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Laying the Foundations for Algebra

Laying the Foundations for Algebra. 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. Lancashire Mathematics Team. Algebra in Primary School.

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Laying the Foundations for Algebra

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  1. Laying the Foundations for Algebra

  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. Lancashire Mathematics Team

  3. Algebra in Primary School Focus for this session: • Forming equations • Solving equations • Using inverses • Identifying number patterns • Expressing relationships

  4. Forming equations = 9 + 3 6

  5. Solving Equations 6 +  = 9 6 + 3 = 

  6. Solving Equations 6 + 3 =  + 4

  7. Solving Equations 6 + 3 =  + 

  8. Solving Equations What do we know about the missing number? What could we do next to find the missing number?

  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.

  10. Using inverses ( - 10) x2 = 44 ( - 10) = 44 ÷ 2 ( - 10) = 22  = 22 + 10  = 32

  11. ? +10 Answer 35 Using inverses Emily chooses a number. She halved the number then added 10 to the result. Her answer was 35. What was the number she started with?

  12. Using inverses Ben thinks of a number. He adds half of the number to a quarter of the number. The result is 60. What was the number Ben first thought of? What images might you draw to answer this question?

  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 ? ? ?

  15. Identifying number patterns

  16. Identifying number patterns 2, 7, 12, 17….. Can you continue the pattern? What would the 20th term 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

  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 Make the fourth shape using multilink. Describe the shape to your partner. Can you explain how the pattern is developing?

  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.

  22. Patterns, sequences and rules Here is a sequence of patterns made from squares and circles. How many squares will be in the pattern that has 25 circles?

  23. Patterns, sequences and rules 3 5 7

  24. Patterns, sequences and rules

  25. Patterns, sequences and rules

  26. Snowflake sequences

  27. Cops and Robbers

  28. Cops and Robbers Area ITP

  29. Money bags Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p, without opening any bag. How many pennies did Ram put in each bag?

  30. “The radical mistake of algebra teaching is in jumping from Particular Arithmetic to Symbolic Algebra and omitting work on Generalised Arithmetic……for generalised arithmetic is the simplest form of algebra.” B. Branford 1908

  31. 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.

  32. 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

  33. Links 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?

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

  35. Partner numbers Every number has a partner number. Make up a rule. Fill in some partner numbers. Give it to a friend to try to complete. 9 4 9 3 2 27

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

  37. 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. Playtrain

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