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.

2 4 6 8 10 _ _ _1)

1 3 5 7 9 _ _ _2)

25 50 75 100 125 _ _ _3)

1 4 9 16 25 _ _ _4)

5 9 13 17 21 _ _ _5)

8 14 20 26 32 _ _ _6)

1 3 6 10 15 _ _ _8)

12 14 16

11 13 15

150 175 200

36 49 64

38 44 50

25 29 33

21 28 36

Perfect Squares

Triangular Numbers

Even Numbers

OddNumbers

Multiples 25

Add 4

Add 6

15 24 35 48 63 _ _ _7) 80 99 120 Add next odd numberRectangular

Numbers

Add next integer

n 1 2 3 4 5 6 7 8

Square NumbersTerm Value

1st 1

2nd 4

3rd 9

4th 16

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 n2

The sequence 3, 8, 15, 24, . . . is a rectangular number pattern. How many squares are there in the 50th rectangular array?

Rectangular Numbers

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 rectangleStep 2: write a linear sequence rule for the base then the heightStep 3: Area = b*h; use this to write the rule for the entire rectangular sequence

*Base 3, 4, 5, 6, … (n+2)

*Height 1, 2, 3, 4, … (n)

Rectangular sequence = base * height = (n+2)(n)

1432

34

56

Add the next odd integer: +5, 7, 9,..

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

Step 1: write in the base and height of each rectangle

1*2 2*3 3*4 4*5 5*6 6*7

Step 2: write a linear sequence rule for the base then the height

Base = 1, 2, 3, 4, … n

Height = 2, 3, 4, 5, … n+1

Step 3: Area = b*h; use this to write the rule for the entire rectangular sequence

nth term rule n(n+1)

30 42 n(n+1)

STEPS to write the rule for a Triangular SequenceStep 1: double each number in the value row

create rectangular numbersStep 2: write in the base and height of each rectangleStep 3: write a linear sequence rule for the base then the heightStep 4: Area = b*h; use this to write the rule for the entire rectangular sequenceStep 5: undo the double in Step 1 by dividing the rectangular rule

by 2.

1 3 6 10

n 1 2 3 4 5 nthvalue 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 sequenceStep 5: undo the double in Step 1 by dividing the rectangular rule by 2.

1

3

6

10

Find the next 5 and describe the pattern

Triangular Numbers

15, 21, 28, 36, 45…….n ?

1st 1 * 2 = 2

2nd 2 * 3 = 6

3rd 3 * 4 = 12

4th 4 * 5 = 20

Does this help?Can you see a pattern yet?

Try this to help write the nth term.

(4 * 5) = 20 = 10 2 2

So what about the nth number in the sequence?

4 * 5 = 20

This is the 4th in the sequence

n (n +1)

2

15 24 35 48 63 7)

2 4 6 8 101)

1 3 5 7 92)

25 50 75 100 1253)

1 4 9 16 254)

5 9 13 17 215)

8 14 20 26 326)

1 3 6 10 15 8)

2n

(2n) - 1

25n

(4n) + 1

(6n) + 2

n2

nth term1 2 3 4 5

(n+2)(n+4)

A RuleWe 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) = 18302

Linear Sequences: add/subtract the common difference

Square/ rectangular Sequences: add the next even/odd integer

Triangular Sequences: add the next integer

How to identify the type of sequence

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.

9

9+1=10

9x10 = 90Take half.

Each Triangle has 45.

459 n