1. Activities (from Guidelines in Number)

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1 Teach Early Years Number page Count all to add (two collections) Targets Children usually start to add by recounting both numbers of objects as an entirely new set to be counted. The next step is to build confidence and develop counting on. To add two numbers, the quickest strategy is to start at the larger (requiring an understanding of order) and then count forward the correct number of counts. (ENRP Add & subtract #1) Evidence Mathematics Online Interview Question Activities (from Guidelines in Number) 2. Numbers to 10 using 10-frames A basic skill involves great familiarity with combinations that total 10 or less. The best tool for this is the 10-frame. Students can explore the many combinations possible and learn about odd and even numbers. 16

2 Teach Early Years Number page Count on to add Targets Develop improved strategies, such as Complements to ten (counting up to the next ten, and then using the rest of the number being added as the ones digit), for example is (30) + 2 more to make 32. (ENRP Add & subtract #2) Evidence Mathematics Online Interview Question Activities (from Guidelines in Number) 2. Numbers to 20, with 20-frames As a logical extension of 10-frames, make use of 20-frames. Use them for finding and understanding the combinations of single digits that go over 10. These can be understood as making up to 10 and then some over. MORE> 17

3 Teach Early Years Number page Tens and ones Try to stress the place value idea that a number can be broken into tens and ones. It is the start of many similar approaches with larger numbers, and is also the basis of subtraction when the process is reversed. Here is an example: Find the first number, and count on. Note that you can think of 7 as 2 (to make 10 with the 8) and 5 Although children can find the answers by counting, they should get used to regrouping into ten and what is left. Children can answer by colouring over the first number in one colour and the part added in another. 4. Flashcards Use flashcards if you have them. Confident automatic response to these additions is vital. 5. Double any single digit. Treat this as adding. Students should learn to recognise that doubling 5, 6, 7, 8 or 9 will produce a tens digit. Instant recall of the pairs of numbers that add to 9 and to 10 is particularly useful later. 18

4 Teach Early Years Number page Count back / count down to / count up from to subtract Targets This is exactly the reverse of the process of adding two single digit numbers. This method stresses the idea that a number in tens and ones can be broken into separate parts. It is the start of many similar approaches with larger numbers, and is also the basis of subtraction when the process is reversed. (ENRP Add & subtract #3) Evidence Mathematics Online Interview Question Activities (from Guidelines in Number) 2. Subtract using only numbers to 10 using 10-frames A basic skill involves great familiarity with combinations that total 10 or less. The best tool for this is the 10-frame. Students can learn to see many combinations and represent them as additions or subtractions. This shows 9 4 = 5 This shows 9 5 = 4 MORE > 19

5 Teach Early Years Number page Find differences by counting backwards Here is an example: Start at the first number. Count backwards the amount of the second number. Note that you can count back to 10 by taking off the amount of the ones digit, then take off the rest. 4. Learn differences from 10 Because taking single digits from the teens can be done by subtracting back to 10 then from 10 (see 4 below), memorising differences from 10 is an essential skill. 5. Subtract from numbers to 20, with 20-frames As a logical extension of 10-frames, make use of 20-frames. Use them for finding and understanding the subtractions that go under 10 when taking single digits from the teens. This is done by breaking the number being subtracted into two parts. 6. Use flashcards if you have them. 20

6 Teach Early Years Number page Basic strategies for adding or subtracting (doubles, commutativity, adding 10, tens facts, other known facts) Targets Build confidence with mental computation, by the memorisation of a basic set of facts (already understood) and their extensions. Commutativity means that adding can be done in either order, and this is the essence of the flexibility for adding. For example : tens first (80 then add 14); ones first (14 then add 80); mixed order (e.g ); shifting numbers about (e.g. add , or ). (ENRP Add & subtract #4) Evidence Mathematics Online Interview Questions 21, 25 Adding 1. Activities (from Guidelines in Number) 2. Use a number chart Here is an example: Start at the first number. Count the second number from the end of the first. MORE > 21

7 Teach Early Years Number page Lead towards an appreciate of the extensions of each basic addition fact The idea is that children will recognise patterns like these: = = = = = = = = = = = = = = = = 47 and so on. If you know one, you also know the others. 4. Use flashcards if you have them. 5. Skip counting Children should be able to skip count, starting from any number and counting in steps of any single digit number. (Example 3, 12, 21, 30, 39, 48,...) This should be done with a 100-chart. If children put markers on the numbers they reach, they will see the patterns: numbers are in lines. It will assist with addition and subtraction as it reinforces the regrouping that occurs at each ten. 6. Use flashcards if you have them. 7. Play the dice game. Each child rolls a die (6 sided or 10 sided) and adds that number of ones to their current total. This means that single digit numbers are being added, mentally. If any child needs help, use the number chart or the material. The first child to reach 100 or more wins the game. 8. Sums This should be regarded as extension material for those who are faster than the others. Here are the answers. a How many ways can you add two different numbers to get 11? (0 + 11, , 2 + 9, 3 + 8, 4 + 7, There are six different ways.) b How many ways can you add two different numbers to get 12? (0 + 12, , , 3 + 9, 4 + 8, There are six different ways. Kids are usually surprised; they think it should be seven, because they think it goes up in 1s is not allowed, because the numbers must be different.) c How many ways can you add two different numbers to get 13? (0 + 13, , , , 4 + 9, 5 + 8, There are seven different ways.) d Continue, and include predicting using the pattern, and testing predictions. e Try to get the kids to describe and explain the pattern. Here is the completed table. Total Number of ways f You can easily extend them to adding three single digits. There are fewer combinations than you might think. Here is the completed table. Its pattern is not so simple! Total Number of ways

8 Subtracting Activities (from Guidelines in Number) 1. Bundles of sticks and Base 10 blocks Teach Early Years Number page 23 We are subtracting single digit numbers from double-digit numbers. There are two types of problem: simple ones without any regrouping, and those involving regrouping of bundles of tens (or longs). We are also subtracting double-digit numbers from double-digit numbers without any regrouping, 2. Mental computation This time the process is similar, but there are some extra tens involved. There are two cases, one when there is no carrying (no ten needs to be regrouped) and one when we regroup a ten so we can take away the ones. Here is an example: Start with the first number and count backwards. Note that the 8 can be split into 5 (to take from 35) and 3 (to take from 30 leaving 27). Lead towards an appreciation of the extensions of each basic subtraction fact The idea is that children will recognise patterns like these: 15 6 = = = = = = = = = = = = = = = = 37 and so on. If you know one, you also know the others. Use flashcards if you have them. Quick mental response is very valuable. 23

9 Teach Early Years Number page Skip counting backwards Children should be able to skip count, starting from any number and counting backwards in steps of any single digit number. (Example 55, 48, 41, 34, 27,...) This should be done with a number chart (to check answers). It will assist with addition and subtraction as it reinforces the regrouping that occurs at each ten. The tough part of skip counting backwards is going back past the previous ten. Suppose we are counting backwards by 3s. So 45 3 = 42 is easy, but 42 3 = 39 is much harder. Each case should be seen as an extension of the idea between 0 and 20. For example, 42 3 is just an extension of 12 3 = 9, so 42 3 = 39. This idea is usefully shown on a number line. Use flashcards if you have them. Quick mental response is very valuable. 4. Play the dice game, backwards. Each child starts with 100. On their turn they roll a die (6 sided or 10 sided) and take that number of ones from their current total. This means that single digit numbers are being subtracted, mentally. If any child needs help, use the number chart. The first child to reach 0 or less wins the game. 5. Differences This should be regarded as extension material for children who are ahead. a How many different pairs of numbers (0 to 19) have a difference of 9? (9 0, 10 1, 12 3, 13 4, 14 5, 15 6, 16 7, 17 8, 18 9, There are ten ways.) b How many different pairs of numbers (0 to 19) have a difference of 8? (8 0, 9 1, 10 2, 11 3, 12 4, 13 5, 14 6, 15 7, 16 8, 17 9, 18 10, There are 11 ways.) Start a table of results. Difference Number of ways c Continue, and include predicting using the pattern, and testing predictions. d Try to get the kids to describe and explain the pattern. Here is the completed table. Difference Number of ways

10 Teach Early Years Number page Derived strategies for adding or subtracting (near doubles, adding 9, build to next ten, fact families, intuitive strategies) Targets Build awareness of fact families, patterns and other strategies for adding and subtracting. (ENRP Add & subtract #5) Evidence Mathematics Online Interview Question 30 Adding Activities (from Guidelines in Number) 1. Use bundles of sticks 2. Use Base 10 blocks 25

11 Teach Early Years Number page Mental computation Here is an example: a Add the tens first ( = 50). b Then add the two ones digits, which should be a quick mental calculation at this stage (5 + 8 = 13). c These will be in addition to the 50 we have already, so that makes Doubling Doubling any single tens digit. Students should learn to recognise that doubling 50, 60, 70, 80, 90 will produce a hundreds digit. Doubling small two-digit numbers, e.g. double 13 (for = 26), 16 (for = 32) Students should learn to recognise that doubling numbers ending in 5 or more etc. will produce an extra tens digit. Extend this to doubling any two digit number. 5. Use flashcards if you have them. Aim for quick mental response. 26

12 Subtracting Activities (from Guidelines in Number) 1. Use bundles of sticks. Teach Early Years Number page Use Base 10 blocks 27

13 3. Mental computation Teach Early Years Number page 28 Mostly this is very similar to the situation above. As well as subtracting the ones, we also subtract tens. It is important to recognise that when operating mentally with a regrouping question (such as 45 28) very few people actually subtract the ones first. Many people use alternatives that feel comfortable to them. For example, here are three legitimate strategies with 45 28: a subtract 20 first to get 25, then take 8 to get 17. b subtract the 8 first to get 37, then subtract 20. c add 2 to both numbers to change the question to = 17. (This strategy makes the subtracted number up to a multiple of ten. In so doing the difficulty with the ones disappears.) Here is an example for strategy (a): a Start with 45. First take away 20 to get 25. b Split the 45 into 30 and 15, in your head. c Take the 8 from the 25, by splitting 8 into two parts (5 and 3) so 25 8 = = 20 3 = Use flashcards if you have them. 28

14 Teach Early Years Number page Extending and applying addition and subtraction using basic, derived and intuitive strategies mentally Targets Using the mental image of the 100 chart, developed from its frequent use in the classroom, students should be able to add and subtract numbers fluently. (ENRP Add & subtract #6) Evidence Mathematics Online Interview Question 36 Activities Adding 1. Teach the extensions The idea is that children will recognise patterns like these: if the ones digits go over 10, then add an extra = = = = 17 Add any tens: = = = = = = = = = = = = 97 and so on. If you know one, you also know the others. 2. Play the dice game. Each child rolls a die (6 sided or 10 sided) and adds that number of ones to their current total. This means that single digit numbers are being added, mentally. If any child needs help, use the number chart or the material. The first child to reach 100 or more wins the game. 29

15 Subtracting Teach Early Years Number page Teach the extensions The idea is that children will recognise patterns like these: 15 6 = = = = = = = = = = = = = = = = 7 and so on. If you know one, you also know the others. 2. Play the dice game, backwards. Each child starts with 100. On their turn they roll a die (6 sided or 10 sided) and take that number of ones from their current total. This means that single digit numbers are being subtracted, mentally. If any child needs help, use the number chart. The first child to reach 0 or less wins the game. 3. Skip counting backwards Children should be able to skip count, starting from any number and counting backwards in steps of any single digit number. (Example 55, 48, 41, 34, 27,...) This should be done with a number chart (to check answers). It will assist with addition and subtraction as it reinforces the regrouping that occurs at each ten. The tough part of skip counting backwards is going back past the previous ten. Suppose we are counting backwards by 3s. So 45 3 = 42 is easy, but 42 3 = 39 is much harder. Each case should be seen as an extension of the idea between 0 and 20. For example, 42 3 is just an extension of 12 3 = 9, so 42 3 = 39. This idea is usefully shown on a number line. 30

16 19 Subtracting many-digit numbers There are four kinds of problems that can arise here. Teach Early Years Number page 31 a problems with no regrouping needed, such as b problems with one regrouping, such as or c problems with two regroupings, such as d problems with zeros as the tens digit, such as This has always been difficult for students, particularly the last type. Regrouping and crossing out numbers can get students very confused, particularly if they have no concepts of place value. The best approach is to use the Base 10 blocks, then move to diagrams, as shown below. Suggested activities 1. Subtract using Base 10 blocks a The approach is to make the larger number and physically take away the smaller number, leaving the answer. This is easy when there is no regrouping required. (Type A) b When there are not enough minis to subtract, one of the longs must be traded for minis. This matches the renaming that usually occurs. (Type B) c A similar method is obvious for when there are not enough longs. (Type B) d It is only a little more complex when you need to regroup both a long and a flat. (Type C) 31

17 Teach Early Years Number page 32 e The chief difficulty with Base 10 blocks occurs when there is a zero in the tens column, so no longs can be regrouped. students need to learn that a flat may be regrouped directly (in one step) into 9 longs and 10 minis. 2. Subtract using diagrams The approach suggested here tries to help them make sense of all this. This approach offers a mental image of what is going on. We draw the number first, and then shade in what is being subtracted. What is left unshaded is the answer. The hundreds are shown as squares containing ten strips, as in the 100-squares we have been using. Here are examples of each type, with some explanation of how the material and diagram might help. Students colour in the number being subtracted. It can be done in any order, even left to right! 32