Report

Maths Workshop for Year 6 Parents and Carers 12 January 2015 Mrs Claire Searle – Maths Leader What is a fraction? Talk to someone else – what do you think? Why do children find fractions difficult? Difficulties with fractions often stem from the fact that they are different from natural numbers in that they are relative rather than a fixed amount - the same fraction might refer to different quantities and different fractions may be equivalent (Nunes, 2006). Would you rather have one quarter of £20 or half of £5? The fact that a half is the bigger fraction does not necessarily mean that the amount you end up with will be bigger. The question should always be, 'fraction of what?'; 'what is the whole?'. Fractions can refer to objects, quantities or shapes, thus extending their complexity. What do Year 6 pupils need to know and do with fractions? Numerators and Denominators •A fraction is made up of 2 numbers. The top number is called the NUMERATOR and the bottom number is called the DENOMINATOR. In the fraction ¾, 3 is the numerator and 4 is the denominator. •DENOMINATOR This number shows how many equal ‘pieces’ something has been divided into. In the fraction ¾, 4 is the denominator showing that there are 4 equal pieces making up the whole. •NUMERATOR This number shows how many of those pieces there are. In the fraction ¾ there are 3 pieces out of the total of 4. Numerators and Denominators For example, if a pizza is cut into 4 equal slices there will be 4 pieces on the plate. This makes a fraction of 4/4 (1 whole). If I eat one of those pieces, ( ¼) then there are 3 pieces left. ( ¾ ). The denominator stays the same, there are still 4 parts that made up the whole pizza, but the numerator has changed, as there are only 3 parts of the pizza left. Simplifying fractions Some fractions can be made simpler by finding the highest common factor . (The highest number that will go into both parts of the fraction.) Eg for 8/10 both the numerator and denominator can be divided by 2 to give 4/5. 16/24 Both the numerator and denominator can be divided by 2, 4 and 8. The highest common factor (HCF) is 8, so this fraction can be simplified to give 2/3. Try this! Simplify 16/36 These can be divided by 2 and 4. The HCF is 4, so the answer is 4/9 Exploring equivalence using a tangram What fraction is each part of the whole? What other fractions can you make? What equivalences can you find? Equivalent fractions ½ = 2/4 = 3/6 = 4/8 = 5/10 = 6/12 = ... ¼ = 2/8 = 3/12 = 4/16 = 5/20 = ... 1/3 = 2/6 = 3/9 = 4/12 = 5/15 = ... Make fraction strips showing quarters, thirds, sixths, eighths, tenths Fraction strips Use your strips of paper to: Make some different fraction strips. What fractions can you find that are equivalent to 1/3? Which is larger, 5/8 or ¾? Fraction strips How can fraction strips help children make sense of problems like this? Comparing and ordering fractions Putting fractions in order of size can be difficult. It’s easiest to convert them (temporarily) to fractions with the same denominator if you are unsure. Try putting these fractions in order: 3/4 9/10 1/3 4/5 15/8 5/16 1/3 ¾ 4/5 9/10 1¼ 5/16 1¼ 15/8 Addition and Subtraction Addition and subtraction need to be done with common denominators. Addition and subtraction Add or subtract these fractions. Remember to convert them into fractions with the same denominator first. •Look for the smallest number that the denominators will all go into. Eg for 3/7 + 2/5 the smallest number that both 7 and 5 will go into is 35. •For 3/7, there are 5 lots of 7 in 35, so multiply both parts of 3/7 by 5 = 15/35. •For 2/5, there are 7 lots of 5 in 35, so multiply both parts of 2/5 by 7 = 14/35. • Now you can add the fractions easily. 15/35 + 14/35 = 29/35. ½+¾= 2/4 + ¾ = 5/4 = 1 ¼ 2/3 + 1/6 = 4/6 + 1/6 = 5/6 ¾ - 2/3 = 9/12 – 8/12 = 1/12 9/10 – 3/5 = 9/10 – 6/10 = 3/10 3/8 + 5/6 + ¾ = 9/24 + 20/24 + 18/24 = 47/24 = 1 23/24 Multiplication by a whole number ½x3= To multiply a fraction by a whole number, first convert the whole number to an improper fraction. ½ x 3/1 = • Now multiply both numerators together and then both denominators giving 3/2. •Finally divide the numerator by the denominator, giving a mixed fraction 1 ½ •So the answer to ½ x 3 is 1 ½ . •You can also think of it as ½ + ½ + ½ also giving 1 ½ Try this: 2/3 x 6 = 2/3 x 6/1 = 12/3 = 4 Multiplication by a fraction ½x¾= •It is useful to imagine the multiplication sign means ‘of’ so this calculation can be expressed as ‘what is ½ of ¾?’ and ‘what is ¾ of ½?’ • Multiply the numerators together and the denominators together. ½ x ¾ = 3/8 This answer is the same for both calculations above, as multiplications can always be done either way round and will give the same answer. Try this: ¾ of 5/8 = 15/32 Division Children need to be able to divide proper fractions by whole numbers, Eg ¼ ÷ 2 = 1/8 . To do this, turn the whole number into a fraction : 2/1 Then turn the fraction upside down: 1/2 Then multiply it by the first fraction ¼ x 1/2 = 1/8 2= The denominator has been doubled, so the value has been halved. Try this! 1/3 ÷ 4 = ? 1/3 ÷ 4/1 1/3 x ¼ = 1/12 Decimal fractions Finding decimal fractions What is 1/5 as a decimal? To convert a fraction to a decimal, simply divide the denominator (bottom part) into the numerator (top part). So to find 1/5 as a decimal, divide 1 by 5 which gives 0.2 1/5 = 1 ÷ 5 = 0.2 ¾ = 3 ÷ 4 = 0.75 Try this! What is 4/5 as a decimal? 4/5 = 4 ÷ 5 = 0.8 Converting decimals to fractions First make the fraction’s denominator (its bottom part) 10, 100, 1000 and so on for every digit after the decimal point. 0.75 Decimal number with 2 places after the decimal point 75 3 100 4 Count the decimal places; if there is 1 digit, the denominator is 10, if there are 2 then it is 100. The numerator is the number after the decimal point. Now divide both numbers by the highest number that goes into both 25. . Have a go! Change 0.6 into a fraction. 0.6 6 10 3 5 Equivalences between fractions, decimals and percentages •Converting between decimals and percentages is easy if the decimal number is below 1. Percentage just means out of 100. •So 0.8 is 80% which is 8 tenths or 80 hundredths. • •0.65 is 65% which is 65 hundredths. •Children need to be sure about place value in decimals to be able to do this conversion easily. They also need to be able to know and use equivalences between fractions decimals and percentages. Which of these fractions are the same? 70% 4/5 80% 3/4 0.55 0.45 8/10 34% Finding percentages of whole numbers •To find 10% of any number, divide by 10. 10% of 86 = 8.6 •To find 5% of any number, divide by 10 and then halve that number. 5% of 86 - halve 8.6 to give 4.3 •To find 15% of any number, add 10% and 5% together. So for 86, add 8.6 and 4.3 = 12.9 •To find 1%, divide by 100. 1% of 18 is 0.18 •Using these it is possible to find any percentage of a number. See how quickly you can do these: 30% of 60 15% of 20 7% of 50 110% of 75 Price reduced by 20%! Was £15, now ______ 25% off! Now £60! What was the price before it was reduced? Example SATs questions Fraction terminology •Numerator - the number on the top of a fraction showing the number of equal parts in the fraction eg 3/4 •Denominator - the number on the bottom of the fraction showing the total number of equal parts in the whole eg 3/4 • Proper fraction – the number of parts examined, shown on the top, is less than the whole eg 2/3 •Improper fraction – the larger numerator indicates that the parts come from more than one whole (also called top-heavy fractions) eg 9/5 •Mixed fraction – has a whole number and a fraction eg 8 ½ •Equivalent fraction – the same fraction written in different ways so each one gives the same answer in a calculation, even though they look different eg ½ and 3/6 •Common denominator – a number that can be divided by the denominators of all of the fractions eg 2/3 5/8 7/12 all the denominators divide into 24 so 2/3 becomes 16/24, 5/8 becomes 15/24, 7/12 becomes 14/24. So 24 is the lowest common denominator as this is the smallest number that 3, 8 and 12 will divide into. Ratio and Proportion Ratio and Proportion Ratio compares the size of quantities. Proportion compares the relationship between 2 sets of quantities. Ratios show how much bigger one thing is than another. Two things are in proportion when a change in one causes a related change in the other. A fruit bowl with a ratio of 6 apples to 2 bananas can be written like this 6:2 This can be divided by 2 and simplified to there is 1 banana. 3:1 meaning that for every 3 apples How many apples would there be if there were 6 bananas in the fruit bowl? Tomato soup! 6 tomatoes make 1 bowl of soup. How would you write the ratio? 6:1 How many tomatoes do you need to make 2 bowls of soup? 3 bowls of soup? 6 bowls of soup? What operation did you use to find the answers? multiplying by 6 How many bowls of soup could you make with 48 tomatoes? What about with 120 tomatoes? 6 million tomatoes? What operation did you use this time? 12 18 36 8 20 1 million! Dividing by 6 The tomatoes and the bowls of soup are in direct proportion. The ratio between them is always the same. Proportion What proportion of the stick is blue? Proportion means ‘fraction’ or ‘percentage’. 6/10 or 3/5 or 60% of the stick is blue. For every 6 blue cubes there are 4 yellow cubes. If the stick had 9 blue cubes, how many yellow cubes would there be? If the stick had 60 blue cubes how many yellow cubes would there be? What is the ratio of blue cubes to yellow cubes? 3:2 Fish Pie Omar makes fish pie for 3 people. How many grams of fish should he use? Mary used 2kg of potato to make a fish pie. How many people did her fish pie feed? How much butter was in her fish pie? How much fish was in her fish pie? From the Nrich website Example SATs questions Useful websites Fractions http://www.bbc.co.uk/skillswise/topic/fractions http://www.bbc.co.uk/skillswise/factsheet/ma17frac-l1-f-fraction-wall http://www.bbc.co.uk/bitesize/ks2/maths/number/fractions/read/1/ http://primarygamesarena.com/fractions Ratio and Proportion http://www.bbc.co.uk/education/topics/zsq7hyc http://www.11plusforparents.co.uk/Maths/ratio1.html http://nrich.maths.org/8959 http://resources.woodlands-junior.kent.sch.uk/maths/fractions/