By the time the unit on quadratic equations is complete, there is not much time left in the school year. The rate at which a class progresses through an Algebra 1 course depends on the ability level of the class. For most classes that I’ve had, I come to the decision that many teachers face in the last month and a half: what topics do I omit? And like most teachers, I feel guilty doing so. I try to minimize any damage by picking the topics to include that I think will do the most good.
In making my selection I consider not only how much material has been covered, but what material will be repeated in Algebra 2. I realize that not every student will be taking Algebra 2, however—it depends on the school district as well as state requirements. Nevertheless, I make the general assumption that students taking algebra in eighth grade will are likely to take Algebra 2, and then move on to the next step.
The topics covered in Algebra 2 vary depending on the textbook. Common Core does not break down the standards by Algebra 1 and 2—rather there are standards for the entire sequence of algebra. Thus, some topics may appear in Algebra 1 textbooks, and others in Algebra 2—and sometimes in neither. For example, not all textbooks address inverse variation.
Most Algebra 1 textbooks aligned to Common Core address some aspect of statistics. They also emphasize the difference between linear, quadratic and exponential functions, showing how rules of translation apply to all of them. I cover neither of these topics in algebra. These topics are covered to some extent in Algebra 2. Also, the basics of statistics and probability are covered in seventh grade, so students have had some initial exposure.
The issue of Common Core-aligned state exams therefore remains as the elephant in the room. I do not view this as cause for concern for the following reasons:
1) The bulk of Algebra standards under Common Core are confined to high school. Therefore, state tests given in eighth grade address only those aspects of algebra that are in the Common Core eighth grade standards, which include basic equations and expressions, slope, and linear systems of equations.
2) After eighth grade, the next state test under Common Core is not until eleventh grade. At this point, students will have had additional math courses which will probably include Algebra 2 and Pre-Calculus. Therefore topics that teachers may choose to omit for an Algebra 1 course will be covered later in some form and at a time when students possess more experience and mathematical maturity.
Given all this, in my eighth grade algebra course I feel safe addressing neither statistics nor extensive forays into transformations (e.g. vertical and horizontal translations) and comparisons of linear, quadratic and exponential graphs. I choose therefore to spend more time with rational expressions (i.e., algebraic fractions), factoring, polynomial multiplication (and division if time), and various traditional types of word problems.
In the time remaining after quadratic equations—a time when attention is waning as the June horizon and summer vacation comes more clearly into view—these are generally the topics that I am able to complete:
Quadratic Functions and Graphing. This unit immediately follows the unit on quadratic equations. In it we look at quadratics in functional form. That is, rather than an equation to be solved, we consider the function y = ax² + bx + c, expressed as f(x) = ax² + bx + c, and how they are to be graphed via the axis of symmetry, (defined as –b/2a), finding the vertex, and reflecting points over the axis of symmetry, defining domain and range and identifying the intervals over which the function increases or decreases. The vertex form of quadratic functions is also addressed; i.e., f(x) = a(x – h)² + k, and how the graphs are transformed by changes in a, h, and k.
Finding the maximum/minimum points of a parabola is covered, and is used to solve word problems of which the following is typical:
Functions and Variation. This unit presents the definition of function, and defines direct variation functions (i.e., f(x) = kx) and inverse variation functions (i.e., f(x) = k/x).
For direct variation, k is called the constant of proportionality or constant of variation. We show how to find k if x and y are known. We also show that the graph of a proportional function passes through the origin. Students have had this in seventh grade, but I tend not to put too much emphasis on it then, since students at that point need more of the fundamentals of proportional relationships. In algebra, those foundations are then built upon as just described.
Typical problems for direct variation are similar to those students have solved in seventh grade but also include translating statements into algebra such as:
For inverse proportion students solve problems based on the relationship between two ordered pairs of an inverse variation:
Typical problems that students solve include:
If 8 men do a job in 9 days, how long does it take 24 men to do the job?
Exponential Functions. Students learn the basic form for an exponential function:
Also covered are exponential growth and decay functions and how those are used to solve problems. Typical growth and decay problems include compound interest for which this formula is given:
A typical problem for compound interest includes:
“Find the value of $6.000 compounded monthly at 10% annual interest.”
A typical problem for exponential decay includes:
“A car valued at $21,500 when new loses 12% of its initial value every year. Find its value after two years.”
FINAL WORD
This book has been a description of how math is taught in a traditional manner. It is a glimpse based on the teaching styles of two of us. As we were writing this and showing it to various teachers to get opinions on various matters, we heard “That’s exactly how I do it.” This is not to say that what we have written embodies the manner and technique of all traditional-minded math teachers. Everyone has their own particular style and approach.
Nevertheless, we feel that we have recreated the look and feel of explicit teaching in the classroom, as well as typical reactions of students. Our hats are off to the teachers who maintain the structures of such a classroom in the face of forces saying to do otherwise.
Also, we would be remiss if we did not show our appreciation to the people most affected by traditional teaching: our students. We know what it is like to be confronted with new information, and the effort it takes to keep things straight. We rely on the occasional outburst of a frustrated student who announces “I’m confused” or “Slow down”. It keeps us honest and makes us better teachers.