Liz Mattarazzo
Direct and Inverse Variation 4
Are you a Belieber?
I.
Content:
Justin Bieber is coming to Worcester and everyone is super excited, especially since his new single “As
Long As You Love Me” is a big hit. You’ve been trying to get tickets but they’re all sold out. But wait,
there’s another option! They’re looking for volunteers to work as part of Justin’s stage crew which will
earn you and five of your friends back stage passes for the concert!!! WOOO!!! 
The problem is that they have lost the directions for putting the stage together. Oh no! They need your
help! The only requirement they have is that it must be a rectangle whose area is 24 square meters, per
Justin’s request, of course. In your mind, you probably have an idea right away; Ms. Shepard taught you
all about area in middle school, piece of cake. BUT we need multiple options; it may not be the most
obvious; Justin is a unique individual, he likes to mix things up sometimes.
This lesson, involving Justin Bieber, is designed to give students practice with inverse variation. There is
this situation, as cited above in the intro and another situation that they will have to make a table for,
graph, and find a rule for.
II.
Learning Goal(s):
Students will know and be able to:
III.
Make a table using the information given (filling in missing points).
Make a graph using data points given.
Find an equation to model a real life situation.
Recognize that this is NOT a direct variation relationship.
Take the above knowledge and translate it to other problems.
Rationale:
Direct variation is a piece of cake compared to inverse variation when trying to teacher ninth grade
students. They see a relationship with numbers and they want to make a proportion, even if it doesn’t
make sense. If it takes Ms. M 1 hour and 15 minutes to drive a certain distance going 70 mph, how long
does it take her to drive the same distance if she’s going 75 mph? They WILL set up a proportion and get
a longer amount of time even though that makes NO sense. This activity is designed to fix that. It begins
with area of a rectangle (or stage) because the students are familiar with area, they know that formula.
The next scenario (volunteers x hours) is a bit more difficult for them but with the first situation under
their belts, they should be able to transfer the knowledge and figure it out. This is a concept they need
to understand. Hopefully, using real life situations, they will be able to see the difference – that it can’t
possibly take more people more time to do something.
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IV.
Assessment:
Students will be working in groups and I will be monitoring them during class. In addition, I will be
collecting their classwork as the conclusion of each day to see how they are doing. The worksheet has
multiple parts (scenario 1, 2 and applications). The students will be evaluated on how well they
completed the worksheet. At the end of the activity they will answer this question:
You are now asked to build a stage for another band, your favorite band, not just an individual singer,
such as Justin Bieber. This stage will have to be bigger to accommodate, let’s say 60 square meters.
Come up with at least 5 possible dimensions it could be. In addition, include which band you are building
it for, which is your favorite?
V.
Personalization:
For this lesson I will allow students to pick their own groups, yes it may backfire but it is worth a try. In
addition, the lesson deals with a “real life” scenario involving Justin Bieber. Most of the students love
Justin Bieber. I try to make these activities interesting for them. This is a variation of an NCTM lesson
that dealt with mowing the lawn and typing a paper… I think this is more interesting to a 14 year old. It
may even be more interesting to an adult, like me, in my own opinion.
If students finish the 2 scenarios, there are problems on the back of the packet that involve transferring
the knowledge they’ve learned through the Justin Bieber situations. After that, I have another
worksheet for them that has more proportional reasoning situations and a bit more writing as well. For
those students that need extra help, they will be working in groups so their groups will be their first
resource. However, a lot of students are going to struggle with this concept of “more volunteers, less
hours”. I am going to be circulating like crazy between the groups trying to catch them if they make
mistakes before they get too far.
VI.
Activity description and agenda:
Materials: computer, powerpoint, worksheets
Grouping: choose your own groups!
Day 1
0-15 Entrance Slip
Students will respond to the
prompt in their POD books.
3 Tell me about three things you learned this
week in math class. Use as much detail as
possible.
2Name two things that you LIKED or DIDN’T
LIKE about class this week AND EXPLAIN
WHY!! If you are choosing things you didn’t
like, be constructive and include ways to
improve the activity, group, situation, etc so
that you would have liked it.
1What is one thing you are still wondering
about or didn’t understand that we did this
2
week in class?
You have to have AT LEAST 6 things (3+2+1)
in your POD book to get an A for this
classwork grade. We’re doing low stakes
today. A or F. You pick.
Notice: I am letting you choose your own
groups today. Show me that you can handle
it. Yes, you are allowed to talk, but you need
to get your work done! You also need to be
respectful when I and your classmates are
talking. If you prove to me you can’t handle
it, this won’t be happening again… ever.
15-22 Justin Bieber
22-55 Scenario 1
A student will volunteer to read
the intro. We will address any
questions.
The students will begin working
on scenario 1 in their groups.
Students will ask me if they need
help.
I will show “As Long as You Love
Me” and we will read the intro
as a class.
In general, I have noticed it takes
students FOREVER to graph data
so they will most likely get stuck
on that part for a while. I
anticipate only a few students
getting to the second scenario
today. In addition, they will most
likely struggle trying to find a
rule. I will prompt them with
guiding questions and ask them
to look back at their work. How
did you find these numbers?
How do length, width and area
relate?
Day 2
0-6 Entrance Slip
6-55 Justin
Students will answer these
questions in their POD books.
If they finish, they will study for
the conversions quiz they have
coming up.
Students will pick up where they
left off on their Justin packets.
They will make it to scenario 2
and most likely struggle, needing
help.
Any students that finish the
scenarios will move onto the
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How many kilometers are in one mile? How
many millimeters make up one centimeter?
How many quarts are in one gallon? How
many meters are in one kilometer? 32 ounces
are how many pounds? 9 feet are how many
yards?
I will be circulating around the
room like crazy today as most of
the students will be reaching
scenario 2, volunteers and hours.
This is going to be a stretch for
them. They are going to want to
do it out like a proportion. I’m
going to need to guide them a
back page – applications. If they
finish that, there is an extension:
proportional thinking worksheet.
VII.
lot.
Massachusetts Learning Standards:
N-Q.1: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units
consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
N-Q.2: Define appropriate quantities for the purpose of descriptive modeling.
A-SSE.1: Interpret expressions that represent a quantity in terms of its context.
A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on
coordinate axes with labels and scales.
A-REI.10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane,
often forming a curve (which could be a line).
F-IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of
the quantities, and sketch graphs showing key features given a verbal description of the relationship.
F-BF.1: Write a function that describes a relationship between two quantities
VIII.
Sources:
http://illuminations.nctm.org/LessonDetail.aspx?id=L729
IX.
Reflection:
This lesson took so much longer than I anticipated, I am sensing a theme. That seems to be my biggest
problem with most of these activities, time. However, in addition, the students struggled a lot with the
idea of inverse variation here. They tried to make proportions even when it did not make sense logically.
Another problem with this lesson in particular was that they would not talk to each other about the
math. I let them pick their own groups for the first day at least; they had no problem talking about off
topic things. However, they would not talk about the math; they will do the problem individually and
just not talk about it in their groups. To fix that, I threw a Socratic Seminar in the middle of this activity
to get them talking about their work. I do not understand what the problem was, they learned to talk
and work things out within groups in middle school; it is not a new thing. In general, they do not usually
have problems with it. I am not sure what it was about this particular lesson but it happened.
In the future, I am not sure if I would change anything. The kids really liked the Justin Bieber theme to
the activity, it made it more interesting to them and easy to contextualize. The only thing was that when
they all got to the second part, they immediately tried to use a proportion instead of logically reasoning
out the problem. Therefore, maybe if I could have incorporated a whole class demonstration or scenario
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that modeled the idea of inverse variation without solving it for them, that would have helped. But in
general, it was good for them to struggle a little bit and have to start over. This was our first real new
concept, because while they never called direct variation, “direct variation”, they solved proportions and
worked with linear situations in middle school.
However, while I let them pick their own groups for the first day, I warned that if there were any issues I
would be reassigning. Of course there were some issues, so I kept some groups the same and moved
others around. And while they love working in groups and talking to their friends; they do not talk about
the math involved in the problem. The second day, they were barely talking at all. Therefore, I added in
a Socratic Seminar on the following day to force them to talk.
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