Learning Math: Patterns, Functions, and Algebra
Nonlinear Functions Homework
Session 7, Homework
You’ve already seen examples of exponential growth. When you have an exponential function where the constant multiple between outputs is less than 1, you have a decreasing function. Instead of exponential growth, you have exponential decay. Here’s an example:
The brightness of a light can be described with a unit called a lumen. A certain type of mirror reflects 3/5 of the light that hits it. Suppose a light of 2,000 lumens is shined on a series of several mirrors.
Complete the table to indicate how much light would be reflected by each of the first three mirrors. (Mirror 0 represents the original light.)
One mirror in the series reflects about 12 lumens of light. What mirror number is it? How did you find your answer?
Which mirror number reflects about 1/10 the original amount of light? Does it depend on the starting amount of light?
The graph below shows the amount of light reflected by another series of mirrors. The amount of light reflected by this type of mirror is different from the amount reflected by the other mirrors you investigated. The intensity of the light being shone on the first mirror is also different.
- What is the intensity of the light being shone on the first mirror?
- How much light was reflected by the first mirror?
- What fraction of light does the first mirror reflect?
- Do the other mirrors reflect the same fraction of light? That is, does this graph show exponential decay?
Suppose you have the function y = (1/5)x.
- As the value of the inputs increases, what happens to the outputs?
- Is this an exponential growth function, or an exponential decay function?
- Will you ever get 0 as an output? Explain your answer.
- Will you ever get negative numbers as outputs? Explain your answer.
- How does your answer to (c) relate to Problems H1-H4?
Homework problems taken from IMPACT Mathematics Course 3, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p.175-176. www.glencoe.com/sec/math
Here is the completed table. One formula is L = 2,000(0.6)m.
The best way to do this, without using more advanced math such as logarithms, is to continue following the table. This is particularly easy with a spreadsheet. The 10th mirror will reflect about 12 lumens.
We’d be looking for the mirror that reflects closest to 200 lumens, which is the 5th mirror (156 lumens). The 4th mirror reflects 259 lumens; this does not depend on the starting amount of light. This means that the 6th mirror reflects about one-tenth the light of the first mirror, the 7th mirror reflects one-tenth the light of the 2nd mirror, and so on.
- The initial intensity is 1,000 lumens.
- The 1st mirror reflects 700 lumens.
- This fraction is 700 / 1000 = 7/10, or 70 percent.
- Yes, the 2nd mirror appears to reflect about 7/10 of 700 lumens, which is 490 lumens. The 3rd mirror reflects 7/10 of 490 lumens, which is 343 lumens. This pattern appears to continue indefinitely at a constant ratio, so it is an exponential decay situation.
- The outputs keep getting smaller, but remain positive, because at each stage a positive number is divided by 5.
- This is an exponential decay function, because successive outputs are getting smaller, and the base is between 0 and 1.
- Zero will never be an output, even though the outputs will become increasingly close to zero. This happens because the numerator remains 1, no matter what the value of x is, while the denominator becomes increasingly larger as x increases.
- Because two positive numbers are used in the division, a negative number can never result.
- In Problems H1-H4, this implies that even after 100 or more mirrors, some light will still be reflected, although the amount of light reflected will become increasingly small.
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.