Unit 1 - Linear Relationships and Regression

What is mathematical modeling and extrapolation? Use this unit to motivate the definition of functions, and use linear relationships as model functions.

Modeling: Student wingspan against height?

1. Linear Equations (1 and 2 variable)

2. Direct and Inverse Variation

78 - Regression Analysis

Unit 2 - Functions

Functions are at the center of mathematical modeling. Before working on specific cases of modeling, we need the language of functions.

27 - Function Notation

28 - Definition of Function, 1:1, onto

29 - Domain and Range

30 - Composition of Functions

31 - Inverse Functions

Unit 3 - Exponential Functions

Don’t move into quadratic functions, because there’s a whole mess of issues that we have to deal with those functions. Start this unit with actual data with percent increase and decrease, talk about exponential regression, and solve equations because extrapolation or prediction requires it.

Modeling: Population growth?

32 - Percent Increase and Decrease

33 - Compounding Intervals

34 - Continuous Growth

35 - Exponent Rules

36 - Fractional Exponents

37 - Power Equations

38 - Exponential Equations

Unit 4 – Logarithms

Move into logarithmic functions with more actual data.

39 - Definition of a Logarithm

40 - Log Laws

41 - Solving Log Equations

42 - Solving Exponential Equations

43 - Graphing Log Equations

44 - Natural Logs

45 - Applications of Logs

Unit 5 - Systems of Equations

Solve a system of exponential and linear functions, as well as a system of linear equations, for different situations. That will work fine graphically.

Modeling: Population grows exponentially, food grows linearly?

25 - Solve a System by Graphing

26 - Solve a System Algebraically

Unit 6 --  Radicals and Complex Numbers

We’re taking a break from modeling for a unit so that we can better model quadratic functions. This unit isn’t answering the question “How do we make predictions?” but instead “What is a number?”

7 - Multiply and Divide Radicals

8 - Rationalize Binomial Denominator

11 - Powers of i

12 - Multiply Complex Numbers

13 - Divide by a Complex Number

Push quadratics for later, since it’s really only good for modeling one thing, and it’s a thing that 11th graders will be learning in the early part of the year. Also, by this stage equation solving has been clearly set forth as useful for extrapolation, and we can talk more easily about what roots are and why we’d be looking for them instead of just saying “You know how we can find some number for linear equations? We can do that with quadratics too, but it’s way more annoying.”

15 - Graph a Parabola in Vertex Form

18 - Quadratic Shortcuts (discriminant, sum and product of roots) (I found myself with kids who were freaked out by the full quadratic formula. Why not start with calculating the discriminant after talking about solving quadratics graphically, and then build the quadratic formula around the discriminant?)

17 - Completing the Square

19 - Write a Quadratic Given the Roots

24 - Solve a Higher Order Equation

Unit 8 -- Rational Functions

This unit is tough, and it doesn’t really fit into my general story. Not sure how to approach this unit, in general. Why do we want them to learn this stuff, again?

51 - Solve Rational Equations - algebraically and graphically

46 - Simplify Rational Expressions

47 - Undefined Rational Expressions

48 - Multiply/Divide Rational Expressions

50 - Complicated Fractions

Unit 9 – Absolute Value and Piece-wise functions

My students often have trouble understanding that the “rules” for functions don’t have to be straightforward. They can arbitrarily defined by a table, and part of what makes circular functions difficult is that they use they find looking up outputs on the Unit Circle to be somewhat arbitrary. It doesn’t feel like following an easy rule. Maybe we can talk about absolute value functions as practice for that, drawing out the piecewise-ness of them.

3. Solve Absolute Value Equations

5. Graphing Absolute Value Functions

Unit 10 – Inequalities

I think that it would be good to teach a general method for dealing with inequalities instead of having kids memorizing “OK, in a quadratic inequality I do this, but in an absolute value inequality I do this…” All inequalities can be handled graphically, and this unit will take us through a lot of the inequalities that we already know and love.

20 - Solve a Quadratic Inequality

21 - Graph a Quadratic Inequality

4. Solve Absolute Value Inequalities

52 - Solve Rational Inequalities - by graphing only