Learning Math: Patterns, Functions, and Algebra
More Nonlinear Functions Part B: Inverse Proportions (60 minutes)
Session 8, Part B
In This Part
- Splitting a Prize
- Ms. Anwar’s Backyard
- Another Inverse Proportion
Splitting a Prize
Imagine that the teachers in your school decide to play the lottery together. If they win, the prize is $800,000. Problem B1 is a table that shows how much each teacher will get, depending on how many of them contribute to buy the tickets.
Fill in the rest of this table:
Describe a rule that fits the table. Try to find more than one rule. Explain why your rule will work if the table is continued.
Graph the rule that shows how much each teacher will receive. Describe how this graph is different from other functions you’ve seen.
Tip: Make sure you distinguish this function from all the other functions you’ve seen — linear, exponential, quadratic, and cyclic.
What would a value of y = 0 mean in terms of the lottery problem? There is an important difference between this type of function — a decreasing curve — and a decreasing line.
You can find this segment on the session video, approximately 10 minutes and 15 seconds after the Annenberg Media logo.
Functions like this are called inverse proportions. The relationship between the two variables in this function is called inverse variation. Inverse proportions are another example of nonlinear functions.
You can think of inverse proportions in two ways:
- output = some constant / input, or
- input * output = some constant
There are many applications of these kinds of functions. If you’re getting paid a fixed amount of money to do a job, for example, your hourly rate depends on how quickly you complete the work. The shorter the time, the larger the hourly rate.
Ms. Anwar's Backyard
Ms. Anwar is considering renting a house that has a large rectangular backyard. She wants to figure out if there will be room for her children’s play equipment. The owner told her, “The backyard has an area of 2,000 square feet.” Ms. Anwar thought about what he said and tried to imagine what the actual dimensions of the yard might be.
Fill in the table below to show some possibilities for the dimensions of the yard if the area is 2,000 square feet.
|Length||Width||Area = length * width|
Find an equation relating the length (x) and the width (y) in the table above.
Graph the length vs. width in the table above.
As x changes, what happens to y? Try to describe this relationship as clearly as possible. These problems may help you:
- Complete the table below. Round values for y to one decimal place, or use the exact fractional value.
x y Decrease
20 100 — 30 66.7 33.3 40 50 60 70 80 90 100
- As x increases by 10, what happens to y? Does y go up a fixed amount? An increasing amount? A decreasing amount?
- As x doubles, what happens to y? As x triples, what happens to y?
- If x is very small, what can you say about y? What if x is very big?
“Ms. Anwar’s Backyard” problem taken from IMPACT Mathematics Course 3, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p.110. www.glencoe.com/sec/math
Another Inverse Proportion
Consider a new function: (x)(y) = 3. Answer these questions:
Find five negative and five positive x values between -10 and 10. Record them in a table with the corresponding y-values. Be creative! Try some non-integers.
Tip: Values like 1/2 and 1/10, -1/2 and -1/10 are interesting to try
If x = 0, what happens to y? What happens on a calculator if you ask for the value of y = 3 / x when x = 0?
Create a graph of this function. What happens as the graph gets near the y-axis? Will it ever cross the y-axis?
Tip: Make sure this graph includes points where x is positive, and other points where x is negative. See Problem B8 if you’re unsure what will happen if x = 0.
Take It Further
In the equation (x)(y) = 3, what does the 3 represent, graphically?
Tip: Looking back at the Interactive Activity may help.
You’ve seen two kinds of functions that have similar names: proportion (or direct variation) and inverse proportion (or inverse variation). Compare and contrast these two functions. What does the word “inverse” indicate in this case?
Take It Further
The equation (x)(y) = 1 is a special kind of inverse variation. How are x and y related in this equation? What does this relationship have to do with solving equations?
Problems B8-B10 adapted from IMPACT Mathematics Course 3, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p. 111.
The two ways to think of inverse proportions are as input * output = constant and as output = constant / input. Both are useful for different situations.
Groups: Work through Problems B1-B3, and then discuss the terms “inverse proportion” and “inverse variation.” Move on to Problems B4-B12. Discuss these problems as a group, especially how inverse proportions are different from other functions we’ve seen.
Unlike other functions we’ve seen, increasing the input (in magnitude) decreases the output by the same factor (in magnitude). As long as the numerator is positive, as x gets bigger, y gets smaller. This is a striking difference from other functions.
Groups: Discuss Problem B6, particularly what’s going on in the table. Some observations that may arise are, “As x increases, y decreases, but by less and less each time.” Or, “As y decreases, so does the amount by which it decreases.” Or, “As y gets smaller, it does so at a slower and slower rate.”
It’s also important to think about the graph and whether it will cross either of the axes. Going back to the contexts of the problems (sharing money among several people, for example) should help make sense of the asymptotic nature of the graphs.
Groups: Take 10 minutes to jot down answers to Problems B11 and B12, then share answers.
Take time to think about the how the word “reciprocal” describes the relationship (x) (y) = 1. There are some important and often confused ideas here. For example, 1 has a reciprocal (itself). The fraction 1/5 also has a reciprocal: 5. Too often, students and teachers think of whole numbers as the “regular” numbers and the “reciprocals” as 1/n. Reciprocals can be used in solving equations. When you solve 3x = 9 by “dividing each side by 3,” another description is “multiplying each side by the reciprocal of 3.” This method generalizes to other systems (equations involving matrices or equations involving modular systems, which we will see in Session 9), whereas the “divide by 3” method is specific to the real numbers.
Groups: End this part by adding inverse proportions to the list of functions.
Here is the completed table:
The last two entries will vary.
One rule is that the product of the input and the output is always 800,000. If M = the money each teacher receives, and T = the number of teachers, then
(M) x (T) = 800,000. This is because the total prize of 800,000 is split evenly among the teachers. This also leads to a second rule: M = 800,000 / T, because the money received by each teacher is 800,000 divided by how many teachers split the prize. Additionally, T = 800,000 / M.
The amount of money decreases as the number of teachers increases, but it is not an exponential decay because the ratio between consecutive outputs is not constant. Looking at differences between outputs guarantees that the graph is neither linear nor quadratic. The graph is not cyclic, because there is no point where the outputs begin to repeat.
Here is the completed table:
The last four entries will vary.
One equation is x * y = 2,000. Note that Area = length * width is always constant at 2,000. Other possible equations are y = 2,000 / x and x = 2,000 / y.
Overall, if x is multiplied by a number, y is divided by the same number. If x is divided by a number, y is multiplied by the same number.
- Here is the completed table. All y-values are rounded to one decimal place.
x y Decrease in y 20 100 — 30 66.7 33.3 40 50 16.7 50 40 10 60 33.3 6.7 70 28.6 4.7 80 25 3.6 90 22.2 2.8 100 20 2.2
- As x increases by 10, y continuously decreases, but the rate of decrease lessens as x grows.
- If x doubles, y is cut in half. If x triples, y is divided by three.
- If x is very small, y must be very large, since the product of x and y is always the same. By the same argument, if x is very large, then y must be very small.
- Here is the completed table. All y-values are rounded to one decimal place.
Let’s try it:
If x = 0, y is undefined, which means that no value of y will make zero times y equal to 3. Using a calculator, 3 / 0 will fail to return a number; the calculator will give an error message.
The graph will not cross the y-axis, because if it did, it would mean that some y-value would be assigned for x = 0 for the graph, and, as explained in the solution to Problem B9, there is no such value. Another way to look at it is to examine the behavior of the graph near x = 0. If x is a little more than zero, y is a very large, positive number, but if x is a little less than zero, y is a very large, negative number. These portions of the graph do not meet.
The 3 represents the area of a rectangle drawn with (0, 0) as one corner and any point on the graph as the opposite corner. Compare this to the Interactive Activity, where there were rectangles of area equaling 2,000 feet — length times width always equaled 2,000. Here, x is the length, and y is the width.
In a direct variation function, an increase in x creates a proportional increase in y; if x is multiplied by 5, y is also multiplied by 5. But in inverse variation, the opposite is true: If x is multiplied by 5, y is divided by 5. The word “inverse” refers to the inverse operations of multiplication and division.
In this particular equation, x is the reciprocal of y, because x and y multiply together to make 1. The reciprocal is used to solve equations like 5n = 16 — multiplying both sides of the equation by the reciprocal of 5 produces two numbers (5 and 1/5) which, when multiplied, make 1. So, multiplying by 1/5 will remove the 5 from the left side, leaving variable n by itself. This is the key to solving equations by backtracking.
Session 1 Algebraic Thinking
In this initial session, we will explore algebraic thinking first by developing a definition of what it means to think algebraically, then by using algebraic thinking skills to make sense of different situations.
Session 2 Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is.
Session 3 Functions and Algorithms
In Session 1, we looked at patterns in pictures, charts, and graphs to determine how different quantities are related. In Session 2, we used patterns and variables to describe relationships in tables and in situations like toothpicks and triangles. This session extends the exploration of relationships to include the concepts of algorithm and function. Note1
Session 4 Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.
Session 5 Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.
Session 6 Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.
Session 7 Nonlinear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations.
Session 8 More Nonlinear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.
Session 9 Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the K-2 grade band.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 3-5 grade band.
Session 12 Classroom Case Studies, Grades 6-8
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 6-8 grade band.