Maths ‘Fractions’ Workshop

The question that all parents dread… Can you help me with my maths homework please? The question that all parents dread… In this workshop: Aims for the new curriculum and our school Progression of fractions Teaching methods and resources Ways in which you can help your child with their day to day maths

Aims of the New National Curriculum The national curriculum for mathematics aims to ensure that all pupils: become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

In other words… Fluent recall of mental maths facts: e.g. times tables, number bonds etc. To reason mathematically: children need to be able to explain the mathematical concepts with number sense; they must explain how they got the answer and why they are correct. Problem solving: applying their skills to real-life contexts.

Good practice in mathematics All children need to learn maths in a real life context. As well as knowing 7x7=49. Children need to be able to do the following: There are 7 fields, each field has 7 sheep in them. How many sheep are there in total? Children need to be able to explain how they have calculated or solved a problem. In the new curriculum, written calculations are taught at an earlier age. The mental methods are essential for supporting pupils understanding of these written calculations. Two points are clear about what mathematical skills children need in life. Educationalists say that children today need to learn 2 key skills. The ability to calculate mentally and the ability to estimate. Mental calculation skills are vital. Just have a think how often you use mental maths in your own lives. Shopping- working out change. Working out how many packets of biscuits or crisps you need to buy for a children’s birthday party. Working out how long it is before you need to leave to pick up children from school. Working out how many days left to do all your Christmas shopping? A lot of maths in life is done in your head. But within that I hope that you’re also estimating. When working out how long you have until you need to come to school, you round to the nearest hour or half hour. So in school, we encourage children to estimate. If they are faced with a problem. I have 18 sweets in 1 bag and 33 in another bag. How many in total? Children in maths lessons today would be encouraged to round to the nearest 10 in their head and work out 20 add 30 to approximate an answer. Alongside the ability to estimate, Educationalists today also say that children need to develop other key skills in maths. Childrern need to learn maths in a context. Therefore in school we aren’t just giving children lists of sums to complete. We are asking them to really think. Research shows that many children who can tell you what 7 x 7 = 49 cannot answer a question in a real life context. E.g. There are 7 fields, each has 7 sheep in them. How many sheep are there altoghether. Children need to be able to explain. What they are doing. You may well say well what about the written calculations. Well these are still taught, but there is a balance. Research shows that teaching children written procedures at too early a stage in their mathematical development can have an adverse effect upon their ability to operate mentally. In line with many other countries, mental calculation skills are being taught and focussed upon, and the introduction of written methods are delayed until children are ready. At St Luke’s we certainly would expect children by the time they leave school to know, understand and use a written strategy for more complex maths calculations but emphasis early on is placed on mental calculations. If we look at the next slide, this will hopefully illustrate the point in hand.

Key Differences in the new Maths Curriculum: Simple fractions (1/4 and 1/2) are taught from Year 1, and by the end of primary school, children should be able to convert decimal fractions to simple fractions and multiply and divide fractions. By the age of nine, children are expected to know times tables up to 12×12 (used to be 10×10 by the end of primary school). End of KS2 mental maths test has been replaced with an arithmetic test.

Expectations Expectations in the new curriculum for each year group are now much higher. I have created a progression map with the statutory requirements for fractions for each year group (see hand-out).

FDP Progression in School Halving of sets of objects begins as early as EYFS and Year 1. It is vital that children know halves and quarters must be equal in size. In Year 2 children begin to use ½, ¼, ¾ and 1/3; find simple fractions of amounts (½ of £12 = £6) and know equivalence of 1/2 = 2/4. http://resources.hwb.wales.gov.uk/VTC/ngfl/ngfl-flash/fractions/fractions.html http://www.topmarks.co.uk/Flash.aspx?f=WhatFractionv3

Progression in KS2 In Year 3 we begin to use the terms denominator and numerator in writing proper fractions. We recognise, find and write fractions of objects and begin to compare and order them. We begin to add and subtract them and solve problems. In year 4 we read and write fractions, ordering them and recognising equivalent fractions. Children also find fractions of amounts (3/5 of 25Kg = 15Kg). They continue to add and subtract fractions and understand more equivalences, including tenths and hundredths.

Progression By Year 5 children simplify fractions. They relate fractions to decimals and percentages, and begin to multiply them. They begin to convert mixed numbers to improper and vice versa. In Year 6 we ask children to find common factors in numerators and denominators. They multiply and divide fractions and associate them with division. Children convert between F, D and P and recall and use equivalences.

So what is a fraction? What do you think?

Fractions - what are they? 1 2 Part of a whole. When an object or number is divided into a number of equal parts, then each part is called a fraction.

Fractions A fraction is a part of a whole Slice a pizza, and you will have fractions: 1/2 1/4 3/8 (One-Half) (One-Quarter) (Three-Eighths) The top number tells how many slices you have The bottom number tells how many slices the pizza was cut into.

The denominator is downstairs! Parts of a Fraction Numerator 2 5 How many parts you have. Denominator The number of parts the whole is divided into (total). The denominator is downstairs!

Lowest Common Denominator Language of fractions Language of fractions is used all around children, “here’s my half.” “I’ll cut this cake into equal parts for the four of us.” etc We would encourage correct use of terms from early on - not “My half is bigger than yours!” Mixed Number Simple Fraction Equivalent Fraction Lowest Common Denominator Improper Fraction Simplifying

Equivalent fractions 1 = 2 2 4 Equivalent fractions are fractions that are equal in size but have different denominators or numerators. 1 = 2 2 4 There are many more! We use a fraction wall as well as lots of images and models to aid understanding.

Equivalent Fractions Some fractions may look different, but are really the same, for example: 4/8 = 2/4 = 1/2 (Four-Eighths)   (Two-Quarters)   (One-Half) It is usually best to show an answer using the simplest fraction ( 1/2 in this case ). That is called Simplifying the Fraction.

Ordering fractions To order fractions of the same denominator we need to look at the numerator. 6 1 4 3 7 8 8 8 8 8 Smallest to largest 1 3 4 6 7 8 8 8 8 8

Ordering Fractions 3 1 1 1 7 becomes 6 1 4 2 7 4 8 2 4 8 8 8 8 8 8 When fractions are to be ordered or added than they need to have the same denominator (which links to equivalent fractions) e.g. 3 1 1 1 7 becomes 6 1 4 2 7 4 8 2 4 8 8 8 8 8 8

Changing fractions to a common denominator 1) Change all the fractions to the same denominator. (Find a common multiple)   In this case we will use 12 because 2, 4, 6, and 3 all go into it. Whatever you do to the top (numerator), you do to the bottom (denominator). 1 x 6 1x3 5 x2 2 x4 2 x 6 4 x3 6 x2 3 x4 Your fractions will now be: 6 3 10 8 12 12 12 12 5) Now put your fractions in order (smallest to biggest.) 3 6 8 10 6) Change them back, keeping them in order. 1 1 2 5 4 2 3 6

Ordering fractions When ordering fractions with different denominators, it’s best to convert all fractions to the lowest common denominator. 3 1 1 1 7 becomes 6 1 4 2 7 4 8 2 4 8 8 8 8 8 8 Smallest to largest 1 2 4 6 7 and then 1 1 1 3 7 8 8 8 8 8 8 4 2 4 8

How do I Simplify a Fraction ? Method 1 Try dividing both the top and bottom of the fraction until you can't go any further (try dividing by 2,3,5,7,... etc). Example: Simplify the fraction 24/108 : ÷2 ÷2 ÷3 24 12 6 2 108 54 27 9

Method 2 Divide both the top and bottom of the fraction by the Greatest Common Factor, (you have to work it out first!).

Example: Simplify the fraction 8/12 : 1) The largest number that goes exactly into both 8 and 12 is 4, so the Greatest Common Factor is 4. 2) Divide both the top and bottom by 4: ÷ 4 8 2 12 3

Finding fractions of an amount or quantities. As a fraction is a division of a whole then we use division to find a fraction of amounts. Finding a half means dividing by 2 to make two equal sized groups. 1 ÷ 4 1 ÷ 3 1 ÷ 5 4 3 5 When it is a unit fraction (1 as the numerator) just divide by the denominator.

“Divide by the bottom and times by the top!” Finding fractions of an amount or quantities. To find fractions such as 2/3 then we need to find a third and then multiply this by 2. 2 of 12 = 1 (12 ÷ 3) = 4 then x 2 = 8 3 3 Divide by denominator and then multiply this by the numerator. “Divide by the bottom and times by the top!”

Mixed Numbers & Improper fractions An improper fraction has a numerator that is bigger than its denominator, for example 10 7 9 is also an improper fraction. It means nine 4 quarters. If you think of this as cakes, nine quarters are more than two whole cakes. It is 2 1/4 cakes.

Mixed Numbers & Improper fractions Mixed numbers and improper fractions show values where there are more than one whole being shown. Two and a half pieces is the same as 5 halves. 1 5 2 = 2 2

Changing a mixed number to an improper fraction Multiply the whole number by the denominator. Add the numerator. This gives you the new numerator; the denominator stays the same. e.g. 2 ¼ = 2 x 4 = 8. 8 + 1 = 9 = 9/4

Adding Fractions You can add fractions easily if the bottom number (the denominator) is the same: 1/4 + 1/4 = 2/4 = 1/2 (One-Quarter)   (One-Quarter)   (Two-Quarters)   (One-Half) Another example: 5/8 + 1/8 = 6/8 = 3/4

Adding Fractions with Different Denominators But what if the denominators are not the same? As in this example: 3/8 + 1/4 = ?     You must somehow make the denominators the same.

So find an equivalent fraction so they have they same denominator. In this case it is easy, because we know that 1/4 is the same as 2/8 : 3/8 + 2/8 = 5/8 This is the same when subtracting fractions. Make sure they have the same denominator, then just subtract one numerator from the other.

2 2 x 1 = 8 x 1 = 8 = 1 2 = 1 1 3 2 3 2 6 6 3

2 2 ÷ 1 = 8 x 2 = 16 = 5 1 3 2 3 1 3 3

Dividing Fractions To divide fractions you need to use KFC! Keep the first fraction. Flip the second one. Change it to a multiplication. E.g. 1 ÷ 1 = 1 x 6 = 6 2 6 2 1 2 Simplify the fraction if you can so 6 = 3 2

Fractions, Decimals and Percentages Decimals, Fractions and Percentages are just different ways of showing the same value: A Half can be written...     As a fraction: 1/2 As a decimal: 0.5 As a percentage: 50%

A Quarter can be written...     As a fraction: 1/4 As a decimal: 0.25 As a percentage: 25%

Why do children find fractions difficult 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. 

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 numerator is less than the denominator eg 2/3 Improper fraction: the numerator is larger than the denominator indicating that the parts come from more than one whole (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.

Place Value In the number 327: the "7" is in the Ones position, meaning just 7 (or 7 "1"s), the "2" is in the Tens position meaning 2 tens (or twenty), and the "3" is in the Hundreds position, meaning 3 hundreds

As we move right, each position is 10 times smaller As we move right, each position is 10 times smaller. From Hundreds, to Tens, to Ones   But what if we continue past Ones? What is 10 times smaller than One? 1/10 ths (Tenths) are!

Decimals But we must first write a decimal point, so we know exactly where the Ones position is: "three hundred and twenty seven and four tenths“ but we usually just say "three hundred and twenty seven point four"

The decimal point is the most important part of a Decimal Number The decimal point is the most important part of a Decimal Number. It is exactly to the right of the Ones position. Without it, we would be lost ... and not know what each position meant. Now we can continue with smaller and smaller values, from tenths, to hundredths, and so on, like in this example:

Percentages When you say "Percent" you are really saying "per 100" So 50% means 50 per 100 (50% of this box is green)

And 25% means 25 per 100 (25% of this box is green)

How can you help?

Number facts need to be learned! Children should learn number bonds/facts to 10, 20 and beyond. They need regular rehearsal to retain these so practise, practise, practise!

They need be able to rapidly recall: Number facts They need be able to rapidly recall: 4 + 6 = 10 5 + 2 = 7 8 + 2 = 10 10 – 3 = 7 10 + 90 = 100 12 + 8 = 20 35 + 65 = 100

Times Tables are also vital! Nothing has changed... Children need to learn their times tables. Without their times tables they may know the method for multiplication, but still arrive at a wrong answer. Children need to know them upside down, inside out and back to front!

They must be able to rapidly recall: Times Tables Counting in multiples e.g. 3, 6, 9, 12… is not good enough. They must be able to rapidly recall: 4 x 6 8 x 2 5 x 7 12 x 3 11 x 9 45 ÷ 9

Applying Times Tables 4 x 3 = 12 3 x 4 = 12 12 ÷ 3 = 12 12 ÷ 4 = 12

Everyday ways to improve your child’s (and your) maths A prominent clock in the kitchen – ideally analogue and digital. Display a traditional calendar. Board games that involve dice and spinners – helps not only with counting but with the concepts of chance. Traditional playing cards – simple games such as snap are a natural way of learning about sorting and chance. Dominoes – to help with number combinations. A calculator. Measuring with scales/kitchen scales – weight, length and capacity. Tape measure and ruler – involve your child in ‘real life’ situations. Dried pasta…or Smarties! – useful for counting large collections to investigate remainders etc. An indoor/outdoor thermometer. Money – coin recognition, working out how much you’ll need/change/saving.

Please don’t… Tell them that they are doing ‘sums’ ‘Sum’ is a mathematical word that means ‘addition’, everything else is a ‘calculation’ or ‘number sentence’. Teach your children that to multiply by 10 you ‘just add a zero’ You ‘move the digits to the left and add zero as a place holder’. e.g. 2.34 x 10 = 2.340 is incorrect Tell them that you can move the decimal point You can’t. You can only move the digits to the left or to the right.

Useful Websites I have printed a list of websites that may be useful. I will put it onto the website so you can click on the hyperlink to get direct access if needed. Remember, if you are ever unsure or need any help/advice, ask your child’s teacher – we’ll try not to bite!

Remember what is important in maths! A focus on mental calculations. The ability to estimate. To use maths in a real life context. To ask children to explain how they have calculated something using a method that suits them. Teach children written calculations, but only when children are ready.