To navigate to a particular chapter, please search this page, using the "Find Text in Page" or similar feature of your web browser. Please see this site's Help page or consult your web browser's documentation for more information.
Chapter 1 - Uses of Variables | Back to Top |
Two important operations with sets are union, denoted by the symbol U, and intersection, denoted by ∩. The union of two sets consists of all elements in the first or the second set (or both). The intersection of two sets consists of those elements common to both sets.
Four uses of variables are described in this chapter. Variables are the language for translating some English expressions and sentences involving numbers into algebraic expressions and sentences. In algebraic expressions, the rules for order of operations are followed: work in parentheses or other grouping symbols first, then do powers, then do multiplications or divisions from left to right, and then additions or subtractions from left to right. Scientific calculators and computer languages usually follow the same rules.
Variables may represent numbers or quantities in formlas like A = lw . Some formulas involve square roots of numbers. The positive square root of a positive number r is written √r.
Variables may stand for unknowns, as in the open sentence x + 5 = 70 or the inequality 3y - 2 > 8.
The solutions to inequalities with one variable are often pictured by a graph on a number line.
Variables may be used to describe patterns. Among the most famous patterns is the Pythagorean Theorem: in a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2. General patterns can sumetimes be found from studying specific instances or arranging information in tables.
Chapter 2 - Multiplication in Algebra | Back to Top |
Area Model: xy is the area of a rectangle with length x and width y.
area Model (discrete version): xy is the number of elements in a rectangular array with x rows and ycolumns.
Rate Factor Model: xy is the result of multiplying a rate x by quantity y.
Multiplication Counting Principal: xy is the number of ways of making a first choice followed by a second choice, if the firest choice can be made in x ways and the second can be made in y ways.
The product of three numbers xyz may be the volume of a box with dimensions, x, y, and z. The product of an odd number of negative numbers is negative; the product of and even number of negative numbers is positive. The product of the integers from 1 to n, written n!, is the number of ways of arranging n objects. The commutative and associative properties allow you to rearrange multiplication expressions.
The numbers 1, 1, and -1 are special in multiplication. Multiplying any number by zero gives the same result: 0. For this reason, equations of the form 0x = b are either true for all real numbers or for none of them. Multiplying any number by 1 yields that number. A conversion factor is the number 1 written using different units, so multiplying by it does not change the value of the quantity. Multiplying any number by -1 changes it to its opposite.
Because of the may uses of multiplication, equations of the form ax = b and inequalities of the form ax < b are quite common. When a ≠ 0, such sentences can be solved by multiplying both sides by the number 1/a, the reciprocal of a. Remember to change the sense of the inequality if a is negative.
Chapter 3 - Addition in Algebra | Back to Top |
The properties of addition can be verified through their uses. For example, putting together quantities in a different order yields the same sum, so addition is commutative.
Graphing provides a picture that can help clarify solutions to a problem or trends in data. If two quantities are being considered, a coordinate graph in a plane can show both values. A two-dimensional slide can be represented as a combination of a horizantal and vertical slide of a coordinate graph.
Sentences of the forms ax + b = c or ax + b < c (in which a ≠ 0) combine multiplication and addition. To solve this type of sentence, first add -b to both sides, and then multiply both sides by 1/a. Sometimes it is necessary to simplify expressions on either side of the inequality first.
When addition and multiplication occur in the same expression, the Distributive Property may provide a link between the operations. It tells how to change between a form where the multiplication is done first, ac + bc, and one where the addition is done first, (a + b)c. Forms of the Distributive Property are used to add fractions, to add like terms, and to remove parentheses from expressions.
Chapter 4 - Subtraction in Algebra | Back to Top |
The Algebraic Definition of Subtraction is in terms of addition: a - b = a + -b. This definition enables you to solve subtraction sentences of the form ax + b = c or ax + b < c by converting them to addition. Such sentences arise from the uses of subtraction. The two major models for subtraction are take-away and comparison. Discounts apply the take-away model. Finding a range of data set is an application of the comparison model.
When an equation has two variables, its solutions may be graphed by making a table of values and plotting the resulting points on a coordinate plane.
Several important geometry properties involve sums and differences. The Triangle Inequality leads to the fact that if two sides of a triangle have lengths x and y, where y > x, the length of the third side is between y - x and y + x. Two angles whose measures add up to 180o are supplementary; two angles whose measures add up to 90o are complementary. The three angle measures of a triangle must equal 180o.
Any of the properties in this and previous chapters can be illustrated on a spreadsheet. Formulas relating the cells of a spreadsheet make spreadsheets powerful tools for storing and obtaining information.
Chapter 5 - Linear Sentences | Back to Top |
The next five lessons of the chapter deal with ways to compare two quantities described by linear expressions such as ax + b and cx + d. One way to compare them is to make a table that shows the value of each expression for different values of x. This can be done with paper and pencil or with a spreadsheet. Questions can then be answered by examining the numbers that appear in the chart.
A second model of comparison is to write an equation to describe each pattern, in the form y = ax + b and y = cx + d. Graph the two equations, and examine where the two graphs cross or where one is below the other. Graphing can be done with paper and pencil or with an automatic grapher.
The third technique discussed is to solve an equation or inequality. To find when the two quantities are equal, solve ax + b = cx + d. To find when the first is less than the second, solve ax + b < cx + d. Solving sentences has the advantage of yielding exact solutions.
The last lessons discuss special techniques involved in solving sentences. Many formulas of equations that contain more than one variable can be "solved" for any of the variables in them. The process is similar to that for equations with just one variable. To solve a sentence containing fractions, find the least common multiple of the denominators, and then multiply each term by it. If the numbers in a sentence are all large, then multiply both sides by a small fraction.
Chunking is a problem-solving technique by which an expression is considered as a single number. Many complicated expressions and their equations can be handled by recognizing their similarities with simpler patterns.
Chapter 6 - Division in Algebra | Back to Top |
Rates and ratios are models for division. A rate compares quantities with different units; ratios compare quantities with the same units. A statement that two fractions are equal is called a proportion. The Means-Extremes Propertycan be used to find missing values in a proportion. One important place proportions appear is in similar figures.
Percent, relative frequency, probability in geometry, and size changes are applications involving rates and ratios. It is possible to translate most percent problems to equations of the form ab = c. Solving the equation then gives an answer to the problem. Since it is impossible to count the infinite number of points in a geometric region, a ratio of lengths or areas is used to compute probabilities in geometric situations. Size changes yield similar figures. If the magnitude is k and k > 1 or k < -1, then the size change is an expansion. If -1 < k < 1, then size change is a contraction.
Chapter 7 - Slopes and Lines | Back to Top |
Constant-increase or constant-decrease situations lead naturally to linear equations of the form y = mx + b. The graph of the set of points (x, y) satisfying this equation is a line with slope m and y-intercept b.
Other situations lead to linear equations in the standard form Ax + By = C. When the = sign in equations of either form is replaced by < or >, the graph of the resulting linear inequality is a half-plane, the set of points on one side of a line. (**Plug (0, 0) into the inequality to see which side to shade.)
A line is determined by any point on it and its slope, and its equation can be found from this information. Likewise, an equation can be found for the line containing two given points. If more than two points are given, then there may be more than one line determined. You can then draw a line that comes close to all the points, and use these points to determine an equation for the line.
Chapter 8 - Exponents and Powers | Back to Top |
| Property Name | Form |
|---|---|
| Zero Exponent | x0 = 1 |
| Negative Exponent | x-1 = 1/xn |
| Product of Powers | xm • xn = xm + n |
| Quotient of Powers | xm/xn = xm - n |
| Power of a Power | (xm)n = xmn |
| Power of a Product | (xy)n = xnyn |
| Power of a Quotient | (x/y)n = xn/yn |
The expression xn can also be the growth model in a period of length n, when the growth factor in each unit period is x. Important applications of exponential growth and decay are population growth and compound interest. In compound interest, the growth factr is the quantity 1 + i, where i is the annual yield. So at an annual yield of i, after n years and amount P grows to P(1 + i)n.
When the growth factor is between 0 and 1, the amount gets smaller, and exponential decay occurs. Graphs of exponential growth or decay are curves.
| Exponential Growth | Exponential Decay |
|---|---|
| y = b • gx, g > 1 | y = b • gx, 0 < g < 1 |
The growth model allows xn to be interpreted when n is not a positive integer. The number x0 is the growth factor for a period of length 0, so x0 is the identity under multiplication. Thus x0 = 1. The number x-n is a growth factor going back in time, and x-n = 1/xn
Chapter 9 - Quadratic Equations and Square Roots | Back to Top |
To determine where this parabola crosses the horizantal line y = k, solve ax2 + bx + c = k.
A quadratic equation (in one variable) is an equation that can be written in the form ax2 + bx + c = 0, where a ≠ 0. The solutions to the equation can be found by the Quadratic Formula, x = (-b ± √b2 - 4ac)/2a
The discriminant of the quadratic equation ax2 + bx + c = 0 is b2 - 4ac. If the discriminant is positive, there are two real solutions; if it is zero, there is one solution; if it is negative, there are no real solutions.
Solutions to quadratic equations involve square roots. Rewriting a square root so it is either a whole number or a product of a whole number and a smaller radical is called simplifying the radical. You can multiply or simplify square roots by using the Product of Square Roots Property.
| If a ≥ 0 and b ≥ 0, then √a • √b = √ab. |
| Number Line | Pythagorean Distance Formula for the Coordinate Plane |
|---|---|
| d = |x2 - x1| | d = √|x2 - x1|2 + |y2 - y1|2 |
Square roots and absolute value are related by the property, for all x, √x2 = |x|.
Chapter 10 - Polynomials | Back to Top |
Polynomials emerge from a variety of situations. Our customary way of writing whole numbers in base 10 can be considered as a polynomial with 10 substituted for the variable. If different amounts of money are invested each year at a scale factor x, the total amount after several years is a polynomial in x. When the dimensions of a geometric figure are given as linear expressions, then areas or volumes related to the figure may be polynomials.
Addition and subtraction of polynomials are based on the Like Terms form of the Distributive Property, which you studied earlier in this book. Multiplication of polynomials is also justified by the Distributive Property. To multiply one polynomial by a second, multiply each term in the second, then add the products. For instance:
| monomial by a polynomial: |
| a(x + y + z) = ax + ay + az |
| two polynomials: |
| (a + b + c)(x + y + z) = ax + ay + az + bx + by + bz + cx + cy + cz |
| two binomials: |
| (a + b)(c + d) = ac + ad + bc + bd |
| perfect square patterns: |
| (a + b)2 = (a + b)(a + b) = a2 + 2ab + b2 |
| (a - b)2 = (a - b)(a - b) = a2 - 2ab + b2 |
| difference of two squares |
| (a + b)(a - b) = a2 - b2 |
The square of the difference of actual and expected values in an experiment, (a - e)2, appears in the calculation of the chi-square statistic. ("chi" is pronounced "ky" as in "sky") This statistic can help you to decide whether the assumptions that led to the expected values are correct.
Chapter 11 - Linear Systems | Back to Top |
One way to solve a system is by graphing. By looking for the intersection point(s) on a graph you can quickly tell if there are any solutions to the system. There are as many solutions as intersection points. Graphing is also a way to describe solutions to systems that have infinitely many solutions. For instance, overlapping half-planes, which arise from systems of linear inequalities, and coincident lines have infinitely many solutions.
However, graphing does not always yeild exact solutions. In this chapter, three strategies are presented for finding exact solutions to systems of linear equations. They are substitution, addition, and multiplication. Substitution is a good method to use if at least one equation is given in y = mx + b form. Addition is appropriate if the same term has opposite signs in the two equations in the system. Multiplication is a good method when both equations are in Ax + By = C form. Each method changes the system into an equivalent system whose solutions are the same as those of the original system.
Any kind of situation that lead to a linear equation can lead to a linear system. All that is needed is more than one condition which must be satisfied.
Chapter 12 - Factoring | Back to Top |
If an integer is a perfect square, then it is the prodeuct of an even number of primes. From this fact it can be deduced that vertain square roots cannot be written as simple fractions. Consequently, such square roots are irrational numbers. More generally, any number that cannot be written as a finite or and infinitely repeating decimal is an irrational number.
If each of two integers has a common factor, then so does their sum, and the common factor can be factored out using the Distributive Property ab + ac = a(b + c). SImilarly, if each of the terms of a polynomial has a commonn monomial factor then so does their sum, and it can be factored out using the Distributive Property.
Factoring the general quadratic trinomial ax2 + bx + c is more difficult, and may require trial and error procedures. HOwever, when a = 1, then the quadratic is relatively easy to factor. Specifically, x2 + bx + c can be factored into (x + p)(x + q) provided there are integers p and q such that p + q = b and pq = c. In general, ax2 + bx + c can be factored over the integers if and only if its discriminant, b2 - 4ac, is a perfect square.
The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. Thus, if ax2 + bx + c = 0 and ax2 + bx + c can be factored , then the solutinos of the equation can be found quickly by setting each factor equal to zero and solving these simpler equations. This is one of the maor uses for factoring, even though most quadratic expressions do not factor over the integers.
About 3700 years ago, Babylonian scribes showe how to solve certain quadratics. They used what we today call substitution to convert a quadratic equation into one of the form x2 = k. In the 18th century, a similar idea was used to derive the Quadratic Formula we know today.
Chapter 13 - Functions | Back to Top |
If a function ƒ contains the ordered pair (a, b), then we write ƒ(a) = b. We say that b is the value of the function at a. The set of possible values of a is the domain of the function. The set of possible values of b is the range of the function. If a and b are real numbers, then the function can be graphed on the coordinate plane and values of the function can be approximated by reading the graph. An automatic grapher saves time when investigating complicated functions or multiple functions. Though convenient, it is not necessary to use ƒ(x) notation for functions; y is often used to stand for the second coordinate. Many of the graphs you studied in earlier chapters describe functions.
| Equation | Graph | Type of Function |
|---|---|---|
| f(x) = mx + b | line | linear |
| f(x) = ax2 + bx + c | parabola | quadratic |
| f(x) = abx | exponential curve | exponential |
| f(x) = |x| | angle | absolute value |
| f(x) = axn + bxn-1 + ... + d | (varied) | polynomial |
In a probability function the domain is a set of outcomes in a situation and the range is the set of probabilities of these outcomes.
An important use of calculators is to evaluate functions at various values of their domains. Trigonometric functions, such as tangent, sine, and cosine, have their own keys on most calculators: tan, sin, cos. The tangent function can be used to determine lengths of sides and measures of angles in right triangles. Other functions on your calculator may include the squaring function x2, factorial function !, and common logarithmic function log. Computer languages often build in functions such as SQR(X) and ABS(X) in BASIC.
All of these are the unabridged chapter summaries from the UCSMP Algebra book, © 1998 by Scott Foresman and Company