new mexico mathematics content standards,...

Prentice Hall Precalculus, 3rd Edition, © 2007, (Blitzer)

C O R R E L A T E D T O

New Mexico Mathematics Standards and BenchmarksGrades 9-12

G r a d e s 9 - 1 2

Publisher: Pearson Prentice Hall Title: PreCalculus 3rd Edition © 2007 (Blitzer) ISBN: 0-13-195993-X, 0-13-188045-4

SE = Student Edition AIE = Annotated Instructor’s Edition *Objectives are clearly stated with measurable outcomes at 90% or above.

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NEW MEXICO MATHEMATICS CONTENT STANDARDS, BENCHMARKS, & PERFORMANCE STANDARDS Publisher Alignment Analyses for Primary Tool of Instruction

This correlation table/matrix is a tool to show alignment with New Mexico’s Content Standards, Benchmarks, & Performance Standards and the proposed instructional material considered for adoption. The purpose is to demonstrate how your material can contribute to student achievement as measured against these Content Standards. Attach a completed copy of this document to each core basal sample you are submitting for review. You will submit 3 copies of each student & teacher edition for each title & other material deemed necessary to provide appropriate instruction, along with these alignment documents at the 2006 June Summer Institute. DO NOT SEND WITH THE RFP.

Mathematics Grades 9–12 Standard 2: ALGEBRA, FUNCTIONS, AND GRAPHS: Students will understand algebraic concepts and applications.

Benchmark

Performance Standards

Publisher Citation (pages)

% Meets Standard*

1. Classify numbers and members of the following sets: • natural • whole • integers • rationals • irrationals

SE/AIE: 6, 7 (Example 5, Check Point 5), 15 (#35–42), 62 (#37), 122 (#8), 125 (#26)

A. Represent and analyze mathematical situations and structures using algebraic symbols.

2. Simplify numerical expressions using the order of operations, including exponents.

SE/AIE: 3 (The Order of Operations Agreement, Example 1, Check Point 1), 14 (#1–16), 16 (#127b, 128b), 17 (#145a), 75 (#87a, 89b), 121 (#1–2)



2


Benchmark



% Meets Standard*

3. Evaluate the numerical value of expressions of one or more variables that are: • polynomial • rational • radical

SE/AIE: 3 (Example 1, Check Point 1), 14 (#1–16), 16 (#127b, 128b), 17 (#145a), 75 (#87a, 89b), 121 (#1–2)

4. Simplify algebraic monomial expressions raised to a power (e.g., [5xy²]³) and algebraic binomial (e.g., [5x² + y]²) expressions raised to a power.

SE/AIE: 18 (The Power Rule), 19–20, 25 (#31–34, 39–40, 43–44), 47 (Squaring a Binomial, Cubing a Binomial), 48 (Example 4b, Check Point 4b), 49 (Example 5b, Check Point 5b, #41–48), 50 (#49–58, 65–68), 62 (#6–7, 24), 123 (#77–80, 82–83), 125 (#13)


5. Compare and order polynomial expressions by degree. SE/AIE: 43, 48, 49 (#5–8), 123 (#72–73)



3


Benchmark



% Meets Standard*

6. Represent and analyze relationships using written and verbal expressions, tables, equations, and graphs, and describe the connections among those representations: • translate from verbal expression to algebraic formulae (e.g.,

“Set up the equations that represent the data in the following equation: John’s father is 23 years older than John. John is 4 years older than his sister Jane. John's mother is 3 years younger than John’s father. John’s mother is 9 times as old as Jane. How old are John, Jane, John’s mother, and John’s father?')

• given data in a table, construct a function that represents these data (linear only)

• given a graph, construct a function that represents the graph (linear only)

SE/AIE: 96–107, 108 (#43–48, 55–61), 124 (#140–149), 126 (#54–59), 171 (Example 3), 178, 180 (#88a), 181 (#91–92), 182 (#110), 242–249, 250 (Check Point 6, #1a, 2a), 251 (#3a, 4a, 5a–5b, 6a–6b, 7a–7b, 8a–8b, 9–13), 252 (#14, 15a, 16a, 17–29), 253 (#30–34, 35a, 36a, 37–38, 43–44), 254 (#45–46), 255 (#63–66), 260 (#58a), 262 (#119a, 120a–120b, 121, 122a, 123), 263 (#124–125), 264 (#27a, 35a, 36–38), 1004 (#43)

7. Know, explain, and use equivalent representations for the same real number including: • integers • decimals • percents • ratios • scientific notation • numbers with integer exponents • inverses (reciprocal) • prime factoring

SE/AIE: 21–22, 23 (Check Point 4), 26 (#65–76), 122 (#32–35), 125 (#29)


8. Simplify algebraic expressions using the distributive property. SE/AIE: 12–13, 14 (Example 9, Check Point 9), 15 (#85–96, 99–100), 32 (Example 4b, Example 5b), 45 (Example 2)



4


Benchmark



% Meets Standard*

9. Explain and use the concept of absolute value. SE/AIE: 9–10, 15 (#51–74), 62 (#38–39), 84 (Example 6, Check Point 6), 93 (#43–54), 94 (#129–130), 95 (#151), 115 (Example 6, Check Point 6), 116–117, 118 (#57–64), 119 (#65–92, 95–96, 111), 122 (#9–12), 124 (#164–165), 126 (#47), 374 (#7, 11, 17), 444 (#1, 8), 568 (#6), 670 (Example 2), 671 (Check Point 2), 780 (#28), 1003 (#15)


10. Know, explain, and use equivalent representations for algebraic expressions.

SE/AIE: 12–13, 14 (Example 9, Check Point 9), 15 (#85–102, 111–118), 19 (Example 1), 20 (Check Point 1), 25 (#23–44), 26 (#45–64), 30 (Example 2b), 31 (Checkpoint 2b, Example 3b, Check Point 3b), 32 (Example 4b, Check Point 4b, Example 5b, Check Point 5b), 38 (Example 12), 39 (Check Point 12, Example 13, Check Point 13, #15–22, 27–32, 35–36, 39–40), 40 (#73–74, 79–80, 91–100, 103–108, 111–114), 44–48, 49 (Example 5, Check Point 5, #9–48), 50 (#49–90), 52–60, 61 (#115a, 116a, 117b, 118b, 119b, 120b, 121–128, 131–134), 62 (#1–12, 15–20, 28–36), 64 (Example 2, Check Point 2), 65–67, 68–73, 74 (#7–64), 75 (#65–86), 76 (#90–93, 96–100, 102–105, 108–110), 122 (#19–22, 28–31, 41–43, 45, 58), 123 (#61, 68–121), 124 (#5, 9–10, 12–25)



5

Benchmark



% Meets Standard*

11. Simplify square roots and cube roots with monomial radicands that are perfect squares or perfect cubes (e.g., 9a²x4)

SE/AIE: 30 (Simplifying √a2)

12. Calculate powers and roots of real numbers, both rational and irrational.

SE/AIE: 25 (#3–12), 28–29, 39 (#1–17, 23–26, 55–58), 122 (#44, 53–54, 56)



6


Benchmark



% Meets Standard*

13. Solve: • formulas for specified variables • radical equations involving one radical

SE/AIE: 83 (Example 5, Check Point 5), 93 (#27–42), 123 (#136–137)

14. Factor polynomials, difference of squares and perfect square trinomials, and the sum and difference of cubes.

SE/AIE: 52–60, 61 (#115a, 116a, 117b, 118b, 119b, 120b, 121–123, 126, 131–137), 62 (#28–36)

15. Simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

SE/AIE: 19 (Example 1c), 20 (Check Point 1c), 26 (#49–58), 62 (#16), 64 (Example 2, Check Point 2), 74 (#7–14), 123 (#108–110), 125 (#5)


16. Manipulate simple expressions with + and – exponents. SE/AIE: 18–20, 25 (#1–44), 26 (#45–64, 107–114), 44–48, 49 (Example 5, Check Point 5, #9–48), 50 (#49–90, 97–100, 103–105), 51 (#108–111), 52–60, 61 (#115a, 116a, 117–123, 125–126, 128, 131–134, 137), 62 (#1–12, 15–20, 22–36), 64 (Example 2, Check Point 2), 65–69, 74 (#7–54), 123 (#62–121), 125 (#1–2, 5, 10, 12–16, 19–25, 30)



7

Benchmark



% Meets Standard*

17. Use the four basic operations (+, –, x, ÷ with: • linear expressions • polynomial expressions • rational expressions

SE/AIE: 44–48, 49 (Example 5, Check Point 5, #9–48), 50 (#49–90, 97–100, 103–105), 51 (#108–111), 62 (#1–2, 5–6, 9–12, 15–17, 19–20, 23–24), 65–69, 74 (#15–54), 75 (#79–86), 76 (#91–92, 103–105, 107–110), 123 (#72–87, 111–117), 125 (#1–2, 10, 12–16)



8


Benchmark



% Meets Standard*

1. Distinguish between the concept of a relation and a function. SE/AIE: 138–140, 153 (#111–112)

2. Determine whether a relation defined by a graph, a set of ordered pairs, a table of values, an equation, or a rule is a function.

SE/AIE: 140–141, 142 (Example 3, Check Point 3), 145, 149 (#1–26), 150 (#55–64), 152 (#99b, 100b, 100d), 193 (#1–8), 259 (#15–20, 25–30), 263 (#1), 886 (#49–54)

3. Describe the concept of a graph of a function. SE/AIE: 144, 153 (#115)

C. Understand patterns, relations, functions, and graphs.

4. Translate among tabular, symbolic, and graphical representations of functions.

SE/AIE: 144, 150 (#39–54), 166 (#73–74), 167 (#103–107), 168 (#108–109, 110a–110b, 115), 171 (Example 3), 172–174, 176 (Example 7), 177 (Check Point 7), 179 (#39–50, 53, 55–56, 59–72), 180 (#81–82, 87a, 88a), 181 (#89b, 90b, 91–92), 182 (#100–103), 206 (#53–88), 207 (#89–118), 258 (#1–4), 260 (#52–56, 58a), 261 (#67–82), 263 (#4), 264 (#6–7, 9, 11–15, 27a), 325 (#12–20), 335–337, 338 (Check Point 7), 391 (Example 6, Check Point 6), 397 (#43–46)



9


Benchmark



% Meets Standard*

5. Explain and use function notation. SE/AIE: 142–144, 145 (Check Point 5), 146–148, 149 (Check Point 8, Identifying Intercepts, #27–33), 150 (#34–54, 65–70), 151, 152 (#93–98, 101), 154 (#121, 122, 124), 155–162, 163 (#1–28), 164 (#45–56), 167 (#88, 94–95, 98, 103–107), 168 (#108–111, 113), 172, 173 (Example 4, Check Point 4), 174, 179 (#41–44, 47–48), 191 (#13–18, 27–30), 195–206, 207 (#89–122, 127), 208 (#128, 135–138), 209 (#139–145), 210–221, 223–230, 231 (#1–28, 39–52)

B. Understand patterns, relations, functions, and graphs.

6. Determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.

SE/AIE: 139 (Example 1, Check Point 1), 147–148, 149 (Check Point 8, #1–10), 193 (#1–6, 10–11), 210–212, 214, 215 (Check Point 3), 218 (Example 5b, Check Point 5b), 219 (#1–28), 220 (#29–48, 65b, 66b, 67b, 68b, 69b, 70b, 71b, 72b, 85–86), 221 (#95–96), 259 (#33a–33b, 34a–34b, 35a–35b), 261 (#83–91), 263 (#2b–2c, 4–5), 264 (#6–15, 19, 30–32), 277–279, 280 (Check Point 4), 284 (#17–38, 39c, 40c, 41c, 42c, 43c, 44c), 325 (#8–11), 326, 327 (Check Point 1), 342 (#1–8), 393 (Example 7, Check Point 7), 395 (Example 10, Check Point 10), 398 (#53–80, 109–112)



10

Benchmark



% Meets Standard*

7. Identify the independent and dependent variables from an application problem (e.g., height of a child).

Opportunities to introduce this standard can be found on the following page: SE/AIE: 141

8. Describe the concept of a graph of an equation. SE/AIE: 129, 137 (#68)



11


Benchmark



% Meets Standard*

9. Understand symmetry of graphs. SE/AIE: 160–161, 164 (#45–57), 165 (#58–60, 61l, 62l), 166 (#64k), 193 (#23, 37a), 260 (#38–40), 263 (#3d), 264 (#23), 274, 502, 511, 523 (The Tangent Curve), 525 (The Cotangent Curve), 527 (The Cosecant Curve, The Secant Curve), 659–665, 666 (#7–40), 667 (#41–44), 668 (#26–32), 705 (#47–55), 877–878, 881–883, 912–915

10. Analyze and describe middle and end (asymptotic) behavior of linear, quadratic, and exponential functions, and sketch the graphs of functions.

SE/AIE: 163 (#29–30, 33–34), 167 (#85), 193 (#14–17), 297 (#17)

11. Work with composition of functions (e.g., find f of g when f(x) = 2x – 3 and g(x) = 3x – 2), and find the domain, range, intercepts, zeros, and local maxima or minima of the final function.

SE/AIE: 215–218, 220 (#49a–49b, 50a–50b, 51a–51b, 52a–52b, 53a–53b, 54a–54b, 55a–55b, 56a–56b, 57a–57b, 58a–58b, 59a–59b, 60a–60b, 61a–61b, 62a–62b, 63a–63b, 64a–64b, 65–72), 221 (#105b–105c), 222 (#106b–106c, 116, 118), 261 (#92a–92b, 93a–93b, 94–95), 264 (#20–21), 374 (#19), 444 (#16), 780 (#29), 1003 (#30)

B. Understand patterns, relations, functions, and graphs.

12. Use the quadratic formula and factoring techniques to determine whether the graph of a quadratic function will intersect the x–axis in zero, one, or two points.

SE/AIE: 275–278, 284 (#17–38), 369 (#13–16), 372 (#5–6)



12

Benchmark



% Meets Standard*

13. Apply quadratic equations to physical phenomena (e.g., the motion of an object under the force of gravity).

SE/AIE: 285 (#59–60), 444 (#18–19), 708 (#14), 1004 (#41), 1054 (#38)



13


Benchmark



% Meets Standard*

1. Model real–world phenomena using linear and quadratic equations and linear inequalities (e.g., apply algebraic techniques to solve rate problems, work problems, and percent mixture problems; solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest; apply quadratic equations to model throwing a baseball in the air ).

SE/AIE: 14 (#19–20), 242–249, 250 (Check Point 6, #1–2), 251, 252 (#14–18, 21–29), 253 (#30–38), 254 (#45–46), 255 (#63–66), 262 (#119–123), 263 (#124–125), 264 (#35–37), 280 (Example 5), 281 (Check Point 5), 283, 285 (#57–60, 65–66), 286 (#67–76, 89), 287 (#96–97), 325 (#27), 370 (#21, 23–24, 29–30), 372 (#8), 444 (#18–19), 708 (#14), 792 (#51–52), 1004 (#41)

C. Use mathematical models to represent and understand quantitative relationships.

2. Use a variety of computational methods (e.g., mental arithmetic, paper and pencil, technological tools).

SE/AIE: 3 (Example 1, Check Point 1), 4 (Example 2, Check Point 2), 10 (Example 7, Check Point 7), 14 (#1–20), 15 (#67–74), 16 (#119–125, 127b, 128b), 23 (Example 5, Check Point 5), 24, 25 (Check Point 7, #1–22), 26 (#87–106), 29 (Example 1, Check Point 1), 30 (Example 2a), 31 (Check Point 2a, Example 3a, Check Point 3a), 32 (Example 4a, Check Point 4a, Example 5a, Check Point 5a), 33–37, 38 (Example 11, Check Point 11), 39 (#1–14, 23–26, 33–34, 37–38, 41–58), 40 (#59–90, 115–119), 41 (#121b, 121b, 122b, 123–124), 42 (#136–138), 62 (#3–4, 21–23, 25–27, 40–42), 122 (#23–27, 36–40, 44, 46–57), 123 (#59–60, 62–67), 125 (#7–8, 30–32), 397 (#21–42)



14

Benchmark



% Meets Standard*

3. Express the relationship between two variables using a table with a finite set of values and graph the relationship.

SE/AIE: 130, 135 (#13–28), 136 (#51–54), 144, 145 (Check Point 5), 150 (#39–54), 174–175, 197, 258 (#1–4), 378, 379 (Check Point 3), 380 (Example 4, Check Point 4), 381 (Example 5, Check Point 5), 384 (#11–18), 391 (Example 6, Check Point 6)

4. Express the relationship between two variables using an equation and a graph: • graph a linear equation and linear inequality in two variables • solve linear inequalities and equations in one variable • solve systems of linear equations in two variables and graph

the solutions • use the graph of a system of equations in two variables to

help determine the solution

SE/AIE: 79–80, 93 (#1–16), 94 (#125), 97–101, 123 (#122–124), 125 (#33), 135 (#15–20), 193 (#25, 27–28, 30, 32–34), 258 (#1–4), 263 (#4), 264 (#7, 13), 712–722, 723 (#19–36), 724 (#37–48, 55–61), 725 (#62–67, 73c, 74c, 75c, 76c, 77b), 726 (#78b, 79–80, 81c, 82c), 727 (#83c, 84c, 85–90), 728 (#91–94, 107–109), 755 (#1–6, 17d, 18–21), 756–758, 759 (Check Point 3), 760, 761 (Check Point 5), 763 (#1–12), 764 (#27–44, 63–66)



15


Benchmark



% Meets Standard*

5. Solve applications involving systems of equations. SE/AIE: 718–723, 724 (#55–61), 725–727, 728 (#91–94, 107–109), 733 (Example 4), 734 (#31), 735, 736 (#36–41, 49), 751 (Example 5), 752 (Check Point 5), 753 (#55–60), 754 (#61–63, 71–72), 755 (#17–23), 776 (#6–10, 13–15), 777 (#36–38), 779 (#12–17), 1003 (#40)

6. Evaluate numerical and algebraic absolute value expressions. SE/AIE: 9–10, 15 (#51–74, 103–106), 62 (#38–39), 122 (#9–12), 670 (Example 2), 671 (Check Point 2), 678 (#1–10)

7. Create a linear equation from a table of values containing co–linear data.

Opportunities to introduce this standard can be found on the following pages: SE/AIE: 171 (Example 3), 172 (Check Point 3), 178, 179 (#25–34), 180 (#87a, 88a), 181 (#89b, 90b, 91–92), 194 (#40), 264 (#27a), 444 (#15), 708 (#19), 780 (#31), 924 (#18), 1003 (#36)

Use mathematical models to represent and understand quantitative relationships.

8. Determine the solution to a system of equations in two variables from a given graph.

SE/AIE: 710 (Example 1), 711 (Check Point 1), 713 (Technology), 716 (Example 5), 717 (Example 6), 722, 725 (#67), 748–751



16


Benchmark



% Meets Standard*

9. Generate an algebraic sentence to model real–life situations. SE/AIE: 97–104, 105 (Check Point 7), 178 (Example 8, Check Point 8), 180 (#87a, 87b), 181 (#89b, 90b, 91–92), 182 (#104c), 242–249, 250 (Check Point 6, #1a, 2a), 251 (#3a, 4a, 5a–5b, 6a–6b, 7a–7b, 8a–8b, 9–13), 252 (#14, 15a, 16a, 17–18, 21–29), 253 (#30–34, 35a, 36a, 37–38), 254 (#45–46), 255 (#63–66), 260 (#58a), 262 (#119a, 120a–120b, 121, 122a, 123), 263 (#124–125), 264 (#27a, 35a, 36–38), 428 (Example 4a), 429 (Check Point 4a), 435 (#31a, 32a), 442 (#88a), 443 (#27), 178 (Example 8, Check Point 8), 180 (#87a, 87b), 181

C. Use mathematical models to represent and understand quantitative relationships.

10. Write an equation of the line that passes through two given points.

SE/AIE: 171 (Example 3), 172 (Check Point 3), 178, 179 (#25–38), 180 (#87b, 88b), 181 (#89b, 90b, 91–92), 194 (#40), 260 (#48), 264 (#24), 444 (#15), 708 (#19), 780 (#31), 924 (#18)



17

Benchmark



% Meets Standard*

11. Understand and use: • such operations as taking the inverse, finding the reciprocal,

taking a root, and raising to a fractional power • the rules of exponents

SE/AIE: 18–20, 25 (#1–44), 26 (#45–64, 107–114), 28–31, 35 (Example 8), 36–38, 39 (Check Point 12, Example 13, Check Point 13, #1–32, 55–58), 40 (#59–74, 83–119), 62 (#7–8, 16–18), 122 (#24–31, 40–44, 53–58), 123 (#62–71), 125 (#5, 9, 30), 226–227, 230 (Example 7, Check Point 7), 231 (#11a, 12a, 13a, 14a, 15a, 16a, 17a, 18a, 19a, 20a, 21a, 22a, 23a, 24a, 25a, 26a, 27a, 28a, 39a, 40a, 41a, 42a, 43a, 44a, 45a, 46a, 47a, 48a, 49a, 50a, 51a, 52a), 261 (#100a, 101a, 102a), 262 (#108–109)

12. Verify that a point lies on a line, given an equation of the line, and be able to derive linear equations by using the point–slope formula.

SE/AIE: 170–171, 172 (Check Point 3), 179 (#11–38), 181 (#89b, 90b), 260 (#47–50), 264 (#24–25), 444 (#15)



18


Benchmark



% Meets Standard*

1. Analyze the effects of parameter changes on these functions: • linear (e.g., changes in slope or coefficients) • quadratic (e.g., f[x–a] changes coefficients and constants) • exponential (e.g., changes caused by increasing x[x + c] or

[a*]) • polynomial (e.g., changes caused by positive or negative

values of a, or in a constant c)

SE/AIE: 150 (#39–44), 195–197, 199 (Example 3), 201, 205, 206 (#53–66), 207 (#95–106), 208 (#135–136), 261 (#67–70, 77–79), 380, 381 (Example 5, Check Point 5), 384 (#25–30), 385 (#31–46), 439 (#5–9)

2. Solve routine two–and three–step problems relating to change using concepts such as: • exponents • factoring • ratio • proportion • average • percent

SE/AIE: 4 (Example 2, Check Point 2), 16 (#119–126), 50 (#93–96), 62 (#42), 75 (#89), 95 (#143–146), 98 (Example 2), 99 (Check Point 2), 106 (#5–8), 122 (#23), 124 (#141), 125 (#32), 126 (#51–53), 178, 180 (#87–89), 181 (#90–92), 186–190, 191 (#13–20, 27–30), 192 (#31–36, 44–45), 194 (#44–45), 260 (#58–59), 261 (#60–61), 262 (#119), 264 (#27, 35), 285 (#57–60)

D. Analyze changes in various contexts.

3. Calculate the percentage of increase and decrease of a quantity. Not addressed in this text.



19


Benchmark



% Meets Standard*

4. Analyze the general shape of polynomial expressions and equations for different degree polynomials (e.g., positive and negative general shapes for third–, fourth–, and fifth–degree polynomials).

SE/AIE: 209 (The Leading Coefficient Test, Example 1, Check Point 1), 295, 297 (#15–21), 298 (#22–24, 41a, 42a, 43a, 44a, 45a, 46a, 47a, 48a, 49a, 50a, 51a, 52a, 53a, 54a, 55a, 56a, 57a, 58a, 59a, 60a, 61a, 62a, 63a, 64a), 299 (#75–76), 370 (#25–28, 30, 34a, 35a, 36a, 37a, 38a, 39a)

5. Estimate the rate of change of a function or equation by finding the slope between two points on the graph.

SE/AIE: 186–190, 191 (#13–20), 192 (#35–36), 193 (#44–45), 260 (#59a–59b)

6. Evaluate the estimated rate of change in the context of the problem.

SE/AIE: 186–187, 189–190, 191 (#19–20), 192 (#35–36), 260 (#59a–59b)

D. Analyze changes in various contexts.

7. Know Pascal's triangle and use it to expand binomial expressions that are raised to positive integer powers.

SE/AIE: 971 (The Universality of Mathematics)



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Mathematics Grades 9–12 Standard 2: ALGEBRA, FUNCTIONS, AND GRAPHS Guidance for Further Study 9–12 Topics for Further Study Solving equations, inequalities and systems (As students encounter ever more sophisticated mathematical situations, they will need to be able to generate and solve a variety of equations, inequalities, and systems. They begin by studying more complex linear and quadratic equations and systems.) Students will be able to:

• solve three–by–three linear systems • solve two–by–two linear quadratic and quadratic–quadratic systems • solve and graph equations and inequalities involving absolute value • solve quadratic inequalities by factoring

Polynomials (Students will extend the concept of solving linear equations to higher degree polynomials. These polynomials can be used to more accurately describe real–world phenomena.) Students will be able to:

• factor polynomials of degree higher than two using the fundamental theorem of algebra (e.g. an nth degree polynomial has at most n distinct linear factors), integral and rational zero theorems, and factor and remainder theorems

• perform the four basic operations on complex numbers • factor polynomials using complex numbers • graph polynomials using the intermediate value theorem • graph and interpret the conic sections

Functions (The language and properties of functions are essential to understanding the components of higher mathematics. Functions are the fundamental objects on which students operate in some higher mathematics and are among the building blocks of higher mathematics.) Students will be able to:

• find and use inverse functions involving ordered pairs, graphs, and explicit statements of a function rule • examine and graph piece–wise defined functions, including the use of the properties of continuity and discontinuity • graph rational functions and locate zeros and horizontal and vertical asymptotes



21

Mathematics Grades 9–12 Standard 3: GEOMETRY AND TRIGOMETRY: Students will understand geometric concepts and applications.

Benchmark



% Meets Standard*

1. Interpret and draw two–dimensional objects and find the area and perimeter of basic figures (e.g., rectangles, circles, triangles, other polygons [e.g., rhombi, parallelograms, trapezoids]).

Opportunities to introduce this standard can be found on the following pages: SE/AIE: 100 (Common Formulas), 247, 248 (Check Point 4), 252 (#17–18, 21–26), 253 (#43–44), 263 (#123), 264 (#37), 283, 285 (#65–66), 286 (#67–72), 287 (#97), 325 (#29), 370 (#21–23), 634 (Example 6, Check Point 6), 636 (#33–38), 643 (Example 4, Check Point 4), 644 (#25–30), 668 (#7–8), 705 (#13–16), 707 (#3), 1003 (#39), 1054 (#33)

2. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles with just edges in common.

Opportunities to introduce this standard can be found on the following pages: SE/AIE: 252 (#27–28), 254 (#47–48), 255 (#64)

3. Find and use measures of sides and interior and exterior angles of triangles and polygons to classify figures (e.g., scalene, isosceles, and equilateral triangles; rectangles [square and non–square]; other convex polygons).

SE/AIE: 103

A. Analyze characteristics and properties of two–and three–dimensional geometric shapes and develop mathematical arguments about geometric relationships.

4. Interpret and draw three–dimensional objects and find the surface area and volume of basic figures (e.g., spheres, rectangular solids, prisms, polygonal cones), and calculate the surface areas and volumes of these figures as well as figures constructed from unions of rectangular solids and prisms with faces in common, given the formulas for these figures.

Opportunities to introduce this standard can be found on the following pages: SE/AIE: 100 (Common Formulas), 246, 248 (Example 5), 249 (Check Point 5), 252 (#15a, 16a), 255 (#66), 262 (#122a), 264 (#38), 1054 (#39)



22


Benchmark



% Meets Standard*

5. Demonstrate an understanding of simple aspects of a logical argument: • identify the hypothesis and conclusion in logical deduction • use counterexamples to show that an assertion is false and

recognize that a single counterexample is sufficient to refute an assertion

Opportunities to introduce this standard can be found on the following pages: SE/AIE: 959–966

6. Demonstrate an understanding of inductive and deductive reasoning, explain the difference between inductive and deductive reasoning, and identify and provide examples of each: • for inductive reasoning, demonstrate understanding that

showing a statement is true for a finite number of examples does not show it is true for all cases unless the cases verified are all cases

• for deductive reasoning, prove simple theorems


A. Analyze characteristics and properties of two–and three–dimensional geometric shapes and develop mathematical arguments about geometric relationships.

7. Write geometric proofs (including proofs by contradiction), including: • theorems involving the properties of parallel lines cut by a

transversal line and the properties of quadrilaterals • theorems involving complementary, supplementary, and

congruent angles • theorems involving congruence and similarity • the Pythagorean theorem (tangram proof)

Opportunities to introduce this standard can be found on the following pages: SE/AIE: A2–A3, A5–A6



23


Benchmark



% Meets Standard*

1. Demonstrate understanding of the construction of the coordinate plane, know the names of the origin, coordinate axes and four quadrants, draw and label them correctly, find the coordinates of an indicated point, and plot a point with given coordinates.

SE/AIE: 128, 129 (Example 1, Check Point 1), 135 (#1–12), 137 (#65–67)

2. Determine the midpoint and distance between two points within a coordinate system and relate these ideas to geometric figures in the plane (e.g., find the center of a circle given two endpoints of a diameter of the circle).

SE/AIE: 234–235, 239 (#19–30), 262 (#110–113), 264 (#34)

3. Given two linear equations, determine whether the lines are parallel, perpendicular, or coincide.

Opportunities to introduce this standard can be found on the following pages: SE/AIE: 183–185, 191 (#1–12, 21–26), 192 (#37–38, 43a, 46–47), 194 (#41–43)

B. Specify locations and describe spatial relationships using coordinate geometry and other representational systems.

4. Use basic geometric ideas (e.g., the Pythagorean theorem, area, and perimeter of objects) in the context of the Euclidean Plane, calculate the perimeter of a rectangle with integer coordinates and sides parallel to the coordinate axes and with sides not parallel.




24


Benchmark



% Meets Standard*

1. Describe the effect of rigid motions on figures in the coordinate plane and space that include rotations, translations, and reflections: • determine whether a given pair of figures on a coordinate

plane represents the effect of a translation, reflection, rotation, and/or dilation

• sketch the planar figure that is the result of a given transformation of this type

SE/AIE: 855–856, 860 (#37–56), 869–872, 874 (#33–50), 881–883, 886 (#35–48), 888 (#3–5, 10–13, 18–21), 920 (#5–8), 921 (#19–22, 29–33), 922 (#3–5)

C. Apply transformations and use symmetry to analyze mathematical situations.

2. Deduce properties of figures using transformations that include translations, rotations, reflections, and dilations in a coordinate system: • identify congruency and similarity in terms of

transformations • determine the effects of the above transformations on linear

and area measurements of the original planar figure

Opportunities to introduce this standard can be found on the following pages: SE/AIE: 855–856, 860 (#37–56), 869–872, 874 (#33–50), 881–883, 886 (#35–48), 888 (#3–5, 10–13, 18–21), 920 (#5–8), 921 (#19–22, 29–33), 922 (#3–5)



25


Benchmark



% Meets Standard*

1. Solve real–world problems using congruence and similarity relationships of triangles (e.g., find the height of a pole given the length of its shadow).

Not addressed in this text.

2. Solve problems involving complementary, supplementary, and congruent angles.


3. Solve problems involving the perimeter, circumference, area, volume, and surface area of common geometric figures (e.g., “Determine the surface area of a can of height h and radius r. How does the surface area change when the height is changed to 3h? How does the surface area change when the radius is changed to 3r? How does the surface area change when both h and r are doubled?”).

SE/AIE: 100 (Common Formulas), 246–247, 248 (Check Point 4, Example 5), 249 (Check Point 5), 252 (#15a, 16a, 17–18, 21–26), 253 (#43–44), 255 (#66), 262 (#122a), 263 (#123), 264 (#37–38), 283, 285 (#65–66), 286 (#67–72), 287 (#97), 325 (#29), 370 (#21–23), 634 (Example 6, Check Point 6), 636 (#33–38), 643 (Example 4, Check Point 4), 644 (#25–30), 668 (#7–8), 705 (#13–16), 707 (#3)

D. Use visualization, spatial reasoning, and geometric modeling to solve problems.

4. Solve problems using the Pythagorean theorem (e.g., “Given the length of a ladder and the distance of the base of the ladder from a wall, determine the distance up the wall to the top of the ladder”).

SE/AIE: 103 (Example 6), 104 (Check Point 6), 107 (#31–36), 108 (#55), 124 (#147), 126 (#57), 484 (#1–8), 553 (Example 6a), 554 (Check Point 6a), 559 (#53–54)



26


Benchmark



% Meets Standard*

5. Understand and use elementary relationships of basic trigonometric functions defined by the angles of a right triangle (e.g., “What is the radius of a circle with an inscribed regular octagon with the length of each side being exactly 2 feet?”).

SE/AIE: 482–483, 485 (#29–34, 53–55), 486 (#56–60), 487 (#73), 501 (#29–30), 550–551, 552 (Example 4, Check Point 4), 557 (#29), 558 (#30–36, 41–46), 559 (#47–51, 55–58), 560 (#73), 564 (#38–40), 565 (#42–44), 566 (#114–120, 124b), 567 (#20–23), 568 (#20), 626 (#18, 20)

6. Use trigonometric functions to solve for the length of the second leg of a right triangle given the angles and the length of the first leg. (e.g., “A surveyor determines that the angle subtended by a two–foot stick at right angles to his transit is exactly one degree. What is the distance from the transit to the base of the measuring stick?”).

SE/AIE: 482, 485 (#29–30, 33–34, 53–54), 486 (#57), 487 (#73), 550, 551 (Example 2, Check Point 2), 558 (#41–46), 564 (#38–39), 565 (#42–43), 566 (#115, 118), 567 (#31), 626 (#18, 20)

D. Use visualization, spatial reasoning, and geometric modeling to solve problems.

7. Know and use angle and side relationships in problems with special right triangles (e.g., 30–, 45–, 60–, and 90– degree triangles).

SE/AIE: 479, 480 (Check Point 4, Table 4.2), 484 (#9–20), 486 (#63–64), 501 (#16), 564 (#32–35), 567 (#7)



27

Mathematics Grades 9–12 Standard 3: GEOMETRY AND TRIGONOMETRY Guidance for Further Study 9–12 Topics for Further Study Logs and exponential functions (Logs and exponential functions provide tools for more sophisticated modeling and applications for understanding real–life phenomena. Higher mathematics requires regular and successful use of logs and exponents to move beyond polynomials.) Students will be able to:

• operate with logs and exponential functions on the basis of their inverse relationship • identify the concept of e • use exponential functions and common and natural logs to understand real–life situations (e.g., half–life, amortization, logistic

growth) • use logs and exponents to solve equations

Trigonometry concepts (Trigonometry allows a student to consider periodic functions) Students will be able to:

• graph all six trigonometric functions using radian measure, their domains and ranges, and the exact values of the five angles of the six trigonometric functions

• demonstrate an understanding of trigonometric functions as circular functions using symmetry • solve trigonometric equations • verify trigonometric identities • apply trigonometric functions to solve physical problems, including the use of the laws of sines and cosines

Series and sequences (As students progress toward higher mathematics, they will need an understanding of sequences and functions whose domains are sets of whole numbers as opposed to sets of real numbers [e.g., discrete functions versus continuous functions). Infinite geometric series provide one way to begin a discussion about limits.]) Students will be able to:

• use algebraic techniques to generate the specific formulas for arithmetic and geometric sequences and series • extend the concept of series to infinite geometric series • use the language and notation of limits • use mathematical induction to prove various mathematical statements



28

Mathematics Grades 9–12 Standard 5: DATA ANALYSIS AND PROBABILITY: Students will understand how to formulate questions, analyze data, and

determine probabilities.

Benchmark


Publisher Citation (pages) % Meets

Standard* 1. Understand the differences between the various methods of data

collection. Not addressed in this text.

2. Know the characteristics of a well–designed and well–conducted survey: • differentiate between sampling and census • differentiate between a biased and an unbiased sample


3. Know the characteristics of a well–designed and well–conducted experiment: • differentiate between an experiment and an observational

study • recognize sources of bias in poorly designed experiments

Opportunities to introduce this standard can be found on the following page: SE/AIE: 986

A. Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

4. Understand the role of randomization in well–designed surveys and experiments.




29



Benchmark



Standard* 1. Understand the meaning of measurement data and categorical

data, and of the term “variable”.' Not addressed in this text.

2. Understand the meaning of “univariate” (i.e., one variable) and “bivariate” (i.e., two variable) data.

Opportunities to introduce this standard can be found on the following pages: SE/AIE: 169, 177–178, 181 (#90–92), 182 (#104, 110), 260 (#58–59), 264 (#27), 429–432, 435 (#35–39), 436 (#40, 54–58, 60), 442 (#89–90, 93)

3. For univariate data, be able to display the distribution and describe its shape using appropriate summary statistics, and understand the distinction between a statistic and a parameter: • construct and interpret frequency tables, histograms, stem

and leaf plots, and box and whisker plots • calculate and apply measures of central tendency (mean,

median, and mode) and measures of variability (range, quartiles, standard deviation)

• compare distributions of univariate data using back–to–back stem and leaf plots and parallel box and whisker plots


B. Select and use appropriate statistical methods to analyze data.

4. For bivariate data, be able to display a scatter plot and describe its shape: • fit a linear model to a set of data using technological tools • describe and interpret the relationship/correlation between

two variables using technological tools

SE/AIE: 177 (Technology), 182 (#104, 110), 436 (#56)



30



Benchmark



Standard* 1. Compare and draw conclusions between two or more sets of

univariate data using basic data analysis techniques and summary statistics.

Not addressed in this text. C. Develop and evaluate inferences and predictions that are based on data.

2. Draw conclusions concerning the relationships among bivariate data: • make predictions from a linear pattern in data • determine the strength of the relationship between two sets

of data by examining the correlation • understand that correlation does not imply a cause–and–

effect relationship

SE/AIE: 178, 181 (#90c, 91–92), 182 (#110), 260 (#58c), 264 (#27c)



31



Benchmark



Standard* 3. Use simulations to explore the variability of sample statistics

from a known population and construct sampling distributions. Not addressed in this text.

4. Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference.


C. Develop and evaluate inferences and predictions that are based on data.

5. Evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions.




32



Benchmark



Standard* 1. Explain the concept of a random variable. Not addressed in this text.

2. Understand the concept of probability as relative frequency. SE/AIE: 984

D. Understand and apply basic concepts of probability.

3. Use simulations to compute the expected value and probabilities of random variables in simple cases.




33



Benchmark



Standard* 4. Distinguish between independent and dependent events. Opportunities to introduce this standard

can be found on the following page: SE/AIE: 993

C. Understand and apply basic concepts of probability.

5. Understand how to compute the probability of an event using the basic rules of probability: • complement rule • addition rule (disjoint and joint events) • multiplication rule (independent events) • conditional probability

SE/AIE: 990–993, 994 (Example 11, Check Point 11), 995 (#31–44), 996 (#45–53, 59–63), 997 (#69), 1001 (#83–85, 89–94, 96–97), 1002 (#22–25, 27–30)

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