Julie McPhail
Function Patterns: Finding Models for Messy Real World Data
Length of lesson: One class period (45-50 minutes)
Description of the class:
Name of course:  Algebra 2
Grade level:  10th or 11th (depending on honors or academic level)
TEKS addressed:
(2A.1)  Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations.
(2A.10)  Rational functions. The student formulates equations and inequalities based on rational functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.
(2A.11)  Exponential and logarithmic functions. The student formulates equations and inequalities based on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.
I. Overview
This lesson will give students a tool for determining a good, representative model for their data. Correlation and regression are powerful tools when we're working with linear data, but in the real world we hardly ever come across linear situations. Function patterns provide a starting point for finding a model for non-linear data.
II.  Performance or learner outcomes
Students will be able to:
Identify the domain and range of a function
Explain what a function is
Predict the best model for a given data set using function patterns
III. Resources, materials and supplies needed
Calculators (non-graphing is fine)
Function Patterns handout *Planets Exploration handout
*Function Patterns Homework assignment
* Used with permission from Mark Daniels

Five-E Organization

Teacher Does                    Probing Questions                     Student Responses   

Engage: Draw five or six charts, equations, or graphs on the board, and ask students to identify which things are functions and which things are not. (For example, draw a graph of y=x^2, draw a chart with varying x and y values, and write the equation y=x-4). After looking at various representations of a few functions and a few things that are not functions, talk about what a function is. Let students discuss this in groups of two or three, then write some conjectures on the board. Wrap up this discussion. Hopefully students will recall that to have a function we need to have a domain and a range, and we need to have some kind of rule for assigning (or mapping, or sending) elements of the domain to elements of the range. Basically we need two sets and a relationship. We also have the restriction that we can't send one domain element to two different range elements (like a machine, you put in one thing, you get back out one thing). This should all be review; just making sure everyone is one the same page... and make sure students come away knowing that "It passes the vertical line test!" is not a good answer. Approx. Time:  10 mins

"Which one of these are functions? Why?" "What does it take for something to be a function? What are the necessary ingredients?" After writing some conjectures, ask, "Do you guys agree that this is necessary for a function? Is there anything else that we need that's not written up here?" Have students figure out the domains and ranges of the charts, equations, or graphs on the board.

"That one's not... it doesn't pass the vertical line test." "It passes the vertical line test." "Nope. We got everything."

Explore: Approx. Time:  5 mins

(This lesson will come directly after the lesson on correlation and regression). "Yesterday we talked about how to find an equation that models linear data, but can you think of functions that are not linear? What other kinds of functions are out there?" "How do we make sense of non-linear data? Any ideas?" (The teacher will have mentioned the previous week that linear functions have an add-add pattern and exponential functions have an add-multiply pattern.) "Does anybody remember when we talked about exponential functions last week how we could tell if we had an exponential function or not?" "It turns out, we can use patterns like this to figure out what kind of function best models our data."

"Quadratic" "Exponential" "Power" "Do something similar to the Least Squares method, just with curves instead of lines." "We looked at patterns." "We saw if it was add-multiply."

Explain: Hand out Function Patterns worksheet. Tell students to work on the second page of the Function Patterns worksheet. If they need help, encourage them to ask their peers first. Approx. Time 15 mins

"We're going to talk about power functions, exponential functions, linear functions, and quadratic functions. Get in your groups and see how much you all know about these different functions. You should have seen all of them before, but it may have been a while." Using the chart in the Function Patterns handout, get students to review in groups about the general properties (graph, equation, etc) of these four functions (linear, power, quadratic, exponential). Walk around and help where needed. After a few minutes, get volunteers to come fill in the chart on the overhead or on the board. Discuss what types of domains and ranges each function type has. After discussing general properties, see if students can guess what patterns indicate what function types just by looking at the equations and graphs of the different functions (Add-Add implies linear, Add-Multiply implies exponential, Multiply-Multiply implies Power, and Add-Constant 2nd difference implies Quadratic). Write a couple different tables on the board and show students how to figure out which pattern the data displays. Walk around and observe student's work. See if they seem to grasp the concept of function patterns.

Students are in groups filling out the chart on the first page of the Function Patterns worksheet. Students make conjectures and finish filling in their charts.

Extend / Elaborate: Give a brief (but interesting) background on Johannes Kepler and Tycho Brahe, and hand out the Planets Exploration handout. After students complete the handout, tell them Kepler's Third Law and tell them to see how well their conjecture matches up. It should be very close if not perfect. Approx. Time: 20 mins

"When you're doing your individual projects, how do you think the data you analyze will be different from what we've worked with so far today?" "Is real world data generally messier or cleaner than the stuff we work with in math class?" "Why?" "You're right, it is usually messier, but we can still use the same methods. Here's some practice with real data."

"It won't be as perfect." "Messier" Human error, how many factors affect our experiment, etc. Students are working in groups to come up with one of Kepler's laws.

Evaluate:

Give out the homework assignment.

Approx. Time: 1 min

Name: Date:

Function Patterns

Use the following table to take notes:

Function type	Function pattern	Equation form	Graph
Linear
Quadratic
Exponential
Power

Show all work. Determine whether the following data has an add-add, multiply-multiply, add-constant 2nd difference, or add-multiply pattern. Identify the pattern, the type of function, the domain, and the range.

1.

x	y
2	1800
4	450
6	200
8	112.5
10	72

2.

x	y
2	400
4	100
6	-200
8	-500
10	-800

3.

x	y
2	900
4	100
6	11.111...
8	1.2345...
10	0.1371...

4.

x	y
1	352
3	136
5	64
7	136
9	352

Name: Date:

Planets Exploration

This table was taken from the World Almanac 2001.

Name	Period (years)	Orbit (millions of km from sun)	Relative Mass
Mercury	0.24	57.9	0.06
Venus	0.61	108.2	0.82
Earth	1	149.6	1
Mars	1.88	228	0.11
Jupiter	11.86	778.5	317.8
Saturn	29.46	1433.5	95.16
Uranus	84.01	2872.6	14.5
Neptune	164.79	4495.6	17.15
Pluto	247.68	5870.5	0.002

a) Use function properties (remember, this is real data) to try to decide what type of function this data represents.

Conjecture:

b) Based on your answer to a, do a regression to find the equation for the function that fits this data.

Your equation:

c) Plot the scatter plot as shown above to see how well the regression equation fits the data.

d) Now explore the data and regression equation using residuals. Does the residual plot support your choice for the type of regression that you chose? Why or why not?

e) Make a scatter plot of "period" as a function of relative planet mass. Does there seem to be a correlation between these two variables?

f) Most asteroids are located in the "asteroid belt" about 430 million kilometers from the sun. Some scientists believe that the asteroids originated from the break up of a planet. If this is true, what would the period of the planet have been?

Name: Date:

Function Patterns Homework

Show work on a separate sheet of paper.

The table below shows air pressure p (measured in psi) for a punctured car tire at different times t (measured in seconds).

t (sec)	p (psi)
5	27
10	21
15	16
20	13
25	9
30	7
35	6
40	4
45	3
50	3

a) Make a scatter plot of air pressure as a function of time. Record the graph and make a conjecture as to what type of function this data might represent.

b) Use your function properties in order to justify or question your conjecture above.

c) Find a regression equation based on your findings. What equation did you find?

d) What is the correlation coefficient for your regression equation? Does this value support your conjecture?

e) Now create and record a residual graph for your data and regression equation. Does the residual plot support your conjecture?

f) Based on all the evidence, is your equation a good model for the data?
(If not, try a different conjecture, and show all findings for the questions above).