1 Patterns Learning Outcomes Use tables to represent

1 Patterns – Learning Outcomes Use tables to represent a repeating-pattern situation. Generalise and explain patterns and relationships in words and numbers. Use tables, diagrams, and graphs for representing and analysing linear, quadratic, and exponential patterns and relations. Find the underlying formula written in words from which the data are derived for linear and HL: quadratic relationships. Write arithmetic expressions for particular terms in a sequence.

2 Explain Patterns in Words and Numbers Describe the pattern below: Can you describe just which ones are blue? What about which ones are red? If the pattern continues, can you decide the colour of the 15 th box? What colour is the 20 th box? What colour is the 100 th box?

3 Explain Patterns in Words and Numbers When the length of a pattern is two, it’s easy to make predictions: divide the pattern into odds and evens. For longer patterns, we do the same, but divide the pattern into more groups. e. g. how often does the pattern shown below repeat?

4 Use Tables to Represent Patterns A table may help to make predictions. Number 1 2 3 4 5 6 7 8 9 10 11 12 Instead of grouping them as odds and evens, we group them into three groups – stars, circles, and triangles. What do the numbers in the triangle group have in common? {3, 6, 9, 12…} They’re all multiples of three!

5 Use Tables to Represent Patterns Each number can be sorted into their group by their remainder when divided by three: Number Remainder 1 1 2 2 3 0 4 1 5 2 6 0 7 1 All stars have remainder 1. All circles have remainder 2. All triangles have no remainder. How do we calculate remainders? 8 2 9 0 10 1 11 2 12 0

6 Use Tables to Represent Patterns If they don’t match, it will require further work – we’ll get to that later.

7 Use Tables to Represent Patterns a) what shape is the 4 th symbol? b) what shape is the 5 th symbol? c) what shape is the 6 th symbol? d) what shape is the 21 st symbol? e) what shape is the 22 nd symbol? f) what shape is the 23 rd symbol? g) what shape is the 43 rd symbol? h) what shape is the 53 rd symbol?

8 Use Tables to Represent Patterns e. g. How often does the pattern shown repeat? A B C D E F A B C D E F

9 Number Remainder Use Tables to Represent Patterns A 1 1 B 2 2 C 3 3 D 4 4 E 5 5 F 6 0 iii. What letter is in position 14? iv. What letter is in position 15? v. What letter is in position 16? vi. What letter is in position 26? vii. What letter is in position 28? A 7 1 B 8 2 C 9 3 D 10 4 E 11 5 F 12 0

10 Explain Patterns in Words and Numbers Consider the sequence: 3, 9, 15, 21, 27, … Imagine each number in a box in sequence: 3 Box #1 9 Box #2 15 Box #3 1. Draw out the next three boxes. 2. What number is in box #10? 3. What box contains the number 75? 4. What number is in box n? 21 Box #4 27 Box #5

11 Explain Patterns in Words and Numbers 3 Box #1 9 Box #2 15 Box #3 21 Box #4 27 Box #5 The number in box #1 is called the start term, a. Only linear patterns have a common difference! The difference between boxes is called the common difference, d. The number in box #n is called the nth term, Tn. (e. g. box #5 contains the 5 th term, T 5, box #25 contains the 25 th term, T 25).

12 Explain Patterns in Words and Numbers Write the first five terms of the following sequences: 1. start term = 2, common difference = 5 2. start term = 5, common difference = 2 3. start term = 18, common difference = -2 Describe in words how to find the nth term of each of these sequences. Write an expression in terms of n to describe the nth term of these sequences.

13 Explain Patterns in Words and Numbers

14 Use Tables to Represent Linear Patterns Disc-shaped tiles are placed to form a pattern as shown:

15 2015 FL Q 14 Use Tables to Represent Linear Patterns A pattern is made using white tiles and shaded tiles. Here are the first two terms in the pattern. a) Draw Term 3 of the pattern in the grid above.

16 2015 FL Q 14 [continued] Use Tables to Represent Linear Patterns b) Fill in the fraction boxes below to show what fraction of each term is shaded. c) Draw a table showing the number of white tiles and the number of shaded tiles in the first five terms of the pattern. d) Ciarán draws another term of the pattern. It has 14 white tiles. How many shaded tiles should it have?

17 2014 OL P 1 Q 11 Use Tables to Represent Linear Patterns The first three stages of a pattern are shown below. Each stage is made up of a certain number of shaded discs and a certain number of white discs. a) Shade in appropriate discs below to show the 4 th stage of the pattern.

18 2014 OL P 1 Q 11 [continued] Use Tables to Represent Linear Patterns b) Complete the table below to show the pattern continues.

19 2014 OL P 1 Q 11 [continued] Use Tables to Represent Linear Patterns c) In a particular stage of the pattern, there are 21 white discs. How many shaded discs are there in this stage of the pattern? d) Write down the relation between the number of shaded discs and the number of white discs in each stage of the pattern. State clearly the meaning of any letters you use.

20 2016 HL P 1 Q 8 Use Tables to Represent Linear Patterns John makes a sequence where each stage is made up of a certain number of Xs arranged in a pattern. The first three stages of John’s sequence are shown below. The sequence starts at stage 0. a) Draw the next stage of John’s sequence.

21 Use Tables to Represent Linear Patterns 2016 HL P 1 Q 8 [continued]

22 Use Tables to Represent Linear Patterns 2016 HL P 1 Q 8 [continued]

23 Use Tables to Represent Quadratic Patterns Term 3 1 st Diff +3 2 nd Diff 6 11 +5 +2 18 +7 +2 27 +9 +2

24 Use Tables to Represent Quadratic Patterns Confirm that the following sequences are quadratic and write down the next two terms: a) 1, 4, 9, 16, 25… b) 3, 6, 11, 18, 27… c) 0, 3, 8, 15, 24… d) 2, 8, 18, 32, 50… e) -3, 8, 23, 42, 65… f) 9, 28, 57, 96, 145… g) 3, 12, 27, 48, 75… h) 16, 7, 2, -4…

25 Use Tables to Represent Quadratic Patterns

26 Use Tables to Represent Quadratic Patterns

27 Use Tables to Represent Quadratic Patterns Term 3 1 st Diff +3 2 nd Diff 6 11 +5 +2 18 +7 +2 27 +9 +2

28 Use Tables to Represent Quadratic Patterns Determine the term rule for each of the following quadratic sequences (you can look up the answers from pg 24 to speed this up): a) 1, 4, 9, 16, 25… b) 3, 6, 11, 18, 27… c) 0, 3, 8, 15, 24… d) 2, 8, 18, 32, 50… e) -3, 8, 23, 42, 65… f) 9, 28, 57, 96, 145… g) 3, 12, 27, 48, 75… h) 16, 7, 2, -4…

29 Use Tables to Represent Exponential Patterns Term 2 1 st Diff +4 2 nd Diff 6 18 +12 +8 54 +36 +24 162 +108 +72 Neither the first nor second differences are common, so try factors: Term 2 6 18 54 162 Factor

30 Use Tables to Represent Exponential Patterns Confirm that the following sequences are exponential and write down the next two terms: a) 6, 18, 54, 162… b) 6, 12, 24, 48… c) 10, 20, 40, 80… d) 12, 36, 108, 324… e) -4, -8, -16, -32… f) -9, -27, -81, -243…

31 Use Tables to Represent Exponential Patterns

32 2013 HL P 1 Q 14 Use Tables to Represent Patterns Investigate whether the pattern in the table below is linear, quadratic, or exponential. Explain your conclusion. Term 1 Term 2 Term 3 Term 4 Term 5

33 Use Graphs to Represent Patterns

34 Use Graphs to Represent Patterns e. g. quadratic: 3, 6, 11, 18, 27

35 Use Graphs to Represent Patterns e. g. exponential: 2, 6, 18, 54, 162

36 2012 HL P 1 Q 7 Use Graphs to Represent Patterns Lisa is on a particular payment plan called “Plan A” for electricity. She pays a standing charge each month even if no electricity is used. She also pays a rate per unit used. The table shows the cost, including the standing charge, of using different amounts of units, in a month. Units Used Plan A Cost (€) 100 38 200 56 300 74 400 92 500 110 600 128 700 146 800 164 a) Use the data in the table to show that the relationship between the number of units used and the cost is linear.

37 Use Graphs to Represent Patterns 2012 HL P 1 Q 7 [continued] b) Draw a graph to show the relationship between the number of units used and the cost of electricity. c) Use your graph to estimate the standing charge. d) Write down a different method of finding the standing charge. Find the standing charge using your method. e) Write down a formula to represent the relationship between the number of units used and the cost for any given number of units. f) The table above does not include VAT. One month Lisa used 650 units. Her total bill for that month, including VAT, was € 155. 50. Find the VAT rate on electricity, correct to one decimal place.

38 2012 HL P 1 Q 7 [continued] Use Graphs to Represent Patterns g) Lisa is offered a new plan, “Plan B”, where the standing charge is € 36 and the rate per unit used is 15. 5 cent. Add a column to the table for plan A, showing the costs for plan B. h) Which plan do you think Lisa should choose? Give a reason for your answer. i) On your diagram for part (b), draw a graph to show the relationship between the number of units used and the cost of electricity for Plan B. Label this graph “Plan B”. j) Use your graph to find the number of units for which both plans have the same cost.