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Do I Have to Mow the Whole Thing?

  • Lesson
9-12
1
Algebra
Zoe Silver
Location: unknown

This lesson offers examples of inverse variation. Students collect data and generate graphs before finding specific equations for inverse variation relationships and examining their graphs.

By working through a few examples, students gain a concept of inverse variation. The formal mathematics is introduced in the second activity, although some students may see the relationship and write an equation after the first activity.

Activity: Do I Have to Mow the Whole Thing?  

pdficon Activity Sheet - Do I Have to Mow the Whole Thing? 


In this activity, students will calculate dimensions for a garden of constant area and are introduced to the idea of inverse variation. Students should use both large and small values for the length and width, and find enough values to fill in each pairing in the table.

After obtaining a graph, students are asked to look more carefully at the graph and table for any relationship between the values. The product of the 2 numbers is always 24, so most students will explain this in words and, eventually, in symbols: xy=24. Encourage students to divide both sides by x to get the equation y=24/x. Point out that we'd like to isolate y so that we can eventually use the calculator to see the graph.

The second scenario (on the second page of the activity sheet), involving days and typists, yields the equation xy=36 or y=36/x.

After the examples of measuring with non-tradional units, calculating possible dimensions of the yard and finding typists, students are ready for a formal introduction to Inverse Proportion. undefined  

Notes for Students:  

Inverse Variation: y decreases as x increases or y increases as x decreases.

The equation looks like xy=k or y=k/x where k is the constant of variation.

** Ask students to identify the value of k in each scenario on the Activity Sheet.

Examples:

  • Find the dimensions of a rectangle when the area stays constant.
  • Suppose you have $30 to spend on new CD's. How many CDs can you buy at $5 each? $3 each? $6 each? $10 each? other prices?

Test for Inverse Variation: Multiply the each pair x and y values together and check for the same result. If the product is the same for all pairs in the list, then you have an inverse proportion and the product is your constant of variation (k).

From the tables below, determine which are examples of inverse variation.

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Activity: How Many Pencils Tall? 

pdficon Activity Sheet - Measuring with Pencils 

Students should pair up for this activity, to verify the measurements and share the measuring resources. They measure the length of some large object using non-standard units of measure rather than centimeters or inches. The longer the unit measure, the fewer lengths of the unit measure are needed to measure the length of the large object. An inverse variation is generated from this relationship.

2578 pencil clips
  1. Students choose an object whose height or length they will investigate. Suggestions include the height of a desk or doorway, the height of a window off the ground, or even someone's height.
  2. Students find 4 objects to use as measuring units for the activity.
    • The shortest measuring item should be about an inch long. Reasonable items include a pencap, an eraser or a paperclip.
    • The longest item should be a 1 to 2 feet long. Reasonable items are a shoe, a large book or the measurement from someone's fingertip to their elbow.
    • Students need two or three other measuring items of varying length that are between the longest and shortest measuring items. A pen, someone's hand, a small shoe or a water bottle are examples of other items that can be used to measure.
     
  3. Students measure with the smallest measuring unit first, and complete the first row of the table on the activity sheet.
  4. Clarification on completing the rest of the table may be needed. Each measuring unit needs to be measured relative to the shortest measuring item. That relative length should be recorded in the Column B. For example, if the first (shortest) measuring unit is a paper clip and the second shortest measuring unit is a pencil that is 4.5 times the length of the paper clip, then the second row may look as follows:

    2578 table sample 

  5. After completing all measurements, students graph the data points they have collected.

Ask students to explain if this relationship is an example of an inverse variation and how they can be sure.

Assessments 

  1. Use the following information to find the constant of proportionality. Write an equation that models the information, and graph the relation.
    Bart and Ernest have to drive 100 miles to get to their uncle's hunting cabin.

    a) How many hours will it take them if they

    • drive there using the back roads at 50 mi/hr? Bike there at 20 mi/hr?
    • hike there at 4 mi/hr?
    • drive there on the interstate at 75 mi/hr?
    b)How fast do they need to travel if they want to get there in 1 hour?

    c) Suppose they plan to take 48 hours to get there. What is the average speed they need to maintain if they leave now?

  2. Explain the difficulties graphing xy=25 on a graphing calculator? How would you have to manipulate the equation?
  3. Tell what characteristics you look for in a table, a graph, and an equation to determine if a relation is an example of inverse variation.
 

Questions for Students
1. What happens to the y-values as the x-values get very small (that is when the positive x-values get very close to zero)?
2. What would the y-values be if x was zero?
(This is a great time to get students to think about the concept of approaching zero, even if asymptotes aren't introduced in this lesson.)
3. Why do you think this relationship is called an inverse variation?
4. Identify another example of an inverse variation.
5. The product of 2 positive numbers is positive. The product of 2 negative numbers is also positive. What does the graph of xy=36 look like?
6. Suppose the constant of variation (the product) is negative, such as: xy=—75. What does the graph of xy=—75 look like?

Teacher Reflection 

  • How much were students able to work through without your assistance?
  • How did students respond to the idea of an asymptote? Do they believe that the graph will never touch either axis?

 

Learning Objectives

By the end of this lesson, students will:

  • Use estimates to measure the height of an object
  • Visualize two-dimensional figures and calculate their area
  • Sketch graphs and determine the equations for inverse variation relationships