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Standard 1: Number and ComputationBenchmark 1: Number Sense . The student demonstrates number sense for real numbers and simple algebraic expressions in a variety of situations.KBI1 knows, explains, and uses equivalent representations for rational numbers and simple algebraic expressions including integers, fractions, decimals, percents, and ratios; rational number bases with integer exponents; rational numbers written in scientific notation with integer exponents; time; and money (2.4.K1a) ($). (Positive Number Line , Number Line , Compare Integers , Fractions to Decimals , Decimals To Fractions , Percentages , Proportions 1 , Simplifying Algebraic Expressions , Ratios , Exponent Basics , Exponents Of Fractional Bases , Negative Exponents Of Fractional Bases , Scientific Notation ) KBI2 compares and orders rational numbers, the irrational number pi, and algebraic expressions (2.4.K1a) ($), e.g., which expression is greater 3n or 3n? It depends on the value of n. If n is positive, 3n is greater. If n is negative, 3n is greater. If n is zero, they are equal. (Compare Mixed Values , Compare Mixed Values 2 ) KBI3 explains the relative magnitude between rational numbers, the irrational number pi, and algebraic expressions (2.4.K1a). KBI4 recognizes and describes irrational numbers (2.4.K1a), e.g., √ 2 is a nonrepeating, nonterminating decimal; or π (pi) is a nonterminating decimal. KBI5 ▴ knows and explains what happens to the product or quotient when (2.4.K1a): a. a positive number is multiplied or divided by a rational number greater than zero and less than one, e.g., if 24 is divided by 1/3, will the answer be larger than 24 or smaller than 24? Explain. b. a positive number is multiplied or divided by a rational number greater than one, C c. a nonzero real number is multiplied or divided by zero, (For purposes of assessment, an explanation of division by zero will not be expected.) KBI6 explains and determines the absolute value of real numbers (2.4.K1a). (Absolute Value 1 , Absolute Value 2 ) AI1 generates and/or solves realworld problems using equivalent representations of rational numbers and simple algebraic expressions (2.4.A1a) ($), e.g., a paper reports a company's gross income as $1.2 billion and their total expenses as $30,450,000. What is the company's net profit? AI2 determines whether or not solutions to realworld problems using rational numbers, the irrational number pi, and simple algebraic expressions are reasonable (2.4.A1a) ($), e.g., the city park is putting a picket fence around their circular rose garden. The garden has a diameter of 7.5 meters. The planner wants to buy 20 meters of fencing. Is this reasonable? Benchmark 2: Number Systems and Their Properties . The student demonstrates an understanding of the real number system; recognizes, applies, and explains their properties; and extends these properties to algebraic expressions. KBI1 explains and illustrates the relationship between the subsets of the real number system [natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] using mathematical models (2.4.K1a), e.g., number lines or Venn diagrams. (Positive Number Line , Number Line , Compare Integers ) KBI2 ▴ identifies all the subsets of the real number system [natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] to which a given number belongs (2.4.K1l). (For the purpose of assessment, irrational numbers will not be included.) KBI3 names, uses, and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($): a. commutative, associative, distributive, and substitution properties [commutative: a + b = b + a and ab = ba; associative: a + (b + c) = (a + b) + c and a(bc) = (ab)c; distributive: a(b + c) = ab + ac; substitution: if a = 2, then 3a = 3 x 2 = 6]; b. identity properties for addition and multiplication and inverse properties of addition and multiplication (additive identity: a + 0 = a, multiplicative identity: a . 1 = a, additive inverse: +5 + 5 = 0, multiplicative inverse: 8 x 1/8 = 1); c. symmetric property of equality, e.g., 7 + 2 = 9 has the same meaning as 9 = 7 + 2; d. addition and multiplication properties of equalities, e.g., if a = b, then a + c = b + c; e. addition property of inequalities, e.g., if a > b, then a + c > b + c; f. zero product property, e.g., if ab = 0, then a = 0 and/or b = 0. (Associative Property 1 , Associative Property 2 , Commutative Property 1 , Commutative Property 2 , Distributive Property , Distributive Property 2 , Variable Substitution , Variable Substitution 2 ) AI1 generates and/or solves realworld problems with rational numbers using the concepts of these properties to explain reasoning (2.4.A1a) ($): a. ▴ commutative, associative, distributive, and substitution properties; e.g., we need to place trim around the outside edges of a bulletin board with dimensions of 3 ft by 5 ft. Explain two different methods of solving this problem and why the answers are equivalent. b. ▴ identity and inverse properties of addition and multiplication; e.g., I had $50. I went to the mall and spent $20 in one store, $25 at a second store and then $5 at the food court. To solve: [$50  ($20 + $25 + $5) = $50  $50 = 0]. Explain your reasoning. c. symmetric property of equality; e.g., Sam took a $15 check to the bank and received a $10 bill and a $5 bill. Later Sam took a $10 bill and a $5 bill to the bank and received a check for $15. $15 = $10 + $5 is the same as $10 + $5 = $15 d. addition and multiplication properties of equality; e.g., the total price (P) of a car, including tax (T), is $14, 685. 33. If the tax is $785.42, what is the sale price of the car (S)? e. zero product property, e.g., Jenny was thinking of two numbers. Jenny said that the product of the two numbers was 0. What could you deduct from this statement? Explain your reasoning (Unit Cost , Distance, Rate, and Time ) AI2 analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions (including mixed numbers), or decimals in solving a given realworld problem (2.4.A1a) ($), e.g., in the store everything is 33 1/3% off. When calculating the discount, which representation of 33 1/3% would you use and why? Benchmark 3: Estimation . The student uses computational estimation with real numbers in a variety of situations. KBI1 estimates real number quantities using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.K1a) ($). KBI2 uses various estimation strategies and explains how they were used to estimate real number quantities and simple algebraic expressions (2.4.K1a) ($). (Rounding Large Numbers , Decimal Rounding ) KBI3 knows and explains why a decimal representation of the irrational number pi is an approximate value (2.4.K1c). KBI4 knows and explains between which two consecutive integers an irrational number lies (2.4.K1a). (Estimating Square Roots ) AI1 adjusts original rational number estimate of a realworld problem based on additional information (a frame of reference) (2.4.A1a) ($), e.g., estimate the height of a building from a picture. In another picture, a person is standing next to the building. By using the person as a frame of reference adjust your original estimate. AI2 estimates to check whether or not the result of a realworld problem using rational numbers and/or simple algebraic expressions is reasonable and makes predictions based on the information (2.4.A1a) ($), e.g., you have a $4,000 debt on a credit card. You pay the minimum of $30 per month. Is it reasonable to pay off the debt in 10 years? AI3 determines a reasonable range for the estimation of a quantity given a realworld problem and explains the reasonableness of the range (2.4.A1c) ($), e.g., determine the reasonable range for the weight of a book and explain why this range is reasonable. AI4 determines if a realworld problem calls for an exact or approximate answer and performs the appropriate computation using various computational methods including mental mathematics, paper and pencil, concrete objects, and/or appropriate technology (2.4.A1a) ($), e.g., do you need an exact or an approximate answer when calculating the area of the walls in a room to determine the number of rolls of wallpaper needed to paper the room?. An approximation is appropriate for the area but an exact answer is needed for the number of roles. What would you do if you were wallpapering 2 rooms? (Estimated Multiply Divide Word Problems ) AI5 explains the impact of estimation on the result of a realworld problem (underestimate, overestimate, range of estimates) (2.4.A1a) ($), e.g., you are estimating the total of three large purchases ($489, $553, and $92). If you rounded each to the nearest $10, would your estimate be slightly lower or higher than the actual amount spent? If you rounded each to the nearest $100, would your estimate be slightly lower or higher than the actual amount spent? Benchmark 4: Computation . The student models, performs, and explains computation with rational numbers, the irrational number pi, and algebraic expressions in a variety of situations. KBI1 computes with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a) ($). KBI2 performs and explains these computational procedures with rational numbers (2.4.K1a): a. ▴N addition, subtraction, multiplication, and division of integers b. ▴N order of operations (evaluates within grouping symbols, evaluates powers to the second or third power, multiplies or divides in order from left to right, then adds or subtracts in order from left to right); c. approximation of roots of numbers using calculators; d. multiplication or division to find: i. a percent of a number, e.g., what is 0.5% of 10? ii. percent of increase and decrease, e.g., if two coins are removed from ten coins, what is the percent of decrease? iii. percent one number is of another number, e.g., what percent of 80 is 120? iv. a number when a percent of the number is given, e.g., 15% of what number is 30? e. addition of polynomials, e.g., (3x  5) + (2x + 8). f. simplifies algebraic expressions in one variable by combining like terms or using the distributive property (2.4.K1a), e.g., 3(x  4) is the same as 3x + 12. (Order Of Operations , Distributive Property , Distributive Property 2 , Percentage Change , Percent of Quantity , Integer Addition , Integer Subtraction , Positive Integer Subtraction , Integer Multiplication , Integer Division , Integer Equivalence , Simplifying Algebraic Expressions ) KBI3 finds factors and common factors of simple monomial expressions (2.4.K1d), e.g., given the monomials 10m^{2}n^{3} and 15a^{2}mn^{2} some common factors would be 5m, 5mn^{2}, and n^{2}. (Binomial Fraction Simplification ) AI1 ▴ generates and/or solves one and twostep realworld problems using computational procedures and mathematical concepts (2.4.A1a) with ($): a. ■ rational numbers, e.g., find the height of a triangular garden given that the area to be covered is 400 square feet with a base of 12½ feet; b. the irrational number pi as an approximation, e.g., before planting, a farmer plows a circular region that has an approximate area of 7,300 square feet. What is the radius of the circular region to the nearest tenth of a foot? c. applications of percents, e.g., sales tax or discounts. (For the purpose of assessment, percents greater than or equal to 100% will NOT be used). (Purchases At Stores , Restaurant Bills , Commissions , Simple Interest ) Standard 2: AlgebraBenchmark 1: Patterns . The student recognizes, describes, extends, develops, and explains the general rule of a pattern from a variety of situations.KBI2 identifies, states, and continues a pattern presented in various formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written using these attributes: f. counting numbers including perfect squares, cubes, and factors and multiples with positive rational numbers (number theory) (2.4.K1a). g. rational numbers including arithmetic and geometric sequences (arithmetic: sequence of numbers in which the difference of two consecutive numbers is the same, geometric: a sequence of numbers in which each succeeding term is obtained by multiplying the preceding term by the same number) (2.4.K1a), e.g., 1/4, 1/2, 3/4, ...; c. geometric figures (2.4.K1h); d. measurements (2.4.K1a); e. things related to daily life ($); f. variables and simple expressions, e.g., 1  x, 2  x, 3  x, 4  x, ...; or x, x^{2}, x^{3}, ... (Patterns: Numbers , Patterns: Shapes , Function Tables , Function Tables 2 ) KBI2 generates and explains a pattern (2.4.K1a). KBI3 generates a pattern limited to two operations (addition, subtraction, multiplication, division, exponents) when given the rule for the nth term (2.4.K1a), e.g., the nth term is n^{2}+1, find the first 4 terms beginning with n = 1; the terms are 2, 5, 10, and 17. KBI4 states the rule to find the nth term of a pattern using explicit symbolic notation (2.4.K1a), e.g., given 2, 5, 8, 11, ...; find the rule for the nth term, the rule is 3n  1. (Function Tables , Function Tables 2 ) KBI5 describes the pattern when given a table of linear values and plots the ordered pairs on a coordinate plane (2.4.K1fg), e.g., in the table below, the pattern could be described as the xcoordinates are increasing by three, while the ycoordinates are increasing by 6, or the x is doubled and one is added to find the y. X 2 5 8 11 Y 5 11 17 23 AI1 generalizes numerical patterns using algebra and then translates between the equation, graph, and table of values resulting from the generalization (2.4.A1de,j) ($), e.g., water is billed at $1.00 per 1,000 gallons, plus a basic fee of $10 per month for being connected to the water district. AI2 recognizes the same general pattern presented in different representations [numeric (list or table), visual (picture, table, or graph), and written] (2.4.A1a,j) ($). Benchmark 2: Variable, Equations, and Inequalities . The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in a variety of situations. KBI1 identifies independent and dependent variables within a given situation. (Independent and Dependent Variables ) KBI2 simplifies algebraic expressions in one variable by combining like terms or using the distributive property (2.4.K1a), e.g., 3(x  4) is the same as 3x + 12. (Distributive Property , Distributive Property 2 , Simplifying Algebraic Expressions ) KBI3 solves (2.4.K1a,e) ($): a. ▴ one and twostep linear equations in one variable with rational number coefficients and constants intuitively and/or analytically; b. onestep linear inequalities in one variable with rational number coefficients and constants intuitively, analytically, and graphically; c. systems of given linear equations with whole number coefficients and constants graphically. (Linear Equations , Single Variable Equations , Single Variable Equations 2 , Single Variable Inequalities , System of Equations Substitution , System of Equations Addition ) KBI4 knows and describes the mathematical relationship between ratios, proportions, and percents and how to solve for a missing monomial or binomial term in a proportion (2.4.K1c), e.g., 2/5 = 1/x+2. (Percentages , Proportions 1 ) KBI5 represents and solves algebraically ($): a. the number when a percent and a number are given, b. what percent one number is of another number, c. percent of increase or decrease, e.g., the price of a loaf of bread is $2.00. With a coupon, the cost is $1.00. What is the percent of decrease? (Percentage Change , Percent of Quantity ) KBI6 evaluates formulas using substitution ($). (Variable Substitution , Simple Interest , Variable Substitution 2 ) AI1 represents realworld problems using (2.4.A1d) ($): a. ▴ ■ variables, symbols, expressions, one or twostep equations with rational number coefficients and constants, e.g., today John is 3.25 inches more than half his sister's height. If J = John's height, and S = his sister's height, then J = 0.5S + 3.25. b. onestep inequalities with rational number coefficients and constants, e.g., after Randy paid $38.50 for a watch, he did not have enough money to by a calculator for $5.50. Represent this situation with an inequality. c. systems of linear equations with whole number coefficients and constants, e.g., two students collected the same amount of money for a walkathon. One student received $5 per mile and a donation of $10, while the other student received $2 per mile and a donation of $20. How many miles did they walk? (Algebraic Word Problems , Age Problems , Algebraic Sentences 2 , Algebraic Sentences ) AI2 solves realworld problems with twostep linear equations in one variable with rational number coefficients and constants and rational solutions intuitively, analytically, and graphically (2.4.A1e) e.g., Mike and Albert are friends, but Joe and Albert are not friends. Which of the following diagrams can be used to describe this situation? (Three dots labeled J, M, A: there is a line between J and M and line between M and A, but no line between J and A.) (Algebraic Word Problems ) AI3 generates realworld problems that represent (2.4.A1d) ($): a. one or twostep linear equations, ($), e.g., given the equation 2x + 10 = 30, the problem could be I bought two shirts and a pair of pants for $10. How much was a shirt, if the total bill was $30? b. onestep linear inequalities, e.g., write a realworld situation that represents the inequality x + 10 > 30. The problem could be: If you give me $10, I will have more than $30. AI4 explains the mathematical reasoning that was used to solve a realworld problem using one or twostep linear equations and inequalities and discusses the advantages and disadvantages to various strategies that may have been used to solve the problem, (2.4.A1d) ($), e.g., given the inequality x + 10 > 30, subtract the same number from both sides of the inequality or graph as y1 = x + 10 and y = 30 and find on the graph where y1 is less than y2. The first method gives an exact answer; the second method is a visual representation and can be used to solve more difficult inequalities. Benchmark 3: Functions . The student recognizes, describes, and analyzes constant, linear, and nonlinear relationships in a variety of situations. KBI1 recognizes and examines constant, linear, and nonlinear relationships using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or appropriate technology (2.4.K1a,eg) ($). (Determining Slope , Nonlinear Functions ) KBI2 knows and describes the difference between constant, linear, and nonlinear relationships (2.4.K1g). (Determining Slope , Nonlinear Functions ) KBI3 explains the concepts of slope and x and yintercepts of a line (2.4.K1g). KBI4 recognizes and identifies the graphs of constant and linear functions (2.4.K1g) ($). (Determining Slope ) KBI5 identifies ordered pairs from a graph, and/or plots ordered pairs using a variety of scales for the x and yaxis (2.4.K1g). (Ordered Pairs ) AI1 represents a variety of constant and linear relationships using written or oral descriptions of the rule, tables, graphs, and symbolic notation (2.4.A1df) ($). AI2 interprets, describes, and analyzes the mathematical relationships of numerical, tabular, and graphical representations (2.4.A1j) ($). AI3 ▴ translates between the numerical, tabular, graphical, and symbolic representations of linear relationships with integer coefficients and constants (2.4.A1a), e.g., a fish tank is being filled with water with a 2liter jug. There are already 5 liters of water in the fish tank. Therefore, you are showing how full the tank is as you empty 2liter jugs of water into it. Y = 2x + 5 (symbolic) can be represented in a table (tabular) . X 0 1 2 3 Y 5 7 9 11 and as a graph (graphical) . (Determining Slope , Graphs to Linear Equations , Graphs to Linear Equations 2 ) Benchmark 4: Models . The student generates and uses mathematical models to represent and justify mathematical relationships found in a variety of situations. KBI1 knows, explains, and uses mathematical models to represent and explain mathematical concepts, procedures, and relationships. Mathematical models include: h. process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, and mathematical relationships and to solve equations (1.1.K16, 1.2.K1, 1.2.K3, 1.3.K12, 1.3.K4, 1.4.K12, 2.1.K1ab, 2.1.K1de, 2.1.K24, 2.2.K23, 3.1.K9, 3.2.K14, 3.3.K14, 3.4.K4, 4.2.K45) ($); i. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures ($); j. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.3.K3, 2.3.K6) ($): k. factor trees to model least common multiple, greatest common factor, and prime factorization (1.4.K3); l. equations and inequalities to model numerical relationships (2.2.K3, (3.4.K2) ($); m. function tables to model numerical and algebraic relationships (2.1.K5, 3.4.K2) ($) ; n. coordinate planes to model relationships between ordered pairs and linear equations and inequalities (2.1.K5, 2.3.K15, 3.4.K2 3) ($); o. two and threedimensional geometric models (geoboards, dot paper, nets, or solids) and realworld objects to model perimeter, area, volume, surface area, and properties of twoand threedimensional figures (2.1.K1c, 3.1.K16, 3.1.K8, 3.1.K10, 3.2.K5, 3.3.K45); p. scale drawings to model large and small realworld objects (3.3.K34); q. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (4.1.K15) ($); r. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stemand leaf plots, scatter plots, boxandwhisker plots, and histograms to organize and display data (4.2.K1, 4.2.K6) ($); s. Venn diagrams to sort data and to show relationships (1.2.K2). (Positive Number Line , Number Line , Compare Integers , Percentage Pictures , Ordered Pairs , Determining Slope , Graphs to Linear Equations , Graphs to Linear Equations 2 , Stem And Leaf Plots , Bar Graphs , Line Graphs , Function Tables , Function Tables 2 ) AI1 recognizes that various mathematical models can be used to represent the same problem situation. Mathematical models include: a. process models (concrete objects, pictures, diagrams, flowcharts, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, mathematical relationships, and problem situations and to solve equations (1.1.A12, 1.2.A12, 1.3.A15, 1.4.A1, 2.1.A1, 3.1.A1, 3.2.A12, 3.3.A1, 3.4.A12) ($); b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to model problem situations ($); c. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (3.2.A3) ($); d. equations and inequalities to model numerical relationships ( 2.1.A2, 2.2.A12, 2.3.A1, 3.4.A2) ($); e. function tables to model numerical and algebraic relationships (2.1.A2, 2.3.A2, 3.4.A2) ($); f. coordinate planes to model relationships between ordered pairs and linear equations and inequalities (2.3.A1 3.4.A2) ($); g. two and threedimensional geometric models (geoboards, dot paper, nets, or solids) and realworld objects to model perimeter, area, volume, surface area and properties of two and threedimensional figures (3.3.A3, 3.4.A2); h. scale drawings to model large and small realworld objects (3.1.A12, 3.3.A4); i. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (4.1.A14); j. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stemandleaf plots, scatter plots, boxandwhisker plots, and histograms to describe, interpret, and analyze data (2.1.A12, 2.3.A23, 4.2.A1, 4.2.A3, 4.2.A13) ($); k. Venn diagrams to sort data and to show relationships. (Positive Number Line , Number Line , Compare Integers , Percentage Pictures , Ordered Pairs , Determining Slope , Graphs to Linear Equations , Graphs to Linear Equations 2 , Stem And Leaf Plots , Bar Graphs , Line Graphs , Function Tables , Function Tables 2 ) AI2 ▴ determines if a given graphical, algebraic, or geometric model is an accurate representation of a given realworld situation ($). AI3 uses the mathematical modeling process to analyze and make inferences about realworld situations ($). Standard 3: GeometryBenchmark 1: Geometric Figures and Their Properties . The student recognizes geometric figures and compares their properties in a variety of situations.KBI1 recognizes and compares properties of two and threedimensional figures using concrete objects, constructions, drawings, appropriate terminology, and appropriate technology (2.4.K1h). KBI2 discusses properties of triangles and quadrilaterals related to (2.4.K1h): h. sum of the interior angles of any triangle is 180°; i. sum of the interior angles of any quadrilateral is 360°; a. parallelograms have opposite sides that are parallel and congruent, opposite angles are congruent; b. rectangles have angles of 90°, sides may or may not be equal; c. rhombi have all sides equal in length, angles may or may not be equal; d. squares have angles of 90°, all sides congruent; e. trapezoids have one pair of opposite sides parallel and the other pair of opposite sides are not parallel; f. kites have two distinct pairs of adjacent congruent sides. (Triangle Angles , Quadrilateral Angles , Quadrilateral Types , Triangle Angles 2 , Solving For Angles ) KBI3 recognizes and describes the rotational symmetries and line symmetries that exist in twodimensional figures (2.4.K1h), e.g., draw a picture with a line of symmetry in it. Explain why it is a line of symmetry. KBI4 recognizes and uses properties of corresponding parts of similar and congruent triangles and quadrilaterals to find side or angle measures using standard notation for similarity (~) and congruence (≅ ) (2.4.K1h). (Congruent And Similar Triangles , Proportions 2 ) KBI5 knows and describes Triangle Inequality Theorem to determine if a triangle exists (2.4.K1h). KBI6 ▴ uses the Pythagorean theorem to (2.4.K1h): a. determine if a triangle is a right triangle, b. find a missing side of a right triangle where the lengths of all three sides are whole numbers. (Pythagorean Theorem ) KBI7 recognizes and compares the concepts of a point, line, and plane. KBI8 describes the intersection of plane figures, e.g., two circles could intersect at no point, one point, two points, or all points. KBI9 describes and explains angle relationships: a. when two lines intersect including vertical and supplementary angles; b. when formed by parallel lines cut by a transversal including corresponding, alternate interior, and alternate exterior angles. (Identifying Angles , Angle Measurements , Angle Measurements 2 ) KBI10 recognizes and describes arcs and semicircles as parts of a circle and uses the standard notation for arc (∩) and circle (Θ) (2.4.K1h). AI1 solves realworld problems by (2.4.A1a): a. ▴ ■ using the properties of corresponding parts of similar and congruent figures, e.g., scale drawings, map reading, proportions, or indirect measurements. b. applying the Pythagorean Theorem, e.g., indirect measurements, map reading/distance, or diagonals. (Proportions 2 ) Benchmark 2: Measurement and Estimation . The student estimates, measures, and uses geometric formulas in a variety of situations. KBI1 determines and uses rational number approximations (estimations) for length, width, weight, volume, temperature, time, perimeter, area, and surface area using standard and nonstandard units of measure (2.4.K1a) ($). KBI2 selects and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate real number representations for length, weight, volume, temperature, time, perimeter, area, surface area, and angle measurements (2.4.K1a) ($). KBI3 converts within the customary system and within the metric system. (Distance Conversion , Volume Conversion , Weight Conversion , Area and Volume Conversions ) KBI4 estimates the measure of a concrete object in one system given the measure of that object in another system and the approximate conversion factor (2.4.K1a), e.g., a mile is about 2.2 kilometers; how far is 2 miles? KBI5 uses given measurement formulas to find (2.4.K1h): a. area of parallelograms and trapezoids; b. surface area of rectangular prisms, triangular prisms, and cylinders; c. volume of rectangular prisms, triangular prisms, and cylinders. (Parallelogram Area , Rectangular Solids , Triangular Prisms , Cylinders , Trapezoids ) KBI6 recognizes how ratios and proportions can be used to measure inaccessible objects (2.4.K1c), e.g., using shadows to measure the height of a flagpole. KBI7 calculates rates of change, e.g., speed or population growth. (Distance, Rate, and Time ) AI1 solves realworld problems (2.4.A1a) by ($): a. converting within the customary and the metric systems, e.g., James added 30 grams of sand to his model boat that weighed 2 kg before it sank. With the sand included, what is the total weight of his boat? b. finding perimeter and area of circles, squares, rectangles, triangles, parallelograms, and trapezoids; e.g., Jane jogs on a circular track with a radius of 100 feet. How far would she jog in one lap? c. finding the volume and surface area of rectangular prisms, e.g., how much paint would be needed to cover a box with dimensions of 3 feet by 4 feet by 5 feet? (Perimeter and Area Word Problems ) AI2 estimates to check whether or not measurements or calculations for length, weight, volume, temperature, time, perimeter, area, and surface area in real world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference) (2.4.A1a) ($), e.g., to check your calculation in finding the area of the floor in the kitchen; you count how many footsquare tiles there are on the floor. AI3 uses ratio and proportion to measure inaccessible objects (2.4.A1c), e.g., using the length of a shadow to measure the height of a flagpole. Benchmark 3: Transformational Geometry . The student recognizes and applies transformations on geometric figures in a variety of situations. KBI1 identifies, describes, and performs single and multiple transformations [reflection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on a twodimensional figure (2.4.K1a). KBI2 describes a reflection of a given twodimensional figure that moves it from its initial placement (preimage) to its final placement (image) in the coordinate plane over the x and yaxis (2.4.K1a,i). (Translations and Reflections ) KBI3 draws (2.4.K1a): a. threedimensional figures from a variety of perspectives (top, bottom, sides, corners); b. a scale drawing of a twodimensional figure; c. a twodimensional drawing of a threedimensional figure. KBI4 determines where and how an object or a shape can be tessellated using single or multiple transformations (2.4.K1a). AI1 generalizes the impact of transformations on the area and perimeter of any twodimensional geometric figure (2.4.A1a), e.g., enlarging by a factor of three triples the perimeter (circumference) and multiplies the area by a factor of nine. (Area And Volume Proportions ) AI2 describes and draws a twodimensional figure after undergoing two specified transformations without using a concrete object. AI3 investigates congruency, similarity, and symmetry of geometric figures using transformations (2.4.A1g). (Congruent And Similar Triangles ) AI4 uses a scale drawing to determine the actual dimensions and/or measurements of a twodimensional figure represented in a scale drawing (2.4.A1h). Benchmark 4: Geometry from an Algebraic Perspective . The student uses an algebraic perspective to examine the geometry of twodimensional figures in a variety of situations. KBI1 uses the coordinate plane to (2.4.K1a): a. ▴ list several ordered pairs on the graph of a line and find the slope of the line; b. ▴ recognize that ordered pairs that lie on the graph of an equation are solutions to that equation; c. ▴ recognize that points that do not lie on the graph of an equation are not solutions to that equation; d. ▴ determine the length of a side of a figure drawn on a coordinate plane with vertices having the same x or ycoordinates; e. solve simple systems of linear equations. (Ordered Pairs , Line Segments , Determining Slope ) KBI2 uses a given linear equation with integer coefficients and constants and an integer solution to find the ordered pairs, organizes the ordered pairs using a Ttable, and plots the ordered pairs on a coordinate plane (2.4.K1eg). KBI3 examines characteristics of twodimensional figures on a coordinate plane using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology (2.4.A1g). AI1 represents, generates, and/or solves distance problems (including the use of the Pythagorean theorem, but not necessarily the distance formula) (2.4.A1a), e.g., a student lives five miles west and three miles north of school and another student lives 4 miles south and 7 miles east of school. What is the shortest distance between the students' homes (as the crow flies)? (Line Segments ) AI2 translates between the written, numeric, algebraic, and geometric representations of a realworld problem (2.4.A1a,dg), e.g., given a situation: make a Ttable, define the algebraic relationship, and graph the ordered pairs. The Ttable can be represented as . as an algebraic relationship 2x + 5 = y, and as a graph (graphical) X 0 1 2 3 Y 5 7 9 11 (Function Tables , Function Tables 2 ) Standard 4: DataBenchmark 1: Probability . The student applies the concepts of probability to draw conclusions, generate convincing arguments, and make predictions and decisions including the use of concrete objects in a variety of situations.KBI1 knows and explains the difference between independent and dependent events in an experiment, simulation, or situation (2.4.K1j) ($). (Object Picking Probability ) KBI2 identifies situations with independent or dependent events in an experiment, simulation, or situation (2.4.K1j), e.g., there are three marbles in a bag. If you draw one marble and give it to your brother, and another marble and give it to your sister, are these independent events or dependent events? (Object Picking Probability ) KBI3 ▴ finds the probability of a compound event composed of two independent events in an experiment, simulation, or situation (2.4.K1j), e.g., what is the probability of getting two heads, if you toss a dime and a quarter? (Probability 2 , Object Picking Probability ) KBI4 finds the probability of simple and/or compound events using geometric models (spinners or dartboards) (2.4.K1j). (Probability , Probability 2 ) KBI5 finds the odds of a desired outcome in an experiment or simulation and expresses the answer as a ratio (2/3 or 2:3 or 2 to 3) (2.4.K1j). (Probability , Probability 2 ) KBI6 describes the difference between probability and odds. (Probability , Probability 2 ) AI1 conducts an experiment or simulation with independent or dependent events including the use of concrete objects; records the results in a chart, table, or graph; and uses the results to draw conclusions and make predictions about future events (2.4.A1ij). (Requires outside materials ) AI2 analyzes the results of an experiment or simulation of two independent events to generate convincing arguments, draw conclusions, and make predictions and decisions in a variety of realworld situations (2.4.A1ij). (Requires outside materials ) AI3 compares theoretical probability (expected results) with empirical probability (experimental results) in an experiment or simulation with a compound event composed of two independent events and understands that the larger the sample size, the greater the likelihood that the experimental results will equal the theoretical probability (2.4.A1i). (Probability 2 , Object Picking Probability ) AI4 makes predictions based on the theoretical probability of (2.4.A1a,i): a. ▴ ■ a simple event in an experiment or simulation, b. compound events composed of two independent events in an experiment or simulation. (Batting Averages ) Benchmark 2: Statistics . The student collects, organizes, displays, explains, and interprets numerical (rational) and nonnumerical data sets in a variety of situations. KBI1 organizes, displays and reads quantitative (numerical) and qualitative (nonnumerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays (2.4.K1k) ($): a. frequency tables; b. bar, line, and circle graphs; c. Venn diagrams or other pictorial displays; d. charts and tables; e. stemandleaf plots (single and double); f. scatter plots; g. boxandwhiskers plots; h. histograms. (Stem And Leaf Plots , Bar Graphs , Line Graphs ) KBI2 recognizes valid and invalid data collection and sampling techniques. KBI3 ▴ determines and explains the measures of central tendency (mode, median, mean) for a rational number data set (2.4.K1a). (Mean, Median, Mode ) KBI4 determines and explains the range, quartiles, and interquartile range for a rational number data set (2.4.K1a). (Stem And Leaf Plots ) KBI5 explains the effects of outliers on the median, mean, and range of a rational number data set (2.4.K1a). KBI6 makes a scatter plot and draws a line that approximately represents the data, determines whether a correlation exists, and if that correlation is positive, negative, or that no correlation exists (2.4.K1k). AI1 uses data analysis (mean, median, mode, range) in realworld problems with rational number data sets to compare and contrast two sets of data, to make accurate inferences and predictions, to analyze decisions, and to develop convincing arguments from these data displays (2.4.A1j) ($): a. frequency tables; b. bar, line, and circle graphs; c. Venn diagrams or other pictorial displays; d. charts and tables; e. stemandleaf plots (single and double); f. scatter plots; g. boxandwhiskers plots; h. histograms. (Mean, Median, Mode ) AI5 explains advantages and disadvantages of various data collection techniques (observations, surveys, or interviews), and sampling techniques (random sampling, samples of convenience, biased sampling, or purposeful sampling) in a given situation (2.4.A1j) ($). AI6 recognizes and explains (2.4.A1j): a. misleading representations of data; b. the effects of scale or interval changes on graphs of data sets. AI4 recognizes faulty arguments and common errors in data analysis. Learn more about our online math practice software. 




