MS After School Intervention
Unit: Linear & Nonlinear Relationships
Day 4 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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

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Chart paper and markers
M&Ms
Small cups
“3-2-1 Closure” resource sheets (one per student)
Bacteria Warm-Up (15 min)
Read the following problem:
“Sean is trying to grow bacteria in a Petri dish for the Science Fair. The bacteria double
every week. He starts with 30 thousand bacteria, but he needs to grow over a million
bacteria in order for them to be visible at the science fair. How many weeks will he need
for his project?”
In small groups, students should work together to determine the answer to the problem.
They may wish to set up a table of values. Have each group give their answer and
explain how they determined it. Ask each group to determine if the pattern should be
linear or non-linear. As a class, set up a table of values on chart paper and sketch a graph
of the bacteria growth to show that it is non-linear.
Answer:
Sean needs 6 weeks to grow enough bacteria for the Science Fair.
Week
0
1
2
3
4
Bacteria
30
60
120
240
480
(in thousands)
M&M Lab – Part I (20 minutes)
5
960
6
1920
Separate the class into small groups. Each group should get a small cup and about 100
M&Ms. Instruct the class that these M&Ms represent bacteria in sick person’s body after
an antibiotic has been prescribed. Each day, a certain number of bacteria die. The
students will simulate this situation by counting their initial number of M&Ms and
dumping them all out of the cup onto a clean sheet of paper. The M&Ms that land “Mside” down have been removed by the antibiotic. They should be placed aside and the
remaining M&Ms should be counted and placed back in the cup for another “day of
antibiotic.” Students should continue to dump the M&Ms, removing all that land “Mside” down. Each group should create a table of values showing days on the antibiotic
versus bacteria remaining in the body. Each group should determine how many days
their antibiotic took to remove all bacteria.
When all groups are finished, have each group share their results. As a class, discuss the
pattern of bacteria removal. (Students should recognize that the bacteria do not decrease
by a constant amount. They may need prompting to see that the number of bacteria is
approximately cut in half each time. If they have a basic knowledge of probability,
students may then recognize that the number of bacteria is cut in half because an M&M
has two sides).
M&M Lab: Part II (15 minutes)
Discuss that the pattern is geometric and how the number of bacteria is being multiplied
1
by approximately or 0.5 each day. Also, discuss whether the graph of the data will be
2
linear or non-linear. Each group should graph their table of values on chart paper to
confirm that the pattern is non-linear.
 students have a basic knowledge of probability, discuss how a six-sided die
Note: If the
might change the problem if the numbers 1-5 represented surviving bacteria (it would
take longer to be cured, the graph would be less steep, etc.) or if the numbers 1-5
represented dead bacteria (it would take less time to be cured, the graph would be much
steeper, etc.).
Right Side – Left Side (15 minutes)
This is a variation of Corners, but there will only be three walls. Label the first wall
“Linear,” the second wall “Nonlinear,” and the third wall the third wall “I don’t know.”
Display one problem at a time on the overhead and instruct students to stand by the wall
that correctly describes the given representation. Discuss solutions and when appropriate
ask students what type of sequence would be represented in the table. Ask students to
explain how they knew which wall to stand by.
Problems & Answers:
1.
x
y
1
5
2
6
3
7
4
8
5
9
4
27
5
81
6
243
(linear & arithmetic)
2. y  6x 2
(nonlinear)
3.
x
y
2
3
3
9
(nonlinear & geometric)
4. y  4 x  1 (linear & arithmetic)
5. “sketch of a quadratic” (nonlinear)
6. “bacteria doubling each hour” (nonlinear & geometric)
7. “sketch of a straight line” (linear & arithmetic)
8. y  6 x (nonlinear & geometric)
9. “the temperature dropping two degrees each hour” (linear & arithmetic)
1 x
10. y    (nonlinear & geometric)
2 
11. “a car losing a certain amount of gas for each mile it travels” (linear & arithmetic)
12. “the relationship between the radius of a circle and its area” (nonlinear)

Do the Wave (20 minutes)
Time five groups of different sizes doing “the wave.” Time each group three times and
take the average time. Be sure use the entire class as a group. List the average time
versus the number of people in the group in a table of values on chart paper. Have
students discuss whether they think the pattern will be linear or non-linear. If they are
struggling, you may wish to ask what is happening to make it take longer. (Students may
respond that you are “adding more students”). Graph the data as a class and discuss
why it is linear. (Each student takes about the same amount of time to stand up and sit
back down, which adds time at a relatively constant rate. You may wish to find this rate
as a class.)
3-2-1 Closure (5 minutes)
Have students fill out a “3-2-1 Closure” by listing the following:
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How to determine if a pattern is linear or nonlinear from 1) the graph, 2) the
pattern, and 3) the equation
A description of 1) an arithmetic sequence and 2) a geometric sequence
One question they still have about linear and nonlinear patterns
3-2-1 Closure
In the space below, write:
3 ways to determine if a pattern is linear or nonlinear.
1. From the graph:
2. From the pattern:
3. From the equation:
2 types of sequences.
1. arithmetic:
2. geometric:
1 question you have regarding the lesson.
1.