Transcript STEPS to write the rule for a Rectangular Sequence

*
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