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Overview:
This lesson teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). This lesson plan is based on the activity used by teacher Peggy Lynn in Part I of the Workshop 7 video.
Time Allotment:
One to two 50-minute periods
Subject Matter:
Direct variation
Linear functions
Learning Objectives:
Students will be able to:
Standards:
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000:
NCTM Algebra Standard for Grades 6-8
http://standards.nctm.org/document/chapter6/alg.htm
NCTM Algebra Standard for Grades 9-12
http://standards.nctm.org/document/chapter7/alg.htm
Supplies:
Teachers will need the following:
Students will need the following:
For each group of four students, you will need:
Steps
Introductory Activity:
1. Conduct a brief discussion about oil spills, their effect on the environment, and ways that scientists work to clean them up. The discussion should involve specific oil spills with which the students might be familiar, such as the Exxon Valdez spill in Alaska. Links to information about the Exxon Valdez spill can be found in the Resources section.
The discussion should help students to conclude that oil spills are actually cylindrical in shape, not circular. One way to do this is to ask students to spill a drop of water on the countertop and look at the spill. They should note that it is generally circular in shape, but always has some thickness to it. The thickness represents the height of the “cylindrical” shape.
2. Brainstorm ideas about factors that affect how oil spills spread. That is, how does the shape of the land change the shape of the spill? Would it be possible to estimate the volume of a spill?
3. Depending on students’ prior knowledge, have them solve the following problems involving circles with “messy numbers” (the area of circles occurs repeatedly during this lesson):
Learning Activities:
Have one student read aloud from the bottom of page 162 in the SIMMS handout:
A simulation of a real-world event involves creating a similar, but more simplified, model. In the introduction, for example, you simulated an oil spill on the ocean using a few drops of oil in a pan of water. In this activity, you simulate oil spills on land by placing drops of oil on sheets of paper.
Note: This paragraph refers to an introductory activity (simulating an oil spill in the ocean) that took place prior to this lesson.
1. Explain to students that they will be conducting an exploration using vegetable oil and toilet paper. Describe how the oil will be dropped onto toilet paper tissues to simulate an oil spill on land. Using eight different samples, students will record data for oil spills involving from one to eight drops.
2. Divide students into groups of four and have each group gather the following: an eye dropper, vegetable oil, eight sheets of toilet paper, a large sheet of paper, and a ruler.
3. Have all the students in each group write their names on the large sheet of paper. They should then place each of the eight sheets of toilet paper on the large sheet. Each sheet of toilet paper should be marked with a numeral (1 through 8) to indicate the number of drops of oil for that sample, and a pencil dot should be placed at the center of each sheet.
Students should read and follow the instructions for conducting the experiment: Carefully place 8 drops of oil on the pencil dot on sheet 8. Continue creating oil spills of different volumes by placing 7 drops on sheet 7, 6 drops on sheet 6, and so on.
Students should then drop the appropriate number of drops onto each sheet.
4. Once all groups have placed the oil onto the sheets of toilet paper, reconvene the class to describe how data will be collected and organized.
5. All students should create a chart in their notebooks, as follows:
6. Inform students that they will measure the radius and diameter of the spills to the nearest tenth of a centimeter and that they will use those measurements to calculate the area covered by the spill.
7. Explain to students that they will use the data from their charts to create a scatterplot that gives the volume of the spill (in drops) along the x-axis and the area of the spill (in cm²) along the y-axis.
8. Distribute a transparency sheet and an overhead pen to each group. Explain that they are to create a scatterplot on the transparency, which they may be asked to describe to the class. (If possible, transparency sheets should already contain a coordinate grid.)
9. Allow students to return to their experiments, complete their charts using data from the experiments, and create their scatterplots.
10. Circulate through the classroom as students work. Offer some assistance, as necessary, but be careful not to give students too much information. While walking around, take note of the work of various groups that would be useful to share during the class discussion later in the lesson.
11. When groups have completed their scatterplots, call on two or three groups to share their work with the class using the overhead projector. Students should address these questions:
12. Students should estimate a line of best fit for their data and determine an equation for that line. Call on several groups to explain how they determined the slope for their estimated line of best fit. (Students may suggest several different methods: calculating the slope using two points on the line, determining the “rise over run” graphically by counting squares on the grid, choosing just one point on the graph and dividing the y-coordinate by the x-coordinate, etc.)
13. Question the y-intercept of students’ lines. Ask students what the point represented by the y-intercept means. For instance, if students have a y-intercept of 1, that represents the point (0, 1), which erroneously suggests that 0 drops of oil resulted in a spill area of 1 cm². Ask students to consider what the y-intercept means in the context of the problem. How much area would a spill of 0 drops cover? An important point about direct variation is that the graph will always contain the point (0, 0). In the context of this problem, 0 drops should yield an area of 0 cm². Have the students answer these questions:
14. Each group should complete an additional column on their chart that shows the ratio of area to volume (cm²/drops). This will give the slope of a line that passes through the origin (0, 0), and the point represented by each particular row. For instance, if row 2 has a volume of 2 drops and an area of 10 cm², the slope will be
(10 – 0) | ![]() |
= 5 |
_________![]() |
![]() |
|
(2 – 0) | ![]() |
15. Using the average of the values in the area/volume column, have students find a new line of best fit that passes through the origin. The average of the values in the area/volume column represents the slope of an approximate line. Students should answer these questions:
16. Review the definition of “slope” and reinforce that it should be considered as a “constant rate of change.” This is an important concept for students to understand about direct variation. The line of best fit has a constant slope and passes through the origin, which implies that as the volume increases, the area will increase proportionally. Ask students to consider these questions:
17. Using the data gathered during the lesson, explain that the relationship between two quantities that increase (or decrease) proportionally is known as “direct variation” or a “direct proportion.” Say, “The area varies directly as the volume (number of drops.)”
18. Point out that a direct variation is a special case of a linear function in which the line passes through the origin. The equation for a line that passes through the origin is y = mx, where m represents the slope. Because the slope of a line is constant, explain that in a direct proportion, the value of m is referred to as the constant of proportionality.
19. Choose a group and use the two equations for lines of best fit from that group to draw a comparison. Use T-tables to show how the values relate, based on their equations. A sample comparison is shown below:
20. Have each group make predictions based on a larger spill and answer these questions:
Culminating Activity/Assessment:
Assign several exercises for independent practice and homework. You may assign any of the problems from 2.1 through 2.9 (pages 166-168) in the SIMMS textbook (PDF).
The questions below dealing with direct variation have been selected from various state and national assessments. Although the lesson above may not fully equip students with the ability to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.
The diameter of a tree trunk varies directly with the age of the tree. A 45-year-old tree has a trunk diameter of 18 inches. What is the age of a tree that has a trunk diameter of 20 inches?
A. 47 years
B. 50 years (correct answer)
C. 63 years
D. 90 years
Tina is filling a 45-gallon tub at a rate of 1.5 gallons of water per minute. At this rate, how long will it take to fill the tub?
A. 30.0 minutes (correct answer)
B. 43.5 minutes
C. 46.5 minutes
D. 67.5 minutes
A person who weighs 63 kilograms burns 56 calories in one hour of sleep and burns twice as many calories in one hour of standing. How many calories does a person who weighs 63 kilograms burn in one half-hour of standing?
Solution: In one hour of standing the person will burn 2 � 56 = 112 calories. In one-half hour, the person will burn 112 � 1/2 = 56 calories. The relationship between duration of standing and calories burned is a direct proportion. Solution: In one hour of standing the person will burn
2 � 56 = 112 calories.
In one-half hour, the person will burn
112 � 1/2 = 56 calories.
The relationship between duration of standing and calories burned is a direct proportion.
The table below shows a relationship between x and y.
Which of these equations represents this relationship?A. y = x²
B. y = 2x (correct answer)
C. y = 1/2x
D. y = x – 2
When a diver goes underwater, the weight of the water exerts pressure on the diver. The table below shows how the water pressure on the diver increases as the diver’s depth increases.
a.Based on the table above, what will be the water pressure on a diver at a depth of 60 feet? Show your work or explain how you obtained your answer.
Solution: For every 10-feet increase in depth, the pressure increases by 4.4 psi. Consequently, the pressure at 60 feet will be 22.0 + 4.4 = 26.4 psi.
b.Based on the table above, what will be the water pressure on a diver at a depth of 100 feet? Show your work or explain how you obtained your answer.
Solution: The relationship between depth and pressure is a direct variation, so the pressure at 100 feet will be double the pressure at 50 feet, which is 44.0 psi.
c.Write an equation that describes the relationship between the depth, D, and the pressure, P, based on the pattern shown in the table.
Solution: P = 0.44D
d.Use your equation from part c to determine the depth of the diver, assuming the water pressure on the diver is 46.2 pounds per square inch. Show your work or explain how you obtained your answer.
Solution: Substituting P = 46.2 into the equation from c gives 46.2 = 0.44D. Solving for D yields
, or D = 105. Consequently, the diver experiences 46.2 psi of pressure at a depth of 105 feet.
“Oil: Black Gold” (PDF) from SIMMS Integrated Mathematics: A Modeling Approach Using Technology; Level 1, Volume 2. Simon & Schuster Custom Publishing, 1996. Used with permission.
Teacher Commentary:
NOTE: Questions 6, 7, and 8 from both work samples are discussed below:
I believe question 6 actually gives better insight into the depth of understanding that each student has concerning direct and indirect variation than do questions 7 and 8. When asked about situations totally unrelated to classroom experiences, Haily’s answers in problem 6 demonstrate a clearer comprehension of the two concepts. Her response in problem 8 shows she can perform the necessary steps to obtain a mathematical model for a data set. Although it was interesting that after making the scatterplot in 8, she did not go back to problem 7 and adjust her graph to fit the definition.
On the other hand, Heather was able to express a memorized definition to answer problem 7, and initiate a memorized series of steps to create a mathematical model in problem 8. But she was unable to finish the process in order to write the algebraic equations to fit the data. She also could not apply the concepts to correctly label the situations in problem 6. That indicates to me that she really did not understand the big picture of direct and indirect variations.
Neither of the students discussed the relationship between the independent and dependent variables: both increasing or decreasing (in a direct variation), or as one variable increases, the other decreases (in an inverse variation).
To better assess students’ understanding, changes I plan to incorporate next year include: