Learning Math: Patterns, Functions, and Algebra
More Nonlinear Functions Part C: Different Functions (35 minutes)
Session 8, Part C
We’ve seen many different kinds of functions in the past few sessions. We’ve examined characteristics of functions, looked at their graphs, and explored situations where they arise. In general, people tend to think in terms of linear functions, trying to fit given data into lines. What we’ve seen, however, is that there are many other kinds of functions. Here are examples of their equations and graphs:
y = ax + b
|Exponential Growth Function
y = bx, where b > 1
|Exponential Decay Function
y = bx, where b < 1
y = ax2 + bx + c
y = k / x, or xy = k
It’s important to be familiar with various kinds of functions. Many different functions might fit just a few pieces of data. Here’s an example to show how this might happen.
Fill in the missing entries according to the rules given above.
Here’s a picture of the three functions given above, showing how they share the same two points. A cyclic function also shares those two points.
Which function corresponds to which number?
Take It Further: Problem C3
How many different functions could fit these two data points, (1,2) and (2,4)? Explain your answer. Can you describe one other function that fits these two data points, either with an equation or through some other way?
Tip: Because there is only one line between two points, any different function would have to be nonlinear.
The purpose of this section is to pull together ideas about functions from the past several sessions and to see how many different functions could fit a small set of data. Start by reviewing the different kinds of functions we’ve seen, both linear and nonlinear.
Some of the graphs look similar to each other, at least in places (exponential growth and the parabola, for example). Think about differences in these functions and how the graphs would differ from each other on a larger scale
Groups: Work on Problems C1-C3.
Groups: Share answers to Problem C3, and see how many different functions — and how many kinds of functions — you’ve come up with.
Any linear function that answers the question in Problem C3 must be equivalent to the one given in the table. (For example, the expression 2 * (input + 1) – 2 works, but gives the same output everywhere as 2 * input, so it is the same function.) Through any two points (in this case, two input/output pairs) there is exactly one line, so any two linear functions that match those points must be equivalent.
Groups: Discuss the concepts outlined above.
Consider what quadratic functions could answer Problem C3. These are harder to come up with, but in fact there are an infinite number of quadratic functions that fit the two inputs given. The reason for this is subtle: You need two points to find exactly one line. You need three points to find exactly one parabola (quadratic function). Since there are only two points given, you can pick any third point. So you can pick an output for 3 and find a quadratic that works. Picking a different output for 3 will give a different quadratic. Here are just three others that fit the points given:
- 1 + n (n +1) / 2 (in other words, the outputs are each one more than the triangular number for that input)
- 5n – n2 – 2
- 2n2 – 4n + 4
Think about whether there are any other kinds of functions that fit the data. Note that a “height of tide” type function like that in Part A could also fit, and is a cyclic function. Here’s an example:
Groups: Consider making a poster of the four graphs together. It’s easy to see that you could sketch another function through those same two points that doesn’t match the other four functions given. In fact, you could sketch infinitely many different functions through those same two points. If you have trouble coming up with algebraic representations of the functions, describe them with sketches, words, or input/output tables.
Note that this table was in no way “rigged.” For any finite table, even if it had thousands of entries, there are an infinite number of functions that fit it. Usually, there will be one “simplest” function. But sometimes it’s not clear what’s simple: Is a quadratic function “simpler” than an exponential function? Is a linear function “simpler” than a cyclic function?
Here is the completed table:
An infinite number of functions share these two data points. There are too many rules that turn the input of 1 into the output of 2 and the input of 2 into the output of 4. The rule could be as complicated as Mr. Lewis’s rule from Session 2, or it could be as simple as the house numbers on the left side of your street.
Some examples: y = 3x-1 + 1; y = n (n + 1) / 2 + 1; y = the date number, every other day, starting with January 2.
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.