Seventh Grade Acceleration Intervention
Unit: Linear & Nonlinear Relationships
Day 2 Lesson
Objective
Students will determine whether relationships are linear or nonlinear when represented in
words, a table, symbolically or a graph.
Students will determine the recursive relationship of arithmetic or geometric sequences
represented in words, a table, or a graph.
Materials
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Overhead projector or document camera
“Finding Graphs” resource sheets (one per group)
“Finding Graphs: Answer Key” resource sheet
Graphing calculators
“Make-A-Problem” resource sheets
“Guess My Rule “Operator”” resource sheet
“Silent Board Game” resource sheets
Chart paper and markers
Opener (10 minutes)
Have students complete the following problem with a partner.
“Jake has $30 on June 1st. He earns $6 each day babysitting. Create a table that shows
the amount of money Jake has from days one to 10. Is this sequence arithmetic or
geometric? How do you know?”
Solution:
1
2
3
4
5
6
30
36
42
48
54
60
This sequence is arithmetic. The pattern is to add six.
7
66
8
72
9
78
10
84
Introductory Activity – Graphing Linear & Non-Linear (20 minutes)
Display a graph of each data set from the previous lesson. Recall the properties and
labels: “linear – straight line” and “non-linear – curved line.”
Place students in groups of two or three. Distribute the “Finding Graphs” resource sheets
and at least one graphing calculator per group.
(Note: If students do not know how to graph equations on the calculator – review the
steps: enter equations into “Y=” and use “Zoom 6” to display the graph)
Have students complete this activity with their group. As a class, discuss the different
features of the graphs. Include a discussion of how the different equations resulted in
graphs of different shapes. Think about the following questions:
1. What did the equations look like when a graph was linear?
2. What types of equations resulted in non-linear graphs?
3. Did all non-linear graphs look alike? Do you notice a pattern between the type of
equation and the shape of graph it creates?
Make-A-Problem (20 minutes)
Assign students to pairs and give each pair a graphing calculator. Each pair of students
should create six equations and sketch the graph from the calculator – three linear and
three nonlinear. They may use the previous activity as inspiration for their equations.
After each pair has sketched their six graphs, form groups of four and have students
exchange problems and indicate whether each graph is linear or non-linear. Students
should return problems to their authors to be checked. Monitor groups for accuracy and
creativity with graphs and equations. Share any interesting problems with the class.
Guess My Rule (20 minutes)
Post or draw the “Operator” on the board or overhead. Explain to the students that they
are going to input numbers into the “Operator” and guess what the “Operator” is doing to
that number, based on the output. Draw an input/output table on the board. Using the
equations provided, walk the students through a few numbers. For example, the equation
y  2 x  1 may have the following table:
INPUT
OUTPUT
2
5
5
11
10
21
…
…
Give the students the first few inputs and their corresponding outputs, and then allow
students to give you their own numbers as inputs, filling in the outputs for them.
Eventually, students should be able to talk through what is happening to the input.
Students may say “it’s doubling and then going up by one,” or “you multiply the number
by 2 and add 1.” Eventually, you may be able to give them an input and have them
produce the output on their own. Draw another input/output table on the board and repeat
the process with a new rule. Once it is clear that students are able to explain, in words,
what is happening in the “Operator,” for the different tables, have them talk through how
they might write an equation/expression to represent each rule. For example, 2(input) + 1
= output would be come y = 2x + 1.
Rules: 2x + 1, 3x + 2, 2x – 1 (Note: Tables will vary based on the numbers generated by
students.)
Silent Board Game (15 minutes)
Draw each table on a separate piece of chart paper. Post one of the pieces of chart paper
in the front of the room. Students are to remain silent and examine the rule being
presented. When students feel they have figured out the pattern, they can raise their
hands. When you call on a student, he/she should silently go to the board and write a
new x- and y-value (input/output) that fits the rule. If that student is correct, let them
know and tell them to sit down with their thumbs up. Continue to allow the students to
silently analyze the pattern being formed. As students figure it out and raise their hands,
allow them to come up and add their own x- and y-values. When it seems as though
many of the students have figured out the rule (based on “thumbs-up” and raised hands),
allow one student that already has their thumbs up to explain the pattern/rule to the class.
Post a new piece of chart paper with a new pattern over the old one and start over.
Closure (5 minutes)
Discuss the following questions, as a class, in regards to one of the tables from the Silent
Board Game:
1. Based on the table, what type of graph would you expect? Explain why. (Some
students may notice that the pattern increases by the same amount, others may
note that the equations had the same format as the linear equations in the
“Finding Graphs” activity.)
2. How does the rule you came up with for this particular pattern correspond with
the pattern in the table? (Students may notice that the number they multiply x by
is the same as the amount by which y increases. They also may notice that the
initial value is the other value in the rule.)
If time permits, graph the table of values you have discussed on a piece of chart paper.
Finding Graphs
Directions: Enter each equation into the graphing calculator and sketch the graph that is shown.
Indicate whether each graph is linear or non-linear.
a. y  7 x  8
b. y  2 x 2  1
Linear / Non-Linear
Linear / Non-Linear
d. y  x 3  5
e. y 
Linear / Non-Linear
1
x4
2
Linear / Non-Linear
c. y 
5
x
Linear / Non-Linear
f. y  31x
Linear / Non-Linear
Finding Graphs: Answer Key
Directions: Enter each equation into the graphing calculator and sketch the graph that is shown.
Indicate whether each graph is linear or non-linear.
y  7x  8
y  2x 2 1
Linear
y  x3  5
Non-Linear
y
y
5
x
Non-Linear
Non-Linear
1
x4
2
y  3  2x
Linear
Non-Linear
Make-A-Problem
Equation
Equation
Equation
______________
_____________
___________
Linear / Non-Linear
Linear / Non-Linear
Linear / Non-Linear
Equation
______________
Linear / Non-Linear
Equation
_____________
Linear / Non-Linear
Equation
___________
Linear / Non-Linear
Guess My Rule “Operator”
INPUT
OPERATOR
OUTPUT
Silent Board Game
x
y
0
1
2
7
2
12
3
17
4
5
6
7
8
RULE: y = 5x + 2
x
y
0
1
3
5
2
7
3
9
4
5
6
7
8
RULE: y = 2x + 3
x
y
0
1
–2
1
2
4
3
7
4
5
6
7
8
RULE: y = 3x – 2
x
y
0
1
3
13
2
23
3
33
4
5
6
7
8
RULE: y = 10x + 3