patterns notes.pdf./ Notes on patterns for maths
- 1. Patterns In todays class… 1.Identify patterns and find the next term in a sequence 2. Describe a rule that helps us find any number in a pattern or sequence 3. Draw a linear pattern on a graph
- 3. Z Growing Patterns: Rules ▪Copy the patterns into your book and draw the next two shapes Predict the number of squares in the 6th & 7th shapes without drawing them ▪Below the shapes, write down the number of squares in each ▪Describe the rule for getting the next number of squares 1. 3. 2. Count the number of squares on the perimeter of each shape – what is the rule for each now?
- 4. Z Numbers in the Patterns ▪Here is another pattern of matches Can you predict the table values for the 5th, 6th and 7th shape? What about the 10th? ▪Copy the shapes and the table below Number of Triangles Number of Matches Perimeter ▪Complete the table for the first three shapes ▪Draw the next two shapes and update your table 1 2 3 4 5 6 … 3 3 5 4 7 5 9 6 11 7 13. 8 15 9 17 10
- 5. Maths dictionary Pattern: is a set of objects, numbers, letters, shapes, pictures, symbols or diagrams which repeat in a set way.
- 7. (i)What would the 28th shape be? (ii)What would the 72nd shape be? (iii)What would the 300th shape be? (iv)What would the 2000th shape be? A three-shape repeating pattern:
- 9. Write down the next 4 terms of the sequence: 2, 4, 6, 8, 10,…..... 1.Describe the sequence in words 2.What type of numbers are these numbers? 3.Copy and complete the table 4.Write a rule in words which defines all these numbers Number Pattern 2 2(1) 4 2(2) 6 8 10 …. 24 100
- 10. Number sequence: is an ordered list of numbers separated by commas which are connected by a rule in some way. Eg. 1, 4, 9, 16,…... Term: the numbers in the sequence. Math’s dictionary
- 12. Term by Term rule: Describes how to get from one term to the next A sequence is described in words by giving the first term and then the term to term rule
- 13. Linear Sequences
- 19. How to find any term in a linear sequence Example: what is the 20th term of the sequence: 7,10,13,16,19
- 20. How to find any term in a linear sequence Example: what is the 20th term of the sequence: 7,10,13,16,19
- 23. Finding the nth term of a linear sequence 7, 10, 13, 16, 19 To find the nth term we look at the differences Between the terms
- 24. Since we are adding 3 we can say that 3n will be a part of our formula. Tn will be 3n + or – some number To work out what we add or subtract we compare with Tn= 3n 3, 6, 9, 12, 15 compared to 7, 10, 13, 16, 19. We have to add 4 to each term in 3n to get our sequence Tn = 3n +4
- 32. Arithmetic Sequence • A sequence in which a constant (d) can be added to each term to get the next term is called an arithmetic sequence • The constant (d) is called the common difference • To find the common difference (d), subtract any term from one that follows it. 2 5 8 11 14 3 3 3 3 t1 t2 t3 t4 t5 Here d = 3 We call the first term a a = t1= 2
- 33. Find the first term and the common difference of each arithmetic sequence. 1.) 4,9,14,19,24 First term (a): 4 Common difference (d): 2 1 a a − = 9 – 4 = 5 2.) 34,27,20,13,6, 1, 8,.... − − First term (a): 34 Common difference (d): -7 BE CAREFUL: ALWAYS CHECK TO MAKE SURE THE DIFFERENCE IS THE SAME BETWEEN EACH TERM ! Examples:
- 34. Now you try! Find the first term and the common difference of each of these arithmetic sequences. b) 11, 23, 35, 47, …. a) 1, -4, -9, -14, ….
- 35. The first term of an arithmetic sequence is (a) . We add (d) to get the next term. There is a pattern, therefore there is a formula (explicit formula) we can use to give use any term that we need without listing the whole sequence ☺. The nth term of an arithmetic sequence is given by: The last # in the sequence/or the # you are looking for First term The position the term is in The common difference tn = a + (n – 1) d
- 36. Examples: Find the 14th term of the arithmetic sequence 4, 7, 10, 13,…… (14 1) + − 4 4 (13)3 = + 4 39 = + 43 = tn = a + (n – 1) d t14 = 3 You are looking for the term! The 14th term in this sequence is the number 43!
- 37. Now you try! Find the 10th and 25th term given the following information. Make sure to derive the general formula first and then list what you have been provided. a) 1, 7, 13, 19 …. c) The second term is 8 and the common difference is 3 b) The first term is 3 and the common difference is -21
- 38. Examples: Find the 14th term of the arithmetic sequence with first term of 5 and the common difference is –6. (14 1) + − tn = a + (n – 1) d t14 = You are looking for the term! List which variables from the general term are provided! The 14th term in this sequence is the number -73! a = 5 and d = -6 5 - 6 = 5 + (13) * -6 = 5 + -78 = -73
- 40. Arithmetic Sequence • A sequence in which a constant (d) can be added to each term to get the next term is called an arithmetic sequence • The constant (d) is called the common difference • To find the common difference (d), subtract any term from one that follows it. 2 5 8 11 14 3 3 3 3 t1 t2 t3 t4 t5 Here d = 3 We call the first term a a = t1= 2
- 41. Find the first term and the common difference of each arithmetic sequence. 1.) 4,9,14,19,24 First term (a): 4 Common difference (d): 2 1 a a − = 9 – 4 = 5 2.) 34,27,20,13,6, 1, 8,.... − − First term (a): 34 Common difference (d): -7 BE CAREFUL: ALWAYS CHECK TO MAKE SURE THE DIFFERENCE IS THE SAME BETWEEN EACH TERM ! Examples: