Square/Rectangular Numbers

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2.3 Number Sequences

• Square/Rectangular Numbers
• Triangular Numbers
• HW 2.3/1-5

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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.

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n 1 2 3 4 5
6 7 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
38 44 50
Add 6
80 99 120
Add next odd number
Rectangular Numbers
Triangular Numbers
Add next integer
21 28 36
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Square Numbers
Term
Value
5
Square Numbers
Term
Value
5th 25 or 5 5
6th 36 or 6 6
7th 49 or 7 7
8th 64 or 8 8
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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 bh use this to write the rule for the
entire rectangular sequence
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3, 8, 15, 24, . .
Add the next odd integer 5, 7, 9,..
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5
4
3
4
3
2
1

• Base ? 3, 4, 5, 6, (n2)
• Height ? 1, 2, 3, 4, (n)
• Rectangular sequence base height (n2)(n)

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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 n
value 2 6 12 20
30
42
n(n1)
12 23 34 45 56 67
Step 3 Area bh use this to write the rule
for the entire rectangular
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
Base 1, 2, 3, 4,
n
nth term rule n(n1)
n1
Height 2, 3, 4, 5,
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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 bh 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.
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n 1 2 3 4 5 nth value 1 3 6 10
15 2value 2 6 12 20
30 12 23 34 45
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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 bh 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.
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Triangular Numbers
1
3
6
Find the next 5 and describe the pattern
15, 21, 28, 36, 45.n ?
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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
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(4 5) 20 10 2 2
So what about the nth number in the sequence?
n (n 1) 2
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nth term
1 2 3 4 5
2n
(2n) - 1
25n
n2
4)
1 4 9 16 25
(4n) 1
(6n) 2
(n2)(n4)
 
1 3 6 10 15
8)
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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
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The rectangles are n high and n1 wide (and
remember we doubled the dots) Rule n(n1)
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Example the 5th Triangular Number is 5(51)
15 2
Example the 60th Triangular Number is 60(601)
1830 2
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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
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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.

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9x10 90 Take half. Each Triangle has 45.
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