Insights Into Algebra 1: Teaching for Learning
Direct and Inverse Variation Lesson Plan 1: Be Direct – Oil Spills on Land
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.
One to two 50-minute periods
Students will be able to:
- Describe the primary characteristics of a direct variation.
- Explain why the relationship between the volume of oil and the area of land it covers represents a direct variation.
- Recognize situations in other contexts that are direct variations (such as the relationship between circumference and diameter).
- Give examples of real-world direct variations besides those studied in class.
- Define direct variation and constant of proportionality.
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000:
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
Teachers will need the following:
- chalkboard and overhead projector
Students will need the following:
- notebook or journal
For each group of four students, you will need:
- eye dropper
- large sheet of paper
- 8 small pieces of toilet paper or a paper towel
- vegetable oil
- overhead transparency sheet (preferably with gridlines)
- overhead pen
Teachers Activities and Assignment
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):
- What is the area of a circle with radius 4 meters? (Answer: 50.27 m², or 16m²)
- A circle has a diameter of 2.1 meters. What is the area of the circle?
(Answer: 3.46 m²)
- A circle covers an area of 15.45 square meters. What is the radius of the circle?
(Answer: 2.22 meters)
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:
- What do the points on the scatterplot represent?
- Pick a point (x, y) from the graph, and describe its meaning in the context of this problem.
- As the volume increased, did the size of the spill increase?
- If the points were connected, what type of graph would result?
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:
- What does the y-intercept mean for your line?
- What does the point represented by the y-intercept mean?
- What would the area of the spill be if 0 drops of oil were spilled?
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:
- Using the average from the last column in your table, what line of best fit did you find?
- How well does this new line represent your data?
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:
- What is the slope of your line of best fit?
- What does the slope represent?
- If the volume of oil is doubled, what happens to the area of the spill?
- If the volume of oil is tripled, what happens to the area of the spill?
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:
- Why is the second equation better (more advantageous) than the first equation when modeling the situation?
(Answer: The second equation shows a proportion between numbers; that is, as one quantity doubles or triples, so does the other. In addition, the second equation contains the origin (0, 0), a necessary condition for a direct proportion.)
- How does the constant of proportionality relate to the oil spill?
(Answer: The oil spill involves direct proportions. For instance, if the number of drops increases five times, the area of the spill should increase five times.)
20. Have each group make predictions based on a larger spill and answer these questions:
- There are approximately 25,000 drops in a liter of oil. What would the area be if a liter of oil were spilled? Have students convert the answer to square meters instead of cm². (There are 100 x 100, or 10,000, square centimeters in a square meter).
- How reasonable does your answer seem?
- How accurate do you think your equation is for predicting the size of an oil spill on land?
- Based on the data that you collected, is it reasonable to extrapolate to 1 liter (25,000 drops)?
(Answer: No, it is not reasonable to extrapolate to 25,000 drops when we only collected data up to 8 drops.)
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).
Related Standardized Test Questions
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.
- Taken from the California High School Exit Examination (Spring 2002):
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
- Taken from the California High School Exit Examination (Spring 2001):
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
- Taken from the Florida Comprehensive Assessment Test (Spring 2001):
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.
- Taken from the Maryland High School Assessment Test (Fall 2002):
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
- Taken from the Massachusetts Comprehensive Assessment, Grade 10 (Fall 2002):
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.
Student Work: Oil Module Unit Test Sample 1
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:
- In problem 6, ask them to explain WHY they think each situation represents a direct or indirect variation.
- Have more detailed requirements for graphs, i.e. scatterplot vs. a connected line graph, and including labels and scales for axes. (For this test I did not take off points for not having a scale on their scatterplots. I was more concerned about the general shape of the graph. One of my goals next year is to incorporate higher expectations across the curriculum, at all grade levels, for complete graphs at all times.)
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.