Learning Math: Patterns, Functions, and Algebra
More Nonlinear Functions Part A: Cyclic Functions (50 minutes)
Session 8, Part A
In This Part
The following situation describes a new kind of function:
The ocean has high tides and low tides. The tide comes in for six hours (ending at “high tide”) and then goes out for six hours (ending at “low tide”). This is repeated twice in a day. An approximate rule to describe the motion of the tide is this:
From low to high tide:
In the first hour, 1/12 of the tide comes in.
In the second hour, 2/12 of the tide comes in.
In the third hour, 3/12 of the tide comes in.
In the fourth hour, 3/12 of the tide comes in.
In the fifth hour, 2/12 of the tide comes in.
In the sixth hour, 1/12 of the tide comes in.
From high to low tide:
In the first hour, 1/12 of the tide goes out.
In the second hour, 2/12 of the tide goes out.
In the third hour, 3/12 of the tide goes out.
In the fourth hour, 3/12 of the tide goes out.
In the fifth hour, 2/12 of the tide goes out.
In the sixth hour, 1/12 of the tide goes out.
Let’s say the height at low tide is zero and at high tide is 24 feet.
Fill out this table, showing the height of the tide as it comes in and rolls out.
Tip: Remember that full tide is 24 feet while calculating the tide for each hour.
Make a graph of the hours after low tide vs. the height of the tide. Connect the points with a smooth curve.
Extend your graph to show a full day (24 hours) of tides.
Tip: Think carefully about what would happen during the second 12 hours.
Describe how this graph is different from graphs of linear, exponential, and quadratic functions you’ve seen.
Functions like the one you just graphed here are called cyclic functions, also known as repeating or periodic functions. These functions are characterized by outputs that repeat in a cycle. Cyclic functions are important in astronomy (they’re used to describe the motion of the planets), engineering, and many other fields.
Two important characteristics of cyclic functions are amplitude and period. The amplitude measures the height of the graph. It’s defined this way:
amplitude = (highest point – lowest point) / 2
The period of a cyclic function is how long it takes to complete a cycle.
What is the period of the graph in Problem A3?
In this video segment, the cyclic function presented in Part A is graphed. The class then discusses the definitions of amplitude and period. Watch this segment after you’ve worked on Problems A1-A5. If you get stuck on the problems, you can watch the video segment to help you.
Think of some other cyclic functions you might encounter in everyday life. What are their periods?
You can find this segment on the session video, approximately 4 minutes and 3 seconds after the Annenberg Media logo.
In this video segment, taken from the “real world” example at the end of the Session 8 video, Rick Garnen of the Massachusetts Maritime Academy talks about the causes and effects of tides on currents in Cape Cod Bay and how this relates to the mathematics of cyclic functions.
You can find this segment on the session video, approximately 23 minutes and 8 seconds after the Annenberg Media logo.
Tide problem adapted from Trigonometry, by I. M. Gelfand and Mark Saul (Boston: Birkhauser Publishing Ltd., 2001). ISBN:0-8176-3914-4.M
When your heart beats, it pumps blood throughout your system of arteries. When doctors measure blood pressure, they usually measure the pressure of the blood in the artery of the upper arm.
But your blood pressure isn’t constant. The graph below shows how blood pressure changes over time.
What can you tell about blood pressure just before a heartbeat?
What happens to blood pressure after a heartbeat?
Is this the graph of a cyclic function?
What are the period and amplitude for this graph?
Heartbeat problem taken from Ups and Downs. Mathematics in Context (Chicago: Encyclopedia Britannica, Inc., 1998), p. 26.
Have graph paper readily available. Before starting work on Problems A1-A5, keep in mind that the graph will have the height of the tide on the y-axis. The “change in tide” column in the table is just to help you create the third column. The computations, though they deal with fractions (multiples of 1/12), should not be difficult, because the height of the tide is a multiple of 12.
Groups: Share graphs by tracing them on overheads. Discuss how this problem is an example of a cyclic function. Think of other examples of things that go in cycles. Examples include the motion of the planets and radio waves.
Think about the concepts of amplitude and period, two ideas that are an important part of the study of cyclic functions. Depending on where you are, the amplitude of tides can vary greatly, although the period is always the same.
Groups: If time allows, sketch graphs for tides in different places around the world. Discuss what would vary and what wouldn’t.
Move on to Problems A6-A8. Spend just a few minutes working on these. Take your pulse, and then mark numbers on the time axis as if it were your own heartbeat. Find the period of your own heartbeat function.
Groups: After working on the heartbeat problems, discuss the idea of period again. Finally, add cyclic functions to the list of nonlinear functions you started in the previous session.
The graph is not linear, because it’s not a straight line, and the difference between consecutive outputs is not constant. The graph is also not exponential, because it does not increase or decrease indefinitely, and the ratio between consecutive outputs is not constant. Neither is the graph quadratic, because the difference between consecutive outputs is not linear, nor are the second differences constant.
The period is 12 hours.
Blood pressure drops quickly, and appears to be lowest just before a heartbeat.
After a heartbeat, blood pressure increases drastically and peaks quickly.
Yes, this is a cyclic function, because its outputs are repeated consistently over time. The period is one heartbeat.
The period is the length of one hearbeat, which can be measured in seconds or fractions of a minute. In this graph, the period is about 0.9 seconds. The amplitude is half the difference in pressure from the lowest point to the highest point, usually measured in mm Hg (millimeters of mercury), and in this graph is (140 – 60)/2, or 40 mm Hg.
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.