1. Square/Rectangular Numbers Triangular Numbers HW: 2.3/1-5
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)
Even Numbers
Odd
Numbers
_ _ _ 2)
Multiples 25
_ _ _ 3)
Perfect Squares
_ _ _ 4)
_ _ _ 5)
Add 4
_ _ _ 6)
Add 6
_ _ _ 7)
Add next odd number
Rectangular Numbers
Triangular Numbers
Add next integer
_ _ _ 8)

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

5. Square Numbers nth n * n or n2 Term Value 5th 25 or 5 * 5

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

8. 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 12 20 30 42
n(n+1)
1*2 2* * * * *7
Step 1: write in the base and height of each rectangle
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
Base = 1, 2, 3, 4, … n
nth term rule n(n+1)
Height = 2, 3, 4, 5, …
n+1

9. STEPS to write the rule for a Triangular Sequence
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 nth
value … …
2*value
1* * * * *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. Find the next 5 and describe the pattern
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.
2nd 2 * 3 = 6
Does this help?
Can you see a
pattern yet?
3rd 3 * 4 = 12
4th 4 * 5 = 20

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

14. nth term 2n n2 (n+2)(n+4) 25n (6n) + 2 1 2 3 4 5 2 4 6 8 10 1) 2n 1)
(2n) - 1 2)
25n 3) n2 4)
(4n) + 1 5)
(6n) + 2 6) 7)
(n+2)(n+4) 8)

15. 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+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

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.

20. 9x10 = 90
Take half.
Each Triangle has 45. 9
9+1=10