Learning Math: Patterns, Functions, and Algebra
Functions and Algorithms Part E: Other Kinds of Functions (45 minutes)
Session 3, Part E
In This Part
- Functions and Non-Functions
- More Functions
- A Geometric Function
Functions and Non-Functions
So far you have been thinking about functions as algorithms or machines. They take an input — in the cases you have seen, a number — and give an output.
A function is really any relationship between an input variable and an output variable in which there is exactly one output for each input. Not all functions have to work on numbers, nor do functions need to follow a computational algorithm. Below are some examples of functions and non-functions. Read through them, then answer Problems E1-E4.
The following relationships are functions.
Input: an integer
Output: classification of the input as even or odd
Input: a person’s Social Security number
Output: that person’s birth date
Input: the name of a state
Output: that state’s capital
Input: the side length of a square
Output: the area of that square
Input: a word
Output: the first letter of that word
For each function described above, make a table of 5 or 6 input/output pairs. Explain why for every possible input there is only one possible output.
In any of your tables, do you have repeated outputs? That is, do you have two different inputs that give the same output?
The following relationships are not functions.
Input: a number
Output: some number less than the input
Input: a whole number
Output: a factor of the input
Input: a person
Output: the name of that person’s grandparent
Input: a city name
Output: the state in which that city can be found
Input: the side length of a rectangle
Output: the area of that rectangle
Input: a word
Output: that word with the letters rearranged
For each relationship described above, make a table of 5 or 6 input/output pairs. Explain why for some inputs there may be more than one possible output.
Tip: Be sure to generate pairs of inputs and outputs that show that the relationship is not a function. What property would those pairs have?
Come up with three more examples of relationships that are functions, and three examples of relationships that are not functions. For each relationship, explain why it is or is not a function.
Problems in Part E taken from IMPACT Mathematics Course 3, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p. 489. www.glencoe.com/sec/math
The next function we will explore is called the “Prime?” function. Most of you will remember that a prime number is a whole number that has only itself and one as factors. A few examples of prime numbers are 7, 13, and 29. Can you come up with some other examples?
Tip: The number 6 is not a prime number, because it has 2 and 3 as factors. The number 11 is prime, because its only factors are 1 and itself. A prime number must have exactly two factors — no more, no less.
The “Prime?” function takes positive whole numbers as inputs and produces the outputs yes and no — yes if the input is a prime, and no if the input is not a prime. Use what you know about functions and prime numbers to answer Problems E5-E11.
- If the input is 3, what is the output?
- If the input is 2, what is the output?
- If the input is 100, what is the output?
- If the input is 1, what is the output?
Tip: Remember, the output is either yes or no.
If the output of the “Prime?” function is yes, what could the input have been?
Tip: How many answers are there?
Explain why “Prime?” is a function.
If possible, describe a function that would undo the “Prime?” function. That is, if you put an input into the “Prime?” function and then put the output into your new function, you get back your original input.
Tip: Note the relationship between Problems E6 and E8.
The “3” function takes real numbers as inputs and always outputs the number 3.
- If the input is 17, what is the output?
- If the input is -2, what is the output?
- If the input is 1.5, what is the output?
Tip: The answers to all the parts of Problem E9 are all the same number.
If the output is 3, what could the input have been?
Explain why “3” is a function.
If possible, describe a function that would undo the “3” function. That is, if you put an input into the “3” function and then put the output into your new function, you get back your original input.
Tip: Note the relationship between Problems E10 and E12.
A Geometric Function
Sometimes, functions can be based on an algorithm but still not use numbers as inputs. Here’s an algorithm; let’s call it Algorithm M:
- Start with a polygon
- Find the midpoint of each side of the polygon
- Connect each midpoint to the two midpoints on either side of it
Here’s what the algorithm does to a pentagon:
Try Algorithm M on three different triangles. Describe in words how the output is related to the input.
Tip: Be sure to select triangles that are different in a significant way: acute, obtuse, scalene, isosceles.
How does any new triangle created by Algorithm M relate to the original, in size and in shape?
Try Algorithm M on several different quadrilaterals. Describe anything you notice about the outputs.
Tip: As with Problem E13, select quadrilaterals that are different in a significant way. You might also concentrate on a specific type of special quadrilateral to determine if Algorithm M does something similar to all quadrilaterals of that type.
Does Algorithm M describe a function? Explain how you know.
The final part of this session introduces a more general notation of function, rather than just algorithmic functions with numeric inputs and outputs.
Groups: Work in pairs to describe what you think a function is. Some people may recall struggling with learning or teaching about functions using diagrams like these:
Read the definition of a function and take a look at the examples in the course text.
Groups: Work on Problems E1-E4 in small groups, then as a whole group. Discuss the other examples of functions and non-functions before moving on.
Read about the “Prime?” function and review the definition of a prime. Think about some examples of primes and non-primes and how you could test to see if a number is prime if you aren’t sure.
Work on Problems E5-E8. These problems address common confusion about both prime numbers and functions.
Groups: Summarize these problems in a discussion as everyone completes their work.
Here are some points to consider:
- 2 is a prime number. It is the only even prime.
- 1 is not a prime. This is a convention. The number 1 fits the definition of prime we have given, since it is only divisible by itself — one — and one. However, an important theorem in mathematics, called the fundamental theorem of arithmetic, says that every integer greater than one is either prime or can be expressed as a unique product of prime numbers. If 1 is considered a prime, this would no longer be the case. Consider: 10 = 2 x 5. But 10 = 1 x 2 x 5. But 10 = 1 x1 x 1 x 1 x 1 x 2 x 5. The fundamental theorem of arithmetic is essential for proving many mathematical results, so it would never do to allow 1 to be a prime!
- Two inputs to a function may give the same output. In this case, many numbers produce the output “yes,” and many will produce the output “no.”
- Not every function can be undone. In this case, if the output is “yes,” for example, there’s no way of knowing what the input was. (You may want to discuss how this is related to the point above.)
Read about the “3” function and work through Problems E9-E12. These problems may reinforce many of the points in Note 8.
Look at how Algorithm M works by going through the steps with a pentagon. After finishing the drawing, consider if there was any other way you could have followed the directions. For example, you could connect the midpoints in a different order. No matter how you connect the midpoints, however, the output will be the same. Once this is clear, work on Problems E13-E16.
Groups: If there is time, compare results of this geometric algorithm.
There are several surprising things that some people may notice:
- No matter what shape triangle you start with, you end up with four identical triangles inside your original triangle. Three are oriented the same way as the original triangle, and one is upside down.
- The four triangles are all similar to the original. For example, if you connect the midpoints of a right triangle, you will end up with right triangles inside.
- The areas of each of the triangles are 1/4 the area of the original (since there are 4 of them, and they are identical).
- The inside figure of the quadrilateral is a parallelogram. That is, opposite sides are parallel. It doesn’t necessarily resemble the outer figure at all.
- The inside figure of a quadrilateral contains half the original area (this may be more difficult to see).
In each case there are clear reasons that there can only be one answer. For example, a state can have only one capital city. A word can only have one first letter.
|Integer||Odd or Even|
Sure, but not always. The odd-or-even, date of birth, and letter functions have the possibility of matching outputs.
For certain (not necessarily all!) inputs, there can be more than one correct output. Note how different this is from Algorithms A and B.
|City Name||State Name|
|New York||New York|
|toilets||T. S. Eliot|
Other functions: a circle’s circumference is a function of its radius; the average temperature is a function of the time of year; a TV program’s rating is a function of the number of people watching the show. For each function, there can only be one output for a given input, while a non-function may have more than one output for the same input. For example, people of more than one age can wear size 11 shoes.
a. The output is yes, 3 is a prime number.
b. The output is yes, 2 is a prime number.
c. No, 100 is not a prime (it has lots of factors).
d. No, 1 is not a prime (it needs to have exactly two factors).
It could be any prime number: 2, 3, 5, 7, 11, 13, 17, 19, …
It’s a function because there is exactly one output. The answer is always “yes” or “no,” never both.
It’s a function because there is exactly one output. The answer is always “yes” or “no,” never both.
a. The output is 3.
b. The output is 3.
c. The output is still 3.
It could be any number at all. Since the output is always 3, telling us that the output is 3 doesn’t give any new information. This is the same situation as Problem D5.
There is exactly one value for the output. It’s always 3, but that doesn’t keep it from being a function.
No such function exists.
The output is a triangle whose sides are 1/2 the sides of the original and parallel to the original sides.
All the sides are half as long, and the new triangle’s area is one-fourth of the original.
All the formed quadrilaterals are parallelograms.
Yes, because there is exactly one output polygon for any starting polygon.
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.