Primary Mastery Specialists

The Project – NCP5 Training

TRG

Reasoning and Mathematical Thinking

Assessment

Year 1Year 2Year 3Year 4Year 5Year 6 COUNTING IN FRACTIONAL STEPS Pupils should count in fractions up to 10, starting from any number and using the1/2 and 2/4 equivalence on the number line (Non Statutory Guidance) count up and down in tenths count up and down in hundredths Spot the mistake 7, 7 ½, 8, 9, 10 8 ½, 8, 7, 6 ½, … and correct it What comes next? 5 ½, 6 ½, 7 ½, …., …. 9 ½, 9, 8 ½, ……, ….. Spot the mistake six tenths, seven tenths, eight tenths, nine tenths, eleven tenths … and correct it. What comes next? 6/10, 7/10, 8/10, ….., …. 12/10, 11/10, ….., ….., ….. Spot the mistake sixty tenths, seventy tenths, eighty tenths, ninety tenths, twenty tenths … and correct it. What comes next? 83/100, 82/100, 81/100, ….., ….., ….. 31/100, 41/100, 51/100, ….., ….., Spot the mistake 0.088, 0.089, 1.0 What comes next? 1.173, 1.183, Spot the mistake Identify and explain mistakes when counting in more complex fractional steps

RECOGNISING FRACTIONS recognise, find and name a half as one of two equal parts of an object, shape or quantity recognise, find, name and write fractions 1 / 3, 1 / 4, 2 / 4 and 3 / 4 of a length, shape, set of objects or quantity recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators recognise that tenths arise from dividing an object into 10 equal parts and in dividing one – digit numbers or quantities by 10. recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents (appears also in Equivalence) What do you notice? Choose a number of counters. Place them onto 2 plates so that there is the same number on each half. When can you do this and when can’t you? What do you notice? What do you notice? ¼ of 4 = 1 ¼ of 8 = 2 ¼ of 12 = 3 Continue the pattern What do you notice? What do you notice? 1/10 of 10 = 1 2/10 of 10 = 2 3/10 of 10 = 3 Continue the pattern. What do you notice? What about 1/10 of 20? Use this to work out 2/10 of 20, etc. What do you notice? 1/10 of 100 = 10 1/100 of 100 = 1 2/10 of 100 = 20 2/100 of 100 = 2 How can you use this to work out 6/10 of 200? 6/100 of 200? What do you notice? One tenth of £41 One hundredth of £41 One thousandth of £41 Continue the pattern What do you notice? = = = 0.1 Continue the pattern for the next five number sentences.

Same and Different Y6 Compare × 7 and (31 + 9) × 7 What’s the same? What’s different? What’s the same, what’s different about these number statements?

Deeper with Fractions Y4 Y2

Captain Conjecture

Y2 – Going Deeper

Representation and Structure

Bar modelling The bar model is used to support children in problem solving. It is not a method for solving problems, but a way of revealing the mathematical structure within a problem and gaining insight and clarity as to how to solve it. It supports the transformation of real life problems into a mathematical form and can bridge the gap between concrete mathematical experiences and abstract representations.

Using the bar model

Using the bar model for addition and subtraction The bar model supports understanding of the relationship between addition and subtraction in that both can be seen within the one representation and viewed as different ways of looking at the same relationships. a = b + c a + c + b a – b = c a – c = b

Using the bar model for multiplicative relationships Peter has 4 books Harry has five times as many books as Peter. How many books has Harry? five times four 4 × 5 = 20 Harry has 20 books ?

There are 32 children in the class. There are 3 times as many boys as girls. How many girls? A computer game was reduced in a sale by 20% and it now costs £48. What was the original price? BBB 32 20%£48

Phase 1: Text Phase (T) Children read the information presented in text form. Phase 2: Structural Phase (S) In this phase, children represent the text information in the structure of the model. Children can alternate between text and the model to check that the model accurately depicts the textual information. Phase 3: Procedural-Symbolic Phase (P) Once they have constructed a model, children then use the model to plan and develop a sequence of logical arithmetic equations, which allows for the solution of the problem. Again, alternating between the two representations, structural and procedural (SP), allows for the accuracy of the arithmetic equations to be checked against the model.

Scaling

Child using the bar model to support their understanding of the problem

Child using the bar model to support understanding of ¼ of 100

Using representations Successful representation of word problems requires an integrated and well-organized knowledge base of number facts, conceptual understanding of part-whole relationships and fraction, multiplicative reasoning, and knowledge of the four operations.

Fluency

National Curriculum 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

Discussion What are the key number facts that primary school children need to know?

Teaching strategies not memorisation Children need to be TAUGHT strategies. They will not be fluent or retain the number facts through rote learning. Teachers often assume that children will make the connections but unless these are taught explicitly then many children struggle to become fluent Thorton (1976)

Video

Cognitive Load If children are not fluent in these facts, then when they are solving more complex problems the working memory is taken up by calculating basic facts, and children have less working memory to focus on solving the actual problem. Fluency in basic facts allows children to tackle more complex maths more effectively. Daniel Willingham

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