Grade 8-12

Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations. 1ALG.1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable: 1ALG.1.1 Students use properties of numbers to demonstrate whether assertions are true or false. 1ALG.2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. 1ALG.3.0 Students solve equations and inequalities involving absolute values. 1ALG.4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12. 1ALG.5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. 1ALG.6.0 Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). 1ALG.7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. 1ALG.8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. 1ALG.9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. 1ALG.10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. 1ALG.11.0 Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. 1ALG.12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. 1ALG.13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. 1ALG.14.0 Students solve a quadratic equation by factoring or completing the square. 1ALG.15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. 1ALG.16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. 1ALG.17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. 1ALG.18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. 1ALG.19.0 Students know the quadratic formula and are familiar with its proof by completing the square. 1ALG.20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. 1ALG.21.0 Students graph quadratic functions and know that their roots are the x- intercepts. 1ALG.22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. 1ALG.23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. 1ALG.24.0 Students use and know simple aspects of a logical argument: 1ALG.24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each. 1ALG.24.2 Students identify the hypothesis and conclusion in logical deduction. 1ALG.24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion. 1ALG.25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements: 1ALG.25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions. 1ALG.25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step. 1ALG.25.3 Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.

Comments for Basic Skills for Algebra I 

The first basic skills that must be learned in Algebra I are those that relate to understanding linear equations and solving systems of linear equations. In Algebra I the students are expected to solve only two linear equations in two unknowns, but this is a basic skill. The following six standards explain what is required:

4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x - 5) + 4(x - 2) =12.

5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

6.0 Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).

7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.

15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. Each of these standards can be a source of difficulty for students, but they all reflect basic skills that must be understood so that students can advance to the next level in their understanding of mathematics. Moreover, modern applications of mathematics rely on solving systems of linear equations more than on any other single technique that students will learn in kindergarten through grade twelve mathematics. Consequently, it is essential that they learn these skills well.

Point-Slope Formula Perhaps the most perplexing difficulty that students have is with Standard 7.0. It often seems very hard for them to understand this point. But it is one of the most critical skills in this section. In particular, the following idea must be clearly understood before the students can progress further: A point lies on a line given by, for example, the equation y = 7x + 3 if and only if the coordinates of that point (a, b) satisfy the equation when x is replaced with a and y with b. One way of explaining this idea is to emphasize that the graph of the equation y = 7x + 3 is precisely the set of points (a, b) for which replacing x by a and y by b gives a true statement. (For example, (3, 2) is not on the graph because replacing x with 3 and y with 2 gives the statement 2 = 23, which is not true.) Thus, the graph consists of all points of the form (a, 7a + 3). It also follows from these considerations that the root r of the linear polynomial 7x + 3 is the x-intercept of the graph of y = 7x + 3 because (r, 0) is on the graph. An additional comment about Standard 7.0 is that, although it singles out the point-slope formula, it is understood that students also have to know how to write the equation of a line when two of its points are given. However, the fact that the slope of a line is the same regardless of which pair of points on the line are used for its definition depends on the considerations of similar triangles. (This fact is first mentioned in Algebra and Functions Standard 3.3 for grade 7.) This small gap in the logical development should be made clear to students, with the added assurance that they will learn the concept in geometry. The same comment applies also to the fact that two non-vertical lines are perpendicular if and only if the product of their slopes is <1 (Standard 8.0).

Quadratic Equations The next basic topic is the development of an understanding of the structure of quadratic equations. Here, one repeats the considerations involved in linear equations, such as graphing and understanding what it means for a point (x, y) to be on the graph. In particular, the graphical interpretation of finding the zeros of a quadratic equation by identifying the x-intercepts with the graph is very important and, as was the case with linear equations, is also a source of serious difficulty. Equally important is the recognition that if a, b are the roots of a quadratic polynomial, then up to a multiplicative constant, it is equal to (x < a)(x < b). When the discriminant of a quadratic polynomial is negative, the quadratic formula yields no information at this point because students have not yet been introduced to complex numbers. This deficiency will be remedied in Algebra II. The following standards show which skills students in a first-year algebra course need for solving quadratic equations:

14.0 Students solve a quadratic equation by factoring or completing the square.

19.0 Students know the quadratic formula and are familiar with its proof by completing the square.

20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. 

21.0 Students graph quadratic functions and know that their roots are the x-intercepts.

23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

Additional Comments Students should be carefully guided through the solving of word problems by using symbolic notations. Many students may be so overwhelmed by the symbolic notation that they start to manipulate symbols carelessly, and word problems become incomprehensible. Teachers and publishers need to be sensitive to this difficulty. In addition to Standard 15.0, cited previously, the other relevant standards for solving word problems using symbolic notations are:

10.0 Students add, subtract, multiply and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.

13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. Among the word problems of this level, those involving direct and inverse proportions occupy a prominent place. These concepts, which are often mired in the language of “proportional thinking,” need clarification. A quantity P is said to be proportional to another quantity Q if the quotient P/Q is a fixed constant k. This k is then called the constant of proportionality. Students should be made aware that this is a mathematical definition, and there is no need to look for linguistic subtleties concerning the phrase “to be proportional to.” Similarly, P is said to be inversely proportional to Q if the product PQ is equal to a fixed nonzero constant h. In Standard 13.0 the emphasis should be on formal rational expressions instead of on rational functions. Many of these formal techniques will become increasingly important in Algebra II and trigonometry. The rules of exponents, for example, are fundamental to an understanding of the exponential and logarithmic functions. Many students fail to cope with the latter topics because their understanding of the rules of (fractional) exponents is weak. The skills in the following standards need to be emphasized in a first-year algebra course:

2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. The gist of Standards 16.0 through 18.0 is to introduce students to a precise concept of functions in the language of ordered pairs. Introducing this concept needs to be done carefully because students at this stage of their mathematical development may not be ready for this level of abstraction. However, during a first-year algebra course is the stage at which students should see and use the functional notation f(x) for the first time. In Standard 24.0 students begin to learn simple logical arguments in algebra. They can be taught the proof that square roots of prime numbers are never rational, thereby solidifying to a certain extent their understanding of rational and irrational numbers (grade seven, Number Sense Standard 1.4). In Standard 3.0 students are taught to solve equations and inequalities involving absolute values, but it is not necessary to introduce the interval notation [a,b], (a,b), [a,b), and so forth at this point. However, they should be introduced to the set notation {a, b, c,. . .} and {x: x satisfies property P} and to the empty set f in, for example, Standard 17.0. Finally, students should become familiar with the terminology “solution set” of Standard 9.0—meaning the set of all solutions.