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KS4 Mathematics

KS4 Mathematics. N3 Fractions. N3 Fractions. Contents. A. N3.2 Finding fractions of quantities. A. N3.1 Equivalent fractions. N3.3 Comparing and ordering fractions. A. N3.4 Adding and subtracting fractions. A. N3.5 Multiplying and dividing fractions. A. Equivalent fractions. × 2.

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KS4 Mathematics

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  1. KS4 Mathematics N3 Fractions

  2. N3 Fractions Contents • A N3.2 Finding fractions of quantities • A N3.1 Equivalent fractions N3.3 Comparing and ordering fractions • A N3.4 Adding and subtracting fractions • A N3.5 Multiplying and dividing fractions • A

  3. Equivalent fractions ×2 ×3 3 6 18 = 4 8 24 ×2 ×3 Look at this diagram: =

  4. Equivalent fractions ×3 ×4 2 6 24 3 9 36 ×3 ×4 Look at this diagram: = =

  5. Equivalent fractions ÷3 ÷2 18 6 3 30 10 5 ÷3 ÷2 Look at this diagram: = =

  6. Equivalent fractions spider diagram

  7. Cancelling fractions to their lowest terms Which of these fractions are expressed in their lowest terms? 14 20 3 15 14 32 16 27 13 21 35 15 A fraction is said to be expressed in its lowest terms if the numerator and the denominator have no common factors. 7 5 2 8 7 5 Fractions which are not shown in their lowest terms can be simplified by cancelling.

  8. Mixed numbers and improper fractions 15 15 15 is an improper fraction. 4 4 4 3 3 4 When the numerator of a fraction is larger than the denominator it is called an improper fraction. For example, We can write improper fractions as mixed numbers. can be shown as =

  9. Improper fraction to mixed numbers + + + + = 1 1 1 1 8 8 8 5 8 5 = + + + + 8 8 8 8 8 8 37 37 37 = 8 8 8 4 4 4 4 5 5 5 = 8 8 Convert to a mixed number. This number is the remainder. 4 37 ÷ 8 = 4 remainder 5 This is the number of times 8 divides into 37.

  10. Mixed numbers to improper fractions 3 3 1 1 1 = + + + 7 7 7 2 2 2 2 7 7 7 7 7 7 7 = + + + 23 = 7 7 Convert to a mixed number. … and add this number … To do this in one step, 3 3 … to get the numerator. 2 2 23 = 7 7 Multiply these numbers together …

  11. Find the missing number

  12. N3 Fractions Contents N3.1 Equivalent fractions • A • A N3.2 Finding fractions of quantities N3.3 Comparing and ordering fractions • A N3.4 Adding and subtracting fractions • A N3.5 Multiplying and dividing fractions • A

  13. Finding a fraction of an amount 2 2 What is of £18? 3 3 of £18 We can see this in a diagram: = £18 ÷ 3 × 2 = £12

  14. Finding a fraction of an amount What is of £20? 7 10 7 of £20 10 Let’s look at this in a diagram again: = £20 ÷ 10 × 7 = £14

  15. Finding a fraction of an amount 5 1 5 What is of £24? 6 6 6 of £24 = of £24 × 5 = £24 ÷ 6 × 5 = £4 × 5 = £20

  16. Finding a fraction of an amount To find of an amount we can multiply by 4 and divide by 7. 4 4 1 7 7 7 36 5 kg = kg 7 What is of 9 kg? We could also divide by 7 and then multiply by 4. 4 × 9 kg = 36 kg 36 kg ÷ 7 =

  17. Finding a fraction of an amount 2 3 multiply by the numerator and divide by the denominator of 18 litres When we work out a fraction of an amount we For example, = 18 litres ÷ 3 × 2 = 6 litres × 2 = 12 litres

  18. Finding a fraction of an amount 2 5 To find of an amount we need to add 1 times the amount to two fifths of the amount. 1 2 2 2 7 5 5 5 5 of 3.5 m = 1 so, of 3.5 m = We could also multiply by 1 What is of 3.5m? 1 × 3.5 m = 3.5 m and 1.4 m 3.5 m + 1.4 m = 4.9 m

  19. MathsBlox

  20. N3 Fractions Contents N3.1 Equivalent fractions • A N3.2 Finding fractions of quantities • A N3.3 Comparing and ordering fractions • A N3.4 Adding and subtracting fractions • A N3.5 Multiplying and dividing fractions • A

  21. Using decimals to compare fractions Which is bigger or ? 7 7 20 20 7 20 3 3 3 8 8 8 > We can compare two fractions by converting them to decimals. = 3 ÷ 8 = 0.375 = 7 ÷ 20 = 0.35 0.375 > 0.35 so

  22. Using equivalent fractions 5 5 5 5 12 24 12 12 24 12 Now, write and as equivalent fractions over 24. 3 3 3 3 ×2 ×3 8 8 8 8 = < = ×2 ×3 Which is bigger or ? Another way to compare two fractions is to convert them to equivalent fractions. First we need to find the lowest common multiple of 8 and 12. The lowest common multiple of 8 and 12 is 24. 9 10 and so,

  23. Ordering fractions

  24. Mid-points

  25. N3 Fractions Contents N3.1 Equivalent fractions • A N3.2 Finding fractions of quantities • A N3.4 Adding and subtracting fractions N3.3 Comparing and ordering fractions • A • A N3.5 Multiplying and dividing fractions • A

  26. Adding and subtracting fractions 3 + 1 3 1 4 5 5 5 5 When fractions have the same denominator it is quite easy to add them together and to subtract them. For example, + = = We can show this calculation in a diagram: + =

  27. Adding and subtracting fractions 7 – 3 4 – 8 8 3 1 7 2 8 8 1 = = = 2 Fractions should always be cancelled down to their lowest terms. We can show this calculation in a diagram: = –

  28. Adding and subtracting fractions + + 1 3 4 1 7 9 3 9 9 9 1 + 7 + 4 12 1 1 9 9 1 = = = = 3 Top-heavy or improper fractions should be written as mixed numbers. Again, we can show this calculation in a diagram: = + +

  29. Fractions with common denominators 11 11 4 4 5 , and 12 12 12 12 12 5 11 + 4 + 5 8 2 1 1 20 12 12 12 3 12 Fractions are said to have a common denominator if they have the same denominator. For example, all have a common denominator of 12. We can add them together: = + + = = =

  30. Fractions with different denominators 2 9 What is – ? 15 11 4 15 – 4 18 18 18 18 5 6 Fractions with different denominators are more difficult to add and subtract. For example, We can show this calculation using diagrams: – = – = =

  31. Using diagrams 3 4 What is + ? 12 15 27 7 1 12 + 15 20 20 20 20 20 3 5 + = + = = =

  32. Using diagrams 1 4 7 1 10 25 14 25 – 14 11 20 20 20 20 What is – ? – = – = =

  33. Using a common denominator 3 5 7 1 3 1 + What is + ? 4 12 9 4 1 4 1) Write any mixed numbers as improper fractions. = 2) Find the lowest common multiple of 4, 9 and 12. The multiples of 12 are: 12, 24, 36 . . . 36 is the lowest common denominator.

  34. Using a common denominator 5 7 1 1 3 1 + What is + ? 12 4 9 9 4 ×4 ×9 ×3 5 63 82 10 4 15 12 36 18 36 36 36 36 36 36 36 ×9 ×3 ×4 5 2 2 63 + 4 + 15 = + + = = = 36 3) Write each fraction over the lowest common denominator. 63 4 15 = = = 4) Add the fractions together.

  35. Adding and subtracting fractions

  36. Using a calculator It is also possible to add and subtract fractions using the key on a calculator. a a b b c c we can key in For example, to enter 4 8 = Pressing the key converts this to: 4 8 The calculator displays this as:

  37. Using a calculator 2 4 3 5 a a b b + c c 2 3 + = 5 4 15 7 1 We write this as To calculate: using a calculator, we key in: The calculator will display the answer as:

  38. Fraction cards

  39. N3 Fractions Contents N3.1 Equivalent fractions • A N3.2 Finding fractions of quantities • A N3.5 Multiplying and dividing fractions N3.3 Comparing and ordering fractions • A N3.4 Adding and subtracting fractions • A • A

  40. Multiplying fractions by integers 4 9 multiply by the numerator and 4 9 divide by the denominator This is equivalent to of 54. 54 × When we multiply a fraction by an integer we: For example, = 54 ÷ 9 × 4 = 6 × 4 = 24

  41. Multiplying fractions by integers 4 5 5 7 7 7 What is 12 × ? 12 × 60 = 7 8 = = 12 × 5 ÷ 7 = 60 ÷ 7

  42. Using cancellation to simplify calculations 7 7 7 What is 16 × ? 12 12 12 We can write 16 × as: 16 × 1 3 9 1 = 3 4 28 = 3

  43. Using cancellation to simplify calculations 8 8 What is × 40? 25 25 We can write × 40 as: 8 40 25 1 64 × 5 12 4 = 5 8 = 5

  44. Multiplying a fraction by a fraction 3 3 4 2 8 8 5 5 = × 40 3 = 10 What is × ? To multiply two fractions together, multiply the numerators together and multiply the denominators together: 3 12 10 We could also cancel at this step.

  45. Multiplying a fraction by a fraction 12 5 5 4 What is × ? 25 6 5 35 12 × = 6 25 5 2 = 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 14 1 5

  46. Multiplying fractions

  47. Dividing an integer by a fraction 1 1 1 3 3 3 What is 4 ÷ ? 4 ÷ means, “How many thirds are there in 4?” 4 ÷ = 12 Here are 4 rectangles: Let’s divide them into thirds.

  48. Dividing an integer by a fraction 2 2 2 What is 4 ÷ ? 5 5 5 4 ÷ means, “How many two fifths are there in 4?” 4 ÷ = 10 Here are 4 rectangles: Let’s divide them into fifths, and count the number of two fifths.

  49. Dividing an integer by a fraction 3 3 1 3 3 What is 6 ÷ ? 4 4 4 4 4 6 ÷ means, ‘How many three quarters are there in six?’ 6 ÷ = 6 × 4 6 ÷ = 24 ÷ 3 8 × = 8 ÷ 4 × 3 There are 4 quarters in each whole. = 24 So, = 8 We can check this by multiplying. = 6

  50. Dividing a fraction by a fraction 1 1 1 1 1 1 1 What is ÷ ? 2 2 8 2 8 8 2 ÷ ÷ means, ‘How many eighths are there in one half?’ Here is of a rectangle: Now, let’s divide the shape into eighths. = 4

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