Pupils should be taught to:
•use common factors to simplify fractions; use common multiples to express fractions in the same denomination.
•add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions.
•multiply simple pairs of proper fractions, writing the answer in its simplest form.
•divide proper fractions by whole numbers. A fraction shows a whole thing that has been broken into equal parts . On their own, they show something which is less than one whole e.g. ¼, ¾, ½ etc…
Numerator- this number shows how many parts of the whole is 3 being represented by the fraction.
Denominator- This number shows how many equal parts the 4 whole thing has been split into. Two important concepts for calculating with fractions are factors and multiples. We will see why later on. Using 12 as our example number, here are the definitions for these concepts and 12’s factors and multiples.
FACTORS MULTIPLES A whole number that divides exactly The product of multiplying a number by into another whole number. an integer (whole number). Groups of a given number. 12 1, 2, 3, 4, 6, 12 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, etc … When we calculate with fractions, we can sometimes end up with large numerators and denominators. If we can, we should simplify these to express them with an equivalent fraction that has the smallest numerator and denominator possible. 9 ÷ 3 3
12 ÷ 3 4 There are two main methods of simplifying fractions. See the next page for how to do this… When we calculate with fractions, we can sometimes end up with large numerators and denominators. If we can, we should simplify these to express them with an equivalent fraction that has the smallest numerator and denominator possible. There are two ways to do this. The first is quickest!
2) Repeatedly divide by the smallest 1) Finding Highest Common Factors prime number possible.
9 To simplify , write down all the factors of For harder numbers, or if you find it 12 9 and 12 and then choose the highest one difficult to find factors, divide by 2 as many that they both share to divide by. times as possible, then try 3, then 5 etc… Factors ÷ 2 ÷ 2 ÷ 3 9 12 ÷ 3 1 1 9 3 24 12 6 2 3 2 9 3 = = = 4 = 6 108 54 27 9 12 ÷ 3 12 4 ÷ 2 ÷ 2 ÷ 3 There are two ways that we can use fractions to show a quantity that is greater than a whole. We need to be able to convert from improper fractions to mixed numbers (and sometimes vice versa) when calculating using fractions.
What fraction of these whole shapes are white?
Improper Fractions Mixed Numbers In improper fractions, the numerator is Mixed numbers are used to show where there greater than the denominator. This shows are whole numbers and fractions. The whole that you have more than enough parts to ones are shown in numbers before any make at least one whole. additional parts shown as fractions.
The quantity shown in the diagram above can be written in 15 these two ways… 3 3 4 4 Reciprocals are useful things when dividing fractions. The reciprocal of a number is 1 divided by that number. The reciprocal of a fraction is the fraction turned upside down. Simple! 3 4 So the reciprocal of is 4 3 For whole numbers, to work out the reciprocal of 4: Roll 4 over! We can think of 4 as 1 1 So 4 turned upside down is 4 Adding Fractions with the Same Denominator
Adding Fractions Where One has a Denominator that is a Factor of the Other
Adding Fractions Where One Denominator is Not a Factor of the Other 1 2 + = ? 5 5
Remember! Don’t add the denominators together! Adding fractions is as 1 2 easy as adding + = ? whole numbers! 5 5
Step 1- The only things that are added together 1 2 are the numerators. The denominator will stay + = the same. (Unless we simplify the answer later). 5 5 5
1 + 2 = Step 2- Add the numerators together. + = 5 5 5 1 2 3 Step 3- Show as the answer to the question. + = 5 5 5 1 4 + = ? 5 10
If one of the denominators is a multiple of the other, you can complete the calculation by converting one to the other. (Usually the smaller one converts to the larger.) Adding fractions is as easy as adding 1 4 whole numbers! + = ? x2 5 10
Step 1- Convert one fraction to the same denominator x2 as the other (if it is a factor of it) by multiplying the 2 4 numerator by the same number as the denominator + = ? must be multiplied by. 10 10
2 + 4 = 6 Step 2- Add only the numerators together. + = 10 10 10
Step 3- Simplify the answer where possible by 6 3 dividing the numerator and denominator by = their highest common factor. 10 5 1 4 + = ? 3 10
To add these together, both fractions must be converted. Look for the lowest common multiple of each of the denominators and convert to that fraction by multiplying the numerator by the same numbers as their denominators. Adding fractions is as 1 4 easy as adding + whole numbers! x10 3 10 x3
Step 1- Convert both fractions to the same x10 x3 denominator by finding the lowest common multiple 10 12 of the two numbers. Multiply the numerators by the + same number as the denominator to convert them. 30 30
10 12 22 Step 2- Add only the numerators together. + = 30 30 30 ÷ 2
Step 3- Simplify the answer where possible by 22 11 dividing the numerator and denominator by = their highest common factor. 30 15 ÷ 2 Subtracting Fractions with the Same Denominator
Subtracting Fractions Where One has a Denominator that is a Factor of the Other
Subtracting Fractions Where One Denominator is Not a Factor of the Other
Subtracting Fractions from Mixed Numbers 7 3 − = ? 8 8
Remember! Don’t subtract the denominators away from each other! If you can add 7 3 fractions, you can − = ? also subtract them! 8 8
Step 1- You do not have to alter the fractions if 7 3 the denominators are the same. − = 8 8
- = Step 2- Simply subtract the numerator away 7 3 from the other. − = 8 8 8 7 3 4 Step 3- Show as the answer to the question. − = 8 8 8 9 2 − = ? 15 5
If one of the denominators is a multiple of the other, you can complete the calculation by converting one to the other. (Usually the smaller one converts to the larger.) This is the same process 9 2 as the addition with − similar denominators. 15 5 x3
Step 1- Convert one fraction to the same denominator x3 as the other (if it is a factor of it) by multiplying the 9 6 numerator by the same number as the denominator − must be multiplied by. 15 15
= 9 - 6 3 Step 2- Subtract only the numerators. − = 15 15 15 ÷ 3
Step 3- Simplify the answer where possible by 3 1 dividing the numerator and denominator by = their highest common factor. 15 5 ÷ 3 6 1 − = ? 7 4
If you can add numbers like this, you can subtract numbers just as easy by converting both fractions using the lowest common multiple. Adding fractions is as 6 1 easy as adding − whole numbers! x4 7 4 x7
Step 1- Convert both fractions to the same x4 x7 denominator by finding the lowest common multiple 24 7 of the two numbers. Multiply the numerators by the − same number as the denominator to convert them. 28 28
24 7 17 Step 2- Subtract one numerator away from the − = other in the usual way. 28 28 28
17 Step 3- Simplify the answer where possible. Be aware: it is not always possible. as it isn’t here! 28 1 2 2 − =? 3 3
The easiest way to compete calculations like this is to convert the mixed numbers into improper fractions. Converting from mixed numbers to improper fractions is easy! 1 1
1 3 3 1 7 2 = + + = 2 − = ? 3 3 3 3 3 3 4
Step 1- Convert the mixed number to an improper 7 1 fraction. Think of the whole numbers in terms of the 3 fraction. 2 whole ones equal . − = ? 3 3 4
Step 2- Convert both fractions to the same 28 3 25 denominator by finding the lowest common − = multiple. 12 12 12
Step 3- Convert the improper fraction back to a mixed number: 25 1 25 12 12 1 1 = + + = 2 = 2 12 12 12 12 12 Simplify the fraction, if this is possible. 12 12 Multiplying Fractions by Fractions
Multiplying Fractions by Whole Numbers
Multiplying Mixed Numbers by Whole Numbers
3 1 푥 = ? 9 3
Fractions can be multiplied together easily. Just multiply the numerators together and then the denominators together! If you know your 3 1 tables, you know 푥 = ? how to do this! 9 3 x = Step 1- Multiply the numerators together and 3 1 the denominators together. 푥 = 9 3 x = 3 1 3 Step 2- Write the answers to these calculations as the new numerator and denominator. 푥 = 9 3 27 3 ÷ 3 1 Step 3- Simplify the fraction if possible by dividing the numerator and the denominator = by their highest common factor. 27 ÷ 3 9 3 푥 6 = ? 5
To multiply a fraction by a whole number (integer), just turn the whole number into a fraction and do the same as on the previous page. Multiplying fractions is easy! Follow this simple guide!
Step 1- Convert the whole number to an improper fraction with a denominator of 1.
Step 2- Multiply the numerators together and the denominators together and write the answer as an improper fraction.
Step 3- Convert the improper fraction to a mixed number. 3 2 푥 6 = ? 5
Complete this type of calculation in stages. Partition the whole number and the fraction and then do the same as in the previous question type. Just one more step to do to multiply mixed numbers by whole numbers.
Step 1- Partition the mixed number into a whole number and a fraction. Multiply the whole numbers first.
Step 2- Next, multiply the fraction with the whole number by converting it to an improper fraction with a denominator of 1.
Step 3- Convert the improper fraction to a mixed number.
Step 4- Add the whole number answer to the mixed number answer to recombine. Dividing Fractions by Fractions
Dividing Fractions by Whole Numbers
Dividing Whole Numbers by Fractions 2 1 ÷ = ? 3 3
To divide by a fraction, turn the divisor on its head to generate the reciprocal. Then, just multiply them together! If you can multiply 2 1 fractions, you can ÷ = ? divide them easily! 3 3
Step 1- Use the reciprocal of the divisor (the 2 3 number you divide by) only! The dividend 푥 = ? (the number you are dividing) is kept the 3 1 same. x = 2 3 6 Step 2- Multiply the numerators together and the denominators together and write the 푥 = answer as an improper fraction. 3 x 1 = 3
Step 3- Convert the improper fraction to a 6 mixed number (or whole number in this case). = 2 3 2 ÷ 3 = ? 3
Dividing fractions by whole numbers can be done easily by converting the whole number to a fraction and using the reciprocal. Just one additional 2 step to doing the same job with whole ÷ 3 = ? number divisors. 3 2 3 Step 1- Turn the whole number into a fraction. ÷ = ? 3 1
Step 2- Turn the divisor upside down to use 2 1 the reciprocal. Multiplying by the reciprocal is 푥 = ? the same as dividing by the original number. 3 3 x = Step 3- Multiply the numerator by the 2 1 2 numerator and the denominator by the 푥 = denominator. 3 3 9 x = Step 4- If possible, simplify the fraction by 2 dividing the numerator and the denominator by their highest common factor! 9 1 9 ÷ = ? 3
For this question type, turn the whole number (integer) into a fraction and then use the reciprocal of the divisor to multiply and find your answer. You will notice something odd about 1 the answers to these 9 ÷ = ? questions. The number 3 gets bigger! 9 1 Step 1- Turn the whole number into a fraction. ÷ = ? 1 3
Step 2- Turn the divisor upside down the use 9 3 the reciprocal. Multiplying by the reciprocal is 푥 = ? the same as dividing by the original number. 1 1 x = Step 3- Multiply the numerator by the 9 3 27 numerator and the denominator by the 푥 = denominator. 1 x 1 = 1 27 Step 4- Convert the improper fraction to a = 27 mixed number (or an integer). 1 Use this space to make notes about any other methods that you find useful or any really difficult examples that you come across.