Lesson: A Visual Approach to Generalize Expressions
Connections to Common Core State Standards:
6th Grade – 6-EE.1, 6-EE.2, 6-EE.3, 6-EE.4, 6-EE.6
7th Grade – 7-EE.1, 7-EE.4
Algebra - F-BF.1, F-BF.2
Objectives:
 Students will write an algebraic expression to generalize the pattern in an arithmetic sequence modeled
by a series of tile pictures.
 Students will determine if the different expressions found for the same set of tile pictures are equivalent.
Materials: Color tiles or cubes (optional), Pattern 1 for projection in the phone activity, copies of Activity
Sheet 1: Pattern 1 (for students to use as note sheets), copies of Activity Sheet 2: Developing Rules for
Number Patterns through Pictures
Strategies and Activities:
Part I – Phone Activity
1 – Pair up the students. Assign one to be Student A and the other to be Student B.
2 – Tell Students A the following:
“I am going to project a series of pictures made of tiles on the screen so that you can see it but your
partner can not. Study the pictures and look for a pattern. The pictures will be numbered 1, 2, 3, and so on.
Then you will pretend that you are talking to your partner on the phone – which means that you can’t show
him/her any of the pictures or point anything out to him/her. You also will not be able to see what they are
doing. Your task is to explain to them how to build or draw the 8th tile picture in the pattern without building or
drawing the previous pictures.”
- Tell Students B the following:
“You will be facing away from the screen so you cannot see the pictures. Your task is to listen carefully
to your partner and build or draw the 8th picture in the pattern as he/she is describing it. Remember you are
talking on the phone so you cannot look at anything they write and they cannot see what you are doing.”
3 – Have Students B face away from the projection screen and show Students A a projection of Picture
Pattern 1. Remind Students A to take a few minutes to study the pattern before starting their “phone
conversation.” Provide tiles or cubes for Students B who prefer to build rather than draw the pattern.
4 – As students are completing this activity circulate the room to watch for signs of recursive
thinking. If Student A is asking Student B to build draw the 8th picture by first building or drawing the previous
7 pictures, then they are using a recursive pattern and not an explicit pattern. This type of thinking will not lead
to the explicit rule. Guide them to involve the picture number in some way. Sometimes it helps to ask how
they would explain how to draw the 50th picture. (We certainly wouldn’t want to do that recursively!)
Allow about 10 minutes for this activity.
Part II – Teaching the Visual Method
1 – Hand out Activity Sheet1:
Pattern 1 to the students. Tell them that different people will see the pattern in a
variety of ways. This sheet is for them to take notes as the different approaches are discussed and the visual
method for generalizing the rule for the nth term is taught.
2 – Have one student share how they described the pattern or how it was described to them.
As they explain
their thinking, show how this can be shaded on the activity sheet for Pattern 1.
Next show them how to sketch the 8th picture. Use this sketch to determine the number of tiles in the 8th
picture. Then show them how they could sketch the 50th picture and determine the number of tiles it would take
to build it.
Next show them how to generalize a sketch of the nth picture. From the generalized sketch show them how to
write an expression that will represent the number of tiles in the nth picture.
Show them how this expression can be used to determine the number of tiles in each picture. For example, if
you substitue 3 into their expression for Pattern 1 and evaluate it, you should get 7. It is important that they
understand that 7 represents the number of tiles it takes to build the 3rd picture.
3 – Repeat Step 2 to discuss and generalize 2 or 3 different student descriptions.
Examples of possible student descriptions and how the T-pictures can be shaded to represent their thinking is
attached. Possible sketches and algebraic expressions are also provided. (See Appendix A).
If students did not come up with a variety of ways to see the explicit pattern, then you can use the examples in
Appendix A to show them more than one way.
4 - With different perspectives of the pattern you will get different expressions for the same pattern.
Ask
students “Since we are all looking at the same pattern, why didn’t we all get the same expression? Remember
the expression tells us how many tiles it would take to build that particular picture.”
Some students may recognize equivalent expressions and some may not, depending on their mathematical
level. What you do next may depend on the level of your class.
At the 6th grade you may want them to determine if the different expressions are equivalent by substituting
values in for the variable and showing that you get the same quantity. (number of tiles.)
If students are combining like terms and using the distributive property then you may want them to simplify all
the expressions and show that they simplify to the same expression.
If students have studied linear functions, then you can connect this activity to the function rule.
Part III – Activity Sheet 2
1 – (Possibly done on Day 2)
Assign Activity Sheet 2: Developing Rules for Number Patterns
through Pictures. Circulate to assist students as needed. You may want to let them work in pairs on this.
2 - Have students share the diffeent rules they come up with for each pattern.
different expressions they may find for the same pattern are equivalent.
Guide them to determine if
Appendix A
Different Possible Approaches to Visualizing the T-pattern
Description #1 – The overall shape is like a T. Put 8 tiles vertically down the center, then put 7
tiles horizontally on each side of the top tile.
With this description students are seeing the pattern as shown in Figure 1. Here the vertical column of
tiles down the center is the same as the figure number. To illustrate this manner of seeing the pattern we shade
that vertical column of tiles. The number of tiles on each side of the top tile is one less than the term number.
If we use n to represent the figure number then we have n tiles down the center with n-1 tiles on each side. To
generate the algebraic expression that represents the number of tiles in the nth figure we simply add these parts
together. The expression is n + (n – 1) + (n – 1).
Figure 1
8th
50th
8-1
7
8-1
7
8
50-1
49
50
nth
50-1
49
2(50 – 1) + 50
2(49) + 50
98 + 50
148
2(8 - 1) + 8
2(7) + 8
14 + 8
22
n-1
n-1
n
2(n - 1) + n
Description #2 – You’re going to make a T. Put one tile down. This is the center tile. Then put 7 tiles in a row
to the right of the center tile, 7 tiles in a row to the left, and 7 tiles below the center tile.
Here the focus is on the one tile in the center. See Figure 2. Each of the three branches off of this center
tile has 1 less tile than the figure number. If the figure number is n, then the number of tiles in each branch is
n – 1. Since there are 3 branches we have 3(n – 1). Adding the center tile yields the expression 3(n – 1) + 1.
Some students who see the pattern this way will write this as (n – 1) + (n – 1) + (n – 1) + 1.
Figure 2
50
8
7
1
7
3(7) + 1
21 + 1
22
7
49
1
49
3(49) + 1
147 + 1
148
n
49
n-1
1
n-1
3n + 1
n-1
Description #3 – Put 15 tiles in a row horizontally. Then put 7 tiles vertically beneath the center tile.
This student observed that the number of tiles in the top part of the T can be found by doubling the
figure number and then subtracting one. This can be expressed as 2n – 1. The number of tiles below the center
tile is n – 1. The total number of tiles is the sum of these two expressions, or (2n – 1) + (n – 1). See Figure 3
.
Figure 4
8
2(8) – 1 =
50
15
8-1=7
[2(8) – 1] + (8 – 1)
[16 – 1] + 7
15 + 7
22
2(50) – 1 =
n
99
50-1=49
[2(50) – 1] + (50 – 1)
[100 – 1] + 49
99 + 49
148
2n – 1
n-1
[2n – 1] + (n – 1)