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?”
6 - Simplify Radicals
7 - Multiply and Divide Radicals
8 - Rationalize Binomial Denominator
9 - Solve Radical Equation
10 - Add Complex Numbers
11 - Powers of i
12 - Multiply Complex Numbers
13 - Divide by a Complex Number
Unit 7 – Quadratics
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?)
16 - Solving Quadratics by Factoring and Quadratic Formula
14 - Quadratic Max/Mins
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
49 - Add/Subtract 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