Lesson 1: COMPLETING THE SQUARE
1.1 Setting the Scene- The Key Players of the Story
This section sets the scene for thinking like a mathematician. It discusses the role of area plays throughout arithmetic and algebra, along with the astounding power symmetry can offer.
1.2 Equations that can be solved by the QUADRUS method: Levels 1-3
We begin the journey of solving quadratic equations, through a series of “levels” that introduce new complexities and challenges at each stage. We are developing confidence in problem-solving.
1.3 Equations that can be solved by the QUADRUS method: Levels 4-6
With the play of area and the power of symmetry we now have the means to solve all quadratic equations. (No memorization needed!) We see that a quadratic equation will have with 0, 1, or 2 real solutions.
Lesson 2: THE QUADRATIC FORMULA
2.1 Deriving the Quadratic Formula
We learn here that visual method of completing the picture of a square in the last lesson is the classic quadratic formula. Feel free to memorize the formula if you wish (but the previous lessons shows that there is no need to!)
2.2 Counting Solutions and the Discriminant
Most curricula require students to analyse the discriminant sitting in the quadratic formula to determine the number of solutions of a quadratic equation.
2.3 Lots of Practice
Plenty of practice problems are offered here.
Lesson 3: FACTORING
3.1 Mersenne Primes
Why do mathematicians care about factoring? We present a classic application here as a puzzle.
3.2 Factoring and Breaking Symmetry
In this section we mention the ONE instance where one might be able to solve a quadratic equation without the power of symmetry. Most curricula insist that students attend to this approach even though it is a side-story that distracts from the power of quadrus method.
3.3 More Practice; The Difference of Two squares
As the title suggests we present practice examples. The classic difference of two squares formula arises through the practice examples and is used to solve the opening puzzle.
Lesson 4: THE FACTOR THEOREM, ROOTS, and DIFFERENCE FORMULAS
4.1 An Opening Puzzler
A curious puzzle explains the context of the algebra technique that is used throughout this advanced upper-level lesson.
4.2 The Factor Theorem
We derive and discuss the Factor Theorem presented in upper-level mathematics.
4.3 Practicing the Factor Theorem; The Difference of Two Squares; The Difference (and Sum) of Two Cubes
As an application of the Factor Theorem we encounter the difference of two squares formula in a new context and extend it to cubes.
Here we attend to the upper-level curriculum analysis of the roots of quadratic equations in relation to the factoring of quadratics.
4.5 Rationalising the Denominator and the Numerator
The difference of two squares formula leads to a natural way to rationalize fractional expressions.
Lesson 5: SETTING THE SCENE FOR GRAPHING QUADRATICS
5.1 Introduction to Graphing; A Graphing Puzzle
The “graph” of an equation is simply a scatter plot of the data that naturally arises from that equation. We develop a solid and clear understanding of what a graph actually is.
5.2 Another Graphing Challenge
We examine the horizontal and vertical shifts of (basic quadratic) graphs.
We change the “steepness” of a quadratic graph by changing a leading coefficient.
5.4 Practicing Graphing
We practice the basics of graphing as developed thus far and make a key observation about symmetry.
Lesson 6: USING SYMMETRY FOR GRAPHING QUADRATICS
6.1 The Opening Puzzle; Picking Things Up from Last Time
We start with a curious puzzle and then pick up on the key point made at the end of last lesson.
6.2 The Full Power of Symmetry
Now that we recognize that symmetry is at play, we show how the graphing of quadratics is nothing but an exercise in following natural common sense.
We show how typical textbook problems are natural and easy to think through.
6.4 More Practice! The Opening Puzzle Revisited
Our closing thoughts.