Laying the Foundations for Algebra

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Objectives. To identify a progression in Algebra.To form and solve equationsTo explore patterns, sequences and rules.To generalise number sequences and express relationships algebraically.. Lancashire Mathematics Team. Algebra in Primary School. Focus for this session:Forming equationsSolving

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Laying the Foundations for Algebra

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2. Objectives To identify a progression in Algebra. To form and solve equations To explore patterns, sequences and rules. To generalise number sequences and express relationships algebraically.

3. Algebra in Primary School Focus for this session: Forming equations Solving equations Using inverses Identifying number patterns Expressing relationships Look through the handout, adapted from the introduction to the 1999 Numeracy Framework. We are going to focus on these 5 areas and see how we as teachers can contribute in class and how we can teach these practically. Basically – an equation is a posh way of saying ‘number sentence’ Solving number sentences Look through the handout, adapted from the introduction to the 1999 Numeracy Framework. We are going to focus on these 5 areas and see how we as teachers can contribute in class and how we can teach these practically. Basically – an equation is a posh way of saying ‘number sentence’ Solving number sentences

4. Forming equations Forming equations (or forming number sentences) – from the word go we are using symbols to represent amounts – from before Reception we introduce digits that represent amounts and we call these numbers. We then introduce symbols that tells us what to do with the numbers - + = for example. This amount of orange counters is represented by this digit. The amount of green counters is represented by this digit. The symbol that tells us to combine these amount is + The equals sign introduces the idea of balance ie one side is the same as the other sideForming equations (or forming number sentences) – from the word go we are using symbols to represent amounts – from before Reception we introduce digits that represent amounts and we call these numbers. We then introduce symbols that tells us what to do with the numbers - + = for example. This amount of orange counters is represented by this digit. The amount of green counters is represented by this digit. The symbol that tells us to combine these amount is + The equals sign introduces the idea of balance ie one side is the same as the other side

5. Solving Equations Solving number sentences This is the equation we have just looked at. We have to ask the children to solve missing number questions – often children interpret the = sign as ‘makes’ and not as a ‘balance’. In Reception when children are asked to solve problems practically, such as we have 6 paint brushes, 9 children want to paint, how many more paint brushes do we need? This is the equation that they are actually solving. Often children will add the 6 and 9 because they do not see the = as a balance – they think they just add whatever numbers they see and put the answer where there’s a space. Children need plenty of opportunities to answer these sorts of questions. Solving number sentences This is the equation we have just looked at. We have to ask the children to solve missing number questions – often children interpret the = sign as ‘makes’ and not as a ‘balance’. In Reception when children are asked to solve problems practically, such as we have 6 paint brushes, 9 children want to paint, how many more paint brushes do we need? This is the equation that they are actually solving. Often children will add the 6 and 9 because they do not see the = as a balance – they think they just add whatever numbers they see and put the answer where there’s a space. Children need plenty of opportunities to answer these sorts of questions.

6. Solving Equations Solving number sentences Children need to see and be able to use equations where there is a calculation on either side of the = signSolving number sentences Children need to see and be able to use equations where there is a calculation on either side of the = sign

7. Solving Equations Solving number sentences This can be extended into more than one missing number *Activity – ‘what can you tell me about the two missing numbers?’ (key answers – both numbers are different – they can not be 4.5 + 4.5 Also possible answer – the second number could be a negative number) This also introduces children to the concept that an unknown number is not necessarily ‘fixed’, but could have a number of possible values.Solving number sentences This can be extended into more than one missing number *Activity – ‘what can you tell me about the two missing numbers?’ (key answers – both numbers are different – they can not be 4.5 + 4.5 Also possible answer – the second number could be a negative number) This also introduces children to the concept that an unknown number is not necessarily ‘fixed’, but could have a number of possible values.

8. Solving Equations The ITP number scales is great for generating missing number problems and giving examples of balanced equations. Show a variety of examples – how two different calculations can balance or not balance e.g. 3 + 4 < 9 – 1 Use the ? Feature to promote problem solving skills. If you want to see the value of the ? Click on the Total button. This could be set up before the children come into class or you could use the ‘Freeze’ button on your projector remote, while you look without the children seeing. You need to ‘Unfreeze’ the projector when you want to continue. The recording of this problem solving could take the form 11 < ? > 30, with the range gradually being narrowed until the number is found. This can be downloaded from the Lancashire Mathematics Team website.The ITP number scales is great for generating missing number problems and giving examples of balanced equations. Show a variety of examples – how two different calculations can balance or not balance e.g. 3 + 4 < 9 – 1 Use the ? Feature to promote problem solving skills. If you want to see the value of the ? Click on the Total button. This could be set up before the children come into class or you could use the ‘Freeze’ button on your projector remote, while you look without the children seeing. You need to ‘Unfreeze’ the projector when you want to continue. The recording of this problem solving could take the form 11 < ? > 30, with the range gradually being narrowed until the number is found. This can be downloaded from the Lancashire Mathematics Team website.

9. Using inverses I think of a number, subtract 10 and double the result. The answer is 44. What is my number? Answer the question and discuss the strategies you used. When forming and solving equations, children need an understanding of the relationships between the different operations – this is not something separate – we have been doing inverse in a simple form Get staff to discuss and pool ideas - maybe everyone will have done this the same way – using inverse operations – children possibly will not do this naturally and will need to have opportunity to see the understanding as to why ‘working backwards’ gives us the answer Exemplify the number sentence – ask staff to write the number sentence to go with this calculation When forming and solving equations, children need an understanding of the relationships between the different operations – this is not something separate – we have been doing inverse in a simple form Get staff to discuss and pool ideas - maybe everyone will have done this the same way – using inverse operations – children possibly will not do this naturally and will need to have opportunity to see the understanding as to why ‘working backwards’ gives us the answer Exemplify the number sentence – ask staff to write the number sentence to go with this calculation

10. Using inverses Remember what the = sign means – whatever is on the one side has to balance the other Therefore what ever we do to the one side we have to do to the other. Our aim is to isolate the unknown value on the left of the = sign First step - eliminate the x2. in order to do this we need to divide each side by 2 Next step – eliminate the -10, so we need to +10 to each sideRemember what the = sign means – whatever is on the one side has to balance the other Therefore what ever we do to the one side we have to do to the other. Our aim is to isolate the unknown value on the left of the = sign First step - eliminate the x2. in order to do this we need to divide each side by 2 Next step – eliminate the -10, so we need to +10 to each side

11. Using inverses

12. Using inverses

13. Identifying number patterns Before identifying patterns in number, they need to be able to identify and make repeating patterns using shapes and colours

14. Identifying number patterns We need to extend this into patterns relating to size, remembering to extend the pattern forwards and backwards.We need to extend this into patterns relating to size, remembering to extend the pattern forwards and backwards.

15. Identifying number patterns

16. Identifying number patterns 2, 7, 12, 17….. Can you continue the pattern? What would the 20th term be? Children need to have opportunity to extend, describe patterns of number and sometimes identify why a pattern happens ie looking at the patterns made by multiples of 4 or 5 in a 10x10 multiplication grid. In this example, they need to be able to describe how they would go about finding out what the 20th term would be.Children need to have opportunity to extend, describe patterns of number and sometimes identify why a pattern happens ie looking at the patterns made by multiples of 4 or 5 in a 10x10 multiplication grid. In this example, they need to be able to describe how they would go about finding out what the 20th term would be.

17. Patterns, sequences and rules YR Talk about, recognise and recreate simple repeating patterns. Y1 Describe simple patterns and relationships involving numbers or shapes; decide whether examples satisfy given criteria. Y2 Describe patterns and relationships involving numbers or shapes, make predictions and test these with examples.

18. Patterns, sequences and rules Using this arrangement, we can ask a variety of questions at varying levels, which we will work through now. What colour comes next in the sequence (after the blue cube)? GREEN Why? The sequence of colours is repeating blue green red, so after every blue is green. What shape should have come before the green cube on the left? A LARGE CUBOID Why? The sequence of shapes is repeating cube, triangular prism, small cuboid, large cuboid, so before a cube is always a large cuboid. If the green cube on the left is shape 1, the red triangular prism shape 2 and so on, answer the following questions. 1. What will be the colour of the 12th shape? BLUE Why? The blue shapes are on the multiples of 3 and 12 is a multiple of 3. 2. In what position will the 4th triangular prism be? 14 Why? Triangular prisms are every fourth shape, with the count starting at 2. 3. What colour will the fourth triangular prism be? RED Why? 14 (shapes) divided by 3 (colours) = 4 r 2 i.e. repeatedly subtracting 3 from 14 will leave you with 2 remaining. The colour at position 2 is red. 4. What will be the colour and shape in position 37? Green cube Why? Green because 37 is a multiple of 3 +1. Multiples of 3 are blue, therefore all multiples of 3 +1 are green. Cube because 37 is a multiple of 4 +1. Multiples of 4 are large cuboids, therefore all multiples of 4 +1 are cubes. Hint when creating such sequences – if having 2 variables (colour and shape), ensure the number of each variable is different e.g. 3 colours, 4 shapes. Using this arrangement, we can ask a variety of questions at varying levels, which we will work through now. What colour comes next in the sequence (after the blue cube)? GREEN Why? The sequence of colours is repeating blue green red, so after every blue is green. What shape should have come before the green cube on the left? A LARGE CUBOID Why? The sequence of shapes is repeating cube, triangular prism, small cuboid, large cuboid, so before a cube is always a large cuboid. If the green cube on the left is shape 1, the red triangular prism shape 2 and so on, answer the following questions. 1. What will be the colour of the 12th shape? BLUE Why? The blue shapes are on the multiples of 3 and 12 is a multiple of 3. 2. In what position will the 4th triangular prism be? 14 Why? Triangular prisms are every fourth shape, with the count starting at 2. 3. What colour will the fourth triangular prism be? RED Why? 14 (shapes) divided by 3 (colours) = 4 r 2 i.e. repeatedly subtracting 3 from 14 will leave you with 2 remaining. The colour at position 2 is red. 4. What will be the colour and shape in position 37? Green cube Why? Green because 37 is a multiple of 3 +1. Multiples of 3 are blue, therefore all multiples of 3 +1 are green. Cube because 37 is a multiple of 4 +1. Multiples of 4 are large cuboids, therefore all multiples of 4 +1 are cubes. Hint when creating such sequences – if having 2 variables (colour and shape), ensure the number of each variable is different e.g. 3 colours, 4 shapes.

19. Patterns, sequences and rules Y3 Identify patterns and relationships involving numbers or shapes, and use these to solve problems. Y4 Identify and use patterns, relationships and properties of numbers or shapes; investigate a statement involving numbers and test it with examples. Y5 Explore patterns, properties and relationships and propose a general statement involving numbers or shapes; identify examples for which the statement is true or false. Y6 Represent and interpret sequences, patterns and relationships involving numbers and shapes; suggest and test hypotheses; construct and use simple expressions and formulae in words then symbols. Y6/7 Generate sequences and describe the general term; use letters and symbols to represent unknown numbers or variables; represent simple relationships as graphs.

20. Patterns, sequences and rules

21. Expressing relationships Possible answers: Vertical columns: 1, 2+1, 3+2, 4+3, Pairs added: 1, 1+2, 1+2+2, 1+2+2+2, Complete rectangle 1, (2x2)-1, (2x3)-1, (2x4)-1 Predict what the 10th shape will look like and how many cubes will be used to make it. Formulae: 2n-1 where n is the number of cubes in the left hand column.

23. Recording in a table can help children spot patterns and develop rules. There are 2 patterns here – the up and down pattern – sequential pattern that determines how to continue the pattern. So in this example the sequential generalisation is ‘ adding 2’, however when we ask the children to answer what number of circles are there if there are 100 squares, we realise that this method is inefficient. We need to find out the pattern or relationship between the squares and circles, so in other words what we need to do to the number of squares to get to the number of circles.Recording in a table can help children spot patterns and develop rules. There are 2 patterns here – the up and down pattern – sequential pattern that determines how to continue the pattern. So in this example the sequential generalisation is ‘ adding 2’, however when we ask the children to answer what number of circles are there if there are 100 squares, we realise that this method is inefficient. We need to find out the pattern or relationship between the squares and circles, so in other words what we need to do to the number of squares to get to the number of circles.

24. The rule to find the number of circles is double the number of squares and + 1. However, when given the number of circles we are having to use our knowledge of inverses from earlier. This may be the number sentence to work with (? x 2) + 1 = 25 Again trying to isolate the ? by subtracting 1 from each side of the = Then dividing each side of the = by 2 Formula: C = 2S + 1 The rule to find the number of circles is double the number of squares and + 1. However, when given the number of circles we are having to use our knowledge of inverses from earlier. This may be the number sentence to work with (? x 2) + 1 = 25 Again trying to isolate the ? by subtracting 1 from each side of the = Then dividing each side of the = by 2 Formula: C = 2S + 1

25. We could now pose the question How many squares will be in the pattern that has 25 circles? Remembering that: C = 2S + 1 And we know how to solve equations: 25 = 2S +1 Subtract 1 from 25 = 24 and then divide by 2 or halve = 12We could now pose the question How many squares will be in the pattern that has 25 circles? Remembering that: C = 2S + 1 And we know how to solve equations: 25 = 2S +1 Subtract 1 from 25 = 24 and then divide by 2 or halve = 12

26. Snowflake sequences This programme is a Gordon ITP. There are many options for the range of numbers. Two of the advantages of this programme involve extending sequences forwards and backwards and the potential for children to problem solve. Demonstrate how it can be used. This can be downloaded from the Lancashire Mathematics Team website.This programme is a Gordon ITP. There are many options for the range of numbers. Two of the advantages of this programme involve extending sequences forwards and backwards and the potential for children to problem solve. Demonstrate how it can be used. This can be downloaded from the Lancashire Mathematics Team website.

27. Cops and Robbers

28. Cops and Robbers This can be modelled using Area ITP with the children using the ITP themselves in the ICT suite. This can be modelled using Area ITP with the children using the ITP themselves in the ICT suite.

29. Taken from the Mathematical challenges for able pupils in Key Stages 1 and 2 booklet.Taken from the Mathematical challenges for able pupils in Key Stages 1 and 2 booklet.

30. Integration of using and applying – representing their work systematically, and communicating their reasons. 15 pennies arranged in 4 bags. Answer – 1p bag, 2p bag, 4p bag, 8p bag Discuss pattern – repeated doubling. Integration of using and applying – representing their work systematically, and communicating their reasons. 15 pennies arranged in 4 bags. Answer – 1p bag, 2p bag, 4p bag, 8p bag Discuss pattern – repeated doubling.

31. 31 pennies in 5 bags Answer – 1p bag, 2p bag, 4p bag, 8p bag, 16p bag. Recognising patterns and predicting. Notice the total number of pennies chosen is 16p more than the previous total of 15.31 pennies in 5 bags Answer – 1p bag, 2p bag, 4p bag, 8p bag, 16p bag. Recognising patterns and predicting. Notice the total number of pennies chosen is 16p more than the previous total of 15.

32. The question changes to focus on the number of bags as well as the can you pay for all amounts.The question changes to focus on the number of bags as well as the can you pay for all amounts.

35. Taking the activity to this level will be at the discretion of the teacher.Taking the activity to this level will be at the discretion of the teacher.

38. Generalised Arithmetic Work out the following: 3037 - 258 Use it to find the following: 3037 - 259 3037 – 257 3037 – 268 Write two calculations with the same answer as 6214 – 1989, explain what you did to the numbers. Ask participants to work through and discuss with a partner the following questions on a handout. Ellen buys 16 packets of crisps at 26p each. This costs £4.16. Use this information to work out: The cost of 17 packets The cost of 15 packets The cost of 16 packets at 25p each The cost of 16 packets at 28p each The cost of 32 packets The cost of 16 packets at 52p each What other questions would this information help you to answer? Ask participants to work through and discuss with a partner the following questions on a handout. Ellen buys 16 packets of crisps at 26p each. This costs £4.16. Use this information to work out: The cost of 17 packets The cost of 15 packets The cost of 16 packets at 25p each The cost of 16 packets at 28p each The cost of 32 packets The cost of 16 packets at 52p each What other questions would this information help you to answer?

39. Algebra times table B x H = AE A x H = CF G x H = FD F x H = D CJ x H = HJ I x H = FJ E x H = FH C x H = H H x H = CE D x H = AF Give handout for participants to work out which times table this represents – it has been rearranged and there isn’t a 0 x ? Number sentence. Possible starting points – C x H = H so C must be 1, or identifying that there are only two answers that are single digits so H must be 4.Give handout for participants to work out which times table this represents – it has been rearranged and there isn’t a 0 x ? Number sentence. Possible starting points – C x H = H so C must be 1, or identifying that there are only two answers that are single digits so H must be 4.

40. The links have these values: Red = 2 Yellow = 3 Blue = 4 Can you make a chain worth 18? How many different chains worth 18 can you make? Links

41. Links All of these chains have a total value of 12.

42. Partner numbers Rule is left number -1 then x3Rule is left number -1 then x3

43. The Process Look at it Try to extend it Verbalise it Predict Generalise

44. Key Messages It is important to lay the foundations for Algebra from Reception. Children need to build upon their previous experience. It is important to follow a progression. Encourage children to follow the process – look at a pattern, extend it, verbalise it, predict and generalise.

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