Chapter 1: Arithmetic to Algebra

1A: The Tables of Arithmetic

Students examine patterns in addition and multiplication tables to develop the rules for addition and multiplication, and extend the rules to negative integers. Students are challenged to work like mathematicians and to understand the rules they use.

1B: The Number Line

The basics of the number line are reviewed for integers and extended past the integers to include the rational and real numbers. The process of extension helps students use the number line to visualize addition and multiplication of any two numbers (not just integers), and to see that the basic rules they know carry over into these new sets of numbers.

1C: The Algorithms of Arithmetic

Students reexamine the algorithms they use to add, subtract, multiply, and divide in detail, to see how these algorithms use the basic rules of arithmetic, such as commutativity, associativity, and distributivity. Students are encouraged to look at the study of algebra as not only about finding a method that works but also understanding why that method works.

Chapter 2: Expressions and Equations

2A: Expressions

Students begin to transform algorithms, such as following simple number tricks, into expressions using variables. Variables are defined as placeholders that create a shorthand to express the patterns they find, and for understanding why those patterns occur.

2B: Equations

Equations are introduced to express relationships between expressions. Simple equations are solved using backtracking – a method for solving equations that has students think of the equation as a series of steps applied to a number x, and then undo each step in reverse order to find the initial value of x. This instinctive process will serve as a starting place for more formal equation solution.

2C: Solving Linear Equations

Students begin to formalize the basic method for solving equations. The Basic Rules and Moves of Equations are those operations that can be performed without changing the solution set of the equation, such as adding the same number to both sides of an equation and multiplying both sides of an equation by the same non-zero number. Students also explore why these moves don't change the solutions of the equation.

2D: Word Problems

Students learn the Guess-Check-Generalize method for building equations from situations. The method involves taking several guesses, checking those guesses against the text of the problem, and keeping careful track of the steps followed to check the answer. Finally, students guess with an arbitrary number (the variable) to build the equation. From there, they use the skills from earlier in the investigation to solve those equations.

Chapter 2 Extras

Chapter 3: Graphs

3A: Introduction to Coordinates

The coordinate plane is re-introduced as students experiment with transformations of points and shapes and explore absolute value and distance. Students also see how the two axes can represent different data points, and that graphs can show trends in data, a topic explored further throughout this chapter and in Chapter 4.

3B: Statistical Data

Students learn to use graphs, charts, and tables to summarize and interpret data. They recognize and construct visual representations of data, including box-and-whisker plots and scatter plots, and use their interpretations to make informed conclusions about data.

3C: Equations and Their Graphs

Students are shown that the graph of an equation is just another representation of the set of points that make the equation true. Simple and complex graphs are used to reinforce to students that no matter how difficult an equation is, it can be used as a point-tester to determine whether a particular point lies on the equation's graph.

3D: Basic Graphs and Translations

Students extend the concept of transformations from Investigation 3A, applying it to equations and their graphs. Simple linear translations of six types of equations are reviewed:

y = kx, y = k/x, y = x², y = x³, y = √x, and y = |x|

Transformations provide a link to Chapter 4, where the generic equation of a line will be

(y – k) = m(x – h) — a translation of the simpler equation y = mx.

Chapter 4: Lines

4A: All About Slope

The study of slope begins by defining the slope between two points. They test for collinearity of sets of points by using the idea that three points are collinear if and only if the slope between each pair of them is the same. Ultimately, they prove the corollary that slope is invariant for all pairs of points on a line. They use this invariance as the point tester to see whether some given point is or is not on a line, and eventually (in the next investigation) to develop an equation for the line itself.

4B: Linear Equations and Graphs

The buildup from the previous investigation is resolved by having students use the point-tester concept to develop a general method for finding the equation of a line. This course does not emphasize any particular form of a linear equation, but rather works on the overriding principle that to graph a line, only two points on that line need to be found, and any two points will do.

4C: Intersections

Students learn to solve systems of linear equations using the substitution and elimination methods. While the explanation of these two methods is more or less traditional, the exposition relies on and emphasizes the basic moves and the point-tester concept. The proof that lines with the same slope are parallel introduces students to the concept of proof by contradiction.

4D: Applications of Lines

Students apply their work with lines to solve inequalities and estimate the line of best fit. Inequalities are explored by treating each side of the inequality as an equation to graph, and the inequality solution is found by comparing the y-heights of the two graphs. Fitting lines are found by determining the balance point of the data, and estimating the slope of the line. Students compare their lines with the actual data, calculating simple errors and thinking about how to minimize that error.

5A: Functions - The Basics

Functions are introduced as a machine defined by a specialized rule — one that assigns each input exactly one output. Students create their own rules for given sets of inputs and outputs, and from this foundation generate tables, algebraic expressions, and ultimately graphs. The lessons gradually add more formal algebra for expressing rules (such as f(x) notation and the concept of domains).

5B: Function and Situations

Students learn to fit functions to tables. First, they explore differences in successive outputs of a function, determining that constant differences imply linear functions. Next, recursive rules are introduced to describe some tables. Finally, these recursive rules are used to fit exponential functions to tables with constant ratios.

5C: Functions and Situations

Students extend their work from the end of Chapter 2 to build functions to model situations described in word problems using the Guess-Check-Generalize method.

Chapter 6: Exponents and Radicals

6A: Exponents

Following a similar process as in Chapter 1, students develop the basic rules of exponents, starting with positive integer exponents. The rules are used to find sensible definitions for zero and negative exponents.

6B: Radicals

Although most students are familiar with square roots prior to Algebra 1, the subject is treated more deeply here. Students learn the difference between rational numbers and irrational numbers and basic rules and conventions for calculating with square roots. The final lessons treat other radicals, such as cube roots, fourth roots, and more generally, nth roots.

6C: Exponential Expressions and Functions

Students explore exponential functions by looking at their graphs, exploring quotient tables (similar to the difference tables they saw in Chapter 5), calculating compound interest, and looking at graphs of exponential functions. These topics will be further investigated in Algebra 2.

Chapter 7: Polynomials

7A: The Need for Identities - Equivalent Expressions

The heart of this investigation lies in factors — forming them, expanding them, comparing them, and ultimately using them in the development of the Zero Product Property. The exercises call upon all of the basic moves of equations students have learned and extend them to recognize equivalent expressions and develop algebraic identities.

7B: Polynomials and Their Arithmetic

Monomial and polynomial are introduced, and students explore the features of polynomial expressions. Extra practice is included for adding and multiplying polynomials, combining like terms, and factoring out the greatest common monomial factor of a polynomial.

7C: Factoring to Solve Quadratics

Students learn to factor quadratic expressions. This work has been previewed in earlier lessons, and that work will be reviewed and practiced, but here students study techniques for factoring quadratic expressions in depth and use factoring to solve quadratic equations.

Chapter 8: Quadratics

8A: The Quadratic Formula

The quadratic formula is derived through the process of completing the square on a general quadratic equation. Students develop expertise in solving quadratic equations. They develop a flexible understanding of the relationship between a quadratic equation and its roots and learn to write a quadratic equation so that it has specific roots.

8B: Quadratic Graphs and Applications

Students develop techniques for graphing quadratic equations, paying special attention to the roots and vertex. They use these graphs to solve for maxima and minima in word problems.

8C: Working With Quadratics

Students look again at solving equations and inequalities by graphing each side of the equation/inequality as a distinct function, and comparing the graphs. They also will look at advanced inequalities, such as systems of inequalities, both linear and quadratic. Finally, they revisit the idea of difference tables for quadratics.