counted twice.

More shares means smaller shares. Suppose a fourth of a sandwich were as big as a half of it. Then two fourths would be as big as two halves, which is the whole sandwich. The other two fourths would have nothing left to be made of. So a fourth must be smaller than a half. The same argument works for any two numbers of shares: if the shares with the bigger count were as big as the others, they would add up to more than the whole.

Same size, different shape. Example 3 marks a square into 4 small squares and counts: the rectangle half and the triangle half each cover 2 of them.

A piece’s name depends on its size. “One third” is a promise about size: three pieces like this one make the whole. If three pieces of different sizes make the whole, then three copies of the smallest would be too small and three copies of the biggest would be too big. So the whole is not cut into thirds. (One piece might still happen to be exactly a third in size, the way a piece in Example 6 is a fourth, but the pieces are not thirds.)

Naming the whole

How much a fraction is depends on its whole. Take a large pizza and a small one. The two halves of the large pizza make the large pizza, and the two halves of the small pizza make the small pizza. Suppose a half of the large pizza were no bigger than a half of the small one. Then its two halves together would be no bigger than the small pizza’s two halves, so the large pizza would be no bigger than the small pizza, which is false. So the large pizza’s half is the bigger one. The same argument, with three thirds or four fourths in place of two halves, works for thirds and fourths. If the two pizzas are the same size, cutting both down the middle the same way gives halves that lie exactly on top of each other, so they are the same amount.

A piece’s name depends on the whole. The name of a piece says how many pieces of its size make the whole. Two pieces the size of the shaded piece in Figure 9 make bar A, so it is one half of bar A. Four pieces that size make bar B, so it is one fourth of bar B. Nothing about the piece changed; only the whole it was measured against.

One share gives back the whole. If a piece is one third of a whole, the whole is cut, or could be cut, into 3 equal shares and the piece is one of them. The other two shares are the same size as the piece. So three pieces of that size, put together with no gaps and no overlaps, make exactly the whole.

Comparing needs one whole. The argument above that a fourth is smaller than a half cut each half of one sandwich in two. It used halves and fourths of the same sandwich, and it says nothing about a fourth of a different, bigger sandwich. Example 10 shows that a fourth of a big whole can be more than a half of a small one.

Worked examples

Have paper, scissors and a few real shapes on the table. Folding and laying one piece on another answers most of these questions.

Example 1

Which are cut into thirds?

Look at Figure 4. Each shape is cut into three pieces. Which shapes are cut into thirds?

Figure 4Each shape is cut into three pieces. Only A and C are cut into thirds.

Solution

Thirds means three pieces of the same size, so for each shape ask: are the three pieces the same size?

A: yes. The three pieces are the same width and the same height, so each would lie exactly on top of another. A is cut into thirds.

B: no. There are three pieces, but they are different widths, so they are not the same size. B is not cut into thirds.

C: yes. The three cuts start at the center and are evenly spaced round it, so the three corners at the center are the same and each slice is the same wedge, turned around. Cut them out and they stack on top of each other exactly. C is cut into thirds.

D: no, and this one is worth slowing down for. The two cuts split the width of the circle into three equal parts, but a circle is fatter in the middle than at its sides, so the middle piece holds more than either side piece: measured, it is almost one and a half times as big. Evenly spaced cuts do not always give equal shares. What counts is the size of each piece, not where the cuts are.

So only A and C are cut into thirds.

Example 2

Cutting a pie into fourths

A pie is to be shared equally by four people. Figure 5 shows two ways to cut it. Which one cuts it into fourths?

Figure 5Pie A is cut into fourths. Pie B is cut into four pieces that are not fourths.

Solution

Pie A is cut by two straight lines through the center, one up and down and one across, so they cross square to each other. That makes four slices, each with a square corner at the center, and each is the same wedge turned around. So the four slices are equal shares, and each is one fourth of the pie. The shaded slice is one fourth. Counting round the pie, “one fourth, two fourths, three fourths, four fourths”, reaches the whole pie, so 4 fourths make the whole.

Pie B is also cut by one up-and-down line and one across line, but they cross away from the center. The four pieces are different sizes: the bottom-left piece is nearly half the pie, about six times the top-right piece. Four pieces, but not fourths, so B is not a fair way to share.

There is also a reason B could never work. If its four pieces were fourths, the up-and-down cut would have two of them on each side, so it would cut the pie into halves. But a straight cut that misses the center never halves a circle: the side that holds the center gets a whole half of the pie (the half cut off by a line through the center, alongside the cut) and a strip more.

Example 3

Two halves that look different

A square can be cut in half down the middle, or from corner to corner. Figure 6 shows both, with one half of each shaded. Are the two shaded pieces the same size?

Figure 6Two halves of the same square, a rectangle and a triangle. Each covers 2 of the 4 small squares.

Solution

Each is one half of the same square, so they must be the same size. But they look so different that it is fair to check. The dashed lines split each square into 4 small squares.

The top shaded piece, a tall rectangle, covers 2 small squares.

The bottom shaded piece, a triangle, covers 1 whole small square (bottom left) and half of 2 more (top left and bottom right). Those two halves of a small square together make 1 more small square. So the triangle covers 1 and 1 more, 2 small squares. Both halves are 2 of the 4 small squares. They are the same size and different shapes.

A child can also check the triangle with scissors: cut off the corner that sticks out past the middle line, turn it round, and it fits exactly into the gap, so the triangle becomes the rectangle. That is the idea of area staying the same when the pieces move.

Example 4

Sharing a bar of squares

A chocolate bar has 12 small squares, all the same size, 6 across and 2 down. Share it equally among 4 children, breaking only along the lines. How many squares does each child get, and what is each share called?

Figure 712 squares in 4 equal shares. Each share is 3 squares.

Solution

Four children sharing equally means four equal shares, so each share is one fourth of the bar. One way to break it is shown in Figure 7: one break down the middle and one across, making four pieces, each a row of 3 squares.

Because every small square is the same size, pieces with the same number of squares are the same size. Each piece has 3 squares, and

3 + 3 + 3 + 3 = 12

so the four pieces use up the whole bar. Each child gets 3 small squares, which is one fourth of the bar.

Sharing 12 things equally into 4 groups of 3 is also a division: 12 \div 4 = 3.

Example 5

A half or a fourth?

Two children share one pancake equally. Four children share another pancake of the same size equally. Who gets more pancake: a child in the pair, or a child in the group of four?

Solution

Each child in the pair gets one half of a pancake. Each child in the group of four gets one fourth of a pancake of the same size. Figure 2 shows the same whole cut both ways: the half is bigger.

Here is why it must be. Take the pancake that was cut into halves and cut each half in half again. Now there are four pieces, all the same size, and together they make the whole pancake, so each of them is a fourth. Each half is made of two of those fourths, so a child in the pair gets more, twice as much as a child in the group of four. (This needs the two pancakes to be the same size.)

Example 6

A fourth among pieces that are not all fourths

A rectangle is cut down the middle, and then only the right half is cut across the middle, as in Figure 8. There are three pieces. Are they thirds? What is the shaded piece?

Figure 8One half and two fourths. The dashed line shows the left half would make two more fourths.

Solution

The three pieces are not the same size: the left piece is as big as the two small ones together. So they are not thirds, even though there are three of them.

Now look at one small piece. The cut down the middle made two halves, so the right half is one half of the rectangle. The across cut split that half into two small pieces of the same size. If the same across cut were carried on through the left half, along the dashed line, it would make two more small pieces exactly like them: slide one across and it covers a small piece on the right exactly. So four small pieces of this size make the whole rectangle, with no gaps and no overlaps, and each small piece is one fourth.

The rectangle is cut into one half and two fourths: three pieces, but not thirds. The shaded piece is one fourth of the rectangle. Its name comes from its size, not from how many pieces are on the table.

Example 7

Half of a big bar, half of a small bar

Ava’s chocolate bar has 8 small squares and Ben’s has 4, all the squares the same size. Each of them eats one half of their own bar. How many squares does each eat? Did they eat the same amount?

Solution

One half of Ava’s bar is one of 2 equal shares of 8 squares. Two equal shares that use up 8 squares have 4 squares each, since

4 + 4 = 8

so Ava eats 4 squares. One half of Ben’s bar is one of 2 equal shares of 4 squares, and 2 + 2 = 4, so Ben eats 2 squares.

Both ate one half, but not the same amount: Ava ate 4 squares and Ben ate 2, so Ava ate twice as much. The fraction was the same; the wholes, 8 squares and 4 squares, were not. Each “one half” is a half of its own bar, and a half of the bigger bar is more.

Example 8

One piece, two names

Figure 9 shows two bars, A and B, lined up at their left ends. The shaded piece of A and the shaded piece of B are the same size. What fraction of bar A is the shaded piece? What fraction of bar B?

Figure 9The shaded pieces are the same size. The piece is one half of bar A and one fourth of bar B.

Solution

Bar A is cut into 2 pieces, each the same size as the shaded piece, and together they make the bar with no gaps or overlaps. So bar A is cut into 2 equal shares, and the shaded piece is one half of bar A.

Bar B is cut into 4 pieces, each the same size as the shaded piece, and together they make bar B. So bar B is cut into 4 equal shares, and its shaded piece is one fourth of bar B.

The two shaded pieces are the same size, yet one is a half and the other a fourth. The name of a piece says how many pieces of its size make the whole: 2 make bar A and 4 make bar B, because bar B is twice as long. So the question “what fraction is this piece?” has no answer until the whole is named.

Example 9

Building the whole from one share

Mo has a piece of ribbon that is one third of the ribbon it was cut from. Show the whole ribbon. If Mo’s piece is 4 centimeters long, how long was the whole ribbon?

Solution

One third means the whole ribbon was cut into 3 equal shares, and Mo’s piece is one of them. The other two shares are the same length as Mo’s piece. So put 3 pieces that long end to end, with no gaps and no overlaps, as in Figure 10. That is the whole ribbon.

Figure 10One third of a ribbon, and the whole ribbon, three pieces that size end to end.

If Mo’s piece is 4 centimeters long, the whole ribbon is three pieces of 4 centimeters:

4 + 4 + 4 = 12

so the whole ribbon was 12 centimeters long.

Check: a ribbon 12 centimeters long cut into 3 equal shares gives pieces of 4 centimeters, since 12 \div 3 = 4, so a 4 centimeter piece is one third of it.

For a ribbon there is only one answer, a ribbon three times as long. For a flat piece, the three copies could be put together in more than one shape (in a row, or in an L), but every such whole has the same size, three times the piece. Working backwards from a part to its whole with any fraction is the subject of Finding the whole when only a fraction of it is known.

Example 10

Who ate more?

Kai’s bar has 12 small squares and Zoe’s has 4, all the squares the same size. Kai eats one fourth of his bar, and Zoe eats one half of hers. Zoe says, “I ate more, because one half is bigger than one fourth.” Is she right?

Figure 11One fourth of Kai’s bar is 3 squares. One half of Zoe’s bar is 2 squares.

Solution

Find how much each ate, in squares. One fourth of Kai’s bar is one of 4 equal shares of 12 squares. Four equal shares that use up 12 squares have 3 squares each:

3 + 3 + 3 + 3 = 12

so Kai ate 3 squares. One half of Zoe’s bar is one of 2 equal shares of 4 squares, and 2 + 2 = 4, so Zoe ate 2 squares (Figure 11).

Kai ate 3 squares and Zoe ate 2, so Kai ate more, and Zoe is wrong.

Her rule, “one half is bigger than one fourth”, is true for one whole. If Kai had eaten one fourth of Zoe’s bar, he would have had 1 square, less than her 2. But here the wholes are different: Kai’s bar is 3 times as big as Zoe’s, so even a smaller fraction of it is more. Two fractions can be compared by their names alone only when they are fractions of the same whole, or of wholes of the same size.

Boundaries

Equal means the same size, not the same shape. Pieces that match exactly, one laid on the other, are the same size, but so can pieces of different shapes be (Example 3).

The shares must use up the whole. If a piece is left over, or two pieces overlap, the pieces are not shares of that whole, even if they match each other.

A fraction’s name belongs to one whole. Half of a small pizza and half of a large pizza are both halves, but they are not the same amount (Figure 3). “A half is bigger than a fourth” is true only for halves and fourths of the same whole, or of wholes the same size (Example 10).

Putting shares together needs one whole too. One half of a sandwich and one half of another sandwich are as much as one whole sandwich only when the two sandwiches are the same size. Halves of a big sandwich and a small one together are more than the small sandwich and less than the big one.

Evenly spaced cuts are not always equal shares. On a rectangle, cuts spaced evenly across it, parallel to its sides, do make equal shares. On a circle, one cut through the center makes halves, but two or more parallel cuts, evenly spaced, do not make equal shares (Example 1, shape D): the middle pieces are fatter. Cuts crossing at a point that is not the center do not make fourths either (Example 2, pie B). The test is always the size of the pieces.

Groups of things. A fraction can also be of a group, such as half of 8 apples. Then equal shares means the same number of things in each share, which is a fraction of a set, and the group is the whole that has to be named: half of 8 apples and half of 4 apples are different amounts. When the whole is a length, equal shares are equal lengths, which is a fraction of a length.

Where this goes next

Equal shares are what the rest of fractions stand on. Shading some of the equal shares of a shape is the area model of a fraction, and one share on its own is a unit fraction, written \displaystyle \frac{1}{3} or \displaystyle \frac{1}{4}. Several of them together make fractions such as \displaystyle \frac{2}{3} and \displaystyle \frac{3}{4}. The fact that a half is two fourths grows into equivalent fractions, and “more shares means smaller shares” becomes comparing fractions with the same top number. Cutting a length into equal shares puts fractions on a line, in fractions on the number line, and “four fourths make one whole” is the start of whole numbers as fractions. Sharing equally among children is also the first meaning of division.

Naming the whole travels with all of it. Every comparison of fractions, and every sum such as adding fractions with the same bottom number, is about one whole, and both stop making sense when the wholes differ. Working back from a part to its whole grows into finding the whole when only a fraction of it is known. Much later, percents ask the same question: a markup is a percent of what the shop paid, and the same number of dollars is a different percent of the price it sells for, because the whole is different.

Exercises

Equal shares or not?

Use Figure 12. For each shape, say whether it is cut into equal shares. If it is, name the shares.

Figure 12Shapes for Exercises 1 to 6.

Shape A.
Shape B.
Shape C.
Shape D.
Shape E.
Shape F.

Name the shares

A pancake is cut into 3 equal shares. What is each share called? How many of them make the whole pancake?
How many fourths make one whole?
How many halves make one whole?
Four oranges are each cut into halves. How many halves are there?

Sharing a bar of squares

Each bar is made of small squares, all the same size, and is broken only along the lines.

A bar of 8 squares is shared equally among 4 children. How many squares does each child get, and what is each share called?
A bar of 12 squares is broken into thirds. How many squares are in one third?
A bar of 10 squares is broken into halves. How many squares are in one half?

Bigger or smaller

Which is bigger, one half of a pizza or one third of the same pizza?
Two ribbons are the same length. One is cut into 2 equal pieces and the other into 4 equal pieces. Which ribbon has the longer pieces?

Half of what?

In these, all the small squares are the same size.

One bar has 10 squares and another has 6. How many squares are in one half of each bar? Which half is more?
Eli eats one fourth of a bar of 8 squares. Ruby eats one fourth of a bar of 12 squares. How many squares does each eat?

One piece, different wholes

Use Figure 13. The shaded piece is the same 2 squares in every bar.

Figure 13Bars for Exercises 18 to 20. The shaded piece is the same in all three.

What fraction of bar A is the shaded piece?
What fraction of bar B is the shaded piece?
What fraction of bar C is the shaded piece?

Finding the whole

A piece of 3 squares is one half of a bar. How many squares are in the whole bar?
A piece of 2 squares is one third of a bar. How many squares are in the whole bar?
A piece of ribbon 5 centimeters long is one fourth of the whole ribbon. How long is the whole ribbon?

Who has more?

Kim eats one half of a bar of 6 squares. Raj eats one third of a bar of 12 squares. Who eats more?
Ana eats one third of a bar of 6 squares. Tom eats one half of a bar of 4 squares. Who eats more?

Draw it

Draw two squares the same size. Cut one into halves and the other into halves a different way.
Draw a circle and cut it into fourths.
Draw a short strip and say it is one third of a whole strip. Draw the whole strip.

Explain

A cake is cut into 3 pieces. Explain why that alone does not tell you whether each piece is a third.
A square is cut from corner to corner. Explain how you know the two triangles are halves.
Explain why a fourth of a sandwich is smaller than a half of the same sandwich.
Can one half of one pizza be bigger than one half of another pizza? Explain.
Nia says, “One half is always more than one fourth.” When is she right, and when is she wrong? Explain.
Sam eats one half of his sandwich and Jo eats one half of hers. Between them, did they eat as much as one whole sandwich? Explain.

Answers

Equal shares: fourths. The two corner-to-corner cuts make 4 triangles that are the same, each one the next turned a quarter of the way round, so each is one fourth of the square.
Not equal shares. The cut misses the center, so the piece on the right is much smaller than the piece on the left. The two pieces are not halves.
Equal shares: fourths. The 4 pieces have the same width and the same height.
Not equal shares. The 3 pieces are different widths, so they are not thirds. (The widest piece is as wide as the other two together, so it is in fact one half of the rectangle.)
Equal shares: halves. The two slanting sides of the triangle are the same length, so folding along the cut lays one piece exactly on the other.
Not equal shares. There are 4 slices, but two are narrow and two are wide, so they are not fourths.
Each share is one third. 3 thirds make the whole pancake.
4 fourths make one whole.
2 halves make one whole.
8 halves. Each orange gives 2 halves, and 2 + 2 + 2 + 2 = 8.
2 squares each, and each share is one fourth of the bar: 2 + 2 + 2 + 2 = 8.
4 squares. Three equal shares that use up 12 squares: 4 + 4 + 4 = 12.
5 squares. Two equal shares that use up 10 squares: 5 + 5 = 10.
One half. The same pizza shared among fewer pieces gives bigger pieces.
The ribbon cut into 2 pieces. Each of its pieces is a half, and each piece of the other is a fourth; a half of a ribbon is two fourths of the same length.
5 squares and 3 squares, since 5 + 5 = 10 and 3 + 3 = 6. The half of the 10-square bar is more: both are halves, but of different wholes.
Eli eats 2 squares and Ruby eats 3, since 2 + 2 + 2 + 2 = 8 and 3 + 3 + 3 + 3 = 12. Both ate one fourth, and Ruby ate more because her bar was bigger.
One half. Bar A has 4 squares, and 2 pieces like the shaded one make it: 2 + 2 = 4.
One third. Bar B has 6 squares, and 3 pieces like the shaded one make it: 2 + 2 + 2 = 6.
One fourth. Bar C has 8 squares, and 4 pieces like the shaded one make it: 2 + 2 + 2 + 2 = 8.
6 squares. One half means 2 equal shares make the bar: 3 + 3 = 6.
6 squares. One third means 3 equal shares make the bar: 2 + 2 + 2 = 6.
20 centimeters. One fourth means 4 equal shares make the ribbon: 5 + 5 + 5 + 5 = 20.
Raj. Kim eats 3 squares, since 3 + 3 = 6, and Raj eats 4 squares, since 4 + 4 + 4 = 12. One half is more than one third of the same bar, but Raj’s bar is bigger.
They eat the same amount, 2 squares each: 2 + 2 + 2 = 6 and 2 + 2 = 4.
For example, cut one square down the middle and the other from corner to corner. Any cut that makes two pieces of the same size is right; any straight cut through the center of the square does.
Cut the circle through its center, then cut through the center again, square to the first cut. The 4 slices are the same wedge turned around, so they are fourths.
Draw three strips the size of the short one, end to end with no gaps and no overlaps. One third means 3 equal shares make the whole, and each share is the size of the short strip.
Thirds means three pieces of the same size. Three pieces can be of different sizes, like a big slice and two thin ones, and then they are not thirds. You have to know the pieces are the same size.
Fold the square along the cut: one triangle lies exactly on the other, so they are the same size. Two pieces of the same size that make the whole square are halves.
Cut each half of the sandwich in half again. The four pieces are the same size and make the whole sandwich, so each is a fourth, and each half is two of them. A fourth is only part of a half, so it is smaller.
Yes, if the pizzas are different sizes. A half of a big pizza is more than a half of a small one. A half is always one of two equal shares of its own whole, so its size depends on that whole.
She is right when the half and the fourth are of the same whole, or of wholes of the same size: each half is two fourths, so it is more than one fourth. She is wrong when the wholes are different. For example, one fourth of a bar of 12 squares is 3 squares, more than one half of a bar of 4 squares, which is 2 squares.
Only if the two sandwiches are the same size. Then Jo’s half is the same amount as the other half of Sam’s sandwich, and two halves make one whole sandwich. If Sam’s sandwich is bigger, they ate more than one of Jo’s sandwiches and less than one of Sam’s. Each half is a half of its own whole.