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 #8
Look for and make use of repeated reasoning.
Sequences and Equations
These dot diagrams show the beginning of a
sequence of patterns:
What would the fifth pattern in the sequence look
like?
Sequences and Equations
These 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
white dots
1
1
2
3
3
5
4
7
5
9
n
???
Sequences and Equations
How can we write an algebraic expression for the
number of white dots?
term
1
2
3
4
5
white dots
1
3
5
7
9
n
2n –
1
Sequences and Equations
How can we write an algebraic expression for the
number of black dots?
term
black dots
1
2
2
3
3
4
4
5
5
6
n
???
Sequences and Equations
How can we write an algebraic expression for the
number of black dots?
term
black dots
1
2
2
3
3
4
4
5
5
6
n
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
total dots
1
3
2
6
3
9
4
12
5
15
n
???
Sequences and Equations
How can we write an algebraic expression to find the
total number of dots?
term
total dots
1
3
2
6
3
9
4
12
5
15
n
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
2n -1+ (n +1) = (2n + n)-1+1= 3n
Number of
white dots
2n – 1
+
Number of
black dots
n+1
=
Total number
of dots
3n
Sequences and Equations
These 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
white dots
1
2
2
3
3
4
4
5
n
???
Let’s look at another pattern:
Write an algebraic expression to find the number of
white dots.
term
white dots
1
2
2
3
3
4
4
5
n
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
black dots
1
3
2
4
3
5
4
6
n
n+2
Let’s look at another pattern:
Write an algebraic expression to find the total number
of dots.
Algebraically,
(n +1) + (n + 2)
= (n + n) + (1+ 2) by combining like terms
= 2n + 3 total dots for each term, n
Using a table,
term
total dots
1
5
2
7
3
9
4
11
n
2n + 3