Insights Into Algebra 1: Teaching for Learning
Direct and Inverse Variation Lesson Plan 2: Very Varied – Inverse Variation
This lesson teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers’ bases. This lesson plan is based on the activity used by teacher Peggy Lynn in Part II of the Workshop 7 video.
One to two 50-minute periods
Students will be able to:
- Identify an inverse variation by the shape of its graph.
- Represent an inverse variation algebraically.
- Describe the primary characteristics of an inverse variation.
- Compare and contrast an inverse variation with a direct variation.
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
- cylindrical jars (about eight to 12) of various sizes, each jar identified by a number for easy reference
Students will need the following:
- notebook or journal
For each group of four students, you will need:
- overhead transparency sheet (preferably with gridlines)
- overhead pen
- water (at least 200 ml)
Teachers Activities and Assignment
1. Give students the two sets of data below to review. Have them determine, using whatever method they choose, whether each of the data sets is a direct variation.
- Which of these represents a direct variation?
(Answer: Only set A, because it has a constant rate of change. Although set B passes through the origin, it does not have a constant rate of change.)
- Does the data in set A pass through the origin?
- What is the constant of proportionality for set A?
(Answer: The slope of the line given by , or 2.5.)
1. Have one student read aloud from the bottom of page 168 in the SIMMS handout.
As soon as a quantity of oil is spilled, it starts to spread. If not contained, the resulting slick can cover a very large area. As the oil continues to spread, the depth of the slick decreases. In the following exploration, you investigate the relationship between the depth of a spill and the area it covers.
2. Explain to students that they will be conducting an exploration similar to the one for direct variation. A set amount of water (200 ml) will be placed into cylinders of various sizes. Each group must collect data for all of the cylinders the teacher provides.
3. All students should create a chart in their notebooks, as follows:
4. Inform students that they will measure the diameter of each jar to the nearest tenth of a centimeter and will use that measurement to calculate the area of the base of the jar. Be sure, however, to emphasize that students must measure the inside diameter of the jar, not the outside diameter. That is, they need to ignore the thickness of the jar.
5. Students should calculate the volume of the water in cubic centimeters, based on the area of the jar’s base and the depth of the water.
6. Explain that each group will be required to create a scatterplot that gives the area of the base (in cm²) along the x-intercept and the depth of the water (in cm) along the y-axis. Distribute a transparency sheet and an overhead pen to each group, and tell them that they may be asked to explain their scatterplot to the class.
7. Using each of the jars, have students complete their charts and create scatterplots.
8. As students work, circulate among the groups. Offer assistance as necessary, but do not provide too much information.
9. Once the groups have completed their scatterplots, call on two or three to present them to the class. The students should answer these questions:
- Pick a point (x, y). What is its meaning in the context of this problem?
- Is the relationship between area and depth linear?
- Based on the data you collected, what happens when the area of the base increases? What happens when the area of the base decreases?
10. Involve the rest of the class by asking the following questions.
- Does the relationship in your graph appear to be a direct variation? (Answer: No.)
- What should the number in last column be, approximately?
(Answer: Because 200 ml was added to each cylinder.)
- But this column for volume is measured in cm³, not ml. Why does it still work?
(Answer: Because in the metric system, 1 ml = 1 cm³.)
- Why are the numbers not exactly 200, but only close to it?
(Answer: It’s difficult to measure the depth of the water, the diameter of the cylinder, etc., so there may be a slight margin of error.)
11. Allow groups a few minutes to generate an equation that describes the graph. Encourage students to think about how the graph was generated: the volume of the water is equal to the base of the jar times the depth of the water; that is, V = Bh, or
200 = xy. To graph this function, it should be rewritten in the form y = 200/x.
12. Point out that this is not a direct variation; it’s not linear, and it does not pass through the origin. Ask students what they think this relationship might be called instead. Explain to them that it is known as an “indirect proportion,” or “inverse variation.”
13. Review by comparing and contrasting direct and inverse variations:
- The graph of a direct variation contains the origin (0, 0) and is linear; that is, it has a constant rate of change. It can be written in the form y = mx, where m is the slope or the constant of proportionality.
- The graph of an indirect variation is a curve with no x-intercept or y-intercept; that is, it never crosses the axes. It does not contain the origin, and it can be written in the form , where k is the constant of proportionality.
- Point out that is constant in a direct proportion; direct proportion implies that the quotient of variables is constant.
- On the other hand, xy is constant in an indirect proportion; indirect proportion implies that the product of variables is constant.
14.Ask the class for examples of direct and indirect variations. For each example, have students attempt to explain the corresponding graph. For indirect variation, students may have difficulty generating an example, so it may be necessary to have an example of your own prepared.
Assign several exercises for independent practice and homework. You may assign any of the problems from 3.1 through 3.9 (pages 170-174) in the SIMMS textbook.
Related Standardized Test Questions
The questions below dealing with inverse variation have been selected from state 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 conceptuunderstanding needed to succeed on these and other state assessment questions.
- Taken from New York Regents High School Examination, August 2002:
To balance a seesaw, the distance, in feet, a person is from the fulcrum is inversely proportional to the person’s weight, in pounds. Bill, who weighs 150 pounds, is sitting 4 feet away from the fulcrum. If Dan weighs 120 pounds, how far from the fulcrum should he sit to balance the seesaw?
A. 4.5 ft
B. 3.5 ft
C. 3 ft
D. 5 ft (correct answer)
- Taken from the Massachusetts Comprehensive Assessment, Grade 10 (Spring 2002):
Which of the following equations does not represent a linear relationship?
A. xy = 12 (correct answer)
B. x + y = 12
C. y = 12x
D. x – y = 12
“Oil: Black Gold” from SIMMS Integrated Mathematics: A Modeling Approach Using Technology; Level 1, Volume 2. Simon & Schuster Custom Publishing, 1996.
Student Work: Factoring Assignment
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.