Fractions: Equal Parts of Shapes and Amounts

Start with one whole thing, such as a sheet of paper. The whole is the complete thing before you split it. A part is some of that whole.

What makes a fair share?

Four friends share one sandwich. Cut it into four equal pieces. Each friend gets one piece, so everyone gets the same amount.

Equal parts are exactly the same size. For a flat shape, they cover the same amount of space. For a ribbon, they have the same length. Four pieces alone do not show that the shares are equal. One huge piece and three tiny pieces would be unfair.

Try a fold or count squares

Fold a paper rectangle in half. Line up the edges carefully. Open the paper and look at the two parts. Folding one part over the other checks that they match.

You can also check parts on a grid. Imagine a rectangle made of 12 same-size squares. Split it into three parts. Count the squares in each part:

• Four, four, and four squares give equal parts.
• Five, four, and three squares give unequal parts.

Both lists use all 12 squares. Only the first gives each part the same amount of space.

For a 12 cm ribbon, three pieces of 4 cm are equal. Lay the pieces end to end to check the length:

Equal space can have different shapes

Take two squares that are the same size. Draw a line from corner to corner on the first. It makes two equal triangles. Draw a line straight down the middle of the second. It makes two equal rectangles. Every piece is half of its square. The triangle and rectangle look different, but cover the same amount of space.

A round pizza needs a different check. Make equal slices that reach from the center to the edge. Cutting it into three equally wide strips does not make three equal shares. The middle strip holds more pizza than an edge strip.

A part name tells you how many equal parts make the whole. Split one pancake fairly among three children. Each share is one third, because three equal shares make that pancake.

Choose the name

Say “one half,” not “one second.” For more than one part, use the plural name: “two halves,” “two thirds,” or “seven tenths.” Fourth and quarter name the same kind of share. You can say “three fourths” or “three quarters.”

Count the parts you have

A rectangle is split into three equal strips. One strip is one third. Choose another strip, and you have two thirds. Choose the last strip, and you have three thirds: the whole rectangle.

All the equal parts together rebuild the whole:

• Two halves make one whole.
• Four fourths make one whole.
• Ten tenths make one whole.

You can make fourths of a square in several ways. Draw four equal strips, four small squares, or four triangles. For the triangles, draw both lines from corner to corner. Each drawing has four equal parts, so each part is one fourth.

Keep track of the whole

A half does not have one fixed size. Half of a large pizza gives more food than half of a small pizza. Both are halves of their own pizzas.

The words “the bigger half” do not fit two pieces of one pizza. Each half must be the same size. If one piece is bigger, the two pieces are unequal shares.

To share a circle in thirds, draw three lines from its center to its edge. Spread them so they make three same-size slices. Each child gets one third when three children share it fairly.

You can write “one fourth” with numbers: $\frac{1}{4}$. This is a fraction. It names part of a whole.

Meet the three pieces of a fraction

Look at $\frac{1}{4}$:

• The bottom number, 4, is the denominator. It counts all the equal parts in the whole.
• The top number, 1, is the numerator. It counts the parts you mean.
• The line between them is the fraction bar.

Picture a square split into four equal small squares. Color one small square. The whole has four equal parts, and one is colored. That is why you write 1 on top and 4 on the bottom.

The bottom number includes the uncolored parts too. It describes the whole square, not just the part you colored.

One part is a unit fraction

A unit fraction has 1 on top. It means one equal part. These are unit fractions:

Read them as “one half,” “one third,” “one fourth,” “one fifth,” “one sixth,” “one eighth,” and “one tenth.” Read the top number first. Then say the bottom number's part name.

Write one sixth

A paper strip has six equal sections. One section is blue. To write the blue fraction:

1. Count all six sections. Write 6 below the bar.
2. Count the one blue section. Write 1 above the bar.
3. Read $\frac{1}{6}$ as “one sixth.”

Writing $\frac{6}{1}$ swaps the jobs of the numbers. One sixth needs 6 on the bottom because six equal sections make the strip.

You may also see a fraction written on one line, such as 1/6. The slash does the job of the fraction bar. Both forms mean one sixth.

Check that the parts are equal before counting them. Three pieces of different sizes do not give three equal parts to count as thirds.

One third means one equal part out of three. Two thirds, $\frac{2}{3}$, means two of those parts. The parts keep their size as you choose more of them.

Write several parts

A rectangle has five equal sections. Two are shaded, and three are white.

The shaded fraction is $\frac{2}{5}$. The 5 counts all the sections. The 2 counts the shaded sections. The white fraction is $\frac{3}{5}$, because three of the five sections are white.

A non-unit fraction has a numerator greater than 1. Examples include $\frac{2}{3}$, $\frac{3}{4}$, and $\frac{7}{10}$. Read $\frac{3}{4}$ as “three fourths” or “three quarters.” It means three parts that are each one fourth.

Be careful with $\frac{2}{3}$. It does not mean two shaded parts and three white parts. There are three equal parts in all: two shaded and one white.

Count in fourths

Imagine a square split into four equal parts. Shade one more part each time:

Say “one fourth, two fourths, three fourths, four fourths.” Four fourths covers the whole square. The top number changes because you count more parts. The bottom number stays 4 because the square still has four equal parts.

Count in tenths

Split a strip into ten equal sections. Each section is one tenth. Count the shaded sections from one to ten:

Ten tenths is the whole strip. You can count backward too: ten tenths, nine tenths, eight tenths. Each time, one fewer tenth is chosen.

If three fourths of a pizza is eaten, one fourth is left. The three eaten slices and the one remaining slice fill all four equal places.

A fraction tells you how to make a picture. Start with the whole shape. Use the bottom number to split it. Use the top number to choose parts.

Draw two thirds

To show $\frac{2}{3}$ of a rectangle:

1. Draw the whole rectangle.
2. Add two lines to make three equal strips.
3. Shade two strips.

Count spaces, not dividing lines. Two lines inside the rectangle make three strips. Check their sizes before you shade.

A fraction strip is a long rectangle with equal sections. Split one into eight sections and shade three. You have shown $\frac{3}{8}$ of the strip.

Shading can be spread out

The shaded parts do not have to touch. A rectangle has ten equal squares. Color three squares in different places. Three out of ten are colored, so the fraction is $\frac{3}{10}$.

To check any fraction picture, ask:

• Do the parts have equal size?
• How many equal parts make the whole?
• How many parts are shaded?

Counting four unequal pieces cannot tell you that one is a fourth. You need to know how much of the whole it covers.

Find a part of a length

A ribbon is 12 cm long. Split it into four equal pieces. Each piece is 3 cm long, because:

One piece is $\frac{1}{4}$ of the ribbon. Three pieces are $\frac{3}{4}$ of it. Their length is:

So three fourths of the ribbon is 9 cm. The last piece is 3 cm. All four pieces together measure 12 cm.

Build the missing whole

Sometimes you see just one part. A small strip is one third of a whole strip. Put three copies of that small strip end to end. They make the whole length, with no gaps or overlaps.

For a shape, match the space instead of just the length. If one small square is a fourth, four such squares fill the whole. They could form one row or a larger square.

A fraction can describe a group, also called a set. Imagine five balloons tied to a chair. The whole is all five balloons, not the chair.

Count every object in the whole

Two balloons are red and three are yellow. What fraction is red?

Count all five balloons for the denominator. Count the two red balloons for the numerator. The red balloons are $\frac{2}{5}$ of the balloons.

Writing $\frac{2}{3}$ would count only the yellow balloons on the bottom. The whole includes both colors. You need all five.

A box holds eight crayons. Three are blue and five are other colors. The blue crayons are $\frac{3}{8}$ of the crayons. The other crayons are $\frac{5}{8}$. Both fractions describe the same whole group of eight.

Count objects, not their sizes

A bag holds one beach ball and three tennis balls. There are four balls in all. The beach ball is $\frac{1}{4}$ of the balls when you count them.

It takes up much more space than a tennis ball. That does not change the count: one ball is still one object. For a group, equal shares have the same number of objects. For a flat shape, equal shares cover the same space.

A row can be one part

You can also split a group into equal smaller groups. Twelve eggs sit in three rows, with four eggs in each row.

Think of each row as one part. One row is one of three equal rows, so it holds $\frac{1}{3}$ of the eggs. Two rows hold $\frac{2}{3}$ of the eggs.

Here the denominator counts the three equal rows. It does not count single eggs. Decide what you are counting before writing your fraction.

The whole changes the amount

One third of twelve eggs is four eggs. One third of six eggs is two eggs. In both cases, you take one of three equal groups. The word “third” tells you the number of groups, not the number of eggs in one group.

To find a fraction of some objects, share them into equal groups. Every object needs a place. Then choose the groups named by the fraction.

Find one third of twelve

Put 12 strawberries beside three plates. Give one strawberry to each plate in turn. Keep going until all the strawberries are on plates.

Each plate has four strawberries. The three plates hold equal groups, so one plate holds $\frac{1}{3}$ of the strawberries. One third of 12 is 4.

Check all three plates together:

The 3 in $\frac{1}{3}$ tells you to make three groups. It does not tell you to put three strawberries on each plate. Three plates with only three strawberries each would leave some strawberries unshared.

Find two thirds of twelve

Keep those same three plates of four. Two thirds means two of the three groups. Choose two plates and count their strawberries:

So $\frac{2}{3}$ of 12 is 8. Do not make two groups at the start. The denominator tells you how many groups to make. The numerator tells you how many to choose.

Show three fourths with counters

Lay out eight counters in four equal rows. There are two counters in each row.

1. All four rows show the whole group of eight.
2. One row shows one fourth: two counters.
3. Three rows show three fourths: six counters.

You can add the three chosen rows:

Two counters gives only one fourth. You need all three chosen rows for three fourths.

Check for leftovers

Twelve objects share into three equal groups with none left over. Ten objects do not. After making three groups of three, one object is left.

That leftover means you have not shared the whole group equally. When you are sharing whole objects, check both things: every group has the same count, and every object has been used.

Finding one fourth of 20 means sharing 20 into four equal parts. Division does that sharing without moving counters one at a time.

Use the bottom number

To find a unit fraction of a number, divide by the denominator. The numerator is 1, so you need just one part.

For $\frac{1}{4}$ of 20:

One fourth of 20 is 5. Picture a bar split into four equal boxes. Write 20 over the whole bar and 5 in each box. Each box is one fourth of the whole.

Taking away 4 would give 16. That removes four objects; it does not share the objects into four equal groups.

Let multiplication facts help

To find $\frac{1}{5}$ of 40, ask, “What number times 5 makes 40?”

So $40÷5=8$, and one fifth of 40 is 8. Halves use your 2s facts. Fifths use your 5s facts. Tenths use your 10s facts.

For one third of 18, count by threes: 3, 6, 9, 12, 15, 18. You made six jumps. That tells you $18÷3=6$.

Halve, then halve again

You can find a fourth by halving twice. Half of 20 is 10. Half of 10 is 5. Splitting both halves in half makes four equal parts.

This gives the same answer as dividing 20 by 4. You can use whichever method helps you see the equal shares.

Keep the unit

Parts of time, money, and length work the same way:

• One fourth of an hour is 15 minutes: $60÷4=15$.
• Half of $10 is $5: $10÷2=5$.
• One tenth of 100 cm is 10 cm: $100÷10=10$.

Multiply back to check

If one third of 18 is 6, three parts of 6 must make 18:

Multiplying 18 by 3 would give 54. That builds three whole groups of 18. One third of this positive whole must be smaller than 18.

Three fourths means three parts, each the size of one fourth. First find the size of one part. Then find the amount in all the chosen parts.

Two steps for three fourths

Find $\frac{3}{4}$ of 12. Draw a bar for 12 and split it into four equal boxes.

1. Find one box: $12÷4=3$.
2. Find three boxes: $3\times 3=9$.

So three fourths of 12 is 9. The first 3 is the amount in one box. The other 3 tells how many boxes to take. Adding $3+3+3$ gives the same result.

Do not stop at 3. That is only one fourth of 12. The numerator asks for three of those parts.

Let the denominator choose the groups

For $\frac{2}{3}$ of 15, split 15 into three equal parts:

Each third holds 5. Choose two thirds:

Two thirds of 15 is 10. Dividing by the top number would split the whole into two groups. The bottom number says there must be three equal groups.

Use fifths or tenths

For $\frac{3}{5}$ of 20, draw five boxes for the whole bar. Each box holds $20÷5=4$. Shade three boxes. They hold $4\times 3=12$.

For $\frac{3}{10}$ of 50, one tenth is $50÷10=5$. Three tenths is $5\times 3=15$.

Larger amounts use the same method. Three fifths of 60 has parts of size $60÷5=12$. Three parts hold $12\times 3=36$.

Use the rest as a check

Return to the four-box bar for 12. The three chosen boxes hold 9. The one box left holds 3.

The chosen part and the rest fill the whole bar. Also, three boxes must hold more than one box, but less than all four. An answer of 3 would be too small for three boxes; 36 would be bigger than the whole.

Sometimes a story gives you a part but hides the whole. A bar model shows how many equal parts you need to rebuild.

One fourth is five

One fourth of a pile is five stickers. How many stickers are in the whole pile?

Draw four equal boxes. One box stands for one fourth, so write 5 inside it. The other boxes must hold 5 too. All four boxes together hold:

The pile has 20 stickers. For a known unit fraction, multiply the part by the denominator. You know one part and need all the equal parts.

Dividing 5 by 4 would make a smaller amount. Here 5 is already one part; the missing whole must be larger.

One third of a class

One third of a class is eight children. Make three equal boxes and put 8 in each. The class has $3\times 8=24$ children.

Check by starting with that whole:

Three fourths is fifteen

Now three fourths of a number is 15. Draw four equal boxes. Mark three boxes together as 15. Do not write 15 in each box: 15 belongs to all three chosen boxes together.

First find one box:

Then find all four boxes:

The whole number is 20. With several known parts, divide by the numerator first. That finds one part. Then multiply by the denominator to build the whole.

Check the clue: one fourth of 20 is 5, and three fourths is $5\times 3=15$.

Rebuild a length

A 4 cm strip is one third of a longer strip. Three copies laid end to end make the longer strip:

The whole strip is 12 cm long.

Read the question before you calculate. Find the whole and the fraction. Then ask what is missing: a part, the rest, or the whole amount.

Find the part and the rest

Sam has 20 stickers. He gives one fourth to his sister. How many does he keep?

The whole is 20 stickers. First find the part he gives away:

Now take that part away:

Sam keeps 15 stickers. An answer of 5 tells how many he gives, not how many he keeps.

For a non-unit fraction, find one part before counting several. Mia eats two thirds of 12 strawberries. One third is $12÷3=4$. Two thirds is $4\times 2=8$. She eats eight, so $12-8=4$ strawberries remain.

Use the right unit

A 20 cm ribbon has one fourth cut off. The cut piece is $20÷4=5$ cm. The ribbon left is $20-5=15$ cm long.

A movie lasts 60 minutes. Leo watches three fourths of it. One fourth lasts $60÷4=15$ minutes. Three fourths lasts $15\times 3=45$ minutes. He has watched 45 minutes.

Name a fraction

Not every answer is an amount. A pizza has eight equal slices. Ana eats three. The eaten fraction is $\frac{3}{8}$. Five of the eight slices remain, so $\frac{5}{8}$ is left.

Find the starting amount

Maya spends one fifth of her money on a $4 book. The $4 is one equal part of her money. Five such parts make the whole:

Maya had $20 at the start.

Compare amounts from different wholes

One child gets half of 10 beads. Another gets a fourth of 24 beads. The first gets $10÷2=5$ beads. The second gets $24÷4=6$ beads, so the second child gets one more.

The part names alone do not tell you the amounts. Work out each share from its own whole. Answer with what the question wants, including words such as stickers, centimeters, minutes, or dollars.

A good check goes back to the whole and its equal parts. You can use a drawing, count groups, or work backward with numbers.

Check a picture

Someone says, “There are three pieces, so each is one third.” Look at the sizes before agreeing. Each piece is a third when all three cover the same amount of space.

If the pieces are unequal, their count cannot tell you the fraction of one piece. On a square grid, count how many same-size squares each piece covers.

Check the size of one part

Lena says one fourth of eight counters is four. The 4 in one fourth counts the groups, not the counters in each group.

Draw four circles and share eight counters among them. Each circle gets two. You can check by rebuilding eight:

So one fourth of eight is two. Four groups of four would need 16 counters, which is more than you started with.

Multiply back

One fifth of 15 is 3. Five equal parts of 3 must make 15:

This checks the division using multiplication. A drawing of five groups of three checks it another way.

Account for the rest

If eight strawberries are eaten from a group of twelve, four remain. Put the two amounts back together:

This checks that the story accounts for every strawberry. It does not check the fraction by itself. To check two thirds, also show three equal groups of four and choose two groups.

Explain the parts in words

You could say, “I shared 12 counters into three equal groups. Each group has four. I chose two groups, so I have eight counters.” This tells someone where both steps came from.

Watch for an answer larger than the whole when you choose fewer than all the parts. Two thirds of 15 cannot be 30. One third is $15÷3=5$, and two thirds is $5\times 2=10$. Ten is less than 15, and the last part of 5 rebuilds the whole.