Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Monthly Update sign up
Mailing List signup
Insights Into Algebra 1 - Teaching For Learning
algebra home workshop 1 workshop 2 workshop 3 workshop 4 workshop 5 workshop 6 workshop 7 workshop 8
Topic Overview Lesson Plans Student Work Teaching Strategies Resources
Workshop 7 Direct and Inverse Variation Lesson Plans
Lesson Plans:


Lesson Plan 1: Be Direct - Oil Spills on Land

Lesson Plan 2: Very Varied - Inverse Variation
Download the Workshop 7 Guide

Tool Box
Graphing Calculator
NCTM Standards

Lesson Plan 1: Be Direct - Oil Spills on Land

Overview Procedures For Teachers Related Standardized Test Questions Materials

Kids learning

Teachers will need the following:
  • chalkboard and overhead projector
Students will need the following:
  • notebook or journal
  • pens/pencils
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
  • ruler
  • vegetable oil
  • overhead transparency sheet (preferably with gridlines)
  • overhead pen

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):
  • 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)
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:

Volume (drops) Diameter (cm) Radius (cm) Area (cm²)

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:

y = 5.5x + 1
x y
1   6.5
2 12
3 17.5

y = 5.75x
x y
1   5.75
2 11.5
3 17.25
  • 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.)
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).

back to top
Next: Related Standardized Test Questions
Site MapAbout This Workshop

© Annenberg Foundation 2017. All rights reserved. Legal Policy