Algebraic Patterning

1. Algebraic Patterning Autauga County Inservice Day January 4, 2010 Deanna McKinley DPES

2. Introduction AL COS #3 states Solve problems using numeric and geometric patterns. Bullet states determine a verbal rule for a function given the input and output The study of patterns is a key part of algebraic thinking. They involve relationships and generalizations.

3. NCTM Standards show the progression PRIMARY CLASSROOMS (K-5) • Recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another. Analyze how both repeating and growing patterns are generated • Describe, extend, and make generalizations about geometric and numeric patterns. Represent and analyze patterns and functions, using words, tables, and graphs • Grades 6–8 • Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules

4. Why Patterns? Patterning is critical to the abstraction of mathematical ideas and relationships, and the development of mathematical reasoning in young children. (English, 2004; Mulligan, Prescott & Mitchelmore, 2004; Waters, 2004) The integration of patterning in early mathematics learning can promote the development of mathematical modelling, representation and abstraction of mathematical ideas. (Papic & Mulligan, Preschoolers’ Mathematical Patterning) Patterns are everywhere; we just need to learn to notice them.

5. Patterns progression Copy a pattern and create the next element Predict relationship values by continuing the pattern with systematic counting Predict relationship values using recursive methods e.g. table of values, numeric expression Predict relationship values using direct rules e.g. ? x 3 + 1 Express a relationship using algebraic symbols with structural understanding e.g. m = 6f + 2 or m = 8 + 6(f – 1) Wright (1998). The learning and Teaching of Algebra: Patterns, Problems and Possibilities.

6. Patterns Jargon Number sequences - Number patterns Sequential - Spatial Linear - Geometric Sequential rules - Functional rules Repeating patterns - Growing pattern Trends - Values

7. Activities for Exploring Number Patterns Paper Folding Fold a piece of paper in half, and then in half again, and again, until you make six folds. When you open it up, how many sections will there be? Make a chart. Continue to fold as you look for a pattern. Paper Tearing Tear a piece of paper in half and give half to someone else. Each person then tears the piece of paper in half and passes half on to another person. How many people will have a piece of paper after 10 rounds of tearing paper like this? Continue tearing paper and record on a chart. Look for patterns as you complete each round.

8. What do we do? Spatial repeating patterns

9. What do we do?Repeating patterns with beads

10. What do we do? Spatial growing patterns

11. What do we do? Spatial and number patterns

12. What do we do? Spatial growing patterns Squares Sticks 3n + 1 4n + 2 4n + 4 3n + 1

13. What do we do? Number Machines

14. Some student work...

15. Pre-repeating patterns

16. Post-repeating patterns

17. Pre-Spatial & Number patterns

18. Post-Number patterns

19. Post-Spatial patterns

20. Pre-Number Machines

21. Post-Number Machines

22. And … Post-Functions

23. Pre-Number Machines

24. Post-Number Machines

25. Some points Lots of hands on material based exploration followed by group discussion. Materials can get in the way and we have to move on. Develop understanding by decomposing spatial shapes in a pattern (i.e., finding what is different and similar) We found beads very helpful to elicit discussion leading to functional rules between the colors Some students preferred to work with the numbers than the spatial patterns (they could see patterns easier), therefore keep using the numbers and spatial patterns together. Don't always put the values of a number pattern table in order. Take the number machines to the next level and then require students to connect it to the functional rule for a number pattern.

26. Bellringers and Journal Ideas Start to use all numbers (rational, irrational, weird, negative) and get students to experiment with calculators. (Stacey and MacGregor, Building foundations for Algebra, 1997) Connecting patterns – tables – graphs. SEE HANDOUTS

27. Basic fact patterns Instant recognition of series • 5x 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 • 2x 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 • 4x 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 • 3x 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 • 9x 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 • 6x 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 • 8x 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 • 7x 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

28. Fractions - Decimals - Percentages • Halves, quarters, and eighths 1/2 0.5 50% 1/4 0.25 25% 1/8 0.125 12.5% 1/2 x table 0.5 1.0 1.5 2.0 2.5 … 5x table 1/4 x table 0.25 0.50 0.75 1.00 1.25 … 25x table 1/8 x table 0.125 0.250 0.375 0.500 0.625 … 125x table

29. Patterns Internal patterns • 10x 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 • 5x 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 • 2x 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 • 8x 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 • 9x 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

30. Fractions – Decimals - Percentages • Thirds, ninths, and sixths times table 1/3 0.333 33.3% 1/9 0.111 11.1% 1/6 0.166 16.6% 1/3 x table 0.333 0.666 0.999 (=1!) … 1/9 x table 0.111 0.222 0.333 0.444 …0.999 (=1) 11x table 1/6 x table0.166 0.333, 0.500, 0.666, 0.833, 1.000

31. More basic facts Patterns Instant recognition of series Instant recognition of membership • Power series1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 • Square numbers1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 • Triangular numbers1, 3, 6, 10, 15, 21, 28, 36, 45 • Cubic numbers1, 8, 27, 81, 125

32. A pattern is a pattern is a pattern Kindergarteners are asked to make patterns and to continue a pattern by coloring So by 3rd grade we better be asking a little more… Making the steps toward generalizing

33. create, represent and continue a variety of number patterns and supply missing elements Stage 1

34. build number relationships(addition and subtraction facts to at least 20)

35. make generalizations about number relationships ‘When I add zero it does not change the number’ ‘when I count by fives the last number goes five, zero, five, zero, …’ An odd number plus an odd number always equals an even number’

36. use the equals sign to record equivalent number relationships

37. generate, describe and record number patterns using a variety of strategies Stage 2 • 1, 4, 7, 10, … • 2.2, 2.0, 1.8, 1.6, … • 2 3 4 5 6 7 • 4, 4, 4, 4, 4, 4, 4, … (generate using calculator, materials or mental strategies)

38. build number relationships(relating multiplication and division facts to at least 10 x 10) 2 x 4 = 4 x 2 (applying the commutative property) ‘The multiplication facts for 6 are double the multiplication facts for 3’ ‘6x4 = 24; so 24÷4 = 6 and 24÷6 = 4. (describing the relationships) (relating multiplication and division)

39. Find so that5+ = 13, Find so that 28 = x 7 • complete simple number sentences by calculating the value of a missing number

40. build simple geometric patterns involving multiples Stage 3 - algebra without symbols

41. complete a table of values for geometric and number patterns

42. describe a pattern in words in more than one way (determining a rule to describe the pattern from the table) ‘It looks like the 3 times tables.’ ‘You multiply the top number by three to get the bottom number.’

43. Construct, verify and complete number sentences • completing number sentences: 5 +  = 12 – 4 7 x  = 7.7 • constructing number sentences to match a word problem • checking solutions and describing strategies

44. Stage 4 - algebra with symbols Five outcomes at this Stage: *use letters to represent numbers *describe number patterns with symbols *simplify, expand and factorise algebraic expressions *solve equations and simple inequalities *graph on the number plane

45. A Row of Triangles, Squares, or Pentagons A Row of Triangles If you line up 100 equilateral triangles (like the green ones in Pattern Blocks) in a row, what will the perimeter measure? If you think of this as a long banquet table, how many people can be seated? Create a chart to record the data and look for patterns as you add triangles. A Row of Squares What will the perimeter measure if you line up a row of 100 squares? If this is a long table, how many people can be seated? Create a chart for your data like the one in the Row of Triangles problem. A Row of Pentagons What will the perimeter measure if you line up a row of 100 pentagons? If this is a long table, how many people can be seated? Create a chart for your data like the one in the Row of Triangles and Row of Squares problem.

46. Handshakes Suppose everyone in the room shakes hands with every other person in the room. How many handshakes will that be? (With one person, there will be no handshake. With two people, there will be one handshake. How many handshakes will there be with three people? Four? Continue by creating a chart of the data and look for patterns.

47. Hundred Board Wonders Select a rule from the list below to explore number patterns using a hundred board • Numbers with a two in them • Numbers whose digits have a difference of 1 (Be sure the students always select numbers whose tens-place digit is 1 greater than the ones-place digit.) • Numbers with a 4 in them • Numbers that are multiples of 3 • Numbers with a 7 in them • Numbers that are multiples of 5 • Numbers with a 0 in them • Numbers that are divisible by 6 • Numbers with a 5 in the tens place • Numbers that are multiples of 4 • Numbers having both digits the same • Numbers that are both multiples of 2 and 3 • Numbers that are divisible by 8 • Numbers whose digits add to 9 (example: 63)

48. Tiling a Patio You are designing square patios. Each patio has a square garden area in the center. You use brown tiles to represent the soil of the garden. Around each garden, you design a border of white tiles. Build the three smallest square patios you can design with brown tiles and white tiles for the border. Record the number of each color tile needed for the patios in a table. Continue filling in the table as you design the next two patios.

49. What’s the Best Deal? Your boss at the video game company where you work has given you two choices of salary schemes. You have to decide which of the choices will permit you to reach your goal of $1000 the fastest. You will need to support your choice by showing the data your collected and by describing the graphs for each situation. Choice 1: Your salary will be doubled each day. You will earn $1 the first day, $2 the second day, $4 the third day, $8 the fourth day, and so on. Choice 2: Your salary will increase $3 each day. You will make $3 the first day, $6 the second day, $9 the third day, $12 the fourth day, and so on. Which of these two ways to earn your salary will get you to $1000 the fastest? Complete a table for each plan. On graph paper, draw a graph for the total earnings for each salary plan.

50. Amazing Calendars Calendars provide examples of number patterns to explore. Consider the following: • How does the calendar change as you look across a week? • How does it change as you go up? Down? Across? Diagonal? • What is the sum of all the numbers in a square? Find other sums? Do you notice any patterns? • What other patterns do you notice?