California Algebra I - Hillcrest Math

456 Pages·2008·5.71 MB·English

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California
Algebra I
Student Textbook
California Standards-Driven Program
iiiii
Contents
This Textbook provides comprehensive coverage of all the California Algebra I Standards. The Textbook is divided
into eight Chapters. Each of the Chapters is broken down into small, manageable Topics and each Topic covers a
specific Standard or part of a Standard.
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Topic 1.1.1 The Basics of Sets ............................................................................................................... 2
1.0 Topic 1.1.2 Subsets of the Real Numbers .............................................................................................. 5
Topic 1.1.3 Unions and Intersections...................................................................................................... 7
Topic 1.1.4 Algebraic and Numeric Expressions..................................................................................... 9
Topic 1.1.5 Coefficients and Evaluating.................................................................................................. 12
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Topic 1.2.1 The Number System............................................................................................................. 14
Topic 1.2.2 Identities and Inverses.......................................................................................................... 16
Topic 1.2.3 The Number Line and Absolute Values................................................................................ 18
1.0
Topic 1.2.4 Addition and Multiplication.................................................................................................... 21
2.0
Topic 1.2.5 Subtraction and Division....................................................................................................... 24
Topic 1.2.6 Order of Operations.............................................................................................................. 27
Topic 1.2.7 Properties of Real Numbers................................................................................................. 30
Topic 1.2.8 More on Multiplication........................................................................................................... 32
Topic 1.2.9 The Real Number Axioms..................................................................................................... 34
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Topic 1.3.1 Exponent Laws..................................................................................................................... 37
Topic 1.3.2 Square Roots........................................................................................................................ 40
2.0 Topic 1.3.3 Properties of Roots .............................................................................................................. 43
Topic 1.3.4 Equivalent Fractions ............................................................................................................. 46
Topic 1.3.5 Multiplying and Dividing Fractions ........................................................................................ 49
Topic 1.3.6 Adding and Subtracting Fractions......................................................................................... 52
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24.0 Topic 1.4.1 Mathematical Proofs............................................................................................................. 55
25.0 Topic 1.4.2 Inductive and Deductive Reasoning...................................................................................... 58
Topic 1.4.3 Algebraic Statements............................................................................................................ 61
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Counting Collections..................................................................................................................................... 65
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Topic 2.1.1 Simplifying Algebraic Expressions........................................................................................ 67
4.0
Topic 2.1.2 Getting Rid of Grouping Symbols......................................................................................... 70
Topic 2.1.3 More Simplifying and Checking Answers ............................................................................. 73
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4.0
Topic 2.2.1 Properties of Equality........................................................................................................... 76
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Topic 2.3.1 Removing Fractions.............................................................................................................. 81
4.0
Topic 2.3.2 Fractional Coefficients in Algebraic Expressions.................................................................. 84
Topic 2.3.3 Eliminating Decimal Coefficients.......................................................................................... 86
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4.0
Topic 2.4.1 Applications of Linear Equations.......................................................................................... 89
5.0
Topic 2.4.2 Coin Tasks............................................................................................................................ 92
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4.0
Topic 2.5.1 Consecutive Integer Tasks.................................................................................................... 95
5.0
Topic 2.5.2 Age-Related Tasks................................................................................................................ 99
15.0
Topic 2.5.3 Rate, Time, and Distance Tasks........................................................................................... 103
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Topic 2.6.1 Investment Tasks.................................................................................................................. 108
15.0
Topic 2.6.2 Mixture Tasks........................................................................................................................ 113
Topic 2.6.3 More Mixture Tasks............................................................................................................... 116
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15.0 Topic 2.7.1 Work Rate............................................................................................................................. 121
Topic 2.7.2 Combined Work Rate ........................................................................................................... 124
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3.0 Topic 2.8.1 Absolute Value Equations..................................................................................................... 128
Topic 2.8.2 More on Absolute Value........................................................................................................ 130
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Wildlife Park Trains........................................................................................................................................ 135
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Topic 3.1.1 Inequalities............................................................................................................................ 137
4.0
Topic 3.1.2 Addition and Subtraction Properties of Inequalities ............................................................. 139
5.0
Topic 3.1.3 Multiplication and Division Properties of Inequalities........................................................... 142
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Contents
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4.0
Topic 3.2.1 Multistep Inequalities............................................................................................................ 147
5.0
Topic 3.2.2 Applications of Inequalities................................................................................................... 150
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Topic 3.3.1 Compound Inequalities......................................................................................................... 155
5.0
3.0 SSSSSeeeeeccccctttttiiiiiooooonnnnn 33333.....44444 ————— AAAAAbbbbbsssssooooollllluuuuuttttteeeee VVVVVaaaaallllluuuuueeeee IIIIInnnnneeeeeqqqqquuuuuaaaaallllliiiiitttttiiiiieeeeesssss
Topic 3.4.1 Absolute Value Inequalities................................................................................................... 158
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Mailing Packages.......................................................................................................................................... 162
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Topic 4.1.1 The Coordinate Plane........................................................................................................... 164
6.0 Topic 4.1.2 Quadrants of the Plane ........................................................................................................ 167
Topic 4.1.3 Lines..................................................................................................................................... 170
Topic 4.1.4 Horizontal and Vertical Lines................................................................................................ 173
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Topic 4.2.1 Points on a Line.................................................................................................................... 175
Topic 4.2.2 Graphing Ax + By = C........................................................................................................... 177
6.0
Topic 4.2.3 Using Line Equations............................................................................................................ 179
7.0
Topic 4.2.4 The x- and y-intercepts ......................................................................................................... 182
Topic 4.2.5 Graphing Lines ..................................................................................................................... 186
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7.0 Topic 4.3.1 Slope of a Line ..................................................................................................................... 189
Topic 4.3.2 Point-Slope Formula............................................................................................................. 194
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Topic 4.4.1 Parallel Lines ........................................................................................................................ 197
8.0 Topic 4.4.2 Perpendicular Lines.............................................................................................................. 201
Topic 4.4.3 The Slope-Intercept Form of a Line...................................................................................... 205
Topic 4.4.4 More About Slopes............................................................................................................... 209
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6.0 Topic 4.5.1 Regions Defined by Inequalities ........................................................................................... 212
Topic 4.5.2 Borders of Regions............................................................................................................... 217
9.0
Topic 4.5.3 Regions Defined by Two Inequalities.................................................................................... 222
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Tree Growth .............................................................................................................................................. 227
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Topic 5.1.1 The Graphing Method........................................................................................................... 229
Topic 5.1.2 The Substitution Method....................................................................................................... 232
9.0 Topic 5.1.3 Comparing the Graphing and Substitution Methods............................................................. 234
Topic 5.1.4 Inconsistent Systems............................................................................................................ 237
Topic 5.1.5 Dependent Systems ............................................................................................................. 239
Topic 5.1.6 Systems of Equations — Further Examples ........................................................................ 241
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9.0
Topic 5.2.1 Systems of Equations — Elimination................................................................................... 244
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Topic 5.3.1 Systems of Linear Equations — Applications ...................................................................... 249
9.0
Topic 5.3.2 Systems of Linear Equations — Integer Problems............................................................... 253
15.0
Topic 5.3.3 Systems of Linear Equations — Percent Mix Problems....................................................... 255
Topic 5.3.4 Systems of Linear Equations — Rate Problems.................................................................. 258
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Breaking Even in an Egg Business............................................................................................................... 261
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Topic 6.1.1 Polynomials........................................................................................................................... 263
2.0
Topic 6.1.2 Adding Polynomials .............................................................................................................. 266
10.0
Topic 6.1.3 Subtracting Polynomials ....................................................................................................... 269
Topic 6.1.4 Adding and Subtracting Polynomials.................................................................................... 272
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2.0 Topic 6.2.1 Rules of Exponents.............................................................................................................. 274
10.0 Topic 6.2.2 Polynomial Multiplication....................................................................................................... 276
Topic 6.2.3 Polynomial Multiplication - Area and Volume........................................................................ 280
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Topic 6.3.1 Division by Monomials.......................................................................................................... 283
Topic 6.3.2 Polynomials and Negative Powers........................................................................................ 285
2.0
10.0 Topic 6.3.3 Division by Polynomials — Factoring.................................................................................... 288
Topic 6.3.4 Division by Polynomials — Long Division............................................................................. 291
Topic 6.3.5 Division by Polynomials — Applications............................................................................... 294
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10.0 Topic 6.4.1 Special Products of Two Binomials ...................................................................................... 297
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Contents
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11.0 Topic 6.5.1 Factors of Monomials........................................................................................................... 302
Topic 6.5.2 Simple Factors of Polynomials ............................................................................................. 305
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11.0 Topic 6.6.1 Factoring Quadratics............................................................................................................. 310
Topic 6.6.2 Factoring Quadratics — ax2 + bx + c...............................................................................................313
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11.0
Topic 6.7.1 Factoring Quadratic Expressions in Two Variables............................................................... 318
Topic 6.7.2 Factoring Third-Degree Polynomials .................................................................................... 322
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11.0 Topic 6.8.1 The Difference of Two Squares............................................................................................ 325
Topic 6.8.2 Perfect Square Trinomials..................................................................................................... 327
Topic 6.8.3 Factoring by Grouping .......................................................................................................... 329
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Pascal’s Triangle........................................................................................................................................... 331
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11.0
Topic 7.1.1 Solving Quadratic Equations by Factoring............................................................................ 333
14.0
Topic 7.1.2 Quadratic Equations — Taking Square Roots...................................................................... 338
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Topic 7.2.1 Completing the Square......................................................................................................... 342
14.0
Topic 7.2.2 More on Completing the Square........................................................................................... 347
Topic 7.2.3 Solving Equations by Completing the Square ...................................................................... 351
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19.0
Topic 7.3.1 The Quadratic Formula......................................................................................................... 355
20.0
Topic 7.3.2 Applications of Quadratics ................................................................................................... 359
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21.0 Topic 7.4.1 Graphs of Quadratic Functions ............................................................................................ 362
22.0 Topic 7.4.2 Drawing Graphs of Quadratic Functions.............................................................................. 365
Topic 7.4.3 Quadratic Graphs and Completing the Square..................................................................... 368
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22.0
Topic 7.5.1 Quadratic Equations and the Discriminant ........................................................................... 372
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23.0 Topic 7.6.1 Motion Tasks......................................................................................................................... 379
Topic 7.6.2 Economic Tasks.................................................................................................................... 383
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The Handshake Problem .............................................................................................................................. 386
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12.0 Topic 8.1.1 Fractions and Rational Expressions ..................................................................................... 388
Topic 8.1.2 Equivalent Fractions ............................................................................................................. 390
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13.0 Topic 8.2.1 Multiplying Rational Expressions.......................................................................................... 394
Topic 8.2.2 Dividing Rational Expressions.............................................................................................. 397
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13.0 Topic 8.3.1 Fractions with Identical Denominators.................................................................................. 403
Topic 8.3.2 Fractions with Different Denominators.................................................................................. 406
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13.0 Topic 8.4.1 Solving Fractional Equations ................................................................................................ 412
Topic 8.4.2 Applications of Fractional Equations.................................................................................... 417
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16.0 Topic 8.5.1 Relations............................................................................................................................... 420
17.0 Topic 8.5.2 Functions.............................................................................................................................. 424
18.0 Topic 8.5.3 Function Notation.................................................................................................................. 429
Topic 8.5.4 More on Functions................................................................................................................ 431
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Transforming Functions................................................................................................................................. 435
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Glossary .............................................................................................................................................. 438
Formula Sheet .............................................................................................................................................. 440
Index .............................................................................................................................................. 442
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California Algebra I Standards
The following table lists all the California Mathematics Content Standards for Algebra I with cross references to where each
Standard is covered in this Textbook. Each Topic begins by quoting the relevant Standard in full, together with a clear and
understandable objective. This will enable you to measure your progression against the California Algebra I Standards as you
work your way through the Program.
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Students identify and use the arithmetic properties of subsets of integers and rational, CCCCChhhhhaaaaapppppttttteeeeerrrrr 11111
1.0 irrational, and real numbers, including closure properties for the four basic arithmetic
operations where applicable:
1.1 Students use properties of numbers to demonstrate whether assertions are true or false. CCCCChhhhhaaaaapppppttttteeeeerrrrr 11111
Students understand and use such operations as taking the opposite, finding the CCCCChhhhhaaaaapppppttttteeeeerrrrrsssss 11111,,,,, 66666
2.0 reciprocal, taking a root, and raising to a fractional power. They understand and use the
rules of exponents.
3.0 Students solve equations and inequalities involving absolute values. CCCCChhhhhaaaaapppppttttteeeeerrrrrsssss 22222,,,,, 33333
Students simplify expressions before solving linear equations and inequalities in one CCCCChhhhhaaaaapppppttttteeeeerrrrrsssss 22222,,,,, 33333
4.0
variable, such as 3(2x – 5) + 4(x – 2) = 12.
Students solve multistep problems, including word problems, involving linear equations and CCCCChhhhhaaaaapppppttttteeeeerrrrrsssss 22222,,,,, 33333
5.0
linear inequalities in one variable and provide justification for each step.
Students graph a linear equation and compute the x- and y-intercepts (e.g., graph CCCCChhhhhaaaaapppppttttteeeeerrrrr 44444
6.0 2x + 6y = 4). They are also able to sketch the region defined by linear inequalities
(e.g., they sketch the region defined by 2x + 6y < 4).
Students verify that a point lies on a line, given an equation of the line. Students are able CCCCChhhhhaaaaapppppttttteeeeerrrrr 44444
7.0 to derive linear equations by using the point-slope formula.
Students understand the concepts of parallel lines and perpendicular lines and how their CCCCChhhhhaaaaapppppttttteeeeerrrrr 44444
8.0 slopes are related. Students are able to find the equation of a line perpendicular to a given
line that passes through a given point.
Students solve a system of two linear equations in two variables algebraically and are able CCCCChhhhhaaaaapppppttttteeeeerrrrrsssss 44444,,,,, 55555
9.0 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.
Students add, subtract, multiply, and divide monomials and polynomials. Students solve CCCCChhhhhaaaaapppppttttteeeeerrrrr 66666
10.0 multistep problems, including word problems, by using these techniques.
Students apply basic factoring techniques to second- and simple third-degree polynomials. CCCCChhhhhaaaaapppppttttteeeeerrrrrsssss 66666,,,,, 77777
11.0 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.
Students simplify fractions with polynomials in the numerator and denominator by factoring CCCCChhhhhaaaaapppppttttteeeeerrrrr 88888
12.0
both and reducing them to the lowest terms.
Students add, subtract, multiply, and divide rational expressions and functions. Students CCCCChhhhhaaaaapppppttttteeeeerrrrr 88888
13.0 solve both computationally and conceptually challenging problems by using these
techniques.
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14.0 Students solve a quadratic equation by factoring or completing the square. CCCCChhhhhaaaaapppppttttteeeeerrrrr 77777
Students apply algebraic techniques to solve rate problems, work problems, and percent CCCCChhhhhaaaaapppppttttteeeeerrrrrsssss 22222,,,,, 55555
15.0
mixture problems.
Students understand the concepts of a relation and a function, determine whether a given CCCCChhhhhaaaaapppppttttteeeeerrrrr 88888
16.0 relation defines a function, and give pertinent information about given relations and
functions.
Students determine the domain of independent variables and the range of dependent CCCCChhhhhaaaaapppppttttteeeeerrrrr 88888
17.0
variables defined by a graph, a set of ordered pairs, or a symbolic expression.
Students determine whether a relation defined by a graph, a set of ordered pairs, or a CCCCChhhhhaaaaapppppttttteeeeerrrrr 88888
18.0
symbolic expression is a function and justify the conclusion.
Students know the quadratic formula and are familiar with its proof by completing the CCCCChhhhhaaaaapppppttttteeeeerrrrr 77777
19.0
square.
Students use the quadratic formula to find the roots of a second-degree polynomial and to CCCCChhhhhaaaaapppppttttteeeeerrrrr 77777
20.0
solve quadratic equations.
21.0 Students graph quadratic functions and know that their roots are the x-intercepts. CCCCChhhhhaaaaapppppttttteeeeerrrrr 77777
Students use the quadratic formula or factoring techniques or both to determine whether CCCCChhhhhaaaaapppppttttteeeeerrrrr 77777
22.0
the graph of a quadratic function will intersect the x-axis in zero, one, or two points.
Students apply quadratic equations to physical problems, such as the motion of an object CCCCChhhhhaaaaapppppttttteeeeerrrrr 77777
23.0
under the force of gravity.
24.0 Students use and know simple aspects of a logical argument:
Students explain the difference between inductive and deductive reasoning and identify and
24.1
provide examples of each.
24.2 Students identify the hypothesis and conclusion in logical deduction.
Students use counterexamples to show that an assertion is false and recognize that a
24.3
single counterexample is sufficient to refute an assertion.
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Students use properties of the number system to judge the validity of results, to justify each
25.0 CCCCChhhhhaaaaapppppttttteeeeerrrrr 11111
step of a procedure, and to prove or disprove statements:
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Students use properties of numbers to construct simple, valid arguments (direct and PPPPPrrrrrooooogggggrrrrraaaaammmmm
25.1
indirect) for, or formulate counterexamples to, claimed assertions.
Students judge the validity of an argument according to whether the properties of the real
25.2
number system and the order of operations have been applied correctly at each step.
Given a specific algebraic statement involving linear, quadratic, or absolute value
25.3 expressions or equations or inequalities, students determine whether the statement is true
sometimes, always, or never.
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