Adding and subtracting fractions Adding fractions with different denominators Multiplying fractions by integers Multiplying a fraction by a fraction Dividing a fraction by a fraction

Adding and subtracting fractions When fractions have the same denominator it is quite easy to add them together and to subtract them. For example, 3 5 1 5 3 + 1 5 4 5 + = = Show the example on the slide and emphasise that when the denominator is the same we can add together the numerators. Show that by writing (3 + 1) over 5 using a single bar we can avoid adding the denominators together by mistake. Show the calculation as a diagram.

Adding and subtracting fractions When fractions have the same denominator it is quite easy to add them together and to subtract them. 7 8 – 3 7 – 3 8 4 8 1 1 2 = = = 2 Fractions should always be cancelled down to their lowest terms. Talk through the example on the board and remind pupils that fractions should always be cancelled down to their lowest terms. The 4 and the 8 in 4/8 are both divisible by 4. Cancelling gives us 1/2.

Adding and subtracting fractions 1 9 + 7 4 1 + 7 + 4 9 12 9 1 3 9 1 1 3 = = = = 3 Improper fractions (top-heavy fractions) should be written as mixed numbers. Explain that for the fraction 12/9, 12 is bigger than 9. This is an improper (or top-heavy) fraction. 9/9 make one whole plus 3/9 left over. Again, remind pupils that fractions must be cancelled if possible.

Adding fractions with different denominators What is + 1 9 ? 3 4 5 12 1) Write any mixed numbers as improper fractions. 1 3 4 7 4 = 2) Find the lowest common multiple of 4, 9 and 12. Talk though the step needed to add and subtract fractions without using diagrams. Revise the method for changing mixed numbers to improper fractions. (This step could be left out if we add whole number parts together separately from fractional parts). We need to write the fractions over a common denominator. The lowest common denominator is equal to the lowest common multiple of the denominators. Tell pupils that one way to do this is to go through the multiples of the highest denominator, in this case 12, until we find a multiple of 12 which is divisible by the other denominators, in this case 4 and 9. Going through multiples of 12. For example, 12 is divisible by 4 but not by 9, 24 is divisible by 4 but not by 9, 36 is divisible by both 4 and 9. Therefore, 36 is the lowest common multiple. The multiples of 12 are: 12, 24, 36 . . . 36 is the lowest common denominator.

Adding fractions with different denominators What is + 1 9 ? 3 4 5 12 3) Write each fraction over the lowest common denominator. ×9 ×4 ×3 7 4 63 1 9 4 5 12 15 = = = 36 36 36 ×9 ×4 ×3 Review the method for finding equivalent fractions. Remind pupils that the answer should be written as a mixed number and cancelled down if possible. 4) Add the fractions together. 36 63 + 4 15 = 36 63 + 4 + 15 = 36 82 = 2 36 10 = 2 18 5

Multiplying fractions by integers When we multiply a fraction by an integer we: multiply by the numerator and divide by the denominator For example, This is equivalent to of 54. 4 9 4 9 54 × Remind pupils that an integer is whole number that can be positive or negative or 0. Stress that it does not matter what order we use to multiply and divide. When the denominator divides exactly into the number we are multiplying by, it is easiest to divide first and then multiply. In this example, we would get the same answer if we multiplied 54 by 4 and then divided by 9. However, if we divide first the numbers are smaller and easier to work out mentally. If the denominator does not divide into the number we are multiplying by we can multiply first and then divide to write the answer as a mixed number. Stress that multiplying 54 by 4/9 is the same as finding 4/9 of 54. = 54 ÷ 9 × 4 = 6 × 4 = 24

Using cancellation to simplify calculations 7 12 What is 16 × ? We can write 16 × as: 7 12 4 16 1 × 7 12 28 = If the number we are multiplying by and the denominator of the fraction share a common factor, then we can cancel the common factor before we multiply. Explain that when we multiply two fractions together we can cancel any of the numerators with any of the denominators before multiplying. Point out that if we didn’t cancel in the first step we would still get the same answer, but the numbers would be more difficult to work out mentally and we would still have to cancel at the end. 16 × 7 is 112. We would then have to cancel 112/12 to 28/3. 3 3 = 1 3 9

Multiplying a fraction by a fraction 5 6 What is × ? 12 25 Start by writing the calculation with any mixed numbers as improper fractions. To make the calculation easier, cancel any numerators with any denominators. 7 2 12 25 35 6 × = 14 5 1 5 = 2 4 5

Dividing a fraction by a fraction 2 3 4 5 What is ÷ ? To divide by a fraction we multiply by the denominator and divide by the numerator. Swap the numerator and the denominator and multiply. 4 5 2 3 ÷ 5 4 2 3 × can be written as This is the reciprocal of . 4 5 Explain that when we are dividing by a fraction we can write an equivalent calculation by swapping the numerator and the denominator around (turning the fraction upside-down) and multiplying. This works because when we multiply by a fraction we multiply by the numerator and divide by the denominator. Multiplying by a fraction is straight forward because we simply multiply the numerators together and multiply the denominators together. 5 4 2 3 × = 10 12 = 5 6

Multiplying and dividing by fractions Multiplying and dividing are inverse operations. multiply by the numerator and divide by the denominator When we multiply by a fraction we: When we divide by a fraction we: divide by the numerator and multiply by the denominator