Part a maths
- 1. Part A Consider which other aspects of arithmetic are relevant and the pre-requisite number skills and understanding that are necessary before/to support conceptual work on your chosen area References should be made to literature which explores the learning journey as well as literature which explores the importance of secure conceptual understanding (ofsted, piaget)does not have to be specific to fractions but more general on the learning journey. Learning Journey What knowledge/skills are needed to do fractions? KS1 – yr1 Pupils should be taught – recognise, name and write ½ as one of two equal parts of an object, shape or quantity Recognise, name and write ¼ and ¾ as parts of two equal parts of an object, shape or quantity And find ½, ¼ and ¾ as parts of an object, shape or quantity. Ensure pupils are taught all of the above fractions as objects and then as operators on discrete and continuous quantities. As objects pupils recognise and combine fractions as part of a whole. As operators, pupils recognise and find, for example, half of a length, container, set of objects and shapes. KS1 YR2 NUMBER AND PLACE VALUE -They should be introduced to counting in multiples of 3 to support their understanding of a 3rd Locate fractions on a number line and use them to find fractions of shapes and quantities solve simple problems involving ratio and direct proportion NC Pupils should be taught to recognise, name and write fractions ¼, 1/3, ½, 2/3 and ¾ of a whole Count in halves or quarters to ten Start using the ½ and 2/4 equivalence Reinforce that fractions can add up to more than one 1 ¼, 1 ½, 1 ¾, 2 Division and multiplication is taught through pupils sharing out quantities, finding simple fractions of objects, numbers and quantities, doubling numbers and quantities and find related halves. KS2 YR3 COMPARE & ORDER unit fractions and fractions with the same denominator
- 2. Recognise fractions that are equivalent to 1 and ones that add up to 1 Perform calculations with addition and subtraction with the same denominator e.g. ( 5/7 + 1/7 = 6/7) KS2 YR4 BE AWARE OF MULTIPLICATION AND DIVISION TABLES UP TO 12X12 Reduce fractions to their simplest form Add and subtract two fractions with common denominators within one whole Write equivalent fractions when just given the denominator or numerator for one fraction Pupils practise counting fractions and decimal fractions - should be taught that ½ = 0.5 Ks2 yr 5 Prime numbers Accurately multiply and divide numbers mentally - drawing upon known facts Recognise improper fractions and convert them Writing remainders as a fraction Division using remainders e.g. 94/4 = 24 r2 = 24.5 = 25 and rounding off Read and write decimal numbers as fractions e.g. 0.71 – 71/100 Recognise the % symbol and it relates to number as part of 100 e.g. 1/100 is 1% Write simple fractions as percentages and decimals e.g. ½ = 50% = 0.5). KS2 YR6 – UPPER Use of brackets 2+1 x3 = 5 (2+1) x 3 = 9 Divide numbers using long division (4 digits by 2 digits) and interpret remainders as fractions , decimals or rounding Add and subtract mixed numbers and fractions with different denominators Divide proper fractions by whole numbers Associate fractions with division to calculate decimal fraction equivalents e.g. 0.375 for a simple (3/8) E.g. if ¾ of a length is 36 then the whole length is 36 x 4 = 144cm - pupils should understand the relationship between unit fraction and division to work backwards by multiplying a quantity that represents a unit fraction to find the whole quantity
- 3. WHAT DOES PIAGET SAY??? Sensorimotor skills (link numbers to objects) Preoperitalstage - problem solving with water or blocks – squash www.tlrp.org context specific. The importance of hands-on experiences cannot be overemphasized. (Burns and sibly, 2000,p.60) These activities Gives them the opportunity to test and confirm their learning. Manipulative materials :pattern blocks, paper folding Application –connecting mathematical concepts to real-life situations – pizza, chocolate The numbers and quantities used to teach children should be meaningful to them – piaget children asked to divide objects among the class. Children learn that 1/3 of a pizza and 2/6 of a pizza are equivalent but not understand that 1 glass of orange with 2 glasses of water is the same as 2 glasses of orange is the same as 4 glasses of water Barmby,P., Bilsborough, L., Harries, T and Higgins, S., (2009) Fractions in primary mathematics Teaching for understanding : Maidenhead: OUP Multiplying fractions – 2/5 x3/4 In this case, this is where the notion of repeated addition of starts to break down. It does not mean anything to add 2/5 by ¾ times. The counting technique DOES NOT REQUIRE THE APPLICATION OF ANY CONCEPTS OF FRACTIONS AS PART OF A WHOLE. Number lines – children have difficulties in placing 1/3 at a third of the distance along the number line however long the number line happens to be. Relative = part –whole relationship where parts and wholes are made up of discrete objects Equivalent fractions comparing them e.g. ¾ / 6/8 and 9/12 how do we make comparison with situations easier? 100 counters and 75 of them are shaded we know that that 75%. % means out of 100. This is the percentage format for proportions and we can make a direct link between fractions and percentages using equivalent fractions: Teaching and learning; research briefing - www.tlrp.org Most pupils in year 4 and 5 have not grasped the relative nature of fractions as numbers. This difficulty is primarily conceptual. Teaching pupils fractions must include a focus on the logical relations involved in this concept. Pupils have some intuitive of the relative nature of fractions from their experiences with division. Teaching logical relations should build on pupil’s intuitions.
- 4. The same fraction may refer to different quantities ½ of 6 or ½ of 8 and that different fractions may be equivalent because they refer to the same quantity 1/3 and 3/9.It is not possible for pupils to make further progress in mathematics without a sound grasp of the relative nature of rational numbers. Part whole fractions are used to introduce fractions. Denominator shows the number of equal parts into which a whole was cut and the numerator indicates the number of parts that were taken. E.g. choc bar cut into 4 equal parts and I ate 1 = ¼ Division situations If one chocolate is shared among four children, the number 1 refers to the number of chocolates being shared and 4 refers to the number of recipients. ¼ indicates both the division 1 divided by 4 and the portion that each child receives Children performed better at solving fractions problems about division. They compared fractions and explanations to each other. The arguments were based on the logic of division. Drawings, through group work showed use of logic division. However, a concern with partitioning and perceptual comparison often left children confused due trouble drawing with suffient precision. It moved them away from logic of division. Division situations provide a sound starting point for pupils Counting and natural numbers need to be taught in early yrs. as a basis for fractions. Sharing problems
- 5. Pre – requisite understanding (required as a prior condition for something else to happen.) Concept mapping - find out what they know Mooney , C et al Teaching Theory and practice Division situations provide a sound starting point for pupils understanding of the logic to rational numbers but they must not be seen as the only context in which rational numbers should be taught. Further lessons of fractions should be taught with different situations and do not remain context specific Must focus on logical relations involved in teaching fractions Pupils have some intuitive understanding of the relative nature of fractions from their experiences EYFS – NEED TO KNOW number names in order “stable order principle” pg 112 ONE TO ONE CORRESPONDENCE (Counting each item) Multiplication is inverse to division Number lines help Put half of these 10 animals in the ark Common error with addition fractions is; ½ + 2/3 = 3/5 – because of prior knowledge for addition Rational numbers