Common Core State Standards
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Trusted Content
Common Core State Standards
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ith American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy. — Common Core State Standards Initiative
What is the goal of the Common Core State Standards? The mission of the Common Core State Standards is to provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that students need for success in college and careers.
Who wrote the standards? The National Governors Association Center for Best Practices and the Council of Chief State School Officers worked with representatives from participating states, a wide range of educators, content experts, researchers, national organizations, and community groups.
quations uations of Linear near Functions nctions
At the high school level the Common Core State Standards are organized by conceptual category. To ease implementation four model course pathways were created: traditional, integrated, accelerated traditional and accelerated integrated. Glencoe Algebra 1, Glencoe Geometry, and Glencoe Algebra 2 follow the traditional pathway.
What are the major points of the standards? The standards seek to develop both students’ mathematical understanding and their procedural skill. The Standards for Mathematical Practice describe varieties of expertise that mathematics teachers at all levels should seek to develop in their students. The Standards for Mathematical Content define what students should understand and be able to do at each level in their study of mathematics.
Mathematical Content Unit 2
Linear Relationships
EEq qquatiion of Linear Equations Liiinear Linear near earr FFunctions Functions Mathematical Content
Get Ready for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Explore: Graphing Technology Lab Investigating Slope-Intercept Form . . . . . . . . . . . . 215
How do I implement the standards?
4-1 Graphing Equations in Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . 216
F.IF.7a, S.ID.7
Extend: Graphing Technology Lab The Family of Linear Graphs. . . . . . . . . . . . . . . . . . 224
F.BF.3, S.ID.7
4-2 Writing Equations in Slope-Intercept Form. . . . . . . . . . . . . . . . . . . . . . . . . . . 226
F.BF.1, F.LE.2
4-3 Writing Equations in Point-Slope Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
F.IF.2, F.LE.2
4-4 Parallel and Perpendicular Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
F.LE.2, S.ID.7
Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 4-5 Scatter Plots and Lines of Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Extend: Algebra Lab Correlation and Causation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Lester Lefkowitz/Photographer’s Choice/Getty Images
The Common Core State Standards are shared goals and expectations for what knowledge and skills your students need to succeed. You as a teacher, in partnership with your colleagues, principals, superintendents, decide how the standards are to be met. Glencoe Algebra 1 is designed to help you devise lesson plans and tailor instruction to the individual needs of the students in your classroom as you meet the Common Core State Standards.
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S.ID.6a, S.ID.6c S.ID.9
4-6 Regression and Median-Fit Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
S.ID.6, S.ID.8
4-7 Inverse Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
A.CED.2, F.BF.4a
Extend: Algebra Lab Drawing Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
F.BF.4a
ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Preparing for Standardized Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Standardized Test Practice, Chapters 1–4 . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Virtual Manipulatives
Graphing Calculator
pp. 158, 248
pp. 169, 215
Foldables pp. 152, 214
Self-Check Practice pp. 207, 213
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Domain Names
Abbreviations
The R Reall N Number Th b SSystem t
RN
Quantities
Q
How do I decode the standards?
Seeing Structure in Expressions
SSE
This diagram provides clarity for decoding the standard identifiers.
Arithmetic with Polynomials and Rational Expressions
APR
Creating Equations
CED
A.REI.2 Conceptual Category A = Algebra N = Number and Quantity F = Functions S = Statistics and Probability
REI
Interpreting Functions
IF
Building Functions
BF
Linear, Quadratic, and Exponential Models
LE
Interpreting Categorical and Quantitative Data
ID
Variables and Expressions
Domain Standard
There are numerous tools for implementing the Common Core State Standards available throughout the program, including: Standards at point-of-use in the Chapter Planner and in each lesson of the Teacher Edition,
Reasoning with Equations and Inequalities
Complete standards coverage in Glencoe Algebra 1 ensures that you have all the content you need to teach the standards,
Correlations that show at a glance where each standard is addressed in Glencoe Algebra 1.
You can also visit connectED.mcgraw-hill.com to learn more about the Common Core State Standards. There you can choose from an extensive collection of resources to use when planning instruction.
nequalities Common Core State Standards
Common Core State Standar C C Content Standards A.REI.3 Solve linear equations andLab ib Algebra Al Alg lggebra including equations with coefficients Solving So S olving vi g IIn Inequalities n
1 Focus
Content Standards C A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
A ti it Activity
Step 1
Use a self-adhesive note to cover the equals sign on the equation mat. Then write a ≤ symbol on the note. Model the inequality.
Step 2
Since you do not want to solve for a negative x-tile, eliminate the negative x-tiles by adding 2 positive x-tiles to each side. Remove the zero pairs. -x
-x
≤
-x
1
1
x
1
1
-x
≤
1
1
1
1
x -2x ≤ 4
Step 3
-x
-1 -1
≤
1
-1
1
-1
1
-1
1
-1
x x
Step 4
x x
NewVocabulary
Use algebra tiles to solve each inequality. 4. -6x ≤ -12 {x | x ≥ 2}
5. In Exercises 1–4, is the coefficient of x in each inequality positive or negative? negative 6. Compare the inequality symbols and locations of the variable in Exercises 1–4 with those in their solutions. What do you find? See Ch. 5 Answer Appendix.
7. Model the solution for 3x ≤ 12. How is this different from solving -3x ≤ 12? See Ch. 5 Answer Appendix. 8. Write a rule for solving inequalities involving multiplication and division. (Hint: Remember that dividing by a number is the same as multiplying by its reciprocal.) See Ch. 5 Answer Appendix.
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3 Assess
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Formative Assessment Use Exercises 5 and 6 to assess whether students understand that when they multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes.
From Concrete to Abstract
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Exercise 8 asks students to generalize what happens to the inequality when multiplying or dividing by positive or negative numbers.
5:47 PM
If the x-tiles end up on the right side of the inequality, students may rotate the mat 180 degrees to read the inequality with the variable on the left side.
0291_ALG1_T_C05_EXP2_663924.indd 291
2 Teach
A term that does not have a variable is a constant term.
Scaffolding Questions Have students read the Why? section of the lesson.
xy
x·y
x(y)
(x)y
Ask: How do you find the cost of the hot dogs that Cassie and her friends eat? Multiply the number of hot dogs by $0.10.
(x)(y) xn base
exponent
Example 1 Write Verbal Expressions
a. 3x 4 three times x to the fourth power
b. 5z 2 + 16 5 times z to the second power plus sixteen
GuidedPractice 1A. 16u 2 - 3 16 times u to the second power minus 3
6b one half of a plus the 1 1B. _ a+_ 2 7 quotient of 6 times b and 7
What does the expression 0.10d stand for? 0.10 times d, the number of hot dogs
What other variable could you use to represent the number of hot dogs? Sample answer: h
1 Write Verbal Expressions
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Example 1 shows how to translate an algebraic expression into a verbal expression. 5
Lesson 1-1 Resources Resource
Chapter Resource Masters Other
Approaching Level AL
5
Teacher Edition
Differentiated Instruction, p. 6
Study Guide and Intervention, pp. 5–6 Skills Practice, p. 7 Practice, p. 8 Word Problem Practice, p. 9
On Level OL
Differentiated Instruction, pp. 6, 9
Study Guide and Intervention, pp. 5–6 Skills Practice, p. 7 Practice, p. 8 Word Problem Practice, p. 9 Enrichment, p. 10 5-Minute Check 1-1 Study Notebook
5/9/12 12:35 PMBL Beyond Level
Differentiated Instruction, p. 9
Practice, p. 8 Word Problem Practice, p. 9 Enrichment, p. 10
English Learners ELL
5-Minute Check 1-1 Study Notebook
5-Minute Check 1-1 Study Notebook
Differentiated Instruction, p. 6
Study Guide and Intervention, pp. 5–6 Skills Practice, p. 7 Practice, p. 8 Word Problem Practice, p. 9
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2x + 4
Write a verbal expression for each algebraic expression.
0005_0009_ALG1_S_C01_L1_663923.indd
Practice Have students complete Exercises 3–8.
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After Lesson 1-1 Evaluate algebraic expressions.
4cd ÷ 3mn
An expression like x n is called a power. The word power can also refer to the exponent. The exponent indicates the number of times the base is used as a factor. In an expression of the form x n, the base is x. The expression x n is read “x to the nth power.” When no exponent is shown, it is understood to be 1. For example, a = a 1.
4 Model with mathematics.
Once students have isolated the x-tiles, remind them to separate the 1-tiles into equal groups to correspond to the number of x-tiles.
p·q
In a multiplication expression, the quantities being multiplied are factors, and the result is the product. A raised dot or set of parentheses are often used to indicate a product. Here are several ways to represent the product of x and y.
Mathematical Practices
Make sure the inequality sign on the self-adhesive note is pointed in the correct direction to match the inequality.
6
The term that contains x or other letters is sometimes referred to as the variable term.
Jupiterimages/Comstock Images/Getty Images
3. -5x ≥ 15 {x | x ≤ -3}
Lesson 1-1 Write verbal expressions for algebraic expressions. Write algebraic expressions for verbal expressions.
A term of an expression may be a number, a variable, or a product or quotient of numbers and variables. For example, 0.10d, 2x and 4 are each terms.
A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
Working in Cooperative Groups
-2 ≤ x or x ≥ -2
z 3+_
2x + 4
0.10d
Content Standards
2 Teach
x
Model and Analyze 2. -4x > -4 {x | x < 1}
Before Lesson 1-1 Perform operations on integers.
1
Put students in groups of two or three and demonstrate the Activity. Have groups complete Exercises 1 and 2.
1. -3x < 9 {x | x > -3}
1 Focus VerticalAlignment
Cassie and her friends are at a baseball game. The stadium is running a promotion where hot dogs are $0.10 each. Suppose d represents represen the number of hot dogs Cassie and her friends fri eat. Then 0.10d represents the cost of the hot dogs they eat.
Write Verbal Expressions An algebraic expression consists of sums and/or products of numbers and variables. In the algebraic expression 0.10d, the letter d is called a variable. In algebra, variables are symbols used to represent unspecified numbers or values. Any letter may be used as a variable.
Common Core State Standards
x
-1 -1
Write algebraic expressions for verbal expressions.
algebraic expression variable term factor product power exponent base
algebra tiles, pp. 10–11 equation mat, p. 16
Teaching Tip Have students use a self-adhesive note to cover the equals sign on the equation mat. Write a ≤ symbol on the note. This will allow students to model inequalities with the equation mat and algebra tiles.
≤
-4 ≤ 2x
2
algebra tiles and equation mats self-adhesive blank notes
Teaching Algebra with Manipulatives Templates for:
Separate the tiles into 2 groups.
-1 -1
Write verbal expressions for algebraic expressions.
Easy to Make Manipulatives
-2x + 2x ≤ 4 + 2x
Add 4 negative 1-tiles to each side to isolate the x-tiles. Remove the zero pairs.
-1 x -1
You performed operations on integers.
Materials for Each Group
S l IInequalities Solve nequaliti nequalit lities ies
Solve -2x ≤ 4.
Lesson 1-1
A.SSE.1a InterpretV parts of and Variables ariables and Expressions an expression, such as Why? Then W hyy? ? terms, factors, andNow 1 coefficients.
Objective Use algebra tiles to model solving inequalities.
Common Core State Standards C
You can use algebra tiles to solve inequalities.
Content Standards
Explore 5-2
5-Minute Check 1-1 Study Notebook
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Trusted Content
Common Core State Standards Common Core State Standards, Traditional Algebra I Pathway, Correlated to Glencoe Algebra 1, Common Core Edition Lessons in which the standard is the primary focus are indicated in bold. Student Edition Lesson(s)
Standards
Student Edition Page(s)
Number and Quantity The Real Number System N-RN Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
7-3
406–413
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
7-3, 10-3, Extend 10-3, 10-4
406–413, 635–639, 640–641, 642–646
Use properties of rational and irrational numbers. 3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Extend 10-2
634
Quantities N-Q Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Throughout the text; for example, 2-6, 2-7, 2-8, 2-9, Extend 3-2, 4-5, 7-5
Throughout the text; for example, 111–117, 119–124, 126–131, 132–138, 169–170, 247–253, 424–429
2. Define appropriate quantities for the purpose of descriptive modeling.
Extend 2-6
118
3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Extend 1-3
23–24
Mathematical Modeling Standards
T12
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
Algebra Seeing Structure in Expressions A-SSE Interpret the structure of expressions 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients.
1-1, 1-4, 8-1, 9-1
5–9, 25–31, 465–471, 543– 553
1-2, 1-3, 9-7
10–15, 16–22, 598–605
2. Use the structure of an expression to identify ways to rewrite it.
1-1, 1-2, 1-3, 1-4, 7-1, 7-2, 7-3, 7-4, Explore 8-5, 8-5, Explore 8-6, 8-6, 8-7, 8-8, 8-9
5–9, 10–15, 16–22, 25–31, 391–397, 398–405, 406–413, 414–420, 493, 494–500, 501–502, 503–509, 510–515, 516–521, 522–529
Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines.
8-5, 8-6, 8-7, 8-8, 8-9
494–500, 503–509, 510–515, 516–521, 522–529
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
9-3, 9-4, Extend 9-4
564–571, 574–579, 580–581
c. Use the properties of exponents to transform expressions for exponential functions.
Extend 7-6
437
b. Interpret complicated expressions by viewing one or more of their parts as a single entity.
Arithmetic with Polynomials and Rational Expressions A-APR Perform arithmetic operations on polynomials. 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Explore 8-1, 8-1, 8-2, Explore 8-3, 8-3, 8-4
463–464, 465–471, 472–477, 478–479, 480–485, 486–491
Creating Equations A-CED Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems.
1-5, 2-1, 2-2, 2-3, 2-4, 2-5, 2-9, 3-2, 5-1, 5-2, 5-3, 5-4, 5-5, 7-6, 8-5, 8-6, 8-7, 9-4, 9-5, 10-4, 11-8
33–39, 75–80, 83–89, 91–96, 97–102, 103–109, 132–138, 163–168, 285–290, 292–297, 298–303, 306–311, 312–316, 432–436, 494–500, 503–509, 510–515, 574–579, 583–589, 642–646, 726–732
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Common Core State Standards
Continued
Student Edition Lesson(s)
Student Edition Page(s)
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Extend 1-7, 3-1, 3-4, 3-5, 3-6, 4-1, 4-2, 4-3, 4-4, 4-5, 4-6, 4-7, 6-1, Extend 6-1, 6-2, 6-3, 6-4, 6-5, 7-5, 7-6, 8-6, 8-7, 8-8, 9-1, 9-2, 9-4, 9-5, 10-1, Extend 10-1, 10-4, 11-2, 11-8
55, 155–162, 182–188, 189– 195, 197–202, 216–223, 226–232, 233–238, 239–245, 247–253, 255–262, 263–270, 335–341, 342–343, 344–349, 350–356, 357–362, 364–369, 424–429, 432–436, 503–509, 510–515, 516–521, 543–553, 555–560, 574–579, 583–589, 621–626, 627, 642–646, 684–689, 726–732
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
4-2, 5-6, 6-1, 6-2
226–232, 317–322, 335–341, 344–349
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
2-8, 2-9, 4-1
126–131, 132–138, 216–223
Standards
Reasoning with Equations and Inequalities A-REI Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
1-5, 2-2, 2-3, 2-4, 2-5, 2-6, 2-9, 8-6, 8-7, 8-9
33–39, 83–89, 91–96, 97–102, 102–109, 111–117, 132–138, 503–509, 510–515, 522–529
Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
1-5, Explore 2-2, 2-2, Explore 2-3, 2-3, 2-4, 2-5, 2-6, 2-7, 2-8, 2-9, 5-1, Explore 5-2, 5-2, 5-3, 5-4, 5-5, 7-3
33–39, 81–82, 83–89, 90, 91–96, 97–102, 103–109, 111–117, 119–124, 126–131, 132–138, 285–290, 291, 292–297, 298–303, 306–311, 312–316, 406–413
4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
9-4, 9-5, 10-2
574–579, 583–589, 628–633
8-6, 8-7, 8-8, 9-2, 9-4, Extend 9-4, 9-5
503–509, 510–515, 516–521, 555–560, 574–579, 580–581, 583–589
b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Mathematical Modeling Standards
T14
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
Solve systems of equations. 5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
6-4
357–362
6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
6-1, Extend 6-1, 6-2, 6-3, 6-4, 6-5, Extend 6-5
335–341, 342–343, 344–349, 350–356, 357–362, 364–369, 370–371
7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Extend 9-3
572–573
Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
1-6, 1-7, 3-1, 3-2, 3-4, 7-5, 9-1, 10-1
40–46, 47–54, 155–162, 163–168, 182–188, 424–429, 543–553, 621–626
11. Explain why the x-coordinates of the points where the graphs of the equations y = f (x ) and y = g (x ) intersect are the solutions of the equation f (x ) = g (x ); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f (x ) and/or g (x ) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Extend 6-1, Extend 7-5, Extend 9-3, 9-7, Extend 11-8
342–343, 430–431, 572–573, 598–605, 733–734
12. Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
5-6, Extend 5-6, 6-6, Extend 6-6
317–322, 323, 372–376, 377
Functions Interpreting Functions F-IF Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f (x ) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f (x ).
1-6, 1-7
40–46, 47–54
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
1-7, 3-6, 4-3, 7-5, 7-6, 9-1, 10-1
47–54, 197–202, 233–238, 424–429, 432–436, 543–553, 621–626
3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
3-5, 7-7, 7-8
189–195, 438–443, 445–450
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Common Core State Standards
Continued
Student Edition Lesson(s)
Student Edition Page(s)
Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
1-8, Explore 3-1, 3-1, Extend 4-1, 7-5, 9-1, 9-7, 10-1
56–61, 153–154, 155–162, 224–225, 424–429, 543–553, 598–605, 621–626
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
1-7, 7-5, 7-6, 9-1, 10-1
47–54, 424–429, 432–436, 543–553, 621–626
6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Explore 3-3, 3-3, Extend 7-7, Extend 9-1, 9-6
171, 172–180, 444, 554, 590–595
Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
3-1, 3-2, Extend 3-2, 3-4, 4-1, Extend 4-1, 9-1, 9-2, Explore 9-3, 9-3, Extend 9-3
155–162, 163–168, 169–170, 182–188, 216–223, 224–225, 543–553, 555–560, 562–563, 564–571, 572–573
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
9-7, Extend 9-7, 10-1, Extend 10-1
598–605, 606, 621–626, 627
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Explore 7-5, 7-5
422–423, 424–429
9-2, 9-3, 9-4, Extend 9-4
555–560, 564–571, 574–579, 580–581
7-1, 7-2, 7-5, 7-6, Extend 7-6
391–397, 398–405, 424–429, 432–436, 437
1-7, 3-6, 4-3, 7-8, 9-1, 9-3
47–54, 197–202, 233–238, 445–450, 543–553, 564–571
Standards
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Building Functions F-BF Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. Mathematical Modeling Standards
T16
1-7, 3-1, 3-4, 3-6, 4-1, 4-2, 4-3, 4-4, 4-5, 4-6, 4-7, 7-6, 7-8
47–54, 155–162, 182–188, 197–202, 216–223, 226–232, 233–238, 239–245, 247–253, 255–262, 263–270, 432–436, 445–450
4-2, 7-6, 9-3
226–232, 432–436, 564–571
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
3-5, 7-7, 7-8
189–195, 438–443, 445–450
Build new functions from existing functions. 3. Identify the effect on the graph of replacing f (x ) by f (x ) + k, kf (x ), f (kx ), and f (x + k ) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
Extend 4-1, Explore 7-5, Extend 7-6, Explore 9-3, 9-3, 10-1, Extend 10-1
224–225, 422–423, 437, 562–563, 564–571, 621–626, 627
4. Find inverse functions. a. Solve an equation of the form f (x ) = c for a simple function f that has an inverse and write an expression for the inverse.
4-7, Extend 4-7, Explore 10-1
263–270, 271, 619–620
Linear, Quadratic, and Exponential Models F-LE Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
Explore 3-3, 3-3, 3-5, 7-7, 9-6
171, 172–180, 189–195, 438–443, 590–595
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
3-5, 3-6, 7-6, 7-7, 9-6
189–195, 197–202, 432–436, 438–443, 590–595
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
7-6, 7-7, 9-6
432–436, 438–443, 590–595
2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
3-5, 3-6, 4-2, 4-3, 4-4, 4-5, 4-6, 7-5, 7-6, 7-7, 9-6, Extend 9-6
189–195, 197–202, 226–232, 233–238, 239–245, 247–253, 255–262, 424–429, 432–436, 438–443, 590–595, 596–597
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
9-6, Extend 9-6
590–595, 596–597
Interpret expressions for functions in terms of the situation they model. 5. Interpret the parameters in a linear or exponential function in terms of a context.
3-4, Explore 4-1, 4-1, Extend 4-1, 4-5, 7-5, 7-6, 7-7, 9-6
182–188, 215, 216–223, 224–225, 247–253, 424–429, 432–436, 438–443, 590–595
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Trusted Trusted Content Content
Common Core State Standards
Continued
Student Edition Lesson(s)
Standards
Student Edition Page(s)
Statistics and Probability Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
0-13, 12-3, 12-4
P40–P46, 764–770, 771–778
2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
12-2, 12-3, 12-4, Extend 12-8
757–763, 764–770, 771–778, 810–811
3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
12-3, 12-4
764–770, 771–778
Summarize, represent, and interpret data on two categorical and quantitative variables. 5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
Extend 12-7
801–802
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
4-5, 4-6, Extend 9-6
247–253, 255–262, 596–597
b. Informally assess the fit of a function by plotting and analyzing residuals.
4-6
255–262
c. Fit a linear function for a scatter plot that suggests a linear association.
4-5, 4-6
247–253, 255–262
Interpret linear models. 7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
4-1, Extend 4-1, 4-4, 4-5
216–223, 224–225, 239–245, 247–253
8. Compute (using technology) and interpret the correlation coefficient of a linear fit.
4-6
255–262
9. Distinguish between correlation and causation.
Extend 4-5
254
Mathematical Modeling Standards
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Correlation
Common Core State Standards for Mathematical Practice, Correlated to Glencoe Algebra 1, Common Core Edition 1. Make sense of problems and persevere in solving them. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 0-1, 1-8, 2-4, 3-4, 4-5, 5-4, 6-4, 7-5, 8-8, 9-3, 10-5, 11-1, and 12-4. 2. Reason abstractly and quantitatively. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-3, 2-1, 3-3, 4-1, 5-1, 6-5, 7-2, 8-5, 9-1, 10-3, 11-8, and 12-2. 3. Construct viable arguments and critique the reasoning of others. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-3, 2-5, Extend 3-5, 4-2, 5-5, 6-1, 7-4, 8-1, 9-2, 10-4, 11-2, and Extend 12-1. 4. Model with mathematics. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-1, 2-9, 3-2, 4-5, 5-1, 6-5, 7-6, 8-7, 9-7, 10-4, 11-7, and 12-5. 5. Use appropriate tools strategically. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Extend 1-7, 2-4, Extend 3-2, 4-4, Extend 5-6, Extend 6-1, Extend 7-5, 8-2, Extend 9-6, 10-6, Extend 11-8, and 12-3. 6. Attend to precision. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Extend 1-3, 2-8, 3-4, 4-2, 5-2, 6-6, 7-4, 8-9, 9-5, 10-1, 11-6, and 12-2. 7. Look for and make use of structure. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-2, 2-5, 3-6, Extend 4-1, 5-5, 6-3, 7-7, 8-6, 9-6, 10-2, 11-2 and 12-8. 8. Look for and express regularity in repeated reasoning. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-4, 2-7, 3-1, 4-1, 5-4, 6-1, 7-1, 8-4, 9-3, 10-2, 11-5 and 12-6.
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Trusted Content
Common Core State Standards
W
ith American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy. — Common Core State Standards Initiative
What is the goal of the Common Core State Standards? The mission of the Common Core State Standards is to provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that students need for success in college and careers.
Who wrote the standards? The National Governors Association Center for Best Practices and the Council of Chief State School Officers worked with representatives from participating states, a wide range of educators, content experts, researchers, national organizations, and community groups.
quations uations of Linear near Functions nctions
At the high school level the Common Core State Standards are organized by conceptual category. To ease implementation four model course pathways were created: traditional, integrated, accelerated traditional and accelerated integrated. Glencoe Algebra 1, Glencoe Geometry, and Glencoe Algebra 2 follow the traditional pathway.
What are the major points of the standards? The standards seek to develop both students’ mathematical understanding and their procedural skill. The Standards for Mathematical Practice describe varieties of expertise that mathematics teachers at all levels should seek to develop in their students. The Standards for Mathematical Content define what students should understand and be able to do at each level in their study of mathematics.
Mathematical Content Unit 2
Linear Relationships
EEq qquatiion of Linear Equations Liiinear Linear near earr FFunctions Functions Mathematical Content
Get Ready for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Explore: Graphing Technology Lab Investigating Slope-Intercept Form . . . . . . . . . . . . 215
How do I implement the standards?
4-1 Graphing Equations in Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . 216
F.IF.7a, S.ID.7
Extend: Graphing Technology Lab The Family of Linear Graphs. . . . . . . . . . . . . . . . . . 224
F.BF.3, S.ID.7
4-2 Writing Equations in Slope-Intercept Form. . . . . . . . . . . . . . . . . . . . . . . . . . . 226
F.BF.1, F.LE.2
4-3 Writing Equations in Point-Slope Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
F.IF.2, F.LE.2
4-4 Parallel and Perpendicular Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
F.LE.2, S.ID.7
Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 4-5 Scatter Plots and Lines of Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Extend: Algebra Lab Correlation and Causation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Lester Lefkowitz/Photographer’s Choice/Getty Images
The Common Core State Standards are shared goals and expectations for what knowledge and skills your students need to succeed. You as a teacher, in partnership with your colleagues, principals, superintendents, decide how the standards are to be met. Glencoe Algebra 1 is designed to help you devise lesson plans and tailor instruction to the individual needs of the students in your classroom as you meet the Common Core State Standards.
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S.ID.6a, S.ID.6c S.ID.9
4-6 Regression and Median-Fit Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
S.ID.6, S.ID.8
4-7 Inverse Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
A.CED.2, F.BF.4a
Extend: Algebra Lab Drawing Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
F.BF.4a
ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Preparing for Standardized Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Standardized Test Practice, Chapters 1–4 . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Virtual Manipulatives
Graphing Calculator
pp. 158, 248
pp. 169, 215
Foldables pp. 152, 214
Self-Check Practice pp. 207, 213
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Domain Names
Abbreviations
The R Reall N Number Th b SSystem t
RN
Quantities
Q
How do I decode the standards?
Seeing Structure in Expressions
SSE
This diagram provides clarity for decoding the standard identifiers.
Arithmetic with Polynomials and Rational Expressions
APR
Creating Equations
CED
A.REI.2 Conceptual Category A = Algebra N = Number and Quantity F = Functions S = Statistics and Probability
REI
Interpreting Functions
IF
Building Functions
BF
Linear, Quadratic, and Exponential Models
LE
Interpreting Categorical and Quantitative Data
ID
Variables and Expressions
Domain Standard
There are numerous tools for implementing the Common Core State Standards available throughout the program, including: Standards at point-of-use in the Chapter Planner and in each lesson of the Teacher Edition,
Reasoning with Equations and Inequalities
Complete standards coverage in Glencoe Algebra 1 ensures that you have all the content you need to teach the standards,
Correlations that show at a glance where each standard is addressed in Glencoe Algebra 1.
You can also visit connectED.mcgraw-hill.com to learn more about the Common Core State Standards. There you can choose from an extensive collection of resources to use when planning instruction.
nequalities Common Core State Standards
Common Core State Standar C C Content Standards A.REI.3 Solve linear equations andLab ib Algebra Al Alg lggebra including equations with coefficients Solving So S olving vi g IIn Inequalities n
1 Focus
Content Standards C A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
A ti it Activity
Step 1
Use a self-adhesive note to cover the equals sign on the equation mat. Then write a ≤ symbol on the note. Model the inequality.
Step 2
Since you do not want to solve for a negative x-tile, eliminate the negative x-tiles by adding 2 positive x-tiles to each side. Remove the zero pairs. -x
-x
≤
-x
1
1
x
1
1
-x
≤
1
1
1
1
x -2x ≤ 4
Step 3
-x
-1 -1
≤
1
-1
1
-1
1
-1
1
-1
x x
Step 4
x x
NewVocabulary
Use algebra tiles to solve each inequality. 4. -6x ≤ -12 {x | x ≥ 2}
5. In Exercises 1–4, is the coefficient of x in each inequality positive or negative? negative 6. Compare the inequality symbols and locations of the variable in Exercises 1–4 with those in their solutions. What do you find? See Ch. 5 Answer Appendix.
7. Model the solution for 3x ≤ 12. How is this different from solving -3x ≤ 12? See Ch. 5 Answer Appendix. 8. Write a rule for solving inequalities involving multiplication and division. (Hint: Remember that dividing by a number is the same as multiplying by its reciprocal.) See Ch. 5 Answer Appendix.
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3 Assess
0291_ALG1_S_C05_EXP2_663923.indd
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Formative Assessment Use Exercises 5 and 6 to assess whether students understand that when they multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes.
From Concrete to Abstract
291
5/9/12
Exercise 8 asks students to generalize what happens to the inequality when multiplying or dividing by positive or negative numbers.
5:47 PM
If the x-tiles end up on the right side of the inequality, students may rotate the mat 180 degrees to read the inequality with the variable on the left side.
0291_ALG1_T_C05_EXP2_663924.indd 291
2 Teach
A term that does not have a variable is a constant term.
Scaffolding Questions Have students read the Why? section of the lesson.
xy
x·y
x(y)
(x)y
Ask: How do you find the cost of the hot dogs that Cassie and her friends eat? Multiply the number of hot dogs by $0.10.
(x)(y) xn base
exponent
Example 1 Write Verbal Expressions
a. 3x 4 three times x to the fourth power
b. 5z 2 + 16 5 times z to the second power plus sixteen
GuidedPractice 1A. 16u 2 - 3 16 times u to the second power minus 3
6b one half of a plus the 1 1B. _ a+_ 2 7 quotient of 6 times b and 7
What does the expression 0.10d stand for? 0.10 times d, the number of hot dogs
What other variable could you use to represent the number of hot dogs? Sample answer: h
1 Write Verbal Expressions
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Example 1 shows how to translate an algebraic expression into a verbal expression. 5
Lesson 1-1 Resources Resource
Chapter Resource Masters Other
Approaching Level AL
5
Teacher Edition
Differentiated Instruction, p. 6
Study Guide and Intervention, pp. 5–6 Skills Practice, p. 7 Practice, p. 8 Word Problem Practice, p. 9
On Level OL
Differentiated Instruction, pp. 6, 9
Study Guide and Intervention, pp. 5–6 Skills Practice, p. 7 Practice, p. 8 Word Problem Practice, p. 9 Enrichment, p. 10 5-Minute Check 1-1 Study Notebook
5/9/12 12:35 PMBL Beyond Level
Differentiated Instruction, p. 9
Practice, p. 8 Word Problem Practice, p. 9 Enrichment, p. 10
English Learners ELL
5-Minute Check 1-1 Study Notebook
5-Minute Check 1-1 Study Notebook
Differentiated Instruction, p. 6
Study Guide and Intervention, pp. 5–6 Skills Practice, p. 7 Practice, p. 8 Word Problem Practice, p. 9
291
12-5-19
2x + 4
Write a verbal expression for each algebraic expression.
0005_0009_ALG1_S_C01_L1_663923.indd
Practice Have students complete Exercises 3–8.
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After Lesson 1-1 Evaluate algebraic expressions.
4cd ÷ 3mn
An expression like x n is called a power. The word power can also refer to the exponent. The exponent indicates the number of times the base is used as a factor. In an expression of the form x n, the base is x. The expression x n is read “x to the nth power.” When no exponent is shown, it is understood to be 1. For example, a = a 1.
4 Model with mathematics.
Once students have isolated the x-tiles, remind them to separate the 1-tiles into equal groups to correspond to the number of x-tiles.
p·q
In a multiplication expression, the quantities being multiplied are factors, and the result is the product. A raised dot or set of parentheses are often used to indicate a product. Here are several ways to represent the product of x and y.
Mathematical Practices
Make sure the inequality sign on the self-adhesive note is pointed in the correct direction to match the inequality.
6
The term that contains x or other letters is sometimes referred to as the variable term.
Jupiterimages/Comstock Images/Getty Images
3. -5x ≥ 15 {x | x ≤ -3}
Lesson 1-1 Write verbal expressions for algebraic expressions. Write algebraic expressions for verbal expressions.
A term of an expression may be a number, a variable, or a product or quotient of numbers and variables. For example, 0.10d, 2x and 4 are each terms.
A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
Working in Cooperative Groups
-2 ≤ x or x ≥ -2
z 3+_
2x + 4
0.10d
Content Standards
2 Teach
x
Model and Analyze 2. -4x > -4 {x | x < 1}
Before Lesson 1-1 Perform operations on integers.
1
Put students in groups of two or three and demonstrate the Activity. Have groups complete Exercises 1 and 2.
1. -3x < 9 {x | x > -3}
1 Focus VerticalAlignment
Cassie and her friends are at a baseball game. The stadium is running a promotion where hot dogs are $0.10 each. Suppose d represents represen the number of hot dogs Cassie and her friends fri eat. Then 0.10d represents the cost of the hot dogs they eat.
Write Verbal Expressions An algebraic expression consists of sums and/or products of numbers and variables. In the algebraic expression 0.10d, the letter d is called a variable. In algebra, variables are symbols used to represent unspecified numbers or values. Any letter may be used as a variable.
Common Core State Standards
x
-1 -1
Write algebraic expressions for verbal expressions.
algebraic expression variable term factor product power exponent base
algebra tiles, pp. 10–11 equation mat, p. 16
Teaching Tip Have students use a self-adhesive note to cover the equals sign on the equation mat. Write a ≤ symbol on the note. This will allow students to model inequalities with the equation mat and algebra tiles.
≤
-4 ≤ 2x
2
algebra tiles and equation mats self-adhesive blank notes
Teaching Algebra with Manipulatives Templates for:
Separate the tiles into 2 groups.
-1 -1
Write verbal expressions for algebraic expressions.
Easy to Make Manipulatives
-2x + 2x ≤ 4 + 2x
Add 4 negative 1-tiles to each side to isolate the x-tiles. Remove the zero pairs.
-1 x -1
You performed operations on integers.
Materials for Each Group
S l IInequalities Solve nequaliti nequalit lities ies
Solve -2x ≤ 4.
Lesson 1-1
A.SSE.1a InterpretV parts of and Variables ariables and Expressions an expression, such as Why? Then W hyy? ? terms, factors, andNow 1 coefficients.
Objective Use algebra tiles to model solving inequalities.
Common Core State Standards C
You can use algebra tiles to solve inequalities.
Content Standards
Explore 5-2
5-Minute Check 1-1 Study Notebook
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Trusted Content
Common Core State Standards Common Core State Standards, Traditional Algebra I Pathway, Correlated to Glencoe Algebra 1, Common Core Edition Lessons in which the standard is the primary focus are indicated in bold. Student Edition Lesson(s)
Standards
Student Edition Page(s)
Number and Quantity The Real Number System N-RN Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
7-3
406–413
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
7-3, 10-3, Extend 10-3, 10-4
406–413, 635–639, 640–641, 642–646
Use properties of rational and irrational numbers. 3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Extend 10-2
634
Quantities N-Q Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Throughout the text; for example, 2-6, 2-7, 2-8, 2-9, Extend 3-2, 4-5, 7-5
Throughout the text; for example, 111–117, 119–124, 126–131, 132–138, 169–170, 247–253, 424–429
2. Define appropriate quantities for the purpose of descriptive modeling.
Extend 2-6
118
3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Extend 1-3
23–24
Mathematical Modeling Standards
T12
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
Algebra Seeing Structure in Expressions A-SSE Interpret the structure of expressions 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients.
1-1, 1-4, 8-1, 9-1
5–9, 25–31, 465–471, 543– 553
1-2, 1-3, 9-7
10–15, 16–22, 598–605
2. Use the structure of an expression to identify ways to rewrite it.
1-1, 1-2, 1-3, 1-4, 7-1, 7-2, 7-3, 7-4, Explore 8-5, 8-5, Explore 8-6, 8-6, 8-7, 8-8, 8-9
5–9, 10–15, 16–22, 25–31, 391–397, 398–405, 406–413, 414–420, 493, 494–500, 501–502, 503–509, 510–515, 516–521, 522–529
Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines.
8-5, 8-6, 8-7, 8-8, 8-9
494–500, 503–509, 510–515, 516–521, 522–529
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
9-3, 9-4, Extend 9-4
564–571, 574–579, 580–581
c. Use the properties of exponents to transform expressions for exponential functions.
Extend 7-6
437
b. Interpret complicated expressions by viewing one or more of their parts as a single entity.
Arithmetic with Polynomials and Rational Expressions A-APR Perform arithmetic operations on polynomials. 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Explore 8-1, 8-1, 8-2, Explore 8-3, 8-3, 8-4
463–464, 465–471, 472–477, 478–479, 480–485, 486–491
Creating Equations A-CED Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems.
1-5, 2-1, 2-2, 2-3, 2-4, 2-5, 2-9, 3-2, 5-1, 5-2, 5-3, 5-4, 5-5, 7-6, 8-5, 8-6, 8-7, 9-4, 9-5, 10-4, 11-8
33–39, 75–80, 83–89, 91–96, 97–102, 103–109, 132–138, 163–168, 285–290, 292–297, 298–303, 306–311, 312–316, 432–436, 494–500, 503–509, 510–515, 574–579, 583–589, 642–646, 726–732
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SECTION SECTION 1 1
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Trusted Trusted Content Content
Common Core State Standards
Continued
Student Edition Lesson(s)
Student Edition Page(s)
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Extend 1-7, 3-1, 3-4, 3-5, 3-6, 4-1, 4-2, 4-3, 4-4, 4-5, 4-6, 4-7, 6-1, Extend 6-1, 6-2, 6-3, 6-4, 6-5, 7-5, 7-6, 8-6, 8-7, 8-8, 9-1, 9-2, 9-4, 9-5, 10-1, Extend 10-1, 10-4, 11-2, 11-8
55, 155–162, 182–188, 189– 195, 197–202, 216–223, 226–232, 233–238, 239–245, 247–253, 255–262, 263–270, 335–341, 342–343, 344–349, 350–356, 357–362, 364–369, 424–429, 432–436, 503–509, 510–515, 516–521, 543–553, 555–560, 574–579, 583–589, 621–626, 627, 642–646, 684–689, 726–732
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
4-2, 5-6, 6-1, 6-2
226–232, 317–322, 335–341, 344–349
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
2-8, 2-9, 4-1
126–131, 132–138, 216–223
Standards
Reasoning with Equations and Inequalities A-REI Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
1-5, 2-2, 2-3, 2-4, 2-5, 2-6, 2-9, 8-6, 8-7, 8-9
33–39, 83–89, 91–96, 97–102, 102–109, 111–117, 132–138, 503–509, 510–515, 522–529
Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
1-5, Explore 2-2, 2-2, Explore 2-3, 2-3, 2-4, 2-5, 2-6, 2-7, 2-8, 2-9, 5-1, Explore 5-2, 5-2, 5-3, 5-4, 5-5, 7-3
33–39, 81–82, 83–89, 90, 91–96, 97–102, 103–109, 111–117, 119–124, 126–131, 132–138, 285–290, 291, 292–297, 298–303, 306–311, 312–316, 406–413
4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
9-4, 9-5, 10-2
574–579, 583–589, 628–633
8-6, 8-7, 8-8, 9-2, 9-4, Extend 9-4, 9-5
503–509, 510–515, 516–521, 555–560, 574–579, 580–581, 583–589
b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Mathematical Modeling Standards
T14
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
Solve systems of equations. 5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
6-4
357–362
6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
6-1, Extend 6-1, 6-2, 6-3, 6-4, 6-5, Extend 6-5
335–341, 342–343, 344–349, 350–356, 357–362, 364–369, 370–371
7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Extend 9-3
572–573
Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
1-6, 1-7, 3-1, 3-2, 3-4, 7-5, 9-1, 10-1
40–46, 47–54, 155–162, 163–168, 182–188, 424–429, 543–553, 621–626
11. Explain why the x-coordinates of the points where the graphs of the equations y = f (x ) and y = g (x ) intersect are the solutions of the equation f (x ) = g (x ); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f (x ) and/or g (x ) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Extend 6-1, Extend 7-5, Extend 9-3, 9-7, Extend 11-8
342–343, 430–431, 572–573, 598–605, 733–734
12. Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
5-6, Extend 5-6, 6-6, Extend 6-6
317–322, 323, 372–376, 377
Functions Interpreting Functions F-IF Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f (x ) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f (x ).
1-6, 1-7
40–46, 47–54
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
1-7, 3-6, 4-3, 7-5, 7-6, 9-1, 10-1
47–54, 197–202, 233–238, 424–429, 432–436, 543–553, 621–626
3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
3-5, 7-7, 7-8
189–195, 438–443, 445–450
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T15
SECTION SECTION 1 1
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Trusted Trusted Content Content
Common Core State Standards
Continued
Student Edition Lesson(s)
Student Edition Page(s)
Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
1-8, Explore 3-1, 3-1, Extend 4-1, 7-5, 9-1, 9-7, 10-1
56–61, 153–154, 155–162, 224–225, 424–429, 543–553, 598–605, 621–626
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
1-7, 7-5, 7-6, 9-1, 10-1
47–54, 424–429, 432–436, 543–553, 621–626
6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Explore 3-3, 3-3, Extend 7-7, Extend 9-1, 9-6
171, 172–180, 444, 554, 590–595
Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
3-1, 3-2, Extend 3-2, 3-4, 4-1, Extend 4-1, 9-1, 9-2, Explore 9-3, 9-3, Extend 9-3
155–162, 163–168, 169–170, 182–188, 216–223, 224–225, 543–553, 555–560, 562–563, 564–571, 572–573
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
9-7, Extend 9-7, 10-1, Extend 10-1
598–605, 606, 621–626, 627
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Explore 7-5, 7-5
422–423, 424–429
9-2, 9-3, 9-4, Extend 9-4
555–560, 564–571, 574–579, 580–581
7-1, 7-2, 7-5, 7-6, Extend 7-6
391–397, 398–405, 424–429, 432–436, 437
1-7, 3-6, 4-3, 7-8, 9-1, 9-3
47–54, 197–202, 233–238, 445–450, 543–553, 564–571
Standards
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Building Functions F-BF Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. Mathematical Modeling Standards
T16
1-7, 3-1, 3-4, 3-6, 4-1, 4-2, 4-3, 4-4, 4-5, 4-6, 4-7, 7-6, 7-8
47–54, 155–162, 182–188, 197–202, 216–223, 226–232, 233–238, 239–245, 247–253, 255–262, 263–270, 432–436, 445–450
4-2, 7-6, 9-3
226–232, 432–436, 564–571
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
3-5, 7-7, 7-8
189–195, 438–443, 445–450
Build new functions from existing functions. 3. Identify the effect on the graph of replacing f (x ) by f (x ) + k, kf (x ), f (kx ), and f (x + k ) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
Extend 4-1, Explore 7-5, Extend 7-6, Explore 9-3, 9-3, 10-1, Extend 10-1
224–225, 422–423, 437, 562–563, 564–571, 621–626, 627
4. Find inverse functions. a. Solve an equation of the form f (x ) = c for a simple function f that has an inverse and write an expression for the inverse.
4-7, Extend 4-7, Explore 10-1
263–270, 271, 619–620
Linear, Quadratic, and Exponential Models F-LE Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
Explore 3-3, 3-3, 3-5, 7-7, 9-6
171, 172–180, 189–195, 438–443, 590–595
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
3-5, 3-6, 7-6, 7-7, 9-6
189–195, 197–202, 432–436, 438–443, 590–595
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
7-6, 7-7, 9-6
432–436, 438–443, 590–595
2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
3-5, 3-6, 4-2, 4-3, 4-4, 4-5, 4-6, 7-5, 7-6, 7-7, 9-6, Extend 9-6
189–195, 197–202, 226–232, 233–238, 239–245, 247–253, 255–262, 424–429, 432–436, 438–443, 590–595, 596–597
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
9-6, Extend 9-6
590–595, 596–597
Interpret expressions for functions in terms of the situation they model. 5. Interpret the parameters in a linear or exponential function in terms of a context.
3-4, Explore 4-1, 4-1, Extend 4-1, 4-5, 7-5, 7-6, 7-7, 9-6
182–188, 215, 216–223, 224–225, 247–253, 424–429, 432–436, 438–443, 590–595
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T17
SECTION SECTION 1 1
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Trusted Trusted Content Content
Common Core State Standards
Continued
Student Edition Lesson(s)
Standards
Student Edition Page(s)
Statistics and Probability Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
0-13, 12-3, 12-4
P40–P46, 764–770, 771–778
2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
12-2, 12-3, 12-4, Extend 12-8
757–763, 764–770, 771–778, 810–811
3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
12-3, 12-4
764–770, 771–778
Summarize, represent, and interpret data on two categorical and quantitative variables. 5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
Extend 12-7
801–802
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
4-5, 4-6, Extend 9-6
247–253, 255–262, 596–597
b. Informally assess the fit of a function by plotting and analyzing residuals.
4-6
255–262
c. Fit a linear function for a scatter plot that suggests a linear association.
4-5, 4-6
247–253, 255–262
Interpret linear models. 7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
4-1, Extend 4-1, 4-4, 4-5
216–223, 224–225, 239–245, 247–253
8. Compute (using technology) and interpret the correlation coefficient of a linear fit.
4-6
255–262
9. Distinguish between correlation and causation.
Extend 4-5
254
Mathematical Modeling Standards
T18
Correlation
Common Core State Standards for Mathematical Practice, Correlated to Glencoe Algebra 1, Common Core Edition 1. Make sense of problems and persevere in solving them. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 0-1, 1-8, 2-4, 3-4, 4-5, 5-4, 6-4, 7-5, 8-8, 9-3, 10-5, 11-1, and 12-4. 2. Reason abstractly and quantitatively. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-3, 2-1, 3-3, 4-1, 5-1, 6-5, 7-2, 8-5, 9-1, 10-3, 11-8, and 12-2. 3. Construct viable arguments and critique the reasoning of others. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-3, 2-5, Extend 3-5, 4-2, 5-5, 6-1, 7-4, 8-1, 9-2, 10-4, 11-2, and Extend 12-1. 4. Model with mathematics. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-1, 2-9, 3-2, 4-5, 5-1, 6-5, 7-6, 8-7, 9-7, 10-4, 11-7, and 12-5. 5. Use appropriate tools strategically. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Extend 1-7, 2-4, Extend 3-2, 4-4, Extend 5-6, Extend 6-1, Extend 7-5, 8-2, Extend 9-6, 10-6, Extend 11-8, and 12-3. 6. Attend to precision. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Extend 1-3, 2-8, 3-4, 4-2, 5-2, 6-6, 7-4, 8-9, 9-5, 10-1, 11-6, and 12-2. 7. Look for and make use of structure. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-2, 2-5, 3-6, Extend 4-1, 5-5, 6-3, 7-7, 8-6, 9-6, 10-2, 11-2 and 12-8. 8. Look for and express regularity in repeated reasoning. Glencoe Algebra 1 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-4, 2-7, 3-1, 4-1, 5-4, 6-1, 7-1, 8-4, 9-3, 10-2, 11-5 and 12-6.
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T19
