Activity: Direct and Inverse Variation Using Hershey Bars
Activity Description: Using Hershey bars, direct and inverse variation will be introduced. Follow up with additional examples
Materials Needed: Hershey Bars, Hershey Bar Worksheet (see below), Situation Worksheet
Key Ideas: Students learn the Direct Variation and Inverse Variation
Connection to Michigan HSCE’s:
A3.4.2 Express directly and inversely proportional relationships as functions and recognize their characteristics.
(For Middle School: A.RP.08.01 Identify and represent linear functions, quadratic functions, and other simple functions including inversely proportional relationships (y=k/x) using tables, graphs, and equations.
Teacher Prompts:
NOTES:
1. I started with inverse variation because the students easily saw that there was a pattern to the table. In a direct variation, both values increase or decrease and the students sometimes do not view this as a pattern because it is straight forward.
2. This lesson is written as a lesson that can be done with the class, complete with teacher prompts. You can have the students create the tables individually in their notes as you create a table in front of the class. However, worksheets are also provided (Hershey Bar Direct vs. Inverse) so that you can have your class explore the variations without your prompts.
Before:
Today we are going to look at some special functions. But before we do, I bought a candy bar. Open the candy bar. How many pieces does this candy bar have? (12). I really want to give this candy bar to ______. How many pieces will ______get? (12) I am feeling really bad for the student next to ______, so I think those two students should share the candy bar. How many pieces will each of them get? (6) This is math class, so let’s make a table of the data.
During:
# of students(x) / # of pieces of the candy bar each will receive
(y)
1 / 12
2 / 6
3 / 4
4 / 3
6 / 2
8 / 1.5 or 1 1/2
12 / 1
16 / .75 or 3/4
24 / .5 or ½
32 / .375 or 3/8
36 / .333 or 1/3
/ These two students have to share with ______. How many pieces do they each get now? Add that to our table.
What if I decide that I want to give the entire table to share the candy bar, how many pieces will each student get?
Let’s add more students and complete the table.
What do you notice about the data in the table?
(as 1 goes up, the other goes down, usually get the students to say opposite then ask for the word in math that means opposite) This is an inverse variation.
How did we figure out how many pieces each student would receive? What if I had 48 students, how many pieces would each student receive? How did you figure that out? (divided 12 by 48)
Was that how we figured out each of them?
Let's write a function as an equation.
Y = 12/x
What must be done to x to get y?
(we divided 12 by x)
So what was the same for each line of the table?
(12)
12 was constant. It is the constant of variation. We use a “k” to designate the constant of variation in an inverse variation. This is how the output is related to the input.
Look at the function equation you wrote. What is the constant of variation? How can you describe it? Where in the equation would you look for the constant of variation?
Now let’s solve the function equation for k.
y= k/x
to get k alone we need to multiply both sides by x
xy=k
This is how we can find the constant of variation from a table of an inverse variation or use it to check a table to see if it is an inverse variation.
Check to see if xy is the same for each line of our table.
With all that work, I bet some of you are getting pretty hungry. I am sure that ______would rather have a whole candy bar than 3/8 of one piece of a candy bar. As a matter of fact, I bet both ______and ______would like a whole candy bar. How many pieces would there be all together if they both had a whole candy bar? Let’s make a table.
# of students(x) / Total # of pieces
(y)
0 / 0
1 / 12
2 / 24
3 / 36
4 / 48
5 / 60
/ Let’s add more students and complete the table.
What do you notice about the data in the table? (both are increasing)
This is different from the last table where the numbers were going in opposite directions. The values in this table are going in the same direction. This is a direct variation.
If I had 50 students, how many pieces would I have? (600)
How did you figure that out? (multiplied 50 x 12)
Is that what we did to each x to get y in our table?
What must be done to x to get y? (multiply by 12)
Let's write a function as an equation.
y = 12x
So what was the same for each line of the table?
(12)
12 was constant. It is the constant of variation. We use a “k” to designate the constant of variation in a direct variation. This is how the output is related to the input.
Look at the function equation you wrote. What is the constant of variation? How can you describe it? Where in the equation would you look for the constant of variation?
Now let’s solve the function equation for k.
y/x = k
This is how we can find the constant of variation from a table of a direct variation or use it to check a table to see if it is a direct variation.
Check to see if y/x is the same for each line of our table.
For every function, you have to know three things. What are they? (a table, a graph and an equation)
What piece do we need for the inverse and direct variation we just looked at? (a graph)
Let’s graph the data for both.
What do you notice about the graphs? (get the students to discuss the appearance/shape)
Now in your groups work on the two situations.
After:
Today we looked at direct and inverse variations. Tomorrow, we will look at the graphing calculator.
What comes next?
Graphing calculator piece: Have the students look at sets of equations such as y=3x and y=3/x on the graphing calculator. Go to this activity for more calculator practice.
Modifications:
This lesson was intended to be an introduction to direct variation and inverse variation. Based on your class, you may be able to give the Hershey bar worksheets to groups of students in your class to work on first, come together to discuss the groups’ findings, and then reconvene the groups to work on the situation sheets.
Extensions:
Have the groups of students write two situations (one direct and one inverse) and swap with another group to complete the table, graph, write the equation, and determine the type of variation.
Direct Skill Practice:
Go to the following for more practice:
/ This is a great link:/
Direct Variation Versus Inverse Variation
Situation Worksheets
Group Members:Date:
Situation 1: Kyle has been asked to make 120 paper airplanes to be sold at the school’s science fair. One person can make 15 paper airplanes an hour. Kyle only has 1 hour to complete the airplanes. He decides that he has to get a few friends to help him. Start by making a table to show how many airplanes Kyle and each of his friends would have to make. How many friends would he need to help to complete the airplanes in one hour? Graph your data below.
/ Define the variables:Write the equation:
What is the constant of variation:
Is this a direct variation or an inverse variation? Why?
Situation 2: Each time Sarah receives an “A” on her report card in high school; her grandparents give her $10 for her savings account. Sarah wants to calculate quarterly how much money she can accumulate during high school if she takes 6 classes per quarter. (You are only calculating the amount of money deposited, not the interest.) Graph your data below.
/ Define the variables:Write the equation:
What is the constant of variation:
Is this a direct variation or an inverse variation? Why?
HERSHEY BAR
Direct Variation versus Inverse Variation
Your class is going to share a Hershey Bar. As I am sure you have noticed, a Hershey bar is marked into several pieces or sections. How many pieces are in one bar? ____ Make a table for the number of students and how many pieces (sections) of the candy bar they will receive.
# of students(x) / # of pieces of the candy bar each person will get
(y)
1 / 12
/
What patterns do you notice in the table? ______
If you had to share the candy bar with 36 students, how much of the candy bar would each student receive? What about 48 students?
______
What did you do to figure out how much of the candy bar 48 student would receive if they shared the candy bar? Does that work for each line in your data table?
______
In terms of an equation, what are you doing to x to get y?______
Write your rule as an equation. ______
Notice that you do the same thing to x each time to get y. The number on the same side of the equation as x is constant. This is called the constant of variation, and is designated by the letter k.
For the equation that you wrote, what is the constant of variation?______
Looking back at the data in the table (and any pattern you may see), do you think that this data as well as your equation represents a direct variation or an inverse variation? (Hint: think about what direct and inverse mean)______
HERSHEY BAR
Direct Variation versus Inverse Variation
This time, each student will receive their own Hershey bar. Make a table for the number of students and how many pieces (sections) there will be in the classroom.
# of students(x) / # of pieces of the candy bar there will be in the classroom
(y)
0
1 / 0
12
/
What patterns do you notice in the table? ______
If you had to share the candy bar with 50 students, how many pieces would we have in our classroom?
______
What did you do to figure out how many pieces 50 candy bars would give us? Does that work for each line in your data table?
______
In terms of an equation, what are you doing to x to get y?______
Write your rule as an equation. ______
Notice that you do the same thing to x each time to get y. That number in front of x is constant. This is called the constant of variation, and is designated by the letter k.
For the equation that you wrote, what is the constant of variation?______
Looking back at the data in the table (and any pattern you may see), do you think that this data as well as your equation represents a direct variation or an inverse variation? (Hint: think about what direct and inverse mean)______