Grade Eight (All Disciplines) Mathematics Standards Map Basic Program

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1 1 Publisher Instructions. 1. In the header of this document, please fill in the program s identifying information. A basic program in mathematics for grade eight must address the standards for one of the following four disciplines or, in the alternative, it must be an integrated program that covers portions of several disciplines: Algebra I Geometry Algebra II Probability and Statistics In the header, please indicate the discipline covered in this program (from the preceding list) or indicate that it is an integrated program and note the disciplines it covers in part. 2. Displayed below are all of the content standards adopted by the California State Board of Education for the four disciplines listed above. Following each standard is a box for In this space, please provide information to help the reviewers determine the extent to which the standard is covered in this program. Succinct responses are encouraged and will be appreciated. The following questions are provided to help guide your responses. Is this standard directly addressed in the program? If so where? [If this standard is addressed many places in the program, please indicate the primary place or places where it is covered; an exhaustive list of citations is not necessary.] Is the program aligned with this standard? If so, please cite one or more references illustrating that alignment. [If this standard is directly addressed, it may be unnecessary to illustrate alignment separately.] Does the program contain any material contrary to this standard? If so, where? Is the contrary material incidental or significant? Please explain as necessary. Are there any other comments you wish to make concerning the program in relation to this standard? 3. If this an integrated mathematics program, you are required to provide the information specified above for all parts of the integrated series including those not submitted for adoption. Please distinguish publisher comments pertaining to non- submitted parts of the integrated series by placing those comments in italics. 4. If you find a standard that is not covered in your program, please put in the Publisher Comments box, along with a brief explanation. If this program addresses a single discipline, it should include coverage of essentially all the standards for that discipline. If this program is part of an integrated series, coverage of the standards in a given discipline will vary from level to level, but essentially all of the standards for the disciplines included in the series should be covered. If you find a significant number of the applicable standards are not covered in your program, be it for a single discipline or an integrated series, you should consider withdrawing the submission. If a standard is not applicable to your program, please put N/A in the Publisher Comments box, along with a brief explanation if necessary. 5. Please leave blank the boxes provided for These boxes have been included for the reviewers to keep their own notes as to whether the program s coverage of each applicable standard is exemplary, adequate, or inadequate, and why they have reached that conclusion. IMAP/CRP members will be guided by the following rubric: Exemplary coverage addresses the standard directly, providing substantial material for teaching and learning. Adequate coverage provides general alignment with the standard, no material fundamentally contrary to the standard, and sufficient material for teaching and learning.

2 2 Inadequate coverage reflects little or no alignment with the standard, inclusion of material that is fundamentally contrary to the standard, and/or insufficient material for teaching and learning. Recommendations for or against a program are generally to be based on coverage of the standards as a whole, not each individual standard. However, some individual standards may be determined to be particularly important in the IMAP/CRP deliberations process.

3 3 ALGEBRA I Symbolic reasoning and calculations with symbols are central in algebra. In the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem solving situations. 1. Students identify and use the arithmetic properties of subsets of integers, rational, irrational and real numbers. This includes closure properties for the four basic arithmetic operations where applicable. This concept is addressed throughout Algebra. Some examples are given , 31-36, 72-77, , , , , , , , CA: 37 TR: Lesson Master 1-4A, 1-4B, 1-6A, 1-6B, 2-1A, 2-1B, 2-5A, 2-5B, 2-7A, 2-7B, 3-1A, 3-1B, 3-2A, 3-2B, 3-7A, 3-7B, 4-1A, 4-1B, 4-2A, 4-2B, 6-1A, 6-1B, 12-1A, 12-1B TECH: Technology Sourcebook, Calculator Masters 1, 2, 4; Activity Kit, Activity 5, Students use properties of numbers to demonstrate that assertions are true or false. SE/TE 77, 106, 153, 182, 218, 245, 449, CA: 31, 37, 38-42

4 4 2. Students understand and use such operations as taking the opposite, reciprocal, raising to a power, and taking a root. This includes the understanding and use of the rules of exponents. SE/TE 31-36, 79-83, , , TR: Lesson Masters 1-6A, 1-6B, 2-2A, 2-2B, 2-5A, 2-5B, 3-2A, 3-2B, 8-5A, 8-5B TECH: Technology Sourcebook, Calculator Master 1 3. Students solve equations and inequalities involving absolute values. SE/TE , CA: 2-6 TR: Lesson Master 9-8A, 9-8B 4. Students simplify expressions prior to solving linear equations and inequalities in one variable such as 3(2x-5) + 4(x-2) = 12. SE/TE TR: Lesson Master 4-1A, 4-1B

5 5 5. Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable, with justification of each step. SE/TE 153, , , , , , CA: Students graph a linear equation, and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., sketch the region defined by 2x + 6y < 4). SE/TE , , 446, 463, TR: Lesson Master 4-9A, 4-9B, 7-4A, 7-4B, 7-9A, 7-9B TECH: Technology Sourcebook, Computer Masters 10, Students verify that a point lies on a line given an equation of the line. Students are able to derive linear equations using the point-slope formula. SE/TE , , TR: Lesson Masters 7-2A, 7-2B. 7-3A, 7-3B, 7-8A, 7-8B TECH: Technology Sourcebook, Calculator Master 5

6 6 8. Students understand the concepts of parallel and perpendicular lines and how their slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. SE/TE , CA: TR: Lesson Masters 11-6A, 11-6B TECH: Technology Sourcebook, Computer Master Students solve a system of two linear equations in two variables algebraically, and are able to interpret the answer graphically. Students are able to use this to solve a system of two linear inequalities in two variables, and to sketch the solution sets. SE/TE: , , , , , , TR: Lesson Masters 11-A, 11-B, 11-2A, 11-2B, 11-3A, 11-3B, 11-4A, 11-4B, 11-5A, 11-5B, 11-6A, 11-6B TECH: Technology Sourcebook, Computer Masters 27, 28; Activity Kit, Activity 24, Students add, subtract, multiply and divide monomials and polynomials. Students solve multistep problems, including word problems, using these techniques. SE/TE 510, 527, , , , , 729 TR: Lesson Masters 10-2A, 10-2B, 10-3A, 10-3B, 10-4A, 10-4B TECH: Technology Sourcebook, Computer Master 25; Activity Kit, Activity 22

7 7 11. Students apply basic factoring techniques to second and simple third degree polynomials. These techniques include finding a common factor to all of the terms in a polynomial and recognizing the difference of two squares, and recognizing perfect squares of binomials. SE/TE , , ,, CA: TR: Lesson Masters 10-6A, 10-6B, 12-2A, 12-2B, 12-3A, 12-3B; Activity Kit, Activity Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing to lowest terms. SE/TE 737, CA: Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems using these techniques. SE/TE 85-90, , , CA: 16-21, 28-31, TR: Lesson Masters 2-3A, 2-3B, 3-9A, 3-9B, 6-1A, 6-1A, 6-1B TECH: Technology Sourcebook, Calculator Master 4

8 8 14. Students solve a quadratic equation by factoring or completing the square. SE/TE , , CA: 22-27, 37 TR: Lesson Masters 12-4A, 12-4B, 12-5A, 12-5B, 12-6A, 12-6B, 12-8A, 12-8B TECH: Technology Sourcebook, Computer Students apply algebraic techniques to rate problems, work problems, and percent mixture problems. SE/TE 26, 91-96, , 132, 305, , , 386, 398, 408, 418, 419, 422, 423, 431, 435, 447, 452, 509, 532, CA: 7-10, TR: Lesson Masters 2-4A, 2-4B, 2-6A, 2-6B, 6-2A, 6-2B, 6-3A, 6-3B TECH: Technology Sourcebook, Computer 3; Activity Kit, Activity 3, Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. SE/TE , , , TR: Lesson Masters 13-1A, 13-1B, 13-2A, 13-2B, 13-3A, 13-3B, 13-4A, 13-4B; Activity Kit, Activity 28

9 9 17. Students determine the domain of independent variables, and range of dependent variables defined by a graph, a set of ordered pairs, or symbolic expression. SE/TE 12-16, TR: Lesson Masters 1-2A, 1-2B, 4-2A, 4-2B TECH: Technology Sourcebook, Calculator Students determine whether a relation defined by a graph, a set of ordered pairs, or symbolic expression is a function and justify the conclusion. SE/TE TR: Lesson Masters 13-1A, 13-1B 19. Students know the quadratic formula and are familiar with its proof by completing the square. SE/TE , , CA: TR: Lesson Masters 9-5A, 9-5B, 12-6A, 12-6B

10 Students use the quadratic formula to find the roots of a second degree polynomial and to solve quadratic equations. SE/TE , , , CA: TR: Lesson Masters 9-5A, 9-5B, 9-6A, 9-6B 21. Students graph quadratic functions and know that their roots are the x-intercepts. SE/TE , , 766, CA: TR: Lesson Masters 9-1A, 9-1B, 9-2A, 9-2B TECH: Technology Sourcebook, Computer 22; Activity Kit, Activity Students use the quadratic formula and/or factoring techniques to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. SE/TE , 775, CA: 27 TR: Lesson Masters 12-4A, 12-4B

11 Students apply quadratic equations to physical problems such as the motion of an object under the force of gravity. SE/TE 51, , , 762, 763, CA: 22-2 TR: Lesson Masters 9-4A, 9-4B, 12-4A, 12-B TECH: Technology Sourcebook, Computer Students use and know simple aspects of a logical argument. SE/TE CA: Students explain the difference between inductive and deductive reasoning and identify and provide examples of each. SE/TE CA: Students identify the hypothesis and conclusion in logical deduction. SE/TE CA: 38-42

12 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion. SE/TE 55, 66, 531, , CA: 42 TR: Lesson Masters 8-9A, 8-9B 25. Students use properties of the number system to judge the validity of results, to justify each step of a procedure and to prove or disprove statements. SE/TE 26, 153, 173, 181, 218, 230, 302, 325, 427, 533, , 752, CA: 25-27, Students use properties of numbers to construct simple valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions. SE/TE , CA: 25-27, Students judge the validity of an argument based on whether the properties of the real number system and order of operations have been applied correctly at each step. SE/TE 230, CA: 38-42

13 Given a specific algebraic statement involving linear, quadratic or absolute value expressions, equations or inequalities, students determine if the statement is true sometimes, always, or never. SE/TE CA: 5-6, 9-10, 25-27, 31 GEOMETRY The geometric skills and concepts developed in this discipline are useful to all students. Aside from these skills and concepts, students will develop their ability to construct formal logical arguments and proofs in geometric settings and problems. 1. Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 2. Students write geometric proofs, including proofs by contradiction.

14 14 3. Students construct and judge the validity of a logical argument. This includes giving counter examples to disprove a statement. 4. Students prove basic theorems involving congruence and similarity. SE/TE 55, 66, 531, Students prove triangles are congruent or similar and are able to use the concept of corresponding parts of congruent triangles. 6. Students know and are able to use the Triangle Inequality Theorem.

15 15 7. Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. 8. Students know, derive, and solve problems involving perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. SE/TE 25, 29, 30, 31, 37, 52, 53, 67, 72-78, 107, 118, 177, 187, 325, 347, 764, 805 TR: Lesson Masters 1-4A, 1-4B, 2-1A, 2-1B 9. Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres. SE/TE 29, 67, 72-78, 325, 491, 528, 764, 805 TR: Lesson Masters 2-1A, 2-1B 10. Students compute areas of polygons including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids. SE/TE 5, 31, 37, 63, 72, 118, 177, 187, 347

16 Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids. 12. Students find and use measures of sides, interior and exterior angles of triangles and polygons to classify figures and solve problems. SE/TE Students prove relationships between angles in polygons using properties of complementary, supplementary, vertical, and exterior angles. 14. Students prove the Pythagorean Theorem. SE/TE 45-51, 236, 586 TR: Lesson Masters 1-8A, 1-8B TECH: Technology Sourcebook, Computer 2

17 Students use the Pythagorean Theorem to determine distance and find missing lengths of sides of right triangles. SE/TE 45-51, 236, 586 TR: Lesson Masters 1-8A, 1-8B TECH: Technology Sourcebook, Computer Students perform basic constructions with straightedge and compass such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line. 17. Students prove theorems using coordinate geometry, including the midpoint of a line segment, distance formula, and various forms of equations of lines and circles. 18. Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them, (e.g., tan(x) = sin(x)/cos(x), (sin (x)) 2 + (cos (x)) 2 = 1). 19. Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.

18 18 SE/TE TR: Lesson Masters 13-8A, 13-8B; Activity Kit, Activity Students know and are able to use angle and side relationships in problems with special right triangles such as triangles and triangles. 21. Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. 22. Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections. SE/TE TR: Lesson Masters 3-4A, 3-4B TECH: Technology Sourcebook, Computer 6 ALGEBRA II This discipline complements and expands the mathematical content and concepts of Algebra I and Geometry. Students who master Algebra II will gain experience with algebraic solutions of problems in various content areas, including the solution of systems of quadratic equations, logarithmic and exponential functions, the binomial theorem, and the complex number system.

19 19 1. Students solve equations and inequalities involving absolute value. SE/TE 593, 784, 786, Students solve systems of linear equations and inequalities (in two or three variables) simultaneously, by substitution, graphically, or with matrices. SE/TE , , , TR: Lesson Masters 11-1A, 11-1B, 11-3A, 11-3B, 11-8A, 11-8B, TECH: Technology Sourcebook, Computer 28; Activity Kit, Actviity Kit 24

20 20 3. Students are adept at operations on polynomials, including long division. SE./TE , , TE 641 TR: Lesson Masters 10-2A, 10-2B, 10-4A, 10-4B, TECH: Technology Sourcebook, Computer Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes. SE/TE , 744, 769 TR: Lesson Masters 12-3A, 12-3B Tech: Activity Kit, Activity Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.

21 21 6. Students add, subtract, multiply, and divide complex numbers. SE/TE Students add, subtract, multiply, divide, reduce and evaluate rational expressions with monomial and polynomial denominators, and simplify complicated fractions including fractions with negative exponents in the denominator. 8. Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system. SE/TE , , TR: Lesson Masters 12-4A, 12-4B, 12-5A, 12-5B, 12-6A, 12-6B TECH: Technology Sourcebook, Computer 30

22 22 9. Students demonstrate and explain the effect changing a coefficient has on the graph of quadratic functions. That is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(xb)<math>2</math> + c. SE/TE , 612, 779 TR: Lesson Masters 9-1A, 9-1B TECH: Technology Sourcebook, Computer 22; Activity Kit, Activity Students graph quadratic functions and determine the maxima, minima, and zeros of the function. 11. Students prove simple laws of logarithms Students understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents Students judge the validity of an argument based on whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.

23 Students know the laws of exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. SE/TE , , 775 TR: Lesson Masters 8-4A, 8-4B, 8-9A, 8-9B TECH: Technology Sourcebook, Computer 21; Activity Kit, Activity Students use the definition of logarithms and the product formula for logs to translate between logarithms in any bases. 14. Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and identify their approximate values.

24 Students determine if a specific algebraic statement involving rational expressions, radical expressions, logarithmic or exponential functions, is sometimes true, always true, or never true. 16. Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it. 17. Given a quadratic equation of the form ax 2 + by 2 + cx + dy + e = 0, students can use the method of completing the square to put the equation into standard form and can recognize whether its graph is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation. 18. Students use fundamental counting principles to compute combinations and permutations. SE/TE TR: Lesson Masters 2-9A, 2-9B TECH: Technology Sourcebook, Computer 4; Activity Kit, Activity 4

25 Students use combinations and permutations to compute probabilities. SE/TE , TECH: Technology Sourcebook, Computer 4; Activity Kit, Activity Students know the Binomial Theorem and use it to expand binomial expressions which are raised to positive integer powers. 21. Students apply the method of mathematical induction to prove general statements about the positive integers. 22. Students find the general term and the sums of arithmetic series and both finite and infinite geometric series.

26 Students derive the summation formulas for arithmetic series and both finite and infinite geometric series. 24. Students solve problems involving functional concepts such as composition, inverse, and arithmetic operations on functions. 25. Students use properties from number systems to justify steps in combining and simplifying functions. PROBABILITY AND STATISTICS This discipline is an introduction to the study of probability, interpretation of data, and fundamental statistical problem solving. Mastery of this academic content will provide students with a solid foundation in probability and facility with processing statistical information.

27 27 1. Students know the definition of the notion of independent events, and can use the addition, multiplication, and complementation rules to solve for probabilities of particular events in finite sample spaces. SE/TE TR: Lesson Masters 6-4A, 6-4B TECH: Technology Sourcebook, Computer Students know the definition of conditional probability, and use it to solve for probabilities in finite sample spaces. 3. Students demonstrate understanding of the notion of discrete random variables by using them to solve for the probabilities of outcomes, such as the probability of the occurrence of five heads in fourteen coin tosses. SE/TE , Students are familiar with the standard distributions (normal, binomial, and exponential), and can use them to solve for events in problems where the distribution belongs to these families.

28 28 5. Students determine the mean and standard deviation of a normally distributed random variable. 6. Students know the definitions of the mean, median, and mode of distribution of real valued data, and can compute them in particular situations. SE/TE 367, 460, Students compute the variance and standard deviation of a distribution of data. 8. Students organize and describe distributions of data using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem and leaf displays, scatter plots, and box and whisker plots. SE/TE 142, 143, 147, 418, 422, 423, 449, 457

29 29 9. Students find the line of best fit to a given distribution of data using least squares regression. SE/TE TR: Lesson Masters 7-7A, 7-7B TECH: Technology Sourcebook, Computer 17; Activity Kit, Activity 17