Sequences and Equations

Common Core State Standards

MACC.7.EE.1.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

MACC.7.EE.1.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.

Highlights Mathematical Practice #8Look for and make use of repeated reasoning.

Sequences and EquationsThese dot diagrams show the beginning of a sequence of patterns:

What would the fifth pattern in the sequence look like?

Sequences and EquationsThese dot diagrams show the beginning of a sequence of patterns:

Something look like this?

Sequences and Equations

If we were to continue the pattern, how many dots of each color will there be in the 10th pattern?

Sequences and Equations

How can we write an algebraic expression for the number of white dots?

term 1 2 3 4 5 nwhite dots 1 3 5 7 9 ???

Sequences and Equations

How can we write an algebraic expression for the number of white dots?

term 1 2 3 4 5 nwhite dots 1 3 5 7 9 2n –

1

Sequences and Equations

How can we write an algebraic expression for the number of black dots?

term 1 2 3 4 5 nblack dots 2 3 4 5 6 ???

Sequences and Equations

How can we write an algebraic expression for the number of black dots?

term 1 2 3 4 5 nblack dots 2 3 4 5 6 n +

1

Sequences and Equations

How can we write an algebraic expression to find the total number of dots?

One way to do this is by looking at the patterns again.

Sequences and Equations

How can we write an algebraic expression to find the total number of dots?

term 1 2 3 4 5 ntotal dots 3 6 9 12 15 ???

Sequences and Equations

How can we write an algebraic expression to find the total number of dots?

term 1 2 3 4 5 ntotal dots 3 6 9 12 15 3n

Sequences and Equations

How can we write an algebraic expression to find the total number of dots?

Another way to do this is by looking at the algebraic expressions we found.

Sequences and Equations

Number of white dots +

Number of black

dots =Total

number of dots

2n – 1 n + 1 3n

Sequences and EquationsThese dot diagrams show the beginning of a sequence of patterns:

What would the fifth pattern in the sequence look like?

Let’s look at another pattern:

Write an algebraic expression to find the number of white dots.

term 1 2 3 4 nwhite dots 2 3 4 5 ???

Let’s look at another pattern:

Write an algebraic expression to find the number of white dots.

term 1 2 3 4 nwhite dots 2 3 4 5 n + 1

Let’s look at another pattern:

Write an algebraic expression to find the number of black dots.

Let’s look at another pattern:

Write an algebraic expression to find the number of black dots.

term 1 2 3 4 nblack dots 3 4 5 6 n + 2

Let’s look at another pattern:

Write an algebraic expression to find the total number of dots.

Algebraically,

term 1 2 3 4 ntotal dots 5 7 9 11 2n +

3

Using a table,