Dartington Church of England Primary School and Nursery

Transcription

1 Dartington Church of England Primary School and Nursery Maths calculation policy Mental and written strategies for: * addition * subtraction * multiplication * division Dartington C of E Primary School and Nursery Maths Strategies

2 Early Calculation Skills Before introducing the calculation concepts children need an opportunity to investigate how the number system works for themselves and to develop secure counting skills. These skills will include: 1) Counting Counting with 1:1 correspondence including random arrangements, moving objects around, making sets of different numbers. Children should have opportunities to count objects, pictures, actions and sounds. 2) Comparing Children should have opportunities to compare sets of objects, pictures, sounds and actions. Children should be introduced to a range of language to include more and less, bigger and smaller, greater and fewer. 3) Beginning to understand calculating Children begin finding one more and one less using amounts of objects, pictures, sounds and actions and linking this to adding one more and taking away one. Children share objects and pictures in groups and sets. Dartington C of E Primary School and Nursery Maths Strategies 2

3 Models and Images to Support Calculation Addition and Subtraction Number lines, bead bar, bead strings, fingers, counters and other objects. Dienes (base 10 equipment), Numicon, Cuisenaire. ITPs counting on and back; difference; measuring cylinder; number line Multiplication and Division Number lines, bead bar, bead strings, Cuisenaire, arrays, fingers, counters and other objects. ITPs grouping; multiplication facts; remainders; multi array Excel spreadsheet; array creator Dartington C of E Primary School and Nursery Maths Strategies 3

4 Connective model for maths It is very important to use both structured and unstructured models and images to support calculation at every stage. The models and images you use need to be linked to all aspects of the connective model. Symbols Always referring to the mathematical symbols and the written calculation e.g = 36 Language Using and reinforcing the mathematical vocabulary e.g more than, less than, equals Pictures/ images Mathematical resources that reinforce the children s understanding of the calculation e.g numicon, bead strings, number lines. Context/concrete experience A real life experience for when you would meet this type of calculation and any images resources to reinforce this e.g. people getting off a bus. Dartington C of E Primary School and Nursery Maths Strategies 4

5 Addition add total and sum + plus make altogether more Dartington C of E Primary School and Nursery Maths Strategies 5

6 Addition Stage 1 Counting all with objects Concrete apparatus and lots of talk eg two bears went for a walk and met two more bears. How many bears are there now? Objects Numicon Fingers Stories involving number, concrete objects used to represent. I can count out two groups of objects, and put them together to count all of them and know that I am adding. Stage 2 Counting all with objects/pictures Representing concrete apparatus with symbols eg dots, in books. Use of the + and = signs. I can count two sets of objects and know that I am adding. Stage 3 Counting on from the first number (in 1s) Counting on from first number using a number line, your fingers or an image = 10 Begin to introduce the written calculation with the correct symbols. Numicon Pictures Counters Numicon Number track Number line Picture on head I can count on from one number and know that I m adding and that the numbers are getting bigger. Dartington C of E Primary School and Nursery Maths Strategies 6

7 Stage 4 Counting on from biggest number using number line = Start at 7 and count on 3 Put 7 in your head and count on 3. I can identify the biggest number and count on using a number line (or other images). Stage 5 Counting on using partitioning Adding starting at the largest number and partitioning the smallest, on a structured number line = I can start on the largest number and add on the second number in parts, on a structured number line. Stage 6 Counting on using partitioning on empty number line Adding starting at the largest number and partitioning the smallest, on a unstructured number line = Use same method for adding more than two numbers. Number line Bead bar / string Numicon for 2nd number We consistently only partition one number for any calculation I can start on the largest number and add on the second number in parts, on an unstructured number line. Stage 7 Counting on using partitioning on empty number line Adding starting at the largest number and partitioning the smallest, on a unstructured number line, with larger numbers and decimals = I can add larger and more complicated numbers using an unstructured number line. Dartington C of E Primary School and Nursery Maths Strategies 7

8 Stage 8 Expanded column addition. To be used once children have a really solid conceptual understanding of place value and of addition. Use this with larger numbers reinforce the idea that formal written methods are for when the numbers get too big to work with mentally = 572 Use Base Ten alongside to show magnitude of numbers here then move on to place value counters once children are ready. Reinforce place value alongside. Stage 9 (Upper KS2) Formal column addition ¹ Dartington C of E Primary School and Nursery Maths Strategies 8

9 Subtraction how many more? take away subtract left - less difference between minus leaves Throughout their experience of subtracting, the children need to understand that subtraction is finding the difference and how many more to make as well as taking away. Dartington C of E Primary School and Nursery Maths Strategies 9

10 Subtraction Early subtraction skills to practise Count out: To subtract a set of objects from a group by counting out the whole group and then removing the smaller set and counting how many are left (e.g. subtracting 10-4 by counting out 10 objects, removing 4 of them and then counting how many are left, 6) Count back: To subtract objects from a group of objects or one number from another number by starting the count always at the larger number and counting back from there (e.g. subtracting 10-4, by starting at 10 and counting back 4 numbers/places 9,8,7,6 ) Jump/step backwards: This is usually used when subtracting on a number line to show that we are moving backwards so many jumps or steps (or places) from one number to another (e.g. subtracting 10-4 by starting at 10 and jumping backwards 4 places to get to 6) Dartington C of E Primary School and Nursery Maths Strategies 10

11 Subtraction Taking away Stage 1: Understanding one more and one less. I have ten apples. How many would I have if I have 1 less. What is 1 less than 10 apples? I can find out what is one less. Stage 2: Taking away by removing practical objects (take away). Using concrete objects to solve real life problems involving taking a set number away, Jack has 10 apples, he eats 4, Count how many are left. Physical objects Fingers Looking at a number line. Move onto two less etc. Begin to introduce the written calculation with the correct symbols. Work in different contexts. I have 10p and buy a toy for 7p. How much money do I have left? I can take away an amount from another by removing objects. Stage 3. Taking away by physically moving beads along a bead string. I have 10 beads, I take two away, how many am I left with? I can take away by starting with a number and moving beads along a bead string to find out how many are left. Dartington C of E Primary School and Nursery Maths Strategies 11

12 Subtraction Taking away Stage 4. Taking away by relating the removal of objects to jumping backwards on a structured bead number line I can take away by jumping backwards on a number line. Stage 5. Taking away by using a structured number line, counting the jumps Children may count the numbers rather the jumps they are making as they take away along a number line. Make sure you continue to show the removal of physical objects alongside working on a number line until they are secure that we are counting the jumps. Relate to frogs or rabbits jumping along the number line we count the jumps they make. I can take away by jumping backwards on a number line. Stage 7. Subtracting through counting back, drawing their own number line, partitioning the number they are taking away in order to increase speed I can subtract by counting back, partitioning the number that I am subtracting Dartington C of E Primary School and Nursery Maths Strategies 12

13 Subtraction Difference Step 1. Comparing the difference between two groups (e.g of children) organised into lines Keep this within a true to life context for the children. E.g. 9 children drink lemonade, 5 drink water, what is the difference? I can compare the numerical difference between two groups of children. Step 2. Finding the difference by comparing two groups, this time using objects to represent actual children. Remain within the same context as above. I can find the difference between two sets of objects. Stage 3: Finding the difference by comparing two sets of children organised in a different mathematical way. This time arrange the children in a different way rather than in lines so that now they are beginning to compare the numbers rather than what they can see. I can find the difference between two sets of objects organised in different ways, Dartington C of E Primary School and Nursery Maths Strategies 13

14 Subtraction Difference Stage 4. Finding the difference between numbers represented through a mathematical image alongside the physical objects eg. Numicon and children. Examples of mathematical images: Numicon, Cuisenaire, base ten, I can find the difference between two sets of objects represented by mathematical equipment. Stage 5. Comparing two numbers represented by two mathematical images. E.g. Two bead strings or two pieces of numicon. I can find the difference between two sets of objects represented by mathematical equipment. Stage 6. Comparing numbers on two separate beaded number lines (with bead strings alongside at first), and counting on or back to find the difference. Show bead strings side by side and relate to number line. Use of Difference ITP is crucial here see Maths leader if you don t have a copy. I can find the difference by comparing numbers on two separate beaded number lines. Dartington C of E Primary School and Nursery Maths Strategies 14

15 Subtraction Difference Stage 7. Compare two numbers on the same number line through counting on or back - using a printed number line. Children to look at the numbers and decide whether to count on or back. E.g. When calculating is it quicker to count on or count back? Use beaded numberlines until children have this visual image really clear I can count on or back using a structured number line. Stage 7. Comparing two numbers on the same number line through counting on or back using a landmarked number line Children to look at the numbers and decide whether to count on or back. Use beaded numberlines until children have this visual image really clear I can count on or back using a structured landmarked beaded number line. Stage 8 Comparing two numbers by counting on or back using an unstructured (blank) number line, by children drawing their own number line Model very clearly how to draw number lines I can find the difference by counting on or back, by drawing my own number line. Dartington C of E Primary School and Nursery Maths Strategies 15

16 Subtraction Final stages If, throughout their experience of subtracting, the children are taught that subtraction is finding the difference and taking away, this next bit will be much clearer! Final Stage 1: Answering word problems by identifying an appropriate strategy. Children to identify whether a question is asking them to find the difference or take away and thereby choose an appropriate strategy. They need to understand that whichever method they use, this will lead to the same answer. I can choose an appropriate strategy to answer a subtraction question. I know that a Find the difference question is asking me to compare two numbers through subtraction. Final Stage 2. (Upper KS2) Formal column subtraction/ decomposition. Column subtraction is the most efficient method for subtracting larger unwieldy numbers so the children do need to learn it. However, it should only be taught when children have a solid grasp of place value and number bonds I can subtract using a formal written method. Introduce using base 10 resources, showing how 86 can be 80+6 but also Show that you can exchange the tens for ten ones in order to make the subtraction easier, Children need to be clear that we always take the bottom number away from the top number. Dartington C of E Primary School and Nursery Maths Strategies 16

17 Multiplication arrays lots of multiply groups of x times repeated addition products factors total Dartington C of E Primary School and Nursery Maths Strategies 17

18 Multiplication Stage 1a Counting groups of a number using real life objects Count groups of real life objects e.g straws, apples, pencils. 5, 10, 15, 20, 25. I can count groups of objects using repeated addition. Stage 1b Counting in groups of a number through chanting Chanting in 2s and 10s. 2, 4, 6, , 20, 30,... I can count up in 2s, 10s, 5s. Stage 2 Using arrays to understand repeated addition. Start off by arranging random groups of concrete materials into arrays. Repeated addition. 3 x 2 = 6 or 2 x 3 = (groups) (total) Count the groups we have counted as well as the total each time I can use an array to show my understanding of repeated addition. Use real life objects. Relate chanting to writing down the numbers as you go along or pointing out numbers on a number line. Show the number on number lines and bead strings or simply by writing the numbers. Can use numicon to support. Expect children to answer questions such as How many groups of... have we counted? Objects Images Arrays Always link arrays to the related division number sentences as well as fractions. So: 3 x 2 =6 2 x 3 =6 6 2 = = 2 1/3 of 6 = 2 1/2 of 6 = 3 Dartington C of E Primary School and Nursery Maths Strategies 18

19 Multiplication Stage 3 Repeated addition on a structured number line. Using structured number line to repeatedly add groups of numbers on. Use number stories Knowing times table facts off by heart. Includes using known facts to find other facts. Eg 2 x 3 = 6, so 2 x 6 must = 12 Show alongside an array I can repeatedly add a number on a number line, and know that I am multiplying. Stage 4. Repeated addition on an unstructured number line Using unstructured number line to repeatedly add groups of number on. Show on arrays alongside. 3 x 5 = I can multiply by repeatedly adding a number on an unstructured number line. Stage 5. Multiplication through repeated addition on a number line, using chunking (ie, using known facts to put the groups you count into chunks) 6 x 6 = 6 x 5 6 x So 6 x 5 and 6 x 1 is the same as 6 x 6 I can multiply on numberline by chunking groups of the number. Dartington C of E Primary School and Nursery Maths Strategies 19

20 Stage 6 Introduce grid method using arrays Recap on partitioning numbers and show using an array. 6 x 12 = Cut up into 6 x 10 and 6 x 2 Be clear that you are partitioning the numbers to make them easier to calculate. I can use an array to show how to partition for multiplication. Stage 7 Grid method for larger numbers Grid method using partitioning. 14 x 25 = X = = 70 Children need to have a firm understanding of partitioning before they are introduced to the grid method. = 350 I can multiply using the grid method. Stage 8 (Upper KS2) Column method for larger numbers 14 x X (this is 41 x 3) 820 (this is 41 x 20) 943 Initially these two steps can be done in two separate algorithms. I can multiply using a formal written method. Children need to be shown how this method relates to grid method so teach it alongside grid method. Dartington C of E Primary School and Nursery Maths Strategies 20

21 Division arrays lots of grouping groups of sharing repeated subtraction divisor/ dividend/ divisible remainder divide Dartington C of E Primary School and Nursery Maths Strategies 21

22 Division Stage 1 Grouping and sharing out a small number of physical objects. Number stories, concrete apparatus. Grouping or sharing out concrete materials. 4 2 = I can group or share a small number of objects. Stage 2 Drawing diagrams on paper to show understanding of grouping and sharing = 4 I can draw diagrams/pictures to show sharing and grouping. Always relate to fractions a third of 12 is 4, two third of 12 is 8. It is important to be clear on the difference between grouping and sharing. E.g Grouping would be putting 12 objects into groups of 3 and counting how many groups there are. Sharing is putting the objects into 3 groups, saying one for you, one for you etc putting one object into each group in turn until you run out of objects then counting how many objects are in each group. Stage 3 Using arrays to explain division and relate to number sentences. How many groups of 3 in 12? How many groups of 4 in 12? (groups) = (total) 12 3 = I can use arrays to show my understanding of division. Show this alongside a number line. Dartington C of E Primary School and Nursery Maths Strategies 22 Always link arrays to the related multiplication number sentences as well as fractions. So: 3 x 2 =6 2 x 3 =6 6 2 = = 2 1/3 of 6 = 2 1/2 of 6 = 3

23 Division Stage 4 Divide on a structured number line using repeated addition. Use repeated jumps (groups) to divide on a structured number line = I can divide on a number line using repeated division. Stage 5 Finding out about remainders Investigate what happens when you have some left over. So, 7 2= 3 remainder 1. Recorded as r1. Use in conjunction with the arrays at first. How many lots of 3 are there in 12? Also show this on a number line. I can divide into groups and explain what happens when there are some left over. Stage 6 Finding remainders on a structured number line = r I can use a structured number line to divide and find remainders. Stage 7 (Can in fact come at any later stage.) Relate fractions, percentages and decimals to division. Look at a counting stick showing 100. Where is a tenth? What about 10%? What about 0.1? Show that these all represent the same amount. Relate back to division. Allow the counting stick to represent a different number, what is 10% of this new number? How can we work it out? By dividing... I can relate fractions, decimals and percentages to division. Dartington C of E Primary School and Nursery Maths Strategies 23

24 Stage 8 Divide using an unstructured number line. Use an unstructured to divide numbers and continue to identify remainders = r I can use an unstructured number line to divide (and find remainders where appropriate) Stage 9 Chunking on a number line. Using an unstructured number line, carry out repeated addition in groups eg groups of 10, 5 or = 5 r 2 2x 2x 1x r I can use chunking to divide on a number line. Stage 10 Divide using an unstructured number line, dividing a remainder to find a decimal = 5 r 2 2x 2x 1x r Divide the remainder 2 into 5 groups again. 2 5=0.4 so the answer is 5.4 I can divide a number to find a decimal answer. Stage 11 Formal written method. (Upper KS2) Only to be used with bigger more unwieldy numbers where using a number line would be inefficient. Start on the left hand side, look at how many 4s in 8, then how many 4s in 5, then how many 4s in 4 and write the totals above r However, when looking at how many 4s in 5, there is 1 remainder 1, so this 1 is actually a ten and adds 1o ones to the 4 in the units column. Then how many 4s in 14? 3 remainder 2 I can divide by using a formal written method. Explain that sometimes we need to give an exact answer, not just a remainder. Contexts measurement, etc. Be clear that calculators do not give remainders! Be clear with your language with this example, when people say how many 4s in 8? they are actually looking at how many 400s in 800 and so on. Dartington C of E Primary School and Nursery Maths Strategies 24