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2.3 Number Sequences

2.3 Number Sequences. Square/Rectangular Numbers Triangular Numbers HW: 2.3/1-5. What are we going to learn today, Mrs Krause?. You are going to learn about number sequences. Square, rectangular & triangular. how to find and extend number sequences and patterns.

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2.3 Number Sequences

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  1. 2.3 Number Sequences Square/Rectangular Numbers Triangular Numbers HW: 2.3/1-5

  2. What are we going to learn today, Mrs Krause? • You are going to learn about number sequences. • Square, rectangular & triangular • how to find and extend number sequences and patterns • change relationships in patterns from words to • formula using letters and symbols.

  3. n 1 2 3 4 5 6 7 8 1) 2 4 6 8 10 _ _ _ 2) 1 3 5 7 9 _ _ _ 3) 25 50 75 100 125 _ _ _ 4) 1 4 9 16 25 _ _ _ 5 9 13 17 21 _ _ _ 5) 8 14 20 26 32 _ _ _ 15 24 35 48 63 _ _ _ 6) 7) 1 3 6 10 15 _ _ _ 8) 12 14 16 Even Numbers Odd Numbers 11 13 15 Multiples 25 150 175 200 Perfect Squares 36 49 64 Add 4 25 29 33 Add 6 38 44 50 Add next odd number Rectangular Numbers 80 99 120 Triangular Numbers Add next integer 21 28 36

  4. Square Numbers 1st 1 2nd 4 3rd 9 4th 16 Term Value

  5. Square Numbers nthn * n or n2 Term Value 5th 25 or 5 * 5 6th 36 or 6 * 6 7th 49 or 7 * 7 8th 64 or 8 * 8

  6. Rectangular Numbers The sequence 3, 8, 15, 24, . . . is a rectangular number pattern. How many squares are there in the 50th rectangular array? STEPS to write the rule for a Rectangular Sequence (If no drawings are given, consider drawing the rectangles to represent each term in the sequence) Step 1: write in the base and height of each rectangle Step 2: write a linear sequence rule for the base then the height Step 3: Area = b*h; use this to write the rule for the entire rectangular sequence

  7. Add the next odd integer: +5, 7, 9,.. • Base  3, 4, 5, 6, … (n+2) • Height  1, 2, 3, 4, … (n) Rectangular sequence = base * height = (n+2)(n) 6 5 4 3 4 3 2 1 3, 8, 15, 24, . .

  8. Use the Steps to writing the rule for a Rectangular Sequence to find the rule for the following sequence 2, 6, 12, 20,.. 30 42 n(n+1) 1*2 2*3 3*4 4*5 5*6 6*7 Step 3: Area = b*h; use this to write the rule for the entire rectangular sequence Step 2: write a linear sequence rule for the base then the height Step 1: write in the base and height of each rectangle n Base = 1, 2, 3, 4, … nth term rule n(n+1) n+1 Height = 2, 3, 4, 5, …

  9. 1 3 6 10 STEPS to write the rule for a Triangular Sequence Step 1: double each number in the value row  create rectangular numbers Step 2: write in the base and height of each rectangle Step 3: write a linear sequence rule for the base then the height Step 4: Area = b*h; use this to write the rule for the entire rectangular sequence Step 5: undo the double in Step 1 by dividing the rectangular rule by 2.

  10. n 1 2 3 4 5 nth value 1 3 6 10 15 … … 2*value 2 6 12 20 30 1*2 2*3 3*4 4*5 5*6 Step 1: double each number in the value row  create rectangular numbers Step 2: write in the base and height of each rectangle Step 3: write a linear sequence rule for the base then the height Step 4: Area = b*h; use this to write the rule for the entire rectangular sequence Step 5: undo the double in Step 1 by dividing the rectangular rule by 2.

  11. Triangular Numbers 1 3 6 Find the next 5 and describe the pattern 15, 21, 28, 36, 45…….n ? 10

  12. Try this to help write the nth term. 1st 1 * 2 = 2 2nd 2 * 3 = 6 Does this help? Can you see a pattern yet? 3rd 3 * 4 = 12 4th 4 * 5 = 20

  13. This is the 4th in the sequence 4 * 5 = 20 (4 * 5) = 20 = 10 2 2 So what about the nth number in the sequence? n(n +1) 2

  14. 1) 2 4 6 8 10 2) 1 3 5 7 9 3) 25 50 75 100 125 5 9 13 17 21 5) 8 14 20 26 32 7) 6) 15 24 35 48 63 nth term 1 2 3 4 5 2n (2n) - 1 25n n2 4) 1 4 9 16 25 (4n) + 1 (6n) + 2 (n+2)(n+4) 1 3 6 10 15 8)

  15. A Rule We can make a "Rule" so we can calculate any triangular number. First, rearrange the dots (and give each pattern a number n), like this: Then double the number of dots, and form them into a rectangle:

  16. The rectangles are n high and n+1wide (and remember we doubled the dots): Rule: n(n+1) 2 Example: the 5th Triangular Number is 5(5+1) = 15 2 Example: the 60th Triangular Number is 60(60+1) = 1830 2

  17. How to identify the type of sequence Linear Sequences: add/subtract the common difference Square/ rectangular Sequences: add the next even/odd integer Triangular Sequences: add the next integer

  18. So what did we learn today? • about number sequences. • especially about square , rectangular and triangular numbers. • how to find and extend number sequences and patterns.

  19. 9x10 = 90 Take half. Each Triangle has 45. 9 9+1=10

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