Algebra in eighth grade is generally reserved for students who qualify by having taken accelerated Math 7 and/or qualifying via a placement exam. There are some students who may be placed in Algebra who did not take the accelerated seventh grade math class, but did well on the placement exam and otherwise show promise. But in addition to all that, there are students who could benefit from a year of Math 8, but are placed in algebra in eighth grade at the insistence of parents.
I watch carefully during the first two months of the course to identify students who are continually struggling and appear to be getting lost despite help after school or help with a tutor. I recommend to parents that the student be moved into the regular Math 8 course. In some cases that is not an option that a parent will consider, and so I have a class with students who can handle it, and those who are lost.
I have written elsewhere that I use a 1962 algebra textbook by Dolciani and others, copies of which I obtained from the internet when the price was a lot less than its current one. ($88 as of this writing). I like the book for its sequencing of topics, and for the problems it presents. Some of the explanations in the book are very formal—an artifact of the 60’s New Math sensibility that prevailed when the book was written.
For those students who might be lost for the reasons indicated above, I have often had them use a more algebra-lite type of book, also authored by Dolciani called “Basic Algebra”.
Students have had some experience solving equations in seventh grade; accelerated math students will have had more. In any case, I handle basic equation solving as a review and do not spend a lot of time on the basics. I do spend more time on equations with variables on both sides, with fractional coefficients, and that may require distribution in the equation itself. Similarly, students will have learned how to operate with negative numbers, so that’s another topic that does not need to be repeated.
Many of today’s algebra textbooks have a dearth of word problems, and those that are included are usually of the “real-world” variety which tend to be wordy, and lacking the challenge that traditional word problems offer. I focus quite a bit on word problems, but recognize that students will have difficulty with these. Therefore on tests and quizzes I do not provide a lot of such problems and those that are included are straightforward. The more complex word problems are included for extra credit, so that there is a differentiation of problem types on tests—not with instruction. I provide the same instruction to all students. Those whose ability is greater than others tend to do the extra credit problems. These problems only add to the score on a test or quiz. They carry no penalty so all students are welcome to try them.
In my own experience, my algebra teacher recognized the difficulty students have with word problems. He admitted that he was not very good at solving or teaching them. When I took Algebra 2, however, the teacher provided thorough instruction and I was able to follow. This might have been because of better instruction, but I’ve come to believe that it was also a maturity that comes with consolidation of algebraic procedures and concepts taught in Algebra 1 which have had time to sink in.
In a first course in algebra, students are not likely to make the connections they can make later in solving word problems. For example, there are many types of distance/rate problems, some of which can be solved by a “distance = distance” approach, and others with a “time = time” approach. And even those solved using “distance = distance” have variation; i.e., travel in opposite direction, travel in same direction, and round-trip travel. I will provide a mixture of such problems in Warm-Ups, homework, and quizzes, but I try to keep it simple. The end result has been that my students generally can solve more word problems than others who are not subjected to the variety of traditional math problems.
Following below is the scope and sequence of an eighth grade algebra course. The first ten units are included in this book. The last two are taught but are not included. In fact, I do not spend a great amount of time on these last two topics. They are presented in Algebra 2 and pre-calculus. Furthermore, state exams that are aligned with Common Core do not address algebra until the eleventh grade, so there is plenty of time for students to become proficient in these topics.
Generally I reach these topics at the end of the school year when attention is waning and students are thinking “I can’t take in any more new information.” I strive to reach the unit on quadratic equations by April, which is before the “I can’t take it” syndrome sets in. It is, in my opinion, the pinnacle of an algebra 1 course, particularly the derivation of the quadratic formula.
Scope and Sequence of Topics for Eighth Grade Algebra
(Italicized entries are not described in the book.)
1. Variables and Open Sentences
Lesson 1 Identifying factors, coefficients and exponents
Lesson 2 Combining like terms; addition and subtraction
Lesson 3 Multiplication (Exponents; powers)
Lesson 4 Division (Exponents; powers)
Lesson 5 Solving open sentences (equations)
Lesson 6 Thinking with variables; from words to symbols
Lesson 7: Solving problems with equations
2: Equations
Lesson 1 Basic Axioms
Lesson 2 Distributive Property; Special Properties of 1 and 0
Lesson 3 Review of one and two step equations
Lesson 4 Solving equations with variables on both sides (G:2-4)
Lesson 5 Absolute Value equations (G-:2 G2-5)
3: Solving equations, inequalities and word problems
Lesson 1: Transforming equations
Lesson 2: Solving inequalities
Lesson 3: Solving absolute value inequalities
Lesson 4: Problems about Consecutive Integers
Lesson 5: Uniform motion problems (Opposite directions)
Lesson 6: Uniform motion problems (Same direction)
Lesson 7: Uniform motion problems (Round trip)
4: Polynomials
Lesson 1: Add polynomials
Lesson 2: Subtract polynomials
Lesson 3: Fractions; adding and subtracting
Lesson 4: Product of powers
Lesson 5: Power of products and of powers
Lesson 6: Multiplying Fractions
Lesson 7: Dividing Fractions
Lesson 8: Quotient of Powers
Lesson 9: Multiplying polynomial by a monomial
Lesson 10: Multiplying two polynomials
Lesson 11: Powers of polynomials
Lesson 12: Zero power and negative exponents
5: Factoring and special products
Lesson 1: Factoring in algebra
Lesson 2: Identifying common factors
Lesson 3: Multiplying sum and diff of two numbers
Lesson 4: Factoring diff of two squares
Lesson 5: FOIL Method
Lesson 6: Squaring a binomial
Lesson 7: Factoring perfect square trinomials
Lesson 8: General method of factoring trinomials
Lesson 9: Complete factoring (G 8-8)
Lesson10: Working with factors whose product is zero (G 8-6, 8-7)
Lesson 11: Solving word problems with factoring
6 Algebraic fractions
Lesson 1: Fractions
Lesson 2: Ratio
Lesson 3: Percents
Lesson 4: Adding & subtracting fractions (cont)
Lesson 5: Mixed expressions (G 11-7)
Lesson 6: Equations with fractional coefficients (G 11-8)
Lesson 7: Percent mixture problems
Lesson 8: Fractional equations (G 11-8)
Lesson 9: Work problems
Lesson 10:Motion Problems: Round trip
Lesson 11: Motion Problems: Current
7 Graphing linear equations
Lesson 1: Graph of linear equation in two variables
Lesson 2: Slope of a line
Lesson 3: Slope-Intercept form of linear eqn
Lesson 4: Point slope form of linear equations
Lesson 5: Parallel and perpendicular lines
Lesson 6: Graphing an inequality
8 Systems of linear equations
Lesson 1: Graphing intro
Lesson 2: Elimination method
Lesson 3: Elimination method with multiplication
Lesson 4: Word probs (Mixtures)
Lesson 5: Substitution method
Lesson 6: Linear Inequalities
Lesson 7: Digit problems
Lesson 8: Wind and current problems
9 Real Numbers
Lesson 1: Properties of square roots
Lesson 2: Repeating decimals
Lesson 3: Pythagorean theorem
Lesson 4: Rationalizing denominators
Lesson 5: Adding and subtracting radicals
Lesson 6: Fractional exponents
Lesson 7: Solving radical equations
10 Quadratic Equations
Lesson 1: Square root method of quadratic equations
Lesson 2: Completing the square (G 9-4)
Lesson 3: Solving equations by completing the square
Lesson 4: Quadratic formula
Lesson 5: Derivation of quadratic formula
Lesson 6: Discriminant
Graphing Quadratics
Graphing quadratics
Graphing y=ax^2+c
Graphing y = ax^2 + bx + c
Graphing maximum and minimums of quadratic functions
Vertex form of quadratic
Exponential Functions
Review of square root properties
Review of quotient of powers; negative exponents
Exponential functions Part I
Exponential functions Pt II
Exponential growth
Exponential decay
Geometric sequences