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  1. 1. Pattern sA Function Based Approach to Algebra
  2. 2. Coloured Blocks DiagramIs there a pattern to these colours?Can you use the pattern to predict whatcolour will be at a particular point?How can we investigate if there is a pattern?
  3. 3. Table Block Colour Black Position 1 Block Position 2 1 2 3 2 4 4 3 6 5 4 8 6 5 10Black block is always even.
  4. 4. Words: You double the number of the black block Black Red Block Position Block Position 1 2 1 1 2 4 2 3 3 6 3 5 4 8 4 7 5 10 5 9 n 2n n 2n – 1
  5. 5. Key Outcomes and Words: identify patterns and describe different situations using tables, graphs, words and formulae predict generalise in words and symbols justify relationship
  6. 6. Coloured Blocks Diagram
  7. 7. Tables Yellow Black GreenBlock Position Block Position Block Position 1 1 1 2 1 3 2 4 2 5 2 6 3 7 3 8 3 9 4 10 4 11 4 12 5 13 5 14 5 15 n 3n – 2 n 3n – 1 n 3n
  8. 8. Money Box Problem Mary Money BoxStart €0Growth per day €2We want to investigate the total amountof money in the money box over time.Is the growth of the money a pattern?Can we predict how much money will be inthe box on day 10?
  9. 9. Money Box Problem John Money BoxStart €3Growth per day €2Is there a pattern to the growth of this money?Can we use this pattern to predict how much moneywill be in the box at some future time?How can we investigate if a pattern exists?
  10. 10. Table for John’s Money Box Time/days Money in Box/€ 0 3 1 5 2 7 3 9 4 11 5 13Is there a pattern?Where is the start value in € and growth per dayin this as seen in the table?What do you notice about successive outputs ?
  11. 11. Money Box Problem Bernie Money BoxStart €4 €2 on week daysGrowth per day €5 on Weekend daysIs there a constant rate of change here?
  12. 12. Table for Money BoxTime/days Money in Box/€0 (Tues) 41 (Wed) 62 (Thurs) 83 (Fri) 104 (Sat) 155 (Sun) 20
  13. 13. Identifying variables and constants Money Box Varying Constant John Mary BernieWhat is varying each Day?What is constant?Can you put this into words?
  14. 14. Mary John BernieTime on Horizontal AxisTotal Money on Vertical Axis * Note: in this example day 0 is a Tuesday.
  15. 15. Draw a Graph 22 21 20 Mary 19 John 18 17 BernieAmount of Money Spent 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Days
  16. 16. Pattern of Growth for John’s MoneyTime/days Money/€ Change 0 3 +2 1 5 +2 2 7 +2 3 9 +2 4 11 +2 5 13 +2 6 15
  17. 17. Now, I want you to observe thepattern.Explain in words & numbers, how to findthe total amount of money in John’s boxafter 15 days.Time/days Money/€ Change 0 3 +2 1 5 +2 2 7 +2 Do this on your 3 9 white board +2 4 11 +2 5 13 +2 6 15
  18. 18. Now, I want you to generalise.Explain in words & symbols, how to findthe total amount of money in John’s boxafter any given day.Time/days Money/€ Change 0 3 +2 1 5 +2 2 7 +2 Do this on your 3 9 white board +2 4 11 +2 5 13 +2 6 15
  19. 19. Table for Mary’s Money Box Money inTime/days Box € What is the general 0 0 formula for Mary? 1 2 A = 0 + 2D 2 4 3 6 4 8 5 10
  20. 20. John : A = 3 + 2DMary : A = 2D + 0Only seeing the formula :Can you read the Johns start amount?Can you read Marys rate of change?What are we building up to?y = c + mx or y = mx + c
  21. 21. Sunflower growth Sunflower a b c d Start height/cm 3 6 6 8 Growth per day/cm 2 2 3 2Is there a pattern to the growth of these sunflowers?Can we use this pattern to predict height at some future time?How can we investigate if a pattern exists?
  22. 22. Table for Each SunflowerTime/days Height/cm Change 0 1 2 3 4 5
  23. 23. Pattern of Growth for 4 Different Sunflowers A B C Dt/d h/cm t/d h/cm t/d h/cm t/d h/cm0 3 0 6 0 6 0 81 5 1 8 1 9 1 102 7 2 10 2 12 2 123 9 3 12 3 15 3 144 11 4 14 4 18 4 165 13 5 16 5 21 5 186 15 6 18 6 24 6 20
  24. 24. 22 A and B 21 20 19 18 17 16 Sunflower A 15 14 Sunflower BHeight/cm 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Time/Days
  25. 25. 22 B and C 21 20 19 18 17 16 Sunflower B 15 14 Sunflower CHeight/cm 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Time/Days
  26. 26. 22 C and D 21 20 19 18 17 16 Sunflower C 15 14 Sunflower DHeight/cm 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Time/Days
  27. 27. Formula Representation• Describe in words the height of the sunflower a, on any day.• Describe in symbols the height of the sunflower a, on any day.• Identify the variables and constants in this formula.• Where do the y – intercept and the slope of the graph appear in the formula?
  28. 28. Pattern of Growth for 4 Different Sunflowers A B C D T H T H T H T H Pattern Pattern Pattern Patterndays cm days cm days cm days cm 0 3 3 0 6 6 0 6 6 0 8 8 1 5 3+2 1 8 6+2 1 9 6+3 1 10 8+2 2 7 3+2+2 2 10 6+2+2 2 12 6+3+3 2 12 8+2+2 3 9 3+2+2+2 3 12 6+2+2+2 3 15 6+3+3+3 3 14 8+2+2+2 4 11 3+2+2+2+2 4 14 6+2+2+2+2 4 18 6+3+3+3+3 4 16 8+2+2+2+2 5 13 3+2+2+2+2+… 5 16 6+2+2+2+2+… 5 21 6+3+3+3+3+… 5 18 8+2+2+2+2+… 6 15 3+2+2+2+2+… 6 18 6+2+2+2+2+… 6 24 6+3+3+3+3+… 6 20 8+2+2+2+2+…Describe in words the height of the sunflower a, on any day.Describe in symbols the height of the sunflower a, on any day.Identify the variables and constants in this formula.Where do the y – intercept and the slope of the graph appear in the formula?
  29. 29. Pattern of Growth for 4 Different Sunflowers A B C D T H T H T H T H Pattern Pattern Pattern Patterndays cm days cm days cm days cm 0 3 3 0 6 6 0 6 6 0 8 8 1 5 3+2 1 8 6+2 1 9 6+3 1 10 8+2 2 7 3+2+2 2 10 6+2+2 2 12 6+3+3 2 12 8+2+2 3 9 3+2+2+2 3 12 6+2+2+2 3 15 6+3+3+3 3 14 8+2+2+2 4 11 3+2+2+2+2 4 14 6+2+2+2+2 4 18 6+3+3+3+3 4 16 8+2+2+2+2 5 13 3+2+2+2+2+… 5 16 6+2+2+2+2+… 5 21 6+3+3+3+3+… 5 18 8+2+2+2+2+… 6 15 3+2+2+2+2+… 6 18 6+2+2+2+2+… 6 24 6+3+3+3+3+… 6 20 8+2+2+2+2+… h = 3 + 2t h = 6 + 2t h = 6 + 3t h = 8 + 2ty − int ercept = 3 y − int ercept = 6 y − int ercept = 6 y − int ercept = 8 slope = 2 slope = 2 slope = 3 slope = 2
  30. 30. Multi – Representation Table t/d h/cm 0 3 1 5 2 7 3 9 Graph 4 11 5 13 6 15 Words Height = 3 + 2 times the number of daysh = 3 +2d Symbols
  31. 31. Over to you on your White Boards1. Draw a rough sketch of the graph: y = 2x + 1
  32. 32. Over to you on your White Boards2. I start off with 6 euro in my money box and put in 3 euro each day. Draw a rough sketch of the graph.
  33. 33. Over to you on your White Boards3. The initial speed of a car is 10 m/s and the rate at which it increases its speed every second is 2 m/s 2 . Write down a linear law for the speed of the car after t seconds. Draw a graph of the law. 30 28 Solution : 26 Speed = 10 + 2(number of seconds travelling) 24 v = 10 + 2t 22 or v = 2t + 10 20 18 16 14 12 10 8 6 4 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
  34. 34. slope = 5 slope = 4 slope = 2 slope = 3 slope = 1 1 slope = 2 1 slope = 3 1 slope = 4 1 slope = 5
  35. 35. All the graphs you a have drawn on your whiteboards have been increasing functions. Do this on your white boardAssess the learning:Story :Isabelle has a money box with 20 euro in it. She takes 2euro out each day to buy sweets in the shop.Draw a rough graph of how this might look.Investigate if your graph is close by doing a table.From your table, what is your rate of change?Conclusion: Decreasing graph has a negative slope
  36. 36. Isabelle Box Problem 30 28 26 24 22 20 Amount of Money/€ 18 16 14 12 10 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time/Days
  37. 37. Connections 13
  38. 38. Growing Squares Write down a relationship which defines how many red squares are required for each white square.Hint: You may need more than one representation to help you!
  39. 39. Characteristics : • first change constant • n term • linear graphAre the characteristics of….. A Linear Relationship
  40. 40. Growing RectanglesComplete the next two rectangles in the above pattern.There is squared paper in your handbooks.Is there a relationship between the: height of the rectangle length of the rectangle number of tiles/area of the rectangle?
  41. 41. Table Number of tiles n in Change of Height, h Length, l Area = h x l Change each change rectangle 1 2 2 2 +4 2 3 6 6 +2 +6 3 4 12 12 +2 +8 4 5 20 20 +2 +10 5 6 30 30 +2 +12 6 7 42 42 +2 +14 7 8 56 56Investigate the first change and the second change.What do we notice?If we let n be the height, write a formula for the area in terms of n,on your white board.
  42. 42. Draw a graph of the table Number of tiles n in Change of Height, h Length, l Area = h x l each Change change rectangle 1 2 2 2 2 3 6 6 3 4 12 12 4 5 20 20 5 6 30 30 6 7 42 42 7 8 56 56What do you observe about the shape of your graph?
  43. 43. Characteristics : • first change varies • second change constant • n term 2 • curved graphAre the characteristics of….. A Quadratic Relationship
  44. 44. Story: How to ask for Pocket Money“I only want you to give me pocket money for the month of July. All I want is for you to give me 2 c on the first day of the month, double that for the second day, and double that again for the 3rdday... and so on. On the first day I will get 2 c, on the 2nd day 4 c, on the 3rd day8c and so on until the end of the month. That is all I want.”Is this a good deal for my parents or is it a good deal for me?
  45. 45. Investigate using a Day Table Money in cent Do this on your white board 1 2 2 2x2 3 2x2x2 4 5 6 7 8 9 10If we let n be the number of days, can we write a formulafor the Amount of Pocket Money?
  46. 46. Formula : Amount = 2 nWords : Doubling
  47. 47. GraphMoney/Cents Time/Days
  48. 48. What if your Dad trebled the amount ofmoney each day? Trebling Doubling Money/Cents Time/Days The money would grow even quicker.
  49. 49. Lets look at the Changes in a Table Money in Change of Days Change cent change 1 2 +2 2 4 +2 +4 3 8 +4 +8 4 16 +8 +16 5 32 +16 +32 6 64 +32 +64 7 128 +64 +128 8 256 +128 +256 9 512 +256 +512 10 1024What do you notice about the Change columns……They develop in a ratio.
  50. 50. Characteristics : • change develops in a ratio • Formula: 2 n or 3 n • Words: Doubling or Trebling • curved graph that grows very quicklyAre the characteristics of….. An Exponential Relationship
  51. 51. F = P( 1 + i ) tIgnoring the principal, the interest rate, and the number of years by setting all these variables equal to "1", and looking only at the influence of the number of compoundings, we get: yearly ( 1 + 1) 1 = 2 2  1 semi − annually  1 + ÷ = 2.25  2 4  1 quarterly  1 + ÷ = 2.44140625  4 12  1  monthly  1 + ÷ = 2.61303529022...  12  52  1  weekly  1 + ÷ = 2.692596954444...  52  365  1  daily 1 + ÷ = 2.71456748202...  365  8760  1  hourly 1 + ÷ = 2.71892792154...  8760  525600  1  every minute 1 + ÷ = 2.7182792154...  525600  31536000  1  every second 1 + ÷ = 2.71828247254...  31536000 
  52. 52. Grains of Rice Story

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