World Population Growth Project
Benchmark Lesson
Name: Katy Monwai
Title of lesson: Thinking About Mathematical Relationships
Length of lesson: 50 min.
Description of the class:
Name of course: Pre-Calculus
Grade level: 10th-12th
Honors or regular: Either
Source of the lesson:
Source of idea is Mark Daniel's Functions and Modeling class.
TEKS addressed:
(2) (A) apply basic transformations, including a ¥ f(x), f(x) + d, f(x - c), f(b ¥ x), |f(x)|, f(|x|), to the parent functions;
(3) (C) use properties of functions to analyze and solve problems and make predictions; and
(4) (A) represent patterns using arithmetic and geometric sequences and series;
I. Overview
Students should understand the differences in different kinds of mathematical relationships so that they can make educated guesses when fitting regression lines to data.
II. Performance or learner outcomes
Students will be able to predict mathematical models for collected data.
III. Resources, materials and supplies needed
Worksheets, calculators.
IV. Supplementary materials, handouts.
None.
Five-E Organization
Teacher Does Probing Questions Student Does
| Engage: Yesterday we created scatterplots of data to get a good first look at what it was doing. Tomorrow we're going to learn how to fit a mathematical function to the data to model the relationship. Today we're going to review some of the different kinds of mathematical relationships that we might use to model data. Right, that's a power function. You can also have polynomial
functions like Right, that's another power function. How could power functions be expressed, generally? |
What are some of the kinds of mathematical relationships you've seen before? (Draw a line on the board) What kind of a mathematical relationship is this? What does its equation look like in general form? What other kinds of mathematical relationships are there? What kind of an equation are you thinking of? What else? What others can you remember? |
Huh? Linear! y=mx+b Like, square? Hopefully students will come up with Exponential: and maybe Logarithmic, Trigonometric, etc. |
| Explore: We're going to do an investigation into how some of these functions grow relative to each other. (Hand out worksheets) |
| Explain: |
Which kind of function or relationship has the add-add property? Which has the add-multiply? What about the power function, what kind of property does it have? How would you express that generally? Which of the three functions grows the slowest? And then what? Does that correlate with the properties we found? What did you find when you graphed your three functions? How did your functions to compare to others' in your group? |
Linear Exponential. Multiply-Multiply. f(kx)/f(x) is a constant for any x. Linear. Then power, then exponential is the fastest. Students will probably understand the relationship between linear (add-add) and exponential (add-multiply), but I expect that the multiply-multiply will throw them off a little. |
| Extend / Elaborate: So, different types of functions describe different types of growth. |
What kinds of things might grow linearly or be best described by a linear relationship? What about power functions? Exponential? |
I expect a variety of answers here, with some possible debate. |
| Evaluate: In the next couple of days, we will use what we've learned to create regression lines to fit our scatterplots. |
What does this have to do with taking data and finding a mathematical model for it? |
Understanding the differences between different types of functions will help us to make educated guesses when we want to fit data to a mathematical model. |
Properties of Mathematical Relationships
Come up with a linear equation of the form 
(substitute
values for m and b)
Come up with a power function of the form 
(substitute
values for c and b)
Come up with an exponential function of the form 
(substitute
values for c and b)
| X |
Linear Function |
Power Function |
Exponential Function |
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| 10 |
Fill in the chart with your function values at each x.
1. One of these functions has what we call the add-add property. For which of the functions does f(x+k) – f(x) equal some constant for every value of x, for each k (try k = 1, 2, 3, É)?
2. Another function has the add-multiply property. For which of the functions does
f(kx) – f(x) equal some constant for every value of x, for each k (try k = 1, 2, 3, É)?
3. What kind of property does the remaining function have? How would you express it in general terms, and what would you call it?
4. Compare your findings with others' in your group. Are they finding similar results?
5. How do these properties of linear, power, and exponential functions give us insight into the behavior of the function? Now, graph each of your functions and compare and contrast.