1-Genefhrhrhrhrgewdfhkjtewerhrating-Patterns.pptx

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1. GENERATING PATTERNS DOMINIC DALTON L. CALING Mathematics | Grade 10

2. At the end of this lesson, you are expected to: generates patterns LEARNING OBJECTIVES

3. A pattern is an ordered set of numbers, shapes, or other math objects, arranged according to a rule. What is a pattern?

4. 2, 4, 6, 8, 10, 12, … a, b, c, d, e Is this a pattern? THESE ARE ALL PATTERNS.

5. A number pattern is a sequence of numbers, that follows a specific rule (repeating or growing). Examples: Skip counting 2, 4, 6, 8, 10 Repeating Numbers 1 2 3, 1 2 3, 1 2 3 NUMBER PATTERNS

6. We often need to spot a pattern in order to predict what will happen next. In maths, the correct name for a pattern of numbers is called a SEQUENCE. The first number in a SEQUENCE is sometimes called the FIRST TERM; the second is the SECOND TERM and so on. PATTERNS AND SEQUENCES

7. For any pattern it is important to try to spot what is happening before you can predict the next number. The first 2 or 3 numbers is rarely enough to show the full pattern - 4 or 5 numbers are best. PATTERNS AND SEQUENCES

8. For any pattern it is important to try to spot what is happening before you can predict the next number. 1, 2, …… What’s the next number? PATTERNS AND SEQUENCES

9. For any pattern it is important to try to spot what is happening before you can predict the next number. 1, 2, 4,… Who thought that the next number was 3? What comes next? PATTERNS AND SEQUENCES

10. For any pattern it is important to try to spot what is happening before you can predict the next number. 1, 2, 4, 8, 16, … What comes next? PATTERNS AND SEQUENCES

11. Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … PATTERNS AND SEQUENCES + 3

12. Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … PATTERNS AND SEQUENCES + 3 + 4

13. Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … PATTERNS AND SEQUENCES + 3 + 4 + 5

14. Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … PATTERNS AND SEQUENCES + 3 + 4 + 5 + 6

15. Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, 30, … PATTERNS AND SEQUENCES + 3 + 4 + 5 + 6 + 7

16. Now try these patterns: 3, 7, 11, 15, 19, …, … 128, 64, 32, 16, 8, …, … 1000, 100, 10, 1, …, … 5, 15, 45, 135, …, … Infinite sequence So what is a finite sequence? PATTERNS AND SEQUENCES

17. In general,      

18. 1, 4, 9, 16, 25, … What is the value of a (T 1 ) ? T 5 ? What is the pattern? T 1 = 1 = 1 2 T 2 = 4 = 2 2 T 3 = 9 = 3 2 … T 12 = ? T n = ? T n is called the general term   Sequence is Consider the following sequence:

19. Example 1 : Write a rule for the nth term.     Look for a pattern…

20.   Think: Example 2 : Write a rule for the n th term of 2, 6, 12, 20, … Can be written as: 1(2), 2(3), 3(4), 4(5), … or  

21. Describe the pattern, write the next term, and write a rule for the n th term of the sequence – 1, – 8, – 27, – 64, . . . SOLUTION You can write the terms as (– 1) 3 , (– 2) 3 , (– 3) 3 , (– 4) 3 , . . . . Example 3: The next term is a 5 = (– 5) 3 = – 125. A rule for the n th term is .  

22. Describe the pattern, write the next term, and write a rule for the n th term of the sequence 0, 2, 6, 12, . . . . SOLUTION You can write the terms as 0(1), 1(2), 2(3), 3(4), . . . Example 4: A rule for the n th term is .   The next term is f(5) = 4(5) = 20.

23. Consider this: What is the pattern? How many dots for the next term? What about the 50 th term? Need to find general term or the rule first. …

24. Rearrange the dots: Double the dots:

25. Thus;  

26. Write in general term 5, 8, 11, 14, 17, … 25, 21, 17, 13, … 1, 3, 9, 27, …      

27. A series is the sum of the terms in the sequence and is represented by S n . Example: S n = T 1 + T 2 + T 3 +…+ T n For finite series, 1 + 3 + 5 + 7. For infinite series, 1 + 2 + 3 + 4 +… SERIES

28.   THE SUMMATION SYMBOL

29. Generally, For finite series For infinite series,    

30. Write the following series using the summation symbol.        

31. Finding the values of the summation     1 +2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55     = 62   [2(-1) -3] + [2(0)-3] + [2(1)-3] + [2(2)-3] = -8    

32. Finding the sum for an infinite sequence Infinite sequence Convergent sequence Divergent sequence

33. Convergence  

34.     Divergence

35. Classify the following sequences as finite sequence or infinite sequence . ________________1. {1, 3, 5, 7, 9, …} ________________2. { 2, 4, 6, 8, 10} ________________3. { 2, -4, 6, -8, 10, …} ________________4. { 3, 6, 9, 12, 15 } ________________5. { 1, 4, 7, 10, 13 } Infinite sequence Infinite sequence Finite sequence Finite sequence Finite sequence

36. THANK YOU!