19/09/2018

3 4 = 6 8 18 24 = Equivalent fractions Look at this diagram: ×2 ×3 ×2 Ask pupils what proportion of the diagram is shaded (3/4). Demonstrate that if each part in the diagram is divided into 2 we have eights. The same amount is shaded but we now have 6/8. Establish that by cutting each quarter into two equal parts we have multiplied the number of shaded sections by two (we had three shaded sections; now we have six) and we have also multiplied the number of equal parts by two (we had four; now we have eight). Click to reveal the arrows showing the numerator and the denominator being multiplied by 2. Explain that the numbers have changed but exactly the same proportion of the circle has been shaded. 3/4 and 6/8 are equivalent fractions. Click to show what happens when we divide each of the eights into three equal parts. There are 3 × 8 = 24 equal parts with 3 × 6 = 18 parts shaded. This shows that 6/8 is equivalent to 18/24. Establish that by multiplying the numerator and the denominator by the same amount we can generate infinitely many fractions equivalent to 3/4. 3 4 = 6 8 18 24 = ×2 ×3

2 3 6 9 24 36 = = Equivalent fractions Look at this diagram: ×3 ×4 ×3 Explain this set of equivalent fractions as in the previous slide. 2 3 6 9 24 36 = = ×3 ×4

18 30 6 10 3 5 = = Equivalent fractions Look at this diagram: ÷3 ÷2 ÷3 Ask pupils what proportion of the diagram is shaded (18/30) and explain that we could simplify the diagram by removing the horizontal lines. Click to show the diagram with 10 equal parts. The same proportion is shaded. Explain that we have divided the number of shaded sections by 3 (we had 18 shaded sections; now we have 6) and we have divided the number of equal parts by 3 (we had 30; now we have 10). Click to reveal the arrows showing the numerator and the denominator being divided by 3. 18/30 and 6/10 are equivalent fractions. Tell pupils that by dividing the numerator and the denominator by the same number, we have simplified the fraction. It is simpler because the numbers are smaller. Demonstrate how the fraction can be simplified further and so establish that 6/10 is equivalent to 3/5. State that 3/5 cannot be simplified any further because 3 and 5 have no common factors. Ask pupils how we could have cancelled 18/30 down in one step. 18 30 6 10 3 5 = = ÷3 ÷2

19/09/2018 Task - Which of these fractions are equivalent? Shade in your circles to help you! 3 6 4 6 1 3 1 2 2 3 2 6 1 4 5 12 4 12 1 12 3 12 1 6

19/09/2018 Solutions 3 6 4 6 1 3 1 2 2 12 2 6 1 4 5 12 4 12 1 12 3 12 1 6

Cancelling fractions to their lowest terms A fraction is said to be expressed in its lowest terms if the numerator and the denominator have no common factors. Which of these fractions are expressed in their lowest terms? 7 5 2 14 16 20 27 3 13 15 21 14 35 32 15 8 7 5 Ask pupils what we mean when we say a fraction has no common factors. Establish that there is no number other than 1 that divides into both the numerator and the denominator. For each fraction ask pupils whether or not they think this fraction has been shown in its lowest terms, before revealing the answer. If pupils do not think that the fraction has been shown in its lowest terms, ask them for a number which will divide into both the numerator and the denominator. Explain that when cancelling it is always best to divide both the numerator and the denominator by the highest number that divides into both, that is, the highest common factor. However, if you do not cancel by the highest common factor the first time round, you can always cancel again. Go through the cancellation of each fraction asking what we are dividing by each time. Establish that the last fraction is an improper fraction and ask pupils how we could write this as a mixed number (2 2/15). Fractions which are not shown in their lowest terms can be simplified by cancelling.

SIMPLIFYING FRACTIONS! 19/09/2018 SIMPLIFYING FRACTIONS! On your mini whiteboards…

19/09/2018 Simplify 14 20

9/19/2018 Simplify 9 15

9/19/2018 Simplify 4 22

9/19/2018 Simplify 6 16

9/19/2018 Simplify 5 100

9/19/2018 Simplify 40 100

9/19/2018 Simplify 15 3

9/19/2018 Simplify 200 100

9/19/2018 Simplify 2

9/19/2018 Simplify 1432

19/09/2018 On mini whiteboards! Sort the fractions into the correct circles… Do as many as you can before the time is up 9 15 4 12 3 9 15 25 10 30 20 32 10 16 18 30 6 10 2 6 15 24 12 20 5 15 25 40 6 18 50 80 1 3 3 5 5 8

Mixed numbers and improper fractions When the numerator of a fraction is larger than the denominator it is called an improper fraction. For example, 15 4 is an improper fraction. We can write improper fractions as mixed numbers. 15 4 Talk through the diagrammatic representation of 15/4. Every four quarters are grouped into one whole, and there are three quarters left over. can be shown as 15 4 3 4 =

Improper fraction to mixed numbers 37 8 Convert to a mixed number. 37 8 = 8 + 5 5 8 1 + = = 4 5 8 Explain that to convert an improper fraction to a mixed number we can divide the numerator by the denominator to find the value of the whole number part. Any remainder is written as a fraction. Relate fractions to division. 37/8 means 37 ÷ 8. Talk through the division of 37 by 8. Discuss the meaning of the remainder in this context. We are dividing by 8 and so the 5 represents 5/8. This number is the remainder. 37 8 = 4 5 8 4 5 37 ÷ 8 = 4 remainder 5 This is the number of times 8 divides into 37.

19/09/2018 ¼ ¼ 1 ¼ 2 ¼ 2¾

19/09/2018 1 2 3

written as a Mixed Fraction ? 2 19/09/2018 9 How is written as a Mixed Fraction ? 2

19/09/2018 1) 7 2) 9 3) 11 2 1 2 1 4 3 3 3 5 5 4 4 4) 7 5) 18 6) 24 1 1 3 3 3 6 6 6 7 7 7) 30 8) 17 9) 14 6 5 1 2 3 5 3 11 3 11

Mixed numbers to improper fractions 2 7 3 Convert to a mixed number. 2 7 3 = 2 7 1 + = 7 + 2 = 23 7 We can explain this conversion by asking for the number of 1/7 in 3 whole ones. Explain that to convert a mixed number to an improper fraction in one step we multiply the whole number part by the denominator of the fractional part and add the numerator of the fractional part (refer to the example). This gives us the numerator of the improper fraction. The denominator of the improper fraction is the same as the fractional part of the mixed number. Explain that there are 21 sevenths in three wholes. Two more sevenths makes 23 sevenths altogether. … and add this number … To do this in one step, 3 3 2 2 23 … to get the numerator. = 7 7 7 Multiply these numbers together …

written as an Improper Fraction ? Two lots of 2 plus an extra 1 19/09/2018 2 + 1 written as an Improper Fraction ? × 2 Five halves. Two lots of 2 plus an extra 1 2 x 2 + 1 5 2

19/09/2018 What about : 4 + 2 14 = × 3 3 7 + 1 29 = × 4 4 1 + 2 11 = × 9 9

19/09/2018 3 5 1 1) 2 2) 3 3) 1 17 5 23 4 9 8 5 4 8 2 4 6 4) 4 5) 2 6) 5 14 3 41 6 18 7 7 3 6 8 9 10 7) 1 8) 3 9) 3 33 4 75 8 53 5 4 8 5

Finding a fraction of an amount 2 3 of £18? What is We can see this in a diagram: Stress that 2/3 means 2 lots of 1/3. 1/3 of £18 is £6 so, 2/3 of £18 is 2 x £6. Click to reveal £12. Explain that we divide by 3 to find 1/3 and then multiply by 2 to find 2/3. 2 3 of £18 = £18 ÷ 3 × 2 = £12

Finding a fraction of an amount 7 10 of £20? What is Let’s look at this in a diagram again: Establish that 1/10 of 20 is £2. The diagram shows each 1/10 being shaded until 7/10 is shaded. Establish that to find 7/10 of £20, we divide by 10 (to find a 1/10) and multiply by 7 (to find 7/10). 7 10 of £20 = £20 ÷ 10 × 7 = £14

Finding a fraction of an amount 5 6 of £24? What is 5 6 of £24 = 1 6 of £24 × 5 = £24 ÷ 6 × 5 Remind pupils that 5/6 means ‘5 lots of 1/6’ so we need to work out 1/6 of 24, by dividing 24 by 6, and then multiply that answer by 5. Reveal each stage in the calculation. = £4 × 5 = £20

Finding a fraction of an amount 4 7 What is of 9 kg? To find of an amount we can multiply by 4 and divide by 7. 4 7 We could also divide by 7 and then multiply by 4. 4 × 9 kg = 36 kg Remind pupils that we can multiply by the numerator and divide by the denominator in either order. In this example, point out that 9 kg is not divisible by 7. Let’s try multiplying by the numerator first and then dividing by the denominator. Stress that 36 ÷ 7 is equivalent to 36/7. Writing this as an improper fraction allows us to write the answer as a mixed number. 36 ÷ 7 is 5 remainder 1 or 51/7. We could also write 51/7 as 5.14 (to 2 decimal places). = 36 7 kg 5 1 7 kg 36 kg ÷ 7 =

Finding a fraction of an amount When we work out a fraction of an amount we multiply by the numerator and divide by the denominator For example, 2 3 of 18 litres Tell pupils that it doesn’t make any difference whether you multiply first or divide first. Demonstrate using the numbers in the example, that multiplying 18 by 2 and then dividing by 3 (this give us 36 ÷ 3 = 12) is the same as dividing 18 by 3 and then multiplying by 2 (this give us 6 × 2 = 12). Establish that in this example if we divide first the numbers will be easier to work with. = 18 litres ÷ 3 × 2 = 6 litres × 2 = 12 litres

On your mini whiteboards… 19/09/2018 Fractions of an amount! On your mini whiteboards…

19/09/2018 1 3 𝑜𝑓 12

19/09/2018 2 3 𝑜𝑓 12

19/09/2018 3 4 𝑜𝑓 12

19/09/2018 2 6 𝑜𝑓 12

19/09/2018 2 3 𝑜𝑓 24

19/09/2018 3 4 𝑜𝑓 100

19/09/2018 3 4 𝑜𝑓 200

19/09/2018 3 4 𝑜𝑓 20

19/09/2018 1 4 𝑜𝑓 80

19/09/2018 3 4 𝑜𝑓 80

19/09/2018 1 8 𝑜𝑓 80

19/09/2018 1 2 3 3 4 of £36 4 5 of 30 kg 5 6 of 48 mm 4 5 6 2 3 of 90 cm 7 8 of 56 km 9 10 of 120 l 7 8 9 7 9 of £36 11 12 of 72 g 13 20 of $180

19/09/2018

Finding a fraction of an amount 1 2 5 What is of 3.5m? To find of an amount we need to add 1 times the amount to two fifths of the amount. 2 5 1 2 5 of 3.5 m = 1 × 3.5 m = 3.5 m and 1.4 m Discuss the meaning of 12/5 of an amount. Go through the example on the board. An alternative would be to convert 12/5 to an improper fraction, 7/5. We could then divide by 5 to get 0.7 m and then multiply by 7 to get the answer 4.9m Ask pupils if they can give you an equivalent decimal calculation (1.4 x 3.5 m) or an equivalent percentage calculation. (140% of 3.5 m). so, of 3.5 m = 2 5 1 3.5 m + 1.4 m = 4.9 m We could also multiply by 7 5

Using equivalent fractions 3 8 5 12 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. Now, write and as equivalent fractions over 24. 3 8 5 12 Tell pupils that another way to compare two fractions is to convert them into equivalent fractions with a common denominator. Talk through the example on the board. Tell pupils that the quickest way to find the lowest common multiple of two numbers is to choose the larger number and to go through multiples of this number until we find a multiple which is also a multiple of the smaller number. This method also works for a group of numbers. ×3 ×2 3 8 = 24 9 5 12 = 24 10 3 8 5 12 < and so, ×3 ×2

Using decimals to compare fractions Which is bigger or ? 3 8 7 20 We can compare two fractions by converting them to equivalent fractions. 3 8 7 20 Tell pupils that another way that we can compare two fractions is to write them both as decimals. Ask pupils to convert the two fractions into decimals using their calculators. Reveal these on the board. State that we could also use short division to convert the fractions to decimals.

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