Halves and Quarters

Start with one whole paper square. The whole is the complete thing. Tear it into pieces, and each piece is a part. Put every piece back in place, and you have the whole again.

A whole can also be a group. A box of 8 crayons is one whole group. You can share that group without breaking a crayon.

Check the size of paper pieces

Equal parts are exactly the same size. Fold a paper square down the middle. Match its corners and edges. The two sides cover each other, with nothing sticking out. They are equal parts.

Now think of a cookie broken near one edge. It has one big piece and one small piece. These are unequal parts. Counting two pieces tells you how many pieces there are. It does not tell you whether their sizes match.

The same check matters with four pieces. A pizza with one huge slice and three tiny slices has four pieces. Those pieces are unequal.

Count the things in each share

When you share a group, check the number in each share. Imagine sharing 6 grapes between you and a friend.

• You get 3 grapes. Your friend gets 3 grapes. The shares are equal.
• You get 4 grapes. Your friend gets 2 grapes. The shares are unequal.

A fair share is an equal part. You can put the grapes in two rows. Point to one grape in each row, making pairs. With 3 in each row, every grape has a partner. With 4 and 2, the longer row has grapes left without partners.

Colors do not change the count. One share might have 2 red cars and 1 blue car. Another might have 3 blue cars. Each share has 3 cars, so the groups have equal numbers.

A line needs checking too

A line across paper does not always make equal parts. Draw one across a strip close to an end. It makes a short part and a long part. Fold the short part over the long one. Some of the long part sticks out.

Move the line to the middle. Bring the strip's ends together to check it. Now the two parts match.

You can make equal parts with paper or with a group of things. With paper, line up the edges. With things, take turns giving them out.

Fold a paper shape

Take a paper rectangle.

1. Bring its left edge over to its right edge.
2. Match the corners, then press the fold flat.
3. Open the paper. The fold marks two equal parts.

You can also bring the top edge to the bottom edge. That makes two equal parts a different way. A line from one corner to the opposite corner works too. It makes two matching triangles.

For a paper circle, a straight fold through the center, right in the middle, gives two equal parts. A straight line off to one side gives a bigger part and a smaller part.

To make four equal parts of a square, fold it in half. Keep it folded, and fold it in half again. Match the edges each time. Open it fully and look at the four parts.

Give everyone a turn

Put 10 marbles in the middle and share between two places.

1. Put one marble in the first place.
2. Put one marble in the second place.
3. Keep taking turns until the middle is empty.
4. Count both shares. Each has 5 marbles.

All the marbles are used, and the numbers match. You can check the whole by adding:

For four children, make four places. Share 8 strawberries, one to each child in turn. After one round, each child has 1. After the next round, each has 2. Check all four shares:

Notice anything left over

Try sharing 7 toy cars between two places. Each place gets 3 cars, and 1 stays in the middle. Giving that car to one place would make the shares unequal.

Food that can be split gives another choice. With 7 cookies, share 3 whole cookies each. Split the last cookie into two equal pieces. Each person gets 3 cookies and half a cookie.

A half is one of two equal parts of a whole. You can say "one half" or "a half." Together, the two parts are called halves.

Put both halves of a sandwich together. They fill the whole sandwich, with no piece missing.

Different shapes can be halves

Imagine three paper squares that are all the same size.

• Fold the first across the middle. It makes two rectangles.
• Fold the second down the middle. It makes two rectangles another way.
• Fold the third from one corner to the opposite corner. It makes two triangles.

Each square has two matching parts. Each part is half of its square. A half does not have to be a rectangle or a triangle. What matters is how much of the whole it takes up.

A paper circle makes halves too. Fold it so the round edges match. The fold goes through the center. Open it, and each half looks like half of a round pizza.

Color one of the two equal parts of your square or circle. You have colored half. Color the other part too, and the whole is colored.

Find the middle of a length

Take a paper strip and bring its two ends together. Keep the strip flat, with its edges lined up. Press the fold and open it. The fold marks the middle of the length.

The part from either end to the fold is half as long as the whole strip. You can check by folding again: the two lengths lie on top of each other.

Think of a train made from 10 equal cubes in a straight line. Split it into two matching trains. Each has 5 cubes, so each is half as long as the first train.

Halve means make halves

To halve something is to split it into two equal parts. You halve a square by making two equal pieces. You halve a strip by making two equal lengths.

Two pieces alone are not enough. A sandwich with one large piece and one small piece has unequal pieces. In math, there is no "bigger half." Check that there are two parts and that they are equal.

Half of a group is one of two equal groups. To find it, share everything between two places. Then count the things in one place.

Share 12 stickers

Sam has 12 stickers. He gives half to his sister.

1. Share the stickers between two places, one at a time.
2. Count the stickers in each place. Both have 6.
3. Give one share to his sister. She gets 6 stickers.

Sam keeps the other 6. Check:

There are two groups, but the answer is 6 stickers. The question asks how many stickers are in one half.

Use a double you know

Halving and doubling undo each other. Double 4 is 8, so half of 8 is 4. Double 6 is 12, so half of 12 is 6.

Here are the halves of even numbers up to 20:

You can check any row by adding the half to itself. For example, 7 and 7 make 14.

Work backward from a half

Half of a box of eggs is 6 eggs. How many eggs fill the box?

The other half has 6 eggs too. Put both halves together:

The whole box holds 12 eggs. Here you double the half to find the whole. Halving 6 again would find a smaller part.

A new layout keeps the same half

Spread 10 counters in a row. Half is 5 counters. Move the same counters into a circle. There are still 10, and half is still 5. Moving things around does not add or take any away.

Odd groups leave something over when you share whole things. With 7 marbles, two shares of 3 leave 1 marble. With 7 apples, the last apple can be split into equal halves. Each share is then 3 apples and half an apple.

A quarter is one of four equal parts of a whole. A fourth means the same thing. Four quarters fit together to make the whole again.

Fold a quarter

Start with a paper square.

1. Fold it in half, matching the edges.
2. Keep it folded. Fold that half in half again.
3. Open the square all the way.

The folds make four equal small squares. Color one small square. You have colored a quarter of the original square.

The second fold split each half into two smaller parts. That is why a quarter is half of a half.

Try other quarter shapes

A square can have quarters that look different. One line across its middle and one line down its middle make four small squares. Two corner-to-corner lines make four equal triangles. Three evenly spaced lines across it make four equal strips.

Every time, check the size and count four parts. The shape of a piece can change while it stays a quarter.

For a circle, picture two straight lines through the center. They cross like a plus sign, making square corners. The circle has four equal slices. Each slice is a quarter.

A quarter of a strip

Bring the ends of a paper strip together and fold it in half. Keep it folded and bring the ends together again. Open it fully.

You see three fold marks across the strip. Count the spaces between the marks and the ends. There are four equal parts. Each part is a quarter of the strip's length.

A train of 8 equal cubes can make four shorter trains. Each shorter train has 2 cubes, so a quarter of the long train is 2 cubes long.

Count parts, then check size

Three lines across a strip can make four parts. Count the parts, not the lines.

If a shape has four equal parts and one is colored, a quarter is colored. With unequal parts, counting four pieces does not tell you the size of the colored piece. Check how much of the whole it covers.

To find a quarter of a group, make four equal groups. Count the things in one group. That count is one quarter of the whole.

Share among four friends

Four friends share 20 marbles. Make four places and give one marble to each place in turn. Keep going until all the marbles are shared.

Each friend gets 5 marbles. A quarter of 20 is 5. Check the four shares:

The 4 tells you how many shares to make. It does not tell you how many marbles each friend gets.

Halve, then halve again

You can find a quarter without dealing out every object. Start with 12 counters.

1. Find half of 12. Two equal groups have 6 each.
2. Take one group of 6. Halve it into 3 and 3.
3. One of those smaller groups has 3 counters. That is a quarter of the original 12.

The other half of 12 can split the same way. You would have four groups of 3 altogether.

These whole groups share into quarters:

Making groups of four can give a different answer. With 8 strawberries, two groups of 4 use them all. But a quarter needs four equal groups: 2 in each group.

Build the whole from a quarter

A quarter of a bag is 3 apples. The whole bag has four matching shares. Add all four:

The whole bag has 12 apples. Adding just two shares would give half the bag.

Check for leftovers

Share 10 whole crayons among four places. Each place gets 2, with 2 crayons left over. Giving those to just two places makes the shares unequal.

You can halve 10 into 5 and 5. But 5 whole crayons cannot be halved equally. To get four equal shares of whole crayons, both halving steps must work without leftovers.

A whole split into quarters gives you four equal parts to join. Keep thinking about the same whole as you move the parts.

Two quarters make a half

Fold a square into four equal small squares. Color the two on the left. Together, they cover the left half of the square.

There are two colored pieces, but their total size is one half. Two quarters make one half. You can call them two fourths too.

This works for groups. A quarter of 8 shells is 2 shells. Join two quarter shares:

Half of 8 is also 4. The two names describe the same amount.

Three of the four parts

Three quarters means three of four equal parts. Imagine a sandwich cut into quarters. Eat one quarter, and three quarters are left.

To find three quarters of 12 counters:

1. Find one quarter of 12. It is 3 counters.
2. Take three matching quarter shares.
3. Add them together.

Three quarters of 12 is 9 counters. The fourth share has 3 counters. It brings the total back to 12.

With 8 counters, one quarter is 2. Three quarters is 2 and 2 and 2, which makes 6.

What fills the missing space?

One quarter and three quarters fill a whole. Two quarters and two more quarters fill a whole too.

A half and a quarter make three quarters. You can see why on your folded square. A half covers two small squares. One more quarter covers a third small square. One quarter is still empty.

For a pizza, that tells you both amounts. If you eat a half and a friend eats a quarter, three quarters are eaten. One quarter is left.

Count pieces carefully. Three torn pieces can have many different sizes. To call an amount three quarters, check that it is as much as three of the whole's four equal shares.

Picture one pancake. Split it into two equal parts, and each is a half. Split the same pancake into four equal parts, and each is a quarter.

Each quarter is smaller. The pancake did not grow when you made more pieces. More equal parts of the same whole make smaller parts.

Match a half with quarters

Take two paper squares of the same size. Fold one into halves. Fold the other into quarters.

Put two quarter pieces over one half. They cover it exactly. One quarter covers just half of that space. A half is twice as big as a quarter of the same whole.

The number 4 in "four equal parts" counts pieces. It does not mean one quarter is bigger than one half.

Compare a group by counting

For a group of 12 counters:

• Half is 6 counters.
• A quarter is 3 counters.
• Three quarters is 9 counters.
• The whole is 12 counters.

Count from the smallest amount to the biggest: 3, 6, 9, 12. For this whole, the parts go in this order:

a quarter, a half, three quarters, the whole.

Look at the starting whole

Half of a tiny cookie can be smaller than a quarter of a large pizza. The pieces come from different-sized wholes.

You can check that idea with groups too. Half of 4 grapes is 2 grapes. A quarter of 20 grapes is 5 grapes. Here the quarter share has more grapes because its starting group is much bigger.

So when you say "a half is bigger than a quarter," keep the same whole. Two matching pancakes or two groups of the same number let you compare the shares directly.

More than half or less than half

Half of 10 crayons is 5 crayons. Use 5 as the amount to compare with.

If you have 7 of those 10 crayons, you have more than half. Seven is more than 5. If you have 4, you have less than half. Four is less than 5. If you have exactly 5, you have half.

A story might ask how much someone eats, shares, keeps, or starts with. Picture the story before you count.

Find what the story asks

Lily has 10 grapes and eats half. How many grapes are left?

1. The whole is 10 grapes.
2. Half means two equal shares. Half of 10 is 5.
3. Lily eats one share of 5. The other share is left.
4. She has 5 grapes left. Both shares make 10 again.

Words can help you choose a share. "Cut in half" and "shared equally between two" point to halves. "A fourth" and "shared equally among four" point to quarters.

Used and left can be different

Dad cuts a pie into four equal slices. The family eats three slices. How much pie is left?

Each slice is a quarter. The family eats three quarters. One quarter is left. If you answer "three quarters," you have found how much was eaten, not how much remains.

Find one person's share

Four friends share 16 crayons equally. A quarter of 16 is one friend's share.

Half of 16 is 8. Half of 8 is 4. Each friend gets 4 crayons. Check all four shares:

Ben's train has 12 cars, and half are red. Half of 12 is 6. There are 6 red cars and 6 cars that are not red.

Find what was there at the start

Mia ate half her crackers. She ate 4 crackers. How many did she start with?

The 4 eaten crackers are one half. The other half has 4 too. Put both shares together:

She started with 8 crackers. The number you know can be a part, so look for what it names.

Find halfway along a path

A tree is 10 equal big steps away. Halfway means half of the way. Half of 10 is 5, so you reach halfway after 5 steps.

You still have 5 steps to go. The two equal stretches make the whole path to the tree.

A turn means spinning around while staying in the same place. Turn a toy car on a table. Its nose points a new way, but it does not drive across the table.

All the way or halfway around

Start with the car facing a door. Turn it all the way around until it faces the door again. That is a whole turn, also called a full turn.

Start facing the door again. Turn only halfway around. The car now faces away from the door. That is a half turn. It ends facing the opposite way.

You can try this with your body too. Face the door, then turn until your back is toward it. You have made a half turn.

Turn to a side

A quarter turn is one of four equal parts of a whole turn. From facing the door, turn to face a side wall.

Your old and new facing ways make a square corner. You can see that corner by putting two fingers along two meeting edges of a sheet of paper.

A quarter turn does not take you all the way to facing backward. It takes two quarter turns in the same direction to face the opposite way.

Choose left or right

A quarter turn to your right ends at your right side. A quarter turn to your left ends at your left side. Those are different ways to face.

A half turn to the right and a half turn to the left both end opposite the start. They go around different sides but end facing the same way.

Most of the way around

Make three quarter turns to the right, one after another. That is a three-quarter turn. You have passed the halfway point, with one quarter turn still to go before a whole turn.

You can order turns from smallest to biggest:

a quarter turn, a half turn, a three-quarter turn, a whole turn.

Watch how far the car travels around as it turns. A whole turn brings it back to facing the start, even though it has moved all the way around.

Little turns can join to make a bigger turn. Keep track of which way you turn as well as how far.

Put quarter turns together

Face the door. Make a quarter turn to the right. Make another quarter turn to the right. Now your back is to the door. Two quarter turns in the same direction make a half turn.

Make two more quarter turns to the right. You face the door again. Four quarter turns in the same direction make a whole turn. Two half turns in the same direction make a whole turn too.

Now try a different path. Face the door, turn a quarter to the right, then a quarter back to the left. The second turn undoes the first. You face the door again, but you have not gone all the way around.

Three quarter turns to the right end facing the same way as one quarter turn to the left. Their paths are different lengths.

Follow the minute hand

Look at a clock face. Follow the minute hand from 12 toward 1, then 2, then 3. It makes a quarter turn.

Keep following the same way around:

• From 12 to 6 is a half turn.
• From 12 to 9 is a three-quarter turn.
• From 12 all the way around to 12 is a whole turn.

The numbers 12, 3, 6, and 9 mark four equal parts of the turning path. Each quarter has 3 number spaces. A half has 6 of the clock's 12 number spaces.

Count spaces, not just the numbers you see. From 12 to 3, move through the spaces from 12 to 1, from 1 to 2, and from 2 to 3.

Start at a different number

Follow the hand from 3 through 4 and 5 to 6. It moves 3 spaces, so that is a quarter turn too. From 6 through 7, 8, 9, 10, and 11 to 12 is a half turn. The size of the turn comes from how far the hand moves.

The time words use these turns. From an o'clock time to "quarter past," the minute hand goes from 12 to 3. From an o'clock time to "half past," it goes from 12 to 6.