Liz Mattarazzo
Direct and Inverse Variation 6
Inverse Variation Notes
I.
Content:
Notes again, yes, sometimes they are necessary. These notes will solidify the ideas of inverse variation
the students have been working with. They, of course, don’t know it is called “inverse variation” yet but
they have gotten some of the concept from the activities we have been doing (Telescope Investigation 2
and Justin Bieber). Just like direct variation, the notes will be done in 4 box format – definition (words),
tables, graph and symbols. We will be drawing from our Justin Bieber data as well so they have real life
situations to relate to the abstract concepts.
II.
Learning Goal(s):
Students will know and be able to:
III.
See patterns by comparing and contrasting data from real life situations
Build a function that models a relationship between two quantities
What an inverse variation relationship looks like (in words, graphs, tables, algebraically)
Rationale:
We will have just completed an activity dealing with inverse variation (Justin Bieber) in addition to the
concepts showing up in other places (Telescope, Socratic Seminar, Problem Sets, Pop Quiz). This means
it is time to solidify the concept they have been working with and give it a name. In this four box format
we will approach the concept from all angles, different learning styles and tie the ideas together. I want
them to be able to see an inverse variation relationship from a graph, word problem, table or equation.
While I know taking notes will not completely teach them this, they will not automatically absorb it from
this lesson; it will set them in the right direction. After these notes we will do skills practice on both
concepts (direct and inverse variation). This is just tying together all their thoughts through Where’s
Waldo, Are you a Belieber?, and the various other places. The notes themselves will focus on data from
Are you a Belieber? so they have a real life situation and problem to relate these concepts back to.
IV.
Assessment:
Notes are a difficult thing to assess. However, the graphic organizer I give them will be mostly empty,
therefore, they will be responsible for filling it in as we go along in class. This ensures they are paying
attention and are responsible for their learning. In addition, I will keep track of who is contributing to
the class discussion in forming the notes and give points for that. Of course, this will require me setting
up the class in this manner. I will tell them at the beginning on class in order to hear from everyone, I
will be keeping a list and giving participation points. Also, I do periodic notebook checks where they are
graded on how complete their notebooks are (table of contents, includes all notes, everything is filled in,
etc). This means that these notes will come up later down the line, they will keep reappearing. Lastly, all
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of our quizzes and tests are open notebook, considering the wrap up to this unit will be a test, how well
they do on it may depend on how well they take notes here.
V.
Personalization:
While I will give them a graphic organizer to glue into their notebooks, they will be in charge of filling it
in as we go along in class. They need structure and organization which is why I have created the
organizer for them. A lot of my students are on 504s or IEPs, struggling with organization, use of
organizers when possible is recommended. Also, by giving them this form that they simply need to fill in
and annotate as we go, ideally it will save time. I KNOW for a fact that the second I tell them to divide
their pages into 4 boxes immediately I’ll hear cries of “Miss, I need a ruler!” Yes, 100% predictable. They
love their lines to be straight; it also, of course, wastes class time. This way, their lines will be straight
and I will save that time.
VI.
Activity description and agenda:
Materials: Each student has a graph paper notebook. I will provide the 4 box note sheet for them to glue
into their notebooks.
Classroom set up/grouping: The room will be set up in two horseshoes, six in the middle, the rest on the
outside. This is the format we usually use for taking whole class notes and it seems to work well.
Time
0-6 min – Entrance Slip
What Students Do
Students will respond to the
prompt as they enter the
classroom.
6-11 min – Wrap Up
Students will share out patterns
they found in the activity we
have been working on (Are you a
Belieber?) or how they solved
the starter. Since they have
already shared in groups, this
will be a full class discussion.
11-14 min – TOC
Update Table of Contents
10/24/12 Are you a Belieber?
Notes
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What Teacher Does
You are now asked to build a
stage for another band, your
favorite band, not just an
individual such as Justin Bieber.
This stage will have to be bigger,
maybe 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?
I will ask for volunteers, stressing
that classwork is 50% of their
grade (They have all wanted to
do really well to bring their
grades up).
Including: ideas of constants,
equations, patterns, and
properties of the graph they
made.
Write updated TOC on board
(As the students are taking the
notes below, I will be modeling
14-20 min – Words
(As we are going along students
will be annotating what I am but
also whatever helps them.)
One student will read aloud
what is written in the “words”
box on their note sheet. Others
will be following along,
underlining words that stand out
to them.
Students will be answering
questions that Ms. M poses
dealing with this section. They
will be silent and listening to
each other’s ideas, building off
of them if possible and helping
each other out when needed.
20-26 min – Table
26-32 min – Graph
32-40 min – Symbols
Students will have the tables
filled in (by me previously) so we
will be looking for things we
notice.
Each of the lines will have the
same v x h and from there we
will define it. It is called a
“constant”, as I will tell them.
They will be writing this down
next to or underneath the
column.
The graph will be constructed for
the students on their note sheet.
We will discuss why it never
touches the axes.
Students will write the number
relationships next to the data
points (in the 4th column of their
tables).
Students will figure out what the
general equation should be.
In addition, some of the students
have been using another
method. 9 and 4. So you divide 9
by 3 so you must multiply 4 by 3.
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on the ELMO by taking my own
notes.)
I will ask what students have
underlined; we will pick out key
things.
Ask students how they would
describe the relationship
between volunteers and hours in
their own words. I will be calling
on students with their hands
raised, enforcing that behavior
and having them build off of one
another’s ideas if possible.
Example: As one goes up, the other
goes down.
What do we notice about this
table? We will observe that the v
x h is the same for each. We will
define it as the constant.
Furthermore it is the constant of
proportionality.
I will ask why our graph never
appears to touch the axes.
We will talk about the “I’m not
touching you” game they
probably played as a kid.
We will then move to
constructing a general equation.
First of all, we will look at how
these numbers relate. If needed:
What times what equals what?
What divided by what equals
what?
From there we will generalize
this idea to an algebraic
equation. y=k/x. Ideally, this will
be straightforward after
modeling it with the numbers
We will discuss this method and
write it down as well.
above. However, if they need
more help, we will look at the
number examples and write the
variables over each number.
Then we will see how to find the
time for 5 volunteers, using what
we have.
I will also discuss any other
methods they have been using
to solve these examples.
40-50 Sorting
Students will have to sort these
cards into direct and inverse
variation piles.
50-end Exit Slip
Students will respond to the
prompt in their POD books.
I will give each pair of students
an envelope with cards in it.
These cards will have examples
of direct and inverse variation.
They will be graphs, tables and
word problems.
How do direct and inverse
variation relationships compare?
How are they the same? How are
they different? Explain. Use
examples.
VII.
Massachusetts Learning Standards:
A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
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:
Inverse Variation Notes – Shannon Hammond
IX.
Reflection:
These notes were incredibly similar to the direct variation notes because those went so well. I really like
the format of the four boxes with all the different representations. This concept was a tiny bit harder for
them to grasp though because they are so used to dealing with proportions to solve everything. I also
briefly introduced the idea of hyperbolas and asymptotes which they seemed to get in the moment but
will eventually forget. Though, that is okay because they do not need to know those things concretely in
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the ninth grade, I was just introducing them so when they show up again they will have a frame of
reference.
Behavior wise, Josh Ramos literally sat in the back of my classroom and talked the entire time. I cold
called him for an answer and he refused to answer even though he said he knew. Other students, like
Juan and Gilberto were ready with their hands raised and trying to help out but I did not back down. I
wanted Josh to answer and eventually he did. I know I should have kicked him out of my class that day
but I guess I was still unclear as to what behavior needs to be to do that. And we literally just had the
talk about it being a new term and everyone starts with As; it is easier to keep an A than to earn it back
later. I talked to him after class and had that same conversation with him about his behavior and his
grades. He told me he did not care how he did as long as he passed my class. As long as he got a 65, he
would be happy. For the time being, I did not even have a response, I think I was so shocked and
disappointed. That is what I am up against, good to know.
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