# PROGRESSION THROUGH CALCULATIONS FOR SUBTRACTION

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1 PROGRESSION THROUGH CALCULATIONS FOR SUBTRACTION By the end of year 6, children will have a range of calculation methods, mental and written. Selection will depend upon the numbers involved. Children should not be made to go onto the next stage if: 1) they are not ready. 2) they are not confident. Children should be encouraged to approximate their answers before calculating. Children should be encouraged to check their answers after calculation using an appropriate strategy. Children should be encouraged to consider if a mental calculation would be appropriate before using written methods. MENTAL CALCULATIONS (ongoing) These are a selection of mental calculation strategies: Mental recall of addition and subtraction facts 10 6 = _ = = _ = 2 Find a small difference by counting up from the smaller number = 3 Counting on or back in repeated steps of 1, 10, 100, = 34 (by counting back in tens and then in ones) = 160 (by counting back in hundreds) Subtract the nearest multiple of 10, 100 and 1000 and adjust = = = = 387 Use the relationship between addition and subtraction = = = = 19 MANY MENTAL CALCULATION STRATEGIES WILL CONTINUE TO BE USED. THEY ARE NOT REPLACED BY WRITTEN METHODS.

2 Subtraction Methods KEY VOCABULARY FOR SUBTRACTION: subtract, take away, minus, less, fewer, difference & decrease. Foundation Year 1 Year 2 Say the number that is one less than a number from 1 to 10. In practice activities and discussions, beginning to use the vocabulary involved in subtraction when taking away objects / groups. Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of recording calculation using pictures etc. Language used with children includes: Take (away), leave, how many are left/left over?, How many have gone? One less, two less, ten less, how many fewer is..? difference between, is the same as Oral and practical work Songs and rhymes Dice and number games, counting back Read, write and interpret mathematical statements involving addition (+), subtraction (-) and (=) signs. Represent and use number bonds and related subtraction facts within 20. Subtract one digit and two digit numbers to 20, including zero. Solve one step problems that involve subtraction using concrete objects and pictorial representations, and missing number problems such as 8 [] = 5 Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of recording calculation using pictures ect. They use number lines, numicon, 100 squares and bead strings to support calculations. Children then begin to use numbered lines to support their own calculations - using a numbered line to count back in ones = 8 Solve subtraction problems using concrete objects and pictorial representations, including: *a two digit number and ones. *a two digit number and tens. *two two-digit numbers. Recall and use subtraction facts to 20 fluently, and derive and use related facts up to 100. Show that the subtraction of one number from another cannot be done in any order. Use the inverse relationship between addition and subtraction to check calculations and solve missing number problems. Start to record subtraction in columns. Children will continue to use concrete objects and pictorial representations including numbers, quantities and measures to solve problems. e.g subtract a two digit number and ones 15 7 = 8 Number stories using objects Bead strings can be used to illustrate subtraction including bridging through ten by counting back 3 then counting back 2 Counting back in 10s and ones 37-12= Numicon can be used to show the difference between the size of two numbers:

3 How many are there? How many now? (after some have been removed) Teacher modelling number sentences, 8 take away 3 is 5 Physical and practical work on number tracks and semi structures numberlines Jumping backwards Number stories, 15 people on a bus 3 get off, how many are left on? 7 3 = 7- = 4-3 = 4 Counting back in 10s from multiples of 10s Giving change to 20p Finding the difference between 2 towers of cubes leading to using the ENL for numbers that are close together Count on = Help children to become more efficient with counting on by: -Subtracting the units in one jump; - Subtracting the tens in one jump and the units in one jump; - Bridging through ten. Number lines should also be used to show that 6-3 means the difference between 6 and 3 or the difference between 3 and 6 and how many jumps they are apart. 6 3 = Use addition as the inverse operation to check an empty box problems e.g. 8 = 12 To know that subtraction is not commutative Teachers beware of 4-8 can t do it (you can when you start doing negative numbers) Start to record subtraction in columns, using expanded methods:

4 Year 3 Year 4 Year 5 Year 6 Subtract a range of numbers mentally including: * A three digit number and ones. *Three digit number and tens. *A three digit number and hundreds. Subtract numbers with up to three digits, using formal methods of columnar subtraction. Estimate the answer to a calculation and use inverse operations to check answers. Solve problems, including missing number problems, using number facts, place value and more complex subtraction. Subtract fractions with the same denominator within one whole (for example 5/7 1/7 = 4/7). Children will continue to use concrete materials eg Numicon and other representations with increasingly large numbers. Children will begin to use informal pencil and paper methods (jottings) to support, record and explain partial mental methods building on existing mental strategies. Subtract numbers with up to four digits, using formal methods of columnar subtraction where appropriate. Estimate and use inverse operations to check answers to a calculation. Solve addition and subtraction two step problems in contexts, deciding which operations and methods to use and why. Subtract fractions with the same denominator. Solve simple measures and money problems involving fractions and decimals to two decimal places. Using ENL to both count back and find the difference between two numbers by counting on (up to four digits). Extend to decimals to two decimal places. Support by re-assigning new values to Subtract whole numbers with more than four digits, including using formal written methods (columnar subtraction). Subtract numbers mentally with increasingly large numbers (eg. 10,462-2,300 = 8,162). Use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy. Solve addition and subtraction multistep problems in contexts, including to three decimal places, deciding which operations and methods to use and why. Add and subtract fractions with the same denominator and denominators that are multiples of the same number. Using ENL to both count on and back, including finding the difference between two numbers. Complimentary addition: = Move towards compact decomposition, including Subtract whole numbers with more than four digits, including using formal written methods (columnar subtraction). Perform mental calculations, including with mixed operations and large numbers. Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why. Add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions. Use an ENL to count on and back = Use formal method of compact decomposition.

5 Partitioning and decomposition This process will initially be demonstrated using arrow cards to show the partitioning and base 10 materials to show the decomposition of the number for numbers up to three digits. Links will be made to concrete materials such as Numicon and Deines. 89 = = 32 Initially, the children will be taught using examples that do not need the children to exchange. Expand method of decomposition for numbers too large to do mentally = Decomposition IF they are ready, some children will move onto the concise method, where the above method is refined as follows Children should: - be able to subtract numbers with different numbers of digits; - using this method, children should also begin to find the difference Numicon: Expanded method of decomposition, leading to more compact recording. Decomposition Children should: -be able to subtract numbers with different numbers of digits; -using this method, children should also begin to find the difference between two three-digit sums of money, with or without adjustment from the pence to the pounds; -know that decimal points should line up under each other. Extend to decimals. NB If your children have reached the concise stage they will then continue this method through into year 6. They will not go back to using the expanded methods. decimals. Subtract fractions with the same denominator and multiples of a same number. Model this with different representations (e.g. physical objects, diagrams etc). Apply to problem solving contexts such as money and measures. Subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions. Model this with different representations (e.g. physical objects, diagrams etc).

6 between two three-digit sums of money, with or without adjustment from the pence to the pounds; - know that decimal points should line up under each other.

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