Fractions
By the end of this chapter you will be able to
- Name a tenth as one of ten equal parts, and count up and down in tenths
- Tell unit fractions (1 on the top) from non-unit fractions, and place fractions on a number line from 0 to 1
- Find a unit fraction and a non-unit fraction of a set of objects
- Show, with bars, that two fractions are equivalent, such as 1/2 = 2/4
- Compare and order unit fractions, and fractions with the same bottom number
- Add and subtract fractions with the same bottom number, within one whole
- Solve word problems that use fractions
You should already know
- Halves, quarters and thirds of shapes and sets (Year 2, Chapter 4)
- Multiplication and division facts, including the 3, 4 and 8 times tables (Year 3, Chapter 3)
- Counting in steps, and place value (Year 3, Chapter 1)
Where this leads
- Equivalent fractions, fractions greater than 1 and fractions of amounts (Year 4)
- Tenths and hundredths as decimals, and fractions with bigger denominators (Year 4)
National Curriculum for England, Key Stage 2 mathematics, Year 3. Content: Count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and from dividing one-digit numbers by 10, Recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators, Recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators, Recognise and show, using diagrams, equivalent fractions with small denominators, Compare and order unit fractions, and fractions with the same denominators, Add and subtract fractions with the same denominator within one whole (for example 5/7 + 1/7 = 6/7), Solve problems that involve all of the above.
Exercise P Do you remember?
Non-calculatorKey words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.4 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.1 shaded parts
- Write the fraction. Top number over bottom number.1/4
Check. 1 shaded + 3 not shaded = 4 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Method. Half means two equal parts. Share 12 into 2 equal parts.
- Share 12 into 2 equal parts. Half of a number is one of two equal parts.12 ÷ 2 = 6
- Write the answer. 6
Check. 6 + 6 = 12. ✓
Watch out. Finding half of a number is the same as dividing it by 2.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 4 tells us how many equal groups to make.8 ÷ 4 = 2 in each group
- Take 1 of the 4 groups. The top number 1 tells us how many groups to take.1 × 2 = 2
- Write the answer. The answer is a number of objects.2
Check. 2 + 6 = 8, so 1 of the 4 groups is 2. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Method. Share the counters into equal groups, one counter at a time, or divide.
- Write the division. We share 20 counters equally between 10 children.20 ÷ 10
- Work out the share. Each child gets the same number, so we divide.10 × 2 = 20
- Write the answer. 2
Check. 2 counters for each of 10 children makes 20 counters. ✓
Watch out. Share one at a time, or use a multiplication fact that you know.
Method. Count on in steps of 5.
- Look at the number before the gap. The numbers go up in steps of 5.10
- Add one step. The missing number is one step after the number before it.10 + 5 = 15
- Check the next number. Adding one more step must give the next number in the list.15 + 5 = 20
- Write the answer. 15
Check. 15 − 5 = 10. ✓
Watch out. Check the step is the same all the way along the list.
Method. Count on in steps of 5.
- Look at the number before the gap. The numbers go up in steps of 5.30
- Add one step. The missing number is one step after the number before it.30 + 5 = 35
- Check the next number. Adding one more step must give the next number in the list.35 + 5 = 40
- Write the answer. 35
Check. 35 − 5 = 30. ✓
Watch out. Check the step is the same all the way along the list.
Tenths
A tenth is one of ten equal parts of a whole. If we cut a whole into 10 equal parts, each part is one tenth. We write one tenth as 1/10.
The bottom number, 10, tells us how many equal parts the whole is cut into. The top number tells us how many of those parts we have. Amira has eaten 3 of the 10 equal squares, so she has eaten 3/10 of the bar.
When we count in tenths we go 1/10, 2/10, 3/10, and so on. Ten tenths make one whole: 10/10 = 1.
Tenths also come from sharing. If 4 biscuits are shared equally between 10 friends, each friend gets 4/10 of a biscuit. We write this as 4 ÷ 10 = 4/10.
Key ideas
- A tenth is one of ten equal parts. We write it as 1/10.
- The bottom number is how many equal parts make the whole. The top number is how many parts we have.
- Count in tenths: 1/10, 2/10, 3/10 and so on. 10/10 is one whole.
- Sharing 4 biscuits between 10 friends gives each friend 4/10 of a biscuit: 4 ÷ 10 = 4/10.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.10 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.7 shaded parts
- Write the fraction. Top number over bottom number.7/10
Check. 7 shaded + 3 not shaded = 10 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count on in tenths. The bottom number stays 10 all the way.
- Start at the first fraction. We count in tenths. Each step is one tenth.4/10
- Count on 3 tenths. Each step adds or takes away one tenth.4/10, 5/10, 6/10, 7/10
- Write the answer. The last fraction in the count is the answer.7/10
Check. 3 tenths added to 4/10 gives 7/10. ✓
Watch out. The bottom number stays 10. Only the top number changes.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Each person gets the same share. Write the share as a fraction with 10 on the bottom.
- Write the division. We share 4 things equally between 10 people.4 ÷ 10
- Find the share of one thing. Each person gets 4 of the 10 equal parts, and each part is one tenth.4 tenths = 4/10
- Write the answer as a fraction. Tenths have 10 on the bottom.4/10
Check. 4 tenths of one thing is 4/10. ✓
Watch out. When you divide by 10, the answer is a tenth, so the bottom number is 10.
Watch out
Watch out: the bottom number counts all the equal parts, shaded or not. In 3/10, the 10 is the number of parts in the whole bar. It is not the number of shaded parts.
Exam tip
Read the bottom number first. It tells you how many equal parts make the whole, and so how big each part is.
Exercise 4A Tenths
Non-calculatorKey words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.10 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.6 shaded parts
- Write the fraction. Top number over bottom number.6/10
Check. 6 shaded + 4 not shaded = 10 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.10 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.9 shaded parts
- Write the fraction. Top number over bottom number.9/10
Check. 9 shaded + 1 not shaded = 10 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number. Then count the parts that are left for the top number.
- Count all the equal parts. The bottom number is the whole.10 equal parts
- Count the parts that are not shaded. Take away the 3 parts that do not count, from all 10.10 − 3 = 7
- Write the fraction. The top number is the parts we want. The bottom number is all the parts.7/10
Check. 7 + 3 = 10 equal parts. ✓
Watch out. Use the whole, not the parts you took away, as the bottom number.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count on in tenths. The bottom number stays 10 all the way.
- Start at the first fraction. We count in tenths. Each step is one tenth.1/10
- Count on 2 tenths. Each step adds or takes away one tenth.1/10, 2/10, 3/10
- Write the answer. The last fraction in the count is the answer.3/10
Check. 2 tenths added to 1/10 gives 3/10. ✓
Watch out. The bottom number stays 10. Only the top number changes.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count on in tenths. The bottom number stays 10 all the way.
- Start at the first fraction. We count in tenths. Each step is one tenth.7/10
- Count on 1 tenth. Each step adds or takes away one tenth.7/10, 8/10
- Write the answer. The last fraction in the count is the answer.8/10
Check. 1 tenth added to 7/10 gives 8/10. ✓
Watch out. The bottom number stays 10. Only the top number changes.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count back in tenths. The bottom number stays 10 all the way.
- Start at the first fraction. We count in tenths. Each step is one tenth.10/10
- Count back 3 tenths. Each step adds or takes away one tenth.10/10, 9/10, 8/10, 7/10
- Write the answer. The last fraction in the count is the answer.7/10
Check. 3 tenths taken from 10/10 gives 7/10. ✓
Watch out. The bottom number stays 10. Only the top number changes.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Each person gets the same share. Write the share as a fraction with 10 on the bottom.
- Write the division. We share 5 things equally between 10 people.5 ÷ 10
- Find the share of one thing. Each person gets 5 of the 10 equal parts, and each part is one tenth.5 tenths = 5/10
- Write the answer as a fraction. Tenths have 10 on the bottom.5/10
Check. 5 tenths of one thing is 5/10. ✓
Watch out. When you divide by 10, the answer is a tenth, so the bottom number is 10.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Each person gets the same share. Write the share as a fraction with 10 on the bottom.
- Write the division. We share 7 things equally between 10 people.7 ÷ 10
- Find the share of one thing. Each person gets 7 of the 10 equal parts, and each part is one tenth.7 tenths = 7/10
- Write the answer as a fraction. Tenths have 10 on the bottom.7/10
Check. 7 tenths of one thing is 7/10. ✓
Watch out. When you divide by 10, the answer is a tenth, so the bottom number is 10.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Each person gets the same share. Write the share as a fraction with 10 on the bottom.
- Write the division. We share 2 things equally between 10 people.2 ÷ 10
- Find the share of one thing. Each person gets 2 of the 10 equal parts, and each part is one tenth.2 tenths = 2/10
- Write the answer as a fraction. Tenths have 10 on the bottom.2/10
Check. 2 tenths of one thing is 2/10. ✓
Watch out. When you divide by 10, the answer is a tenth, so the bottom number is 10.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Each person gets the same share. Write the share as a fraction with 10 on the bottom.
- Write the division. We share 9 things equally between 10 people.9 ÷ 10
- Find the share of one thing. Each person gets 9 of the 10 equal parts, and each part is one tenth.9 tenths = 9/10
- Write the answer as a fraction. Tenths have 10 on the bottom.9/10
Check. 9 tenths of one thing is 9/10. ✓
Watch out. When you divide by 10, the answer is a tenth, so the bottom number is 10.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number. Then count the parts that are left for the top number.
- Count all the equal parts. The bottom number is the whole.10 equal parts
- Count the parts that are left. Take away the 4 parts that do not count, from all 10.10 − 4 = 6
- Write the fraction. The top number is the parts we want. The bottom number is all the parts.6/10
Check. 6 + 4 = 10 equal parts. ✓
Watch out. Use the whole, not the parts you took away, as the bottom number.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count the equal steps from 0 to 1 for the bottom number. Then count the steps to the box for the top number.
- Count the equal steps from 0 to 1. The line is cut into 10 equal steps.10 steps
- Count the steps from 0 to the box. Each step is one equal part.3 steps
- Write the fraction. Steps taken over all the steps.3/10
Check. 3 steps + 7 more steps = 10 steps from 0 to 1. ✓
Watch out. Count the spaces between the marks, not the marks. Each space is one equal part.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number. Then count the parts that are left for the top number.
- Count all the equal parts. The bottom number is the whole.10 equal parts
- Count the parts that are not shaded. Take away the 7 parts that do not count, from all 10.10 − 7 = 3
- Write the fraction. The top number is the parts we want. The bottom number is all the parts.3/10
Check. 3 + 7 = 10 equal parts. ✓
Watch out. Use the whole, not the parts you took away, as the bottom number.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Think about what 10 tenths make.
- Count the tenths in one whole bar. A whole bar is cut into 10 equal parts.10 equal parts
- Shade every part. If every one of the 10 parts is shaded, the whole bar is shaded.10 shaded parts = 10/10
- Decide. Sam is right. Ten tenths make one whole.Yes, 10 tenths make one whole
Check. 10/10 = 1 whole. ✓
Watch out. The top and bottom numbers are the same when the whole is shaded.
Unit and non-unit fractions
A fraction is a number. It can be a part of a whole, or a point on a number line from 0 to 1.
A unit fraction has a 1 on the top, such as 1/2, 1/3, 1/4 or 1/8. It is one of the equal parts. Jack ate 1/4 of the pizza.
A non-unit fraction has a bigger number on the top, such as 2/3, 3/4 or 5/8. It is several of the equal parts. Ella ate 3/4 of the pizza.
Key ideas
- A unit fraction has 1 on the top: 1/3, 1/5, 1/8.
- A non-unit fraction has a bigger number on the top: 2/3, 3/5, 5/8.
- A fraction is a number. It is a point on a number line.
- The bottom number tells you how many equal parts make one whole.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.5 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.3 shaded parts
- Write the fraction. Top number over bottom number.3/5
Check. 3 shaded + 2 not shaded = 5 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count the equal steps from 0 to 1 for the bottom number. Then count the steps to the box for the top number.
- Count the equal steps from 0 to 1. The line is cut into 5 equal steps.5 steps
- Count the steps from 0 to the box. Each step is one equal part.2 steps
- Write the fraction. Steps taken over all the steps.2/5
Check. 2 steps + 3 more steps = 5 steps from 0 to 1. ✓
Watch out. Count the spaces between the marks, not the marks. Each space is one equal part.
Key words. A unit fraction has 1 on the top, like 1/4. A non-unit fraction has a bigger number on the top, like 3/4.
Method. A unit fraction has 1 on the top. Look at the top number of each one.
- Look at the top number of 1/6. It is 1, so 1/6 is a unit fraction.1 on the top
- Look at the top number of 3/6. It is 3, so 3/6 is not a unit fraction.3 on the top
- Look at the top number of 5/6. It is 5, so 5/6 is not a unit fraction.5 on the top
- Write the answer. 1/6
Check. Only 1/6 has a 1 on the top. ✓
Watch out. The bottom number does not decide. Look at the top number.
Watch out
Watch out: 3/4 means 3 parts out of 4. It does not mean 4 parts out of 3. The bottom number is always the number of equal parts in the whole.
Exam tip
To check if a fraction is a unit fraction, cover the bottom number. If the top number is 1, it is a unit fraction.
Exercise 4B Unit and non-unit fractions
Non-calculatorKey words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.6 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.1 shaded parts
- Write the fraction. Top number over bottom number.1/6
Check. 1 shaded + 5 not shaded = 6 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.4 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.3 shaded parts
- Write the fraction. Top number over bottom number.3/4
Check. 3 shaded + 1 not shaded = 4 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.6 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.5 shaded parts
- Write the fraction. Top number over bottom number.5/6
Check. 5 shaded + 1 not shaded = 6 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.8 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.1 shaded parts
- Write the fraction. Top number over bottom number.1/8
Check. 1 shaded + 7 not shaded = 8 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.5 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.4 shaded parts
- Write the fraction. Top number over bottom number.4/5
Check. 4 shaded + 1 not shaded = 5 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.3 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.1 shaded parts
- Write the fraction. Top number over bottom number.1/3
Check. 1 shaded + 2 not shaded = 3 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.3 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.2 shaded parts
- Write the fraction. Top number over bottom number.2/3
Check. 2 shaded + 1 not shaded = 3 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.6 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.1 shaded parts
- Write the fraction. Top number over bottom number.1/6
Check. 1 shaded + 5 not shaded = 6 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count the equal steps from 0 to 1 for the bottom number. Then count the steps to the box for the top number.
- Count the equal steps from 0 to 1. The line is cut into 8 equal steps.8 steps
- Count the steps from 0 to the box. Each step is one equal part.6 steps
- Write the fraction. Steps taken over all the steps.6/8
Check. 6 steps + 2 more steps = 8 steps from 0 to 1. ✓
Watch out. Count the spaces between the marks, not the marks. Each space is one equal part.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count the equal steps from 0 to 1 for the bottom number. Then count the steps to the box for the top number.
- Count the equal steps from 0 to 1. The line is cut into 3 equal steps.3 steps
- Count the steps from 0 to the box. Each step is one equal part.2 steps
- Write the fraction. Steps taken over all the steps.2/3
Check. 2 steps + 1 more steps = 3 steps from 0 to 1. ✓
Watch out. Count the spaces between the marks, not the marks. Each space is one equal part.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number. Then count the parts that are left for the top number.
- Count all the equal parts. The bottom number is the whole.8 equal parts
- Count the parts that are left. Take away the 3 parts that do not count, from all 8.8 − 3 = 5
- Write the fraction. The top number is the parts we want. The bottom number is all the parts.5/8
Check. 5 + 3 = 8 equal parts. ✓
Watch out. Use the whole, not the parts you took away, as the bottom number.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 6, so the parts are the same size.1/6 and 2/6
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.1 + 2 = 3
- Write the answer. Keep the bottom number 6.3/6
Check. 3/6 − 2/6 = 1/6. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. A unit fraction has 1 on the top, like 1/4. A non-unit fraction has a bigger number on the top, like 3/4.
Method. A unit fraction has 1 on the top. Look at the top number.
- Look at the top number. The top number of 3/5 is 3, not 1.3 on the top
- Look at the bottom number. The bottom number does not make a fraction a unit fraction.5 on the bottom
- Decide. Mia is not right. 3/5 is a non-unit fraction.No
Check. A unit fraction would be 1/5. ✓
Watch out. Only the top number decides whether a fraction is a unit fraction.
Key words. A unit fraction has 1 on the top, like 1/4. A non-unit fraction has a bigger number on the top, like 3/4.
Method. Pick out the fractions with 1 on the top.
- Look at 2/7. The top number is 2, so it is not a unit fraction.2 on the top
- Look at 1/7 and 1/9. The top number is 1, so both are unit fractions.1 on the top
- Look at 5/7. The top number is 5, so it is not a unit fraction.5 on the top
- Write the unit fractions. 1/7, 1/9
Check. The top numbers of 1/7 and 1/9 are both 1. ✓
Watch out. The bottom numbers can be any size. Only the top number is 1.
Fractions of a set of objects
A set is a group of objects, such as 12 cubes or 20 pencils. We can find a fraction of a set by sharing the objects into equal groups.
To find 1/4 of 12, share the 12 cubes into 4 equal groups. There are 3 cubes in each group, so 1/4 of 12 is 3. The bottom number tells us how many groups to make. It does not tell us the answer.
To find 3/4 of 12, first find 1/4 (3 cubes). Then take 3 of the 4 groups: 3 + 3 + 3 = 9. So 3/4 of 12 is 9.
Key ideas
- To find a unit fraction of a set, divide the set into equal groups. 1/4 of 12 = 12 ÷ 4 = 3.
- The bottom number is the number of equal groups.
- To find a non-unit fraction, find the unit fraction first, then multiply by the top number. 3/4 of 12 = 3 × 3 = 9.
- The answer is a number of objects.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 2 tells us how many equal groups to make.10 ÷ 2 = 5 in each group
- Take 1 of the 2 groups. The top number 1 tells us how many groups to take.1 × 5 = 5
- Write the answer. The answer is a number of objects.5
Check. 5 + 5 = 10, so 1 of the 2 groups is 5. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 3 tells us how many equal groups to make.15 ÷ 3 = 5 in each group
- Take 1 of the 3 groups. The top number 1 tells us how many groups to take.1 × 5 = 5
- Write the answer. The answer is a number of objects.5
Check. 5 + 10 = 15, so 1 of the 3 groups is 5. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 4 tells us how many equal groups to make.12 ÷ 4 = 3 in each group
- Take 3 of the 4 groups. The top number 3 tells us how many groups to take.3 × 3 = 9
- Write the answer. The answer is a number of objects.9
Check. 9 + 3 = 12, so 3 of the 4 groups is 9. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Watch out
Watch out: 1/4 of 12 is 3, not 4. Divide by the bottom number to find the size of one group. The number of groups is not the answer.
Exam tip
Draw the groups if you are not sure. Count the objects in one group, then count the groups you need.
Exercise 4C Fractions of a set of objects
Non-calculatorKey words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 2 tells us how many equal groups to make.14 ÷ 2 = 7 in each group
- Take 1 of the 2 groups. The top number 1 tells us how many groups to take.1 × 7 = 7
- Write the answer. The answer is a number of objects.7
Check. 7 + 7 = 14, so 1 of the 2 groups is 7. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 5 tells us how many equal groups to make.20 ÷ 5 = 4 in each group
- Take 1 of the 5 groups. The top number 1 tells us how many groups to take.1 × 4 = 4
- Write the answer. The answer is a number of objects.4
Check. 4 + 16 = 20, so 1 of the 5 groups is 4. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 3 tells us how many equal groups to make.18 ÷ 3 = 6 in each group
- Take 1 of the 3 groups. The top number 1 tells us how many groups to take.1 × 6 = 6
- Write the answer. The answer is a number of objects.6
Check. 6 + 12 = 18, so 1 of the 3 groups is 6. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 4 tells us how many equal groups to make.16 ÷ 4 = 4 in each group
- Take 1 of the 4 groups. The top number 1 tells us how many groups to take.1 × 4 = 4
- Write the answer. The answer is a number of objects.4
Check. 4 + 12 = 16, so 1 of the 4 groups is 4. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 10 tells us how many equal groups to make.30 ÷ 10 = 3 in each group
- Take 1 of the 10 groups. The top number 1 tells us how many groups to take.1 × 3 = 3
- Write the answer. The answer is a number of objects.3
Check. 3 + 27 = 30, so 1 of the 10 groups is 3. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 6 tells us how many equal groups to make.24 ÷ 6 = 4 in each group
- Take 1 of the 6 groups. The top number 1 tells us how many groups to take.1 × 4 = 4
- Write the answer. The answer is a number of objects.4
Check. 4 + 20 = 24, so 1 of the 6 groups is 4. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 8 tells us how many equal groups to make.32 ÷ 8 = 4 in each group
- Take 1 of the 8 groups. The top number 1 tells us how many groups to take.1 × 4 = 4
- Write the answer. The answer is a number of objects.4
Check. 4 + 28 = 32, so 1 of the 8 groups is 4. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.4 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.3 shaded parts
- Write the fraction. Top number over bottom number.3/4
Check. 3 shaded + 1 not shaded = 4 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 4 tells us how many equal groups to make.12 ÷ 4 = 3 in each group
- Take 3 of the 4 groups. The top number 3 tells us how many groups to take.3 × 3 = 9
- Write the answer. The answer is a number of objects.9
Check. 9 + 3 = 12, so 3 of the 4 groups is 9. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 5 tells us how many equal groups to make.15 ÷ 5 = 3 in each group
- Take 2 of the 5 groups. The top number 2 tells us how many groups to take.2 × 3 = 6
- Write the answer. The answer is a number of objects.6
Check. 6 + 9 = 15, so 2 of the 5 groups is 6. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 4 tells us how many equal groups to make.20 ÷ 4 = 5 in each group
- Take 3 of the 4 groups. The top number 3 tells us how many groups to take.3 × 5 = 15
- Write the answer. The answer is a number of objects.15
Check. 15 + 5 = 20, so 3 of the 4 groups is 15. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 3 tells us how many equal groups to make.9 ÷ 3 = 3 in each group
- Take 2 of the 3 groups. The top number 2 tells us how many groups to take.2 × 3 = 6
- Write the answer. The answer is a number of objects.6
Check. 6 + 3 = 9, so 2 of the 3 groups is 6. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 5 tells us how many equal groups to make.10 ÷ 5 = 2 in each group
- Take 3 of the 5 groups. The top number 3 tells us how many groups to take.3 × 2 = 6
- Write the answer. The answer is a number of objects.6
Check. 6 + 4 = 10, so 3 of the 5 groups is 6. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 8 tells us how many equal groups to make.16 ÷ 8 = 2 in each group
- Take 4 of the 8 groups. The top number 4 tells us how many groups to take.4 × 2 = 8
- Write the answer. The answer is a number of objects.8
Check. 8 + 8 = 16, so 4 of the 8 groups is 8. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 3 tells us how many equal groups to make.24 ÷ 3 = 8 in each group
- Take 1 of the 3 groups. The top number 1 tells us how many groups to take.1 × 8 = 8
- Write the answer. The answer is a number of objects.8
Check. 8 + 16 = 24, so 1 of the 3 groups is 8. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 5 tells us how many equal groups to make.20 ÷ 5 = 4 in each group
- Take 3 of the 5 groups. The top number 3 tells us how many groups to take.3 × 4 = 12
- Write the answer. The answer is a number of objects.12
Check. 12 + 8 = 20, so 3 of the 5 groups is 12. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Share 12 into 4 equal groups. The bottom number 4 tells us how many groups to make.
- Look at the bottom number. The bottom number is 4, so we make 4 groups.12 ÷ 4 = 3
- Check what Ben did. Ben divided by 3, which is not the bottom number.12 ÷ 3 = 4 (not the right division)
- Decide. Ben is not right. 1/4 of 12 is 3.No, 1/4 of 12 is 3
Check. Four groups of 3 make 12. ✓
Watch out. Divide by the bottom number. Each group is one fraction of the set.
Equivalent fractions
Equivalent fractions are different fractions that name the same amount. Priya ate 1/2 of her bar. Noah ate 2/4 of his bar. Both bars are the same size, so they ate the same amount.
Look at the bars below. Each bar is the same length, and the shaded part ends in the same place. So 1/2 = 2/4 = 4/8.
When each part is cut into 2 smaller parts, there are twice as many parts, and twice as many are shaded. The amount does not change.
Key ideas
- Equivalent fractions name the same amount, with different numbers.
- Bars show equivalent fractions: 1/2 = 2/4 = 4/8.
- Other examples: 1/3 = 2/6, and 1/4 = 2/8.
- When each part is cut into two, there are twice as many parts and twice as many are shaded.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?4 ÷ 2 = 2
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 2 = 2
- Write the answer. 2
Check. 1/2 and 2/4 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?6 ÷ 3 = 2
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 2 = 2
- Write the answer. 2
Check. 1/3 and 2/6 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottoms match with a multiplication. Then check whether the tops match too.
- Compare the bottom numbers. Find how many times bigger the second bottom number is.8 ÷ 4 = 2
- Do the same to the first top number. Multiply the top by the same number.1 × 2 = 2
- Decide. 2 is the same as 2, so the fractions are equal.Yes
Check. 1/4 and 2/8 show the same amount. ✓
Watch out. Check both the top and the bottom. Matching bottoms are not enough.
Watch out
Watch out: 2/4 is not the same as 2/3 just because both have a 2 on top. Equivalent fractions have the same amount shaded, not the same top number.
Exam tip
Draw two bars of the same length, one split in 2 and one split in 4. Shade them and see if the ends line up.
Exercise 4D Equivalent fractions
Non-calculatorKey words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?4 ÷ 2 = 2
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 2 = 2
- Write the answer. 2
Check. 1/2 and 2/4 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?6 ÷ 2 = 3
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 3 = 3
- Write the answer. 3
Check. 1/2 and 3/6 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?8 ÷ 2 = 4
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 4 = 4
- Write the answer. 4
Check. 1/2 and 4/8 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?6 ÷ 3 = 2
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 2 = 2
- Write the answer. 2
Check. 1/3 and 2/6 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?8 ÷ 4 = 2
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 2 = 2
- Write the answer. 2
Check. 1/4 and 2/8 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?10 ÷ 5 = 2
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 2 = 2
- Write the answer. 2
Check. 1/5 and 2/10 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Find how many times bigger the top number is. Then divide the bottom number by the same amount.
- Compare the top numbers. How many times bigger is the top number?2 ÷ 1 = 2
- Do the same to the bottom number. Divide the bottom by the same number. The amount stays the same.4 ÷ 2 = 2
- Write the answer. 2
Check. 2/4 and 1/2 both shade the same amount of the bar. ✓
Watch out. The top and bottom must change by the same amount.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Find how many times bigger the top number is. Then divide the bottom number by the same amount.
- Compare the top numbers. How many times bigger is the top number?4 ÷ 2 = 2
- Do the same to the bottom number. Divide the bottom by the same number. The amount stays the same.6 ÷ 2 = 3
- Write the answer. 3
Check. 4/6 and 2/3 both shade the same amount of the bar. ✓
Watch out. The top and bottom must change by the same amount.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Find how many times bigger the top number is. Then divide the bottom number by the same amount.
- Compare the top numbers. How many times bigger is the top number?6 ÷ 3 = 2
- Do the same to the bottom number. Divide the bottom by the same number. The amount stays the same.8 ÷ 2 = 4
- Write the answer. 4
Check. 6/8 and 3/4 both shade the same amount of the bar. ✓
Watch out. The top and bottom must change by the same amount.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottoms match with a multiplication. Then check whether the tops match too.
- Compare the bottom numbers. Find how many times bigger the second bottom number is.6 ÷ 3 = 2
- Do the same to the first top number. Multiply the top by the same number.1 × 2 = 2
- Decide. 2 is the same as 2, so the fractions are equal.Yes
Check. 1/3 and 2/6 show the same amount. ✓
Watch out. Check both the top and the bottom. Matching bottoms are not enough.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottoms match with a multiplication. Then check whether the tops match too.
- Compare the bottom numbers. Find how many times bigger the second bottom number is.8 ÷ 4 = 2
- Do the same to the first top number. Multiply the top by the same number.2 × 2 = 4
- Decide. 4 is not the same as 3, so the fractions are not equal.No
Check. 2/4 and 3/8 show different amounts. ✓
Watch out. Check both the top and the bottom. Matching bottoms are not enough.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottoms match with a multiplication. Then check whether the tops match too.
- Compare the bottom numbers. Find how many times bigger the second bottom number is.6 ÷ 2 = 3
- Do the same to the first top number. Multiply the top by the same number.1 × 3 = 3
- Decide. 3 is the same as 3, so the fractions are equal.Yes
Check. 1/2 and 3/6 show the same amount. ✓
Watch out. Check both the top and the bottom. Matching bottoms are not enough.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?10 ÷ 2 = 5
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 5 = 5
- Write the answer. 5/10
Check. 1/2 and 5/10 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?8 ÷ 4 = 2
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 2 = 2
- Write the answer. 2/8
Check. 1/4 and 2/8 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?9 ÷ 3 = 3
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 3 = 3
- Write the answer. 3/9
Check. 1/3 and 3/9 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom numbers the same. Then compare the amounts.
- Write the first fraction. Sam has 1/2 of his cake.1/2
- Write the second fraction with 2 equal parts in each half. Ella has 2/4 of an identical cake.2/4
- Compare. Each person has the same amount, because 2/4 = 1/2.1/2 = 2/4
- Decide. Neither has more. They have the same amount.Neither, they have the same amount
Check. Both cakes have half of the cake left. ✓
Watch out. Equivalent fractions can look different but be the same amount.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Find how many times bigger the top number is. Then divide the bottom number by the same amount.
- Compare the top numbers. How many times bigger is the top number?2 ÷ 1 = 2
- Do the same to the bottom number. Divide the bottom by the same number. The amount stays the same.6 ÷ 2 = 3
- Write the answer. 1/3
Check. 2/6 and 1/3 both shade the same amount of the bar. ✓
Watch out. The top and bottom must change by the same amount.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?6 ÷ 3 = 2
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.2 × 2 = 4
- Write the answer. 4/6
Check. 2/3 and 4/6 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Compare the two fractions by looking at the bars.
- Look at the first bar. It is cut into 2 equal parts. One part is shaded.1/2
- Look at the second bar. It is cut into 3 equal parts. Two parts are shaded.2/3
- Compare the amounts. The shaded part of 2/3 is bigger than 1/2 of the bar.2/3 is bigger than 1/2
- Decide. Sam is wrong. The fractions are not equal.No, 1/2 is smaller than 2/3
Check. Sam only looked at the top numbers. Equal fractions need the same amount shaded. ✓
Watch out. Two fractions with the same top number are not always equal. Look at the amount shaded.
Comparing and ordering fractions
We can compare fractions to say which is bigger or smaller.
Unit fractions: the bottom number tells how many equal parts the whole is cut into. The more parts, the smaller each part. So 1/2 is bigger than 1/3, and 1/3 is bigger than 1/4. The bigger the bottom number, the smaller the unit fraction.
Same bottom number: the parts are the same size, so the bigger top number is the bigger fraction. 5/8 is bigger than 3/8.
We use the signs > (greater than) and < (less than). The open end of the sign faces the bigger fraction.
Key ideas
- For unit fractions, the bigger the bottom number, the smaller the fraction: 1/2 > 1/3 > 1/4.
- For fractions with the same bottom number, the bigger top number is the bigger fraction: 5/8 > 3/8.
- Use bars to check. Shade them on bars of the same length.
- The sign > means greater than. The sign < means less than.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the two fractions. Use a bar to check if you need to.
- Look at the two fractions. We want to know which fraction is bigger.1/3 and 1/4
- Compare them. Both are unit fractions. The bigger the bottom number, the more parts, so each part is smaller.1/3 is bigger than 1/4
- Write the sign. The open end of the sign faces the bigger fraction.1/3 > 1/4
Check. Shade both on bars of the same length to check. ✓
Watch out. For unit fractions, a bigger bottom number is a smaller fraction. 1/8 is smaller than 1/3.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the fractions, then write them from smallest to largest.
- Write the fractions in a list. Look at the top and bottom numbers.1/2, 1/8, 1/4
- Compare the fractions. For unit fractions, the bigger the bottom number, the smaller the fraction.1/8 < 1/4 < 1/2
- Write the answer in order. Read the list from smallest to largest.1/8, 1/4, 1/2
Check. Each fraction is smaller than the one after it. ✓
Watch out. Do not put them in order by the bottom numbers alone. Think about the size of each part.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the fractions, then write them from largest to smallest.
- Write the fractions in a list. Look at the top and bottom numbers.3/8, 7/8, 5/8, 1/8
- Compare the fractions. With the same bottom number, the bigger top number is the bigger fraction.7/8 > 5/8 > 3/8 > 1/8
- Write the answer in order. Read the list from largest to smallest.7/8, 5/8, 3/8, 1/8
Check. Each fraction is bigger than the one after it. ✓
Watch out. Do not put them in order by the bottom numbers alone. Think about the size of each part.
Watch out
Watch out: many children think 1/8 is bigger than 1/3, because 8 is bigger than 3. It is the other way round. For unit fractions, a bigger bottom number means smaller pieces.
Exam tip
When you compare two fractions with the same bottom number, look only at the top numbers.
Exercise 4E Comparing and ordering fractions
Non-calculatorKey words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the two fractions. Use a bar to check if you need to.
- Look at the two fractions. We want to know which fraction is bigger.1/2 and 1/5
- Compare them. Both are unit fractions. The bigger the bottom number, the more parts, so each part is smaller.1/2 is bigger than 1/5
- Write the sign. The open end of the sign faces the bigger fraction.1/2 > 1/5
Check. Shade both on bars of the same length to check. ✓
Watch out. For unit fractions, a bigger bottom number is a smaller fraction. 1/8 is smaller than 1/3.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the two fractions. Use a bar to check if you need to.
- Look at the two fractions. We want to know which fraction is bigger.1/6 and 1/3
- Compare them. Both are unit fractions. The bigger the bottom number, the more parts, so each part is smaller.1/6 is smaller than 1/3
- Write the sign. The open end of the sign faces the bigger fraction.1/6 < 1/3
Check. Shade both on bars of the same length to check. ✓
Watch out. For unit fractions, a bigger bottom number is a smaller fraction. 1/8 is smaller than 1/3.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the two fractions. Use a bar to check if you need to.
- Look at the two fractions. We want to know which fraction is bigger.3/8 and 5/8
- Compare them. The bottom numbers are the same, so the parts are the same size. The bigger top number is the bigger fraction.3/8 is smaller than 5/8
- Write the sign. The open end of the sign faces the bigger fraction.3/8 < 5/8
Check. Shade both on bars of the same length to check. ✓
Watch out. For unit fractions, a bigger bottom number is a smaller fraction. 1/8 is smaller than 1/3.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the two fractions. Use a bar to check if you need to.
- Look at the two fractions. We want to know which fraction is bigger.7/10 and 4/10
- Compare them. The bottom numbers are the same, so the parts are the same size. The bigger top number is the bigger fraction.7/10 is bigger than 4/10
- Write the sign. The open end of the sign faces the bigger fraction.7/10 > 4/10
Check. Shade both on bars of the same length to check. ✓
Watch out. For unit fractions, a bigger bottom number is a smaller fraction. 1/8 is smaller than 1/3.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the fractions, then write them from smallest to largest.
- Write the fractions in a list. Look at the top and bottom numbers.1/3, 1/6, 1/4
- Compare the fractions. For unit fractions, the bigger the bottom number, the smaller the fraction.1/6 < 1/4 < 1/3
- Write the answer in order. Read the list from smallest to largest.1/6, 1/4, 1/3
Check. Each fraction is smaller than the one after it. ✓
Watch out. Do not put them in order by the bottom numbers alone. Think about the size of each part.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the fractions, then write them from smallest to largest.
- Write the fractions in a list. Look at the top and bottom numbers.1/5, 1/2, 1/10
- Compare the fractions. For unit fractions, the bigger the bottom number, the smaller the fraction.1/10 < 1/5 < 1/2
- Write the answer in order. Read the list from smallest to largest.1/10, 1/5, 1/2
Check. Each fraction is smaller than the one after it. ✓
Watch out. Do not put them in order by the bottom numbers alone. Think about the size of each part.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the fractions, then write them from largest to smallest.
- Write the fractions in a list. Look at the top and bottom numbers.2/7, 6/7, 4/7
- Compare the fractions. With the same bottom number, the bigger top number is the bigger fraction.6/7 > 4/7 > 2/7
- Write the answer in order. Read the list from largest to smallest.6/7, 4/7, 2/7
Check. Each fraction is bigger than the one after it. ✓
Watch out. Do not put them in order by the bottom numbers alone. Think about the size of each part.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the fractions, then write them from largest to smallest.
- Write the fractions in a list. Look at the top and bottom numbers.9/10, 2/10, 5/10
- Compare the fractions. With the same bottom number, the bigger top number is the bigger fraction.9/10 > 5/10 > 2/10
- Write the answer in order. Read the list from largest to smallest.9/10, 5/10, 2/10
Check. Each fraction is bigger than the one after it. ✓
Watch out. Do not put them in order by the bottom numbers alone. Think about the size of each part.
Key words. A unit fraction has 1 on the top, like 1/4. A non-unit fraction has a bigger number on the top, like 3/4.
Method. The biggest unit fraction has the smallest bottom number.
- Look at the bottom numbers. For unit fractions, the bottom number tells how many parts the whole is cut into.1/2, 1/3, 1/6
- Find the smallest bottom number. The fewest equal parts make the biggest piece.2 is the smallest
- Write the biggest fraction. 1/2
Check. The bar with 2 parts has the biggest pieces. ✓
Watch out. 1/2 is bigger than 1/6, even though 6 is a bigger number.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. For unit fractions, the biggest bottom number is the smallest piece.
- Write the amount each friend gets. Ali 1/8, Ben 1/4, Cara 1/6 and Dan 1/3 of the cake.1/8, 1/4, 1/6, 1/3
- Compare the bottom numbers. The bigger the bottom number, the smaller the piece.8 > 6 > 4 > 3
- Pick the smallest piece. The biggest bottom number makes the smallest piece.1/8 is the smallest
- Write the answer. Ali
Check. 1/8 < 1/6 < 1/4 < 1/3, so Ali has the smallest piece. ✓
Watch out. Do not choose the piece with the smallest number. Look at the bottom numbers.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. The same bottom number means the parts are the same size. Compare the top numbers.
- Look at Ella. Ella ran 3/5 of the race.3/5
- Look at Sam. Sam ran 4/5 of the same race.4/5
- Compare the top numbers. The bottom numbers are both 5, so the bigger top number runs further.4 > 3
- Decide. Sam ran further.Sam
Check. 4/5 is bigger than 3/5. ✓
Watch out. The bigger top number is the bigger fraction when the bottoms match.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. The smaller fraction is the one with less shaded.
- Look at Tom. Tom has 2/9 of a pizza.2/9
- Look at Lily. Lily has 7/9 of an identical pizza.7/9
- Compare the top numbers. The bottom numbers are both 9, so the smaller top number is less.2 < 7
- Decide. Tom has less pizza.Tom
Check. 2/9 is smaller than 7/9. ✓
Watch out. Compare the top numbers only when the bottom numbers are the same.
Key words. A unit fraction has 1 on the top, like 1/4. A non-unit fraction has a bigger number on the top, like 3/4.
Method. A unit fraction with a bigger bottom number is smaller. Check it with bars.
- Look at the bars. The 1/5 bar is cut into 5 parts. The 1/6 bar is cut into 6 parts.1/5 and 1/6
- Compare the parts. Each part of the 1/5 bar is bigger than each part of the 1/6 bar.1/5 > 1/6
- Decide. Mia is right. 1/5 is bigger than 1/6.Yes, 1/5 is bigger than 1/6
Check. 5 is smaller than 6, so the piece for 1/5 is bigger. ✓
Watch out. The bigger bottom number makes the smaller piece.
Key words. A unit fraction has 1 on the top, like 1/4. A non-unit fraction has a bigger number on the top, like 3/4.
Method. Test the statement with unit fractions.
- Try 1/2 and 1/4. The bottom number is bigger for 1/4.1/2 > 1/4
- Try 1/3 and 1/8. Again the bigger bottom number gives the smaller fraction.1/3 > 1/8
- Decide. It works every time for unit fractions.Always
Check. A bigger bottom number means more parts, and smaller parts. ✓
Watch out. Check the statement with two or three examples.
Adding and subtracting fractions
When two fractions have the same bottom number, the parts are the same size. We can add them by adding the top numbers. The bottom number stays the same.
2/7 + 3/7 = 5/7. Five sevenths of the jug are full.
We can subtract in the same way. 5/8 − 2/8 = 3/8. Take 2 eighths away from 5 eighths, and 3 eighths are left.
The answer must stay inside one whole. If the top number is the same as the bottom number, the answer is one whole: 8/8 = 1.
Key ideas
- When the bottom numbers are the same, add or subtract the top numbers.
- The bottom number stays the same: 2/7 + 3/7 = 5/7.
- Do not add the bottom numbers. 2/5 + 1/5 is 3/5, not 3/10.
- The answer stays within one whole. 3/8 + 5/8 = 8/8 = 1.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 7, so the parts are the same size.2/7 and 3/7
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.2 + 3 = 5
- Write the answer. Keep the bottom number 7.5/7
Check. 5/7 − 3/7 = 2/7. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Subtract the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 8, so the parts are the same size.5/8 and 2/8
- Subtract the top numbers. Take the smaller top number away from the bigger one. The bottom number stays the same.5 − 2 = 3
- Write the answer. Keep the bottom number 8.3/8
Check. 3/8 + 2/8 = 5/8. ✓
Watch out. Do not subtract the bottom numbers. The parts stay the same size.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 10, so the parts are the same size.3/10 and 4/10
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.3 + 4 = 7
- Write the answer. Keep the bottom number 10.7/10
Check. 7/10 − 4/10 = 3/10. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Watch out
Watch out: 1/3 + 1/3 = 2/3, not 2/6. Adding the top numbers changes how many parts there are. The size of the parts stays the same, so the bottom number stays the same.
Exam tip
When you add or subtract, check that the bottom numbers are the same. If they are, change only the top number.
Exercise 4F Adding and subtracting fractions
Non-calculatorKey words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 5, so the parts are the same size.1/5 and 2/5
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.1 + 2 = 3
- Write the answer. Keep the bottom number 5.3/5
Check. 3/5 − 2/5 = 1/5. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 9, so the parts are the same size.4/9 and 3/9
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.4 + 3 = 7
- Write the answer. Keep the bottom number 9.7/9
Check. 7/9 − 3/9 = 4/9. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Subtract the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 8, so the parts are the same size.7/8 and 2/8
- Subtract the top numbers. Take the smaller top number away from the bigger one. The bottom number stays the same.7 − 2 = 5
- Write the answer. Keep the bottom number 8.5/8
Check. 5/8 + 2/8 = 7/8. ✓
Watch out. Do not subtract the bottom numbers. The parts stay the same size.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Subtract the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 10, so the parts are the same size.7/10 and 3/10
- Subtract the top numbers. Take the smaller top number away from the bigger one. The bottom number stays the same.7 − 3 = 4
- Write the answer. Keep the bottom number 10.4/10
Check. 4/10 + 3/10 = 7/10. ✓
Watch out. Do not subtract the bottom numbers. The parts stay the same size.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 6, so the parts are the same size.2/6 and 3/6
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.2 + 3 = 5
- Write the answer. Keep the bottom number 6.5/6
Check. 5/6 − 3/6 = 2/6. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Subtract the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 9, so the parts are the same size.5/9 and 1/9
- Subtract the top numbers. Take the smaller top number away from the bigger one. The bottom number stays the same.5 − 1 = 4
- Write the answer. Keep the bottom number 9.4/9
Check. 4/9 + 1/9 = 5/9. ✓
Watch out. Do not subtract the bottom numbers. The parts stay the same size.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 4, so the parts are the same size.1/4 and 2/4
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.1 + 2 = 3
- Write the answer. Keep the bottom number 4.3/4
Check. 3/4 − 2/4 = 1/4. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 8, so the parts are the same size.3/8 and 5/8
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.3 + 5 = 8
- Write the answer. Keep the bottom number 8.8/8
Check. 8/8 − 5/8 = 3/8. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Use the inverse. Take the known part away from the total.
- Write the number sentence. The box is the missing part.3/7 + [ ] = 5/7
- Take the known part away from the total. Subtraction undoes addition.5 − 3 = 2
- Write the missing part. Keep the bottom number 7.2/7
Check. 3/7 + 2/7 = 5/7. ✓
Watch out. Subtract the top numbers. The bottom number stays the same.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Use the inverse. The missing part is the total minus what is left.
- Write the number sentence. The box is the part we take away.9/10 − [ ] = 4/10
- Find the difference. The missing part is the total minus what is left.9 − 4 = 5
- Write the missing part. Keep the bottom number 10.5/10
Check. 9/10 − 5/10 = 4/10. ✓
Watch out. Start with the whole amount, then take away. Do not add the two fractions.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 8, so the parts are the same size.2/8 and 3/8
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.2 + 3 = 5
- Write the answer. Keep the bottom number 8.5/8
Check. 5/8 − 3/8 = 2/8. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 10, so the parts are the same size.2/10 and 5/10
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.2 + 5 = 7
- Write the answer. Keep the bottom number 10.7/10
Check. 7/10 − 5/10 = 2/10. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Subtract the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 8, so the parts are the same size.8/8 and 5/8
- Subtract the top numbers. Take the smaller top number away from the bigger one. The bottom number stays the same.8 − 5 = 3
- Write the answer. Keep the bottom number 8.3/8
Check. 3/8 + 5/8 = 8/8. ✓
Watch out. Do not subtract the bottom numbers. The parts stay the same size.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Do not add the bottom numbers.
- Look at the two fractions. Both have the bottom number 3.1/3 + 1/3
- Add the top numbers. Add 1 and 1. The bottom number stays 3.1 + 1 = 2
- Write the answer. The answer is 2/3, not 2/6.No. 1/3 + 1/3 = 2/3
Check. Two thirds is the same as two of the three equal parts. ✓
Watch out. The bottom number is the size of each part. Adding does not make the parts smaller.
Chapter summary
| Skill | Method |
|---|---|
| Tenths | One whole cut into 10 equal parts. Each part is 1/10. Ten tenths make one whole. Sharing 4 things between 10 people gives 4/10 each. |
| Unit and non-unit fractions | A unit fraction has 1 on the top, like 1/4. A non-unit fraction has a bigger top number, like 3/4. Each fraction is a number between 0 and 1 on a number line. |
| Fractions of a set | Divide the set into equal groups using the bottom number. Then take the top number of groups. 3/4 of 12 is 3 × 3 = 9. |
| Equivalent fractions | Equivalent fractions name the same amount. Cut each part into smaller equal parts and shade the same amount: 1/2 = 2/4 = 4/8. |
| Comparing and ordering | For unit fractions, the bigger the bottom number, the smaller the fraction. With the same bottom number, the bigger top number is the bigger fraction. |
| Adding and subtracting | With the same bottom number, add or subtract the top numbers and keep the bottom number. The answer stays within one whole. |
Can you…?
- I can show tenths on a bar and count in tenths.
- I can name unit fractions and non-unit fractions, and place them on a number line.
- I can find a fraction of a set of objects by sharing it into equal groups.
- I can show, using bars, that two fractions are equivalent.
- I can compare and order unit fractions, and fractions with the same bottom number.
- I can add and subtract fractions with the same bottom number, within one whole.
- I can solve problems that use fractions.
Key terms
- Fraction
- A number that names one or more equal parts of a whole, or a point between 0 and 1 on a number line.
- Numerator
- The top number of a fraction. It tells how many parts we have.
- Denominator
- The bottom number of a fraction. It tells how many equal parts make the whole.
- Tenth
- One of ten equal parts of a whole. It is written 1/10.
- Unit fraction
- A fraction with 1 on the top, such as 1/3 or 1/8.
- Non-unit fraction
- A fraction with a number bigger than 1 on the top, such as 3/4 or 2/5.
- Equivalent fractions
- Different fractions that name the same amount, such as 1/2 and 2/4.
- Set
- A group of objects, such as 12 cubes.
- Compare
- To decide which of two numbers is bigger or smaller.
- Order
- To put numbers in a list, from smallest to largest or from largest to smallest.
Exercise 4Z Check-up
Non-calculatorKey words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.10 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.4 shaded parts
- Write the fraction. Top number over bottom number.4/10
Check. 4 shaded + 6 not shaded = 10 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 4 tells us how many equal groups to make.16 ÷ 4 = 4 in each group
- Take 1 of the 4 groups. The top number 1 tells us how many groups to take.1 × 4 = 4
- Write the answer. The answer is a number of objects.4
Check. 4 + 12 = 16, so 1 of the 4 groups is 4. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 5 tells us how many equal groups to make.20 ÷ 5 = 4 in each group
- Take 3 of the 5 groups. The top number 3 tells us how many groups to take.3 × 4 = 12
- Write the answer. The answer is a number of objects.12
Check. 12 + 8 = 20, so 3 of the 5 groups is 12. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?8 ÷ 2 = 4
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 4 = 4
- Write the answer. 4
Check. 1/2 and 4/8 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the two fractions. Use a bar to check if you need to.
- Look at the two fractions. We want to know which fraction is bigger.1/3 and 1/8
- Compare them. Both are unit fractions. The bigger the bottom number, the more parts, so each part is smaller.1/3 is bigger than 1/8
- Write the sign. The open end of the sign faces the bigger fraction.1/3 > 1/8
Check. Shade both on bars of the same length to check. ✓
Watch out. For unit fractions, a bigger bottom number is a smaller fraction. 1/8 is smaller than 1/3.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the fractions, then write them from smallest to largest.
- Write the fractions in a list. Look at the top and bottom numbers.1/3, 1/8, 1/5
- Compare the fractions. For unit fractions, the bigger the bottom number, the smaller the fraction.1/8 < 1/5 < 1/3
- Write the answer in order. Read the list from smallest to largest.1/8, 1/5, 1/3
Check. Each fraction is smaller than the one after it. ✓
Watch out. Do not put them in order by the bottom numbers alone. Think about the size of each part.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 9, so the parts are the same size.5/9 and 2/9
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.5 + 2 = 7
- Write the answer. Keep the bottom number 9.7/9
Check. 7/9 − 2/9 = 5/9. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Subtract the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 10, so the parts are the same size.9/10 and 4/10
- Subtract the top numbers. Take the smaller top number away from the bigger one. The bottom number stays the same.9 − 4 = 5
- Write the answer. Keep the bottom number 10.5/10
Check. 5/10 + 4/10 = 9/10. ✓
Watch out. Do not subtract the bottom numbers. The parts stay the same size.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottoms match with a multiplication. Then check whether the tops match too.
- Compare the bottom numbers. Find how many times bigger the second bottom number is.4 ÷ 2 = 2
- Do the same to the first top number. Multiply the top by the same number.1 × 2 = 2
- Decide. 2 is the same as 2, so the fractions are equal.Yes
Check. 1/2 and 2/4 show the same amount. ✓
Watch out. Check both the top and the bottom. Matching bottoms are not enough.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 4 tells us how many equal groups to make.24 ÷ 4 = 6 in each group
- Take 1 of the 4 groups. The top number 1 tells us how many groups to take.1 × 6 = 6
- Write the answer. The answer is a number of objects.6
Check. 6 + 18 = 24, so 1 of the 4 groups is 6. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Total 19 marks. Allow about 19 minutes.
Exercise 4X Practice test questions
Non-calculatorKey words. The bottom number (the denominator) is how many equal parts the whole is cut into. The top number (the numerator) is how many of those parts we have.
Method. Count all the equal parts for the bottom number, then count the shaded parts for the top number.
- Count all the equal parts. The bottom number tells how many equal parts the whole is cut into.5 equal parts
- Count the shaded parts. The top number tells how many of those parts we have.3 shaded parts
- Write the fraction. Top number over bottom number.3/5
Check. 3 shaded + 2 not shaded = 5 equal parts. ✓
Watch out. The bottom number counts every equal part, shaded or not.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 6 tells us how many equal groups to make.18 ÷ 6 = 3 in each group
- Take 1 of the 6 groups. The top number 1 tells us how many groups to take.1 × 3 = 3
- Write the answer. The answer is a number of objects.3
Check. 3 + 15 = 18, so 1 of the 6 groups is 3. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?6 ÷ 3 = 2
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.2 × 2 = 4
- Write the answer. 4
Check. 2/3 and 4/6 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. Equivalent fractions are equal. They have different numbers, but they name the same amount.
Method. Make the bottom number the new one. Then do the same multiplication to the top number.
- Compare the bottom numbers. How many times bigger is the new bottom number?6 ÷ 3 = 2
- Do the same to the top number. Multiply the top by the same number. The amount stays the same.1 × 2 = 2
- Write the answer. 2/6
Check. 1/3 and 2/6 are equal: both shade the same amount of the bar. ✓
Watch out. Multiply the top and the bottom by the same number. Do not add to them.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 5 tells us how many equal groups to make.25 ÷ 5 = 5 in each group
- Take 2 of the 5 groups. The top number 2 tells us how many groups to take.2 × 5 = 10
- Write the answer. The answer is a number of objects.10
Check. 10 + 15 = 25, so 2 of the 5 groups is 10. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the two fractions. Use a bar to check if you need to.
- Look at the two fractions. We want to know which fraction is bigger.4/10 and 7/10
- Compare them. The bottom numbers are the same, so the parts are the same size. The bigger top number is the bigger fraction.4/10 is smaller than 7/10
- Write the sign. The open end of the sign faces the bigger fraction.4/10 < 7/10
Check. Shade both on bars of the same length to check. ✓
Watch out. For unit fractions, a bigger bottom number is a smaller fraction. 1/8 is smaller than 1/3.
Key words. < means less than. > means greater than. The open end of the sign faces the bigger number.
Method. Compare the fractions, then write them from smallest to largest.
- Write the fractions in a list. Look at the top and bottom numbers.1/4, 1/10, 1/2
- Compare the fractions. For unit fractions, the bigger the bottom number, the smaller the fraction.1/10 < 1/4 < 1/2
- Write the answer in order. Read the list from smallest to largest.1/10, 1/4, 1/2
Check. Each fraction is smaller than the one after it. ✓
Watch out. Do not put them in order by the bottom numbers alone. Think about the size of each part.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 9, so the parts are the same size.5/9 and 3/9
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.5 + 3 = 8
- Write the answer. Keep the bottom number 9.8/9
Check. 8/9 − 3/9 = 5/9. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 8 tells us how many equal groups to make.16 ÷ 8 = 2 in each group
- Take 3 of the 8 groups. The top number 3 tells us how many groups to take.3 × 2 = 6
- Write the answer. The answer is a number of objects.6
Check. 6 + 10 = 16, so 3 of the 8 groups is 6. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. A set is a group of objects. To find a fraction of a set, share the objects into equal groups.
Method. Divide the set into equal groups using the bottom number. Then take the top number of groups.
- Share the set into equal groups. The bottom number 8 tells us how many equal groups to make.16 ÷ 8 = 2 in each group
- Take 5 of the 8 groups. The top number 5 tells us how many groups to take.5 × 2 = 10
- Write the answer. The answer is a number of objects.10
Check. 10 + 6 = 16, so 5 of the 8 groups is 10. ✓
Watch out. Divide by the bottom number first. The answer to the division is one group, not the answer to the question.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Add the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 10, so the parts are the same size.2/10 and 4/10
- Add the top numbers. Only the top numbers are added. The bottom number stays the same.2 + 4 = 6
- Write the answer. Keep the bottom number 10.6/10
Check. 6/10 − 4/10 = 2/10. ✓
Watch out. Do not add the bottom numbers. Adding the tops of 1/5 and 2/5 gives 3/5, not 3/10.
Key words. When fractions have the same bottom number, we add or subtract the top numbers. The bottom number stays the same.
Method. Subtract the top numbers. Keep the bottom number the same.
- Check the bottom numbers. Both fractions have the bottom number 10, so the parts are the same size.10/10 and 6/10
- Subtract the top numbers. Take the smaller top number away from the bigger one. The bottom number stays the same.10 − 6 = 4
- Write the answer. Keep the bottom number 10.4/10
Check. 4/10 + 6/10 = 10/10. ✓
Watch out. Do not subtract the bottom numbers. The parts stay the same size.