6th Grade Math Textbook, Fundamentals

    2   2   2   2

 (1)

    2   2   2   2

 (1)

    2

(1)

    2

 2  3  4  5  6  7  8

 &KDSWHU

13-2

1

1  2  3

(1 2)  3  6

(1  2 3)  4  10

(1  2  3 4)  5  15

(1  2  3  4 5)  6  21

(1  2  3  4  5 6)  7  28

(1  2  3  4  5  6 7)  8  36

Algebraic Patterns and Sequences

Objective To recognize, describe, and extend patterns with more than one constant

• To recognize, describe, and extend number patterns • To recognize patterns related
  to iterations

Danilo drew the visual pattern at the right. Danilo’s pattern

represents triangular numbers. are a

sequence of whole numbers in which each number

corresponds to an arrangement of dots in the shape of a

triangle. How many dots are in the 8th term in this pattern?

To find the 8th term, list the sequence represented by the

dots, then identify and extend the pattern.

1, 3, 6, 10, 15,...

Some sequences do not have a constant difference or a constant ratio.

3  1  2 6  3  3 10  6  4 15  10  5

Notice that for this sequence, the difference increases

by consecutive numbers.

The pattern rule is: Start at 1. Add consecutive numbers.

Triangular numbers

1, 3, 6, 10, 15, 21, 28, 36, ...

So the 8th term in the pattern above will have 36 dots

in the shape of a triangle.

Some sequences have terms that relate directly to the value of their positions

in the sequence rather than to the value of preceding and succeeding terms.

Tables help show this relationship.

Find the next three terms in this pattern: 2, 4, 3, 6, 5, 10, 9, 18, 17,...

2, 4, 3, 6, 5, 10, 9, 18, 17,...

The pattern rule is: Start at 2. Multiply by 2. Then add 1.

2, 4, 3, 6, 5, 10, 9, 18, 17, 34, 33, 66

These terms have a constant ratio of 2.

The rule is multiply by 2.

These terms have a constant difference of 1.

The rule is add 1.

1