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* 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 • change relationships in patterns from words to formula using letters and symbols. n 1 2 3 4 5 6 7 8 1) 2 4 6 8 10 12 _ 14 _ 16 _ 2) 1 3 5 7 9 11 _ 13 _ 15 _ 3) 25 50 75 100 125 150 _ 175 _ 200 _ 4) 1 4 9 16 25 36 _ 49 _ 64 _ Perfect Squares 5) 5 9 13 17 21 25 _ 29 _ 33 _ Add 4 6) 8 14 20 26 32 38 _ 44 _ 50 _ Add 6 7) 15 24 35 48 63 80 _ 99 _ 120 _ 8) 1 3 6 10 15 21 _ 28 _ 36 _ Even Numbers Odd Numbers Multiples 25 Add Rectangular next odd number Numbers Triangular Add next Numbers integer Square Numbers Term Value 1st 1 nd 4 3 rd 9 4th 16 2 Square Numbers Term Value 5th 25 or 5*5 6th 36 or 6*6 7th 49 or 7*7 8th 64 or 8*8 nth n* n or n 2 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 * Add the next odd integer: +5, 7, 9,.. 1 5 4 3 2 *Base 3, 4, 5, 6, … *Height 1, 2, 3, 4, … 6 4 3 (n+2) (n) Rectangular sequence = base * height = (n+2)(n) Use the Steps to writing the rule for a Rectangular Sequence to find the rule for the following sequence 2, 6, 12, 20,.. n 1 2 3 4 5 6 value 2 6 12 20 30 42 2*3 3*4 4*5 5*6 6*7 1*2 … Step1: 3: Area = b*h; use this write the rulebase for the entire 2:write write in a the linear sequence ruleof foreach the then the height Step base andto height rectangle rectangular sequence n Base = 1, 2, 3, 4, … term rule n+1 n(n+1) Height = 2, nth 3, 4, 5, … n n(n+1) … 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. n 1 value 1 2*value 2 1*2 2 3 6 2*3 3 6 12 4 10 20 5 15 30 3*4 4*5 5*6 nth … … 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. Triangular Numbers 1 3 6 Find the next 5 and describe the pattern 10 15, 21, 28, 36, 45…….n ? 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 This is the 4th in the sequence 4 * 5 = 20 (4 * 5) = 20 = 10 2 2 So what about the n th number in the sequence? n (n +1) 2 nth term 2n 1 2 3 4 5 1) 2 4 6 8 10 2) 1 3 5 7 9 (2 ) - 3) 25 50 75 100 125 25n n n 1 2 4) 1 4 9 16 25 5) 5 9 13 17 21 6) 8 14 20 26 32 (6n) + 2 7) 24 35 48 63 (n+2)(n+4) 3 6 10 15 15 8) 1 n) + 1 (4 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: The rectangles are n high and n+1 wide (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 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 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. 9x10 = 90 Take half. Each Triangle has 45. 9 9+1=10 n9 45