Learning Math: Patterns, Functions, and Algebra
More Nonlinear Functions Homework
Session 8, Homework
We said previously that algebra has become very much concerned with operations. So far, the only operations we’ve used are the ones from arithmetic. Let’s take a quick look at another kind of operation that often shows up in algebra. Consider the operation that divides a whole number by 3 and hands you back the remainder. This is usually called the “mod 3” operation.
Example: 17 divided by 3 is 5 with a remainder of 2, so we say 17 mod 3 = 2, or 17 = 2 (mod 3).
If the input is 5, what is the output?
If the input is 12, what is the output?
If the input is 2, what is the output?
Now try some undoing:
- Describe all the numbers that produce an output of 1.
- What is the “pullback” of 2? (The pullback of an output is the collection of inputs that produce it.)
- What numbers produce an output of zero?
- How many possible outputs are there for this function? What are they?
Make an input/output table for this function. What kind of function is it?
Take It Further
Make a list of all the numbers that leave a remainder of 3 when divided by 5 and a remainder of 1 when divided by 3, then give one rule that would find them all.
Take It Further
If my age is divided by 3, the remainder is 2. If my age is divided by 5, the remainder is also 2. If my age is divided by 7, the remainder is 5. How old am I?
Take It Further
Here’s a table showing the first two outputs for the linear function y = 3x – 2. Come up with at least two other functions that match these outputs, and complete the table using your functions.
The output will be 2. Five goes into 3 once, and the remainder is 2.
The output will be 0. Twelve goes into 3 four times, with no remainder.
The output will be 2. Two doesn’t go into 3 at all, so the remainder is still 2.
- There are an infinite number of answers, beginning with 1. The next is 4, then 7, then 10. Any number which is one more than a multiple of 3 produces an output of 1.
- The “pullback” is all numbers which are two more than a multiple of 3: 2, 5, 8, 11, 14, … .
- The numbers producing an output of zero will be all numbers which are zero more than a multiple of 3, or in other words, the multiples of 3: 0, 3, 6, 9, 12, … .
- The only possible outputs are 0, 1, and 2. You might have noticed in H4(a)-H4(c) that every whole number falls into one of the three categories. Another explanation is that the only possible remainders when dividing by 3 are 0, 1, and 2.
It is a cyclic function. Here is a table:
This can be done by looking through the table of numbers which are one more than a multiple of 3 and looking for a pattern of those numbers which are also three more than a multiple of 5. The list consists of: 13, 28, 43, 58, 73, 88, …. The first of these numbers occurs with 13, and they repeat every 15 numbers. One rule to find all the numbers would be that they are all 13 more than a multiple of 15, or all numbers whose output is 13 in the “mod 15” operation.
This is harder to find, but the first two conditions mean that the age is two more than a multiple of 15 — those numbers are 2, 17, 32, 47, 62, 77, … . Among those numbers, the only one which is five more than a multiple of 7 is 47. There are other possible answers, but the others are greater than 100 (the first of these is 152). Overall, the condition is that the age must be equal to 47 (mod 105).
Here is one possibility for a completed table:
Of course, there are lots of other possibilities!
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.