MS After School Intervention
Unit: Linear & Nonlinear Relationships
Day 3 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.
Common Core Standards:
8.F.3 Interpret the equation y  mx  b as defining a linear function, whose graph is a
straight line; give examples of functions that are not linear. For example, the function
A  s 2 giving the area of a square as a function of its side length is not linear because its
graph contains points (1, 1), (2, 4), and (3, 9), which are not on a straight line.
8.F.4 Construct a function to model a linear relationship between two quantities.
Determine the rate of change and initial value of the function from a description of a
relationship or from two (x, y) values, including reading these from a table or from a
graph. Interpret the rate of change and initial value of a linear function in terms of the
situation it models, and in terms of its graph or a table of values.
8.F.5 Describe qualitatively the functional relationship between two quantities by
analyzing a graph (e.g., where the function is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits the qualitative features of a function that has
been described verbally.
Materials
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Overhead projector or document camera
Chart paper and markers
Cheerios
Compass
Circles of different sizes with diameters marked (prepared in advance – one per
group)
Graphing calculator
Revisiting the Oil Spill (10 minutes)
Display the data tables for oil run-off and bacteria from the previous lesson. Recall the
addition and multiplication patterns and sequence terminology that was discussed.
Add variables to the table as shown below.
day (x) 1
gallons 5
(y)
2
8
3 4 5 6 7 8 9 10
11 14 17 20 23 26 29 32
In small groups, students should work together to create a rule for this data, using prior
knowledge and examples from the graphing activities to help them.
Answer: Students should find y = 3x + 2.
Many Representations (20 minutes)
Place students in groups of three or four. Each student should have a role: table maker,
equation writer, or graph sketcher. Post the problems one at a time for the groups to
solve. After two problems, have students switch roles so that they have fulfilled each
task.
Problems & Answers:
1. John buys 10 cookies. Starting on day 1, he eats 2 cookies per day. Find the number
of remaining cookies, c, based on the number of days that passes, d.
d
c
0
10
1
8
2
6
3
4
4
2
5
0
c  2d  10
2. Leah sees 4 crabs on the beach. The next day she sees 7, and the next day she sees
10. This pattern continues for 3 more days. Find the number of crabs, c, based on the
day, d.
d
c
1
4
2
7
3
10
4
13
5
16
6
19
c  3d  1
3. Chickens-R-Us makes 24 sandwiches. They sell 3 sandwiches per hour. Find the
number of remaining sandwiches, s, based on the day, d.
d
s
s  3d  24
0
24
1
21
2
18
3
15
4
12
5
9
6
6
7
3
4. Amanda has $45 in the bank. Each week, she gets $15 in allowance to add to her
bank account. Find the amount of money she has in the bank, m, based on the week, w.
w
m
0
45
1
60
2
75
3
90
4
105
5
120
6
135
7
150
m = 15w +45
5. Jarod weighs 345 pounds. He is on an exercise routine that allows him to lose 7
pounds per month. Find the amount Jarod weighs, w, based on the month, m.
m
w
0
345
1
338
2
331
3
324
4
317
5
310
6
303
7
296
w = –7m + 345
Cheerio Lab: Part I – Table and Graph (15 minutes)
Before the activity begins, use a compass to create circles of different sizes on different
sheets of paper. (Note: the diameters of the circles should range in size from 2.5 inches
to 8 inches.)
Separate the class into small groups. Give a circle of a different size to each group. Have
the groups each measure the diameter and circumference of their circle in cheerios. On
chart paper, post the following table:
Diameter
(in Cheerios)
Circumference
(in Cheerios)
Be sure there are enough rows for every group to record their data. Once all the data has
been collected, make a scatter plot of the data on chart paper. Have students discuss why
diameter should be on the x-axis (circumference depends on the length of the diameter).
Discuss the shape of the graph, which should be roughly linear. In addition, discuss why
the graph isn’t perfectly straight (human error).
Cheerio Lab: Part II – Estimating Pi (20 minutes)
Refer to the table and graph from Part I of the Cheerio Lab. Have students return to their
groups. In their groups, have the students attempt to determine the average change in
circumference Cheerios per every increase in diameter Cheerio. You may have to prompt
students to divide the changes and then average them. Have each group do their work on
a separate piece of chart paper and post them around the room. They should also discuss
anything they notice about the average change.
As a class, discuss what each group determined. Ask the students how they think their
number might relate to the problem situation. You may have to remind them that they
are working with the diameter and circumference of a circle. Students may realize that
the number is close to pi at this point. Otherwise, have the students give you the formula
for the circumference of a circle ( C  d ).
Discuss why this relationship is linear, based on everything they have learned so far (the
format of the equation, the increase by a constant). In addition, discuss what their rule
would be for the pattern (varies based on accuracy of data) and what it means in terms of
the problem situation (for every cheerio added to the diameter, the circumference
increases by about 3.14).
Use the document camera to display the class’ original scatter plot. Graph the equation
found by the class over the original scatter plot to show the fit. If time permits, graph
C  d as well, so the students can see the accuracy of their results graphically.
Cheerio Lab: Part III – Area (20 minutes)
Have students return to their groups with their original circles and Cheerios. Students
should fill their entire circle with Cheerios to determine the area. Have them fill out a
table of radius versus area, similar to the first table. Before creating a scatter plot, discuss
the following as a class:
1. Do you think this graph will be linear? Why or why not?
2. What would differ about the equation for this pattern? Prompt students to think
about the formula for the area of a circle. ( A  r 2 )
3. What might the graph look like based on this new equation? Students may have
to revisit their Finding Graphs activity to determine the shape of a quadratic
function (“u-shaped”).
Graph the relationship on chart paper. The graph should appear non-linear, but not
quadratic. Discuss with the class why the shape is only half of a “u.” You may have to
sketch in the missing half to encourage discussion. (Students may recognize that their
can be no negative radii in the real world. They may also note that it’s impossible for the
area to decrease as the radius increases).
Closure (5 minutes)
Have students respond “linear” or “nonlinear” to each of the following relationships and
discuss any disagreements:
1. The decrease in a bank account as you spend the same amount of money on food each
week. (linear)
2. The increase in population in a small town if it doubles every ten years. (non-linear)
3. The relationship between the amount of outer crust on a pizza and the pizza’s
diameter. (linear)
4. The relationship between the amount of carpet needed for a square room and the
length of that room. (nonlinear)