Standard 1 Number and Computation:
The student uses numerical and computational concepts and procedures in a variety of situations.
Benchmark 1
Number Sense - The student demonstrates number sense for real numbers and algebraic expressions in a variety of situations.
Indicator 1
(K) The student 2. compares and orders real numbers and/or algebraic expressions and explains the relative magnitude between them (2.4.K1a), e.g., Will 3n + 2 always, sometimes, or never be larger than 3n?; (5n)^2 is greater than 5n, if n > 1 and (5n)^2 is smaller than 5n, if 0 < n < 1.; or The square root of 5 is between 2 and 3."
Indicator 2
(K) The student knows and explains what happens to the product or quotient when a real number is multiplied or divided by (2.4.K1a):
a. a rational number greater than zero and less than one,
b. a rational number greater than one,
c. a rational number less than zero.
Indicator 1
(A) The student generates and/or solves real-world problems using equivalent representations of real numbers and algebraic expressions (2.4.A1a) $, e.g., a math classroom needs 30 books and 15 calculators. If B represents the cost of a book and C represents the cost of a calculator, generate two different expressions to represent the cost of 9 math classrooms. 9(30B + 15C) and 270B + 270B.
Indicator 2
(A) The student determines whether or not solutions to real-world problems using real numbers and algebraic expressions are reasonable $, e.g., in January a business gave its employees a 10% raise. The following year, due to the sluggish economy, the employees decided to take a 10% reduction in their salary. Is it reasonable to say they are now making the same wage they made prior to the 10% raise?
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.
Indicator 1
(K) The student names, uses, and describes these properties with the real number system and demonstrates their meaning including the use of concrete objects:
a. 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], and substitution properties (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 (if a = b, then b = a);
d. addition and multiplication properties of equality (if a = b, then a + c = b + c and if a = b, then ac = bc) and inequalities (if a > b, then a + c > b + c and if a > b, and c > 0 then ac > bc);
e. zero product property (if ab = 0, then a = 0 and/or b = 0).
Indicator 1
(A) The student generates and/or solves real-world problems with real numbers using the concepts of these properties to explain reasoning (2.4.A1a):
a. commutative, associative, distributive, and substitution properties, e.g., The chorus is sponsoring a trip to an amusement park. They need to purchase 15 adult tickets at $6 each and 15 student tickets at $4 each. How much money will the chorus need for tickets? The expression that models the problem is (15)(6) + (15)(4). To simplify the expression, the distributive property can be used - (15) 6) + (15)(4) = (15)(6 + 4) = (15)(10) = 150. Therefore, the chorus needs $150 to purchase the tickets.;
b. identity and inverse properties, e.g., ex;
c. symmetric property of equality; e.g., ex;
d. addition and multiplication properties of equality, e.g., ex;
e. zero product property, e.g., ex.
Indicator 2
(A) The student analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions (including mixed numbers), decimals or irrational numbers and their rational approximations in solving a given real-world problem, e.g., ex.
Benchmark 3
Estimation - The student uses computational estimation with real numbers in a variety of situations.
Indicator 1
(K) The student estimates real number quantities using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.K1a) $.
Indicator 1
(A) The student adjusts original rational number estimate of a real-world problem based on additional information (a frame of reference) $, e.g., estimate how long it takes to walk from here to there; time how long it takes to take five steps and adjust your estimate.
Benchmark 4
Computation - The student models, performs, and explains computation with real numbers and polynomials in a variety of situations.
Indicator 1
(K) The student computes with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a) $.
Indicator 1
(A) The student generates and/or solves multi-step real-world problems with real numbers and algebraic expressions using computational procedures (addition, subtraction, multiplication, division, roots, and powers excluding logarithms), and mathematical concepts with:
a. applications from business, chemistry, and physics that involve addition, subtraction, multiplication, division, squares, and square roots when the formulae are given as part of the problem and variables are defined $, e.g., Given F = ma, where F = force in newtons, m = mass in kilograms, a = acceleration in meters per second squared. Find the acceleration if a force of 20 newtons is applied to a mass of 3 kilograms.;
b. volume and surface area given the measurement formulas;
c. probabilities, e.g., If the probability of getting a defective light bulb is 2%, and you buy 150 light bulbs, how many would you expect to be defective?;
d. application of percents $, e.g., compound interest given the formula;
e. simple exponential growth and decay (excluding logarithms) and economics $ e.g., A population of cells doubles every 20 years. If there are 20 cells to start with, how long will it take for there to be more than 150 cells? or If the radiation level is now 400 and it decays by ½ or its half-life is 8 hours, how long will it take for the radiation level to be below an acceptable level of 5? or If $1000 is placed in a savings account with a 6% annual interest rate and is compounded semiannually, how much money will be in the account at the end of 2 years?
The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1
Patterns - The student recognizes, describes, extends, develops, and explains the general rule of a pattern in a variety of situations.
Indicator 1
(K) The student identifies, states, and continues the following patterns using various formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written (2.4.K1a-b,f,h):
a. arithmetic and geometric sequences using real numbers and/or exponents; e.g., radioactive half-lives;
b. patterns using geometric figures;
c. algebraic patterns including consecutive number patterns or equations of functions, e.g., n, n + 1, n + 2, ... or f(n) = 2n - 1;
d. special patterns, e.g., Pascal's triangle and the Fibonacci sequence.
Benchmark 2
Variables, Equations, and Inequalities - The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in variety of situations.
Indicator 1
(K) The student solves (2.4.K1g):
a. linear equations and inequalities both analytically and graphically without the aid of a calculator or computer $, N;
b. quadratic equations with integer solutions (may be solved by trial and error, graphing, quadratic formula, or factoring);
c. systems of linear equations with two unknowns using rational coefficients and constants;
d. radical equations with no more than one inverse operation around the radical expression;
e. equations where the solution to a rational equation can be simplified as a linear equation with a nonzero denominator;
f. equations and inequalities with absolute value quantities containing one variable with a special emphasis on using a number line and the concept of absolute value.
g. exponential equations with the same base without the aid of a calculator or computer, e.g., 3^x + 2 = 3^5.
Indicator 1
(A) The student represents real-world problems using variables, symbols, expressions, equations, inequalities, and simple systems of linear equations (2.4 A1c-d).
Indicator 2
(A) The student represents and/or solves real-world problems with (2.4.A1c-d):
a. linear equations and inequalities both analytically and graphically, $, N e.g., ex
b. quadratic equations with integer solutions (may be solved by trial and error, graphing, quadratic formula, or factoring), e.g., ex;
c. systems of linear equations with two unknowns (2.4.A1c-d) $, e.g., When comparing two cellular telephone plans, Plan A costs $10 per month and $.10 per minute and Plan B costs $12 per month and $.07 per minute. The problem is represented by Plan A = .10x + 10 and Plan B = .07x + 12 where x is the number of minutes.;
d. radical equations with no more than one inverse operation around the radical expression, e.g., ex;
e. a rational equation where the solution can be simplified as a linear equation with a nonzero denominator, e.g.,
Indicator 3
(A) The student explains the mathematical reasoning that was used to solve a real-world problem using equations and inequalities and analyzes the advantages and disadvantages of various strategies that may have been used to solve the problem (2.4.A1c-d).
Benchmark 3
Functions - The student analyzes functions in a variety of situations.
Indicator 1
(K) The student matches equations and graphs of constant and linear functions and quadratic functions limited to y = ax^2 + c.
Indicator 2
(K) The student recognizes how changes in the constant and/or slope within a linear function changes the appearance of a graph.
Indicator 1
(A) The student translates between the numerical, graphical, and symbolic representations of functions (2.4.A1g) $.
Indicator 2
(A) The student interprets the meaning of the x- and y- intercepts, slope, and/or points on and off the line on a graph in the context of a real-world situation (2.4 A1g), e.g., The graph below represents a tank full of water being emptied. What does the y-intercept represent? What does the x intercept represent? What is the rate at which it is emptying? What does the point (2, 25) represent in this situation? What does the point (2,30) represent in this situation?
Benchmark 4
Models - The student develops and uses mathematical models to represent and justify mathematical relationships found in a variety of situations involving tenth grade knowledge and skills.
Indicator 1
(K) The student knows, explains, and uses mathematical models to represent and explain mathematical concepts, procedures, and relationships.
Mathematical models include:
a. process models (concrete objects, pictures, diagrams, number lines, coordinate grids, hundred charts, measurement tools, multiplication arrays, or division sets) to model computational procedures, algebraic relationships, and mathematical relationships and to solve equations (1.1.K1-3, 1.3.K1, 1.4.K1, 2.1 K1, 2.2.K3, 2.3.K1) $;
b. factor trees to model least common multiple, greatest common factor, and prime factorization; (1.4.K2h);
c. algebraic expressions to model relationships between two successive numbers in a sequence or other numerical patterns (2.1.K1-5);
d. equations and inequalities to model numerical and geometric relationships (1.4.K2d, 2.2.K2-3, 3.1.A5b);
e. function tables to model numerical and algebraic relationships (2.1.K1, 2.3.K3, 2.3.K5a)
f. coordinate planes to model relationships between ordered pairs and equations and inequalities and linear and quadratic functions (2.3.K3-4, 3.4.K1, 3.4.K1-3, 3 4.K5b, 3.4.K6-8);
g. constructions to model geometric theorems and properties; (3.1.K2, 3.1.K6)
h. two- and three-dimensional geometric models (geoboards, dot paper, coordinate plane, nets, or solids) and real-world objects to model perimeter, area, volume, and surface area and isometric views of three dimensional figures (3.2.K1, 3.2.K4-5, 3.3.K1-4);
i. scale drawings to model large and small real-world objects;
j. Pascal's Triangle to model binomial expansion and probability;
k. geometric models (spinners, targets, or number cubes), process models concrete objects, pictures, diagrams, or coins), and tree diagrams to model probability;
l. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts tables, single and double stem-and-leaf plots, scatter plots, box and-whisker plots, histograms, and matrices to organize and display data (4.2.K1) $.
m. Venn diagrams to sort data and to show relationships.
Indicator 1
(A) The student 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, coordinate grids, hundred charts, measurement tools, multiplication arrays or division sets) to model computational procedures, algebraic relationships, mathematical relationships, and problem situations and to solve equations (1.1 K1, 2.2.A1-3) $;
b. algebraic expressions to model relationships between two successive numbers in a sequence or other numerical patterns (2.1.A2);
c. equations and inequalities to model numerical and geometric relationships (2.1.A2, 2.2.A1-3);
d. function tables to model numerical and algebraic relationships (3.4.A2) $;
e. coordinate planes to model relationships between ordered pairs and equations and inequalities and linear and quadratic functions (2.2.A2a, 2.3.A1-5, 3.3.A4, 3.4.A1-4);
f. two- and three-dimensional geometric models (geoboards, dot paper, coordinate plane, nets, or solids) and real-world objects to model perimeter, area volume, and surface area and isometric views of three dimensional figures (3.3 A1-4);
g. scale drawings to model large and small real-world objects (3.3.A3, 3.4.A1);
h. geometric models (spinners, targets, or number cubes) and process models coins, pictures, or diagrams) to model probability (4.1.3c);
i. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts tables, single and double stem-and-leaf plots, scatter plots, box and-whisker plots, histograms, and matrices to describe, interpret, and analyze data (4.2.A1) $.
j. Venn diagrams to sort data and show relationships.
Indicator 2
(A) The student uses the mathematical modeling process to analyze and make inferences about real-world situations.
The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1
Geometric Figures and Their Properties - The student recognizes geometric figures and compares properties and concepts of geometric figures, and justifies the properties of geometric figures in a variety of situations.
Indicator 1
(K) The student recognizes and compares properties of two-and three dimensional figures using concrete objects, constructions, drawings, appropriate terminology, and appropriate technology.
Indicator 2
(K) The student discusses properties of regular polygons related to (2.4.K1f):
a. angle measures,
b. diagonals.
Indicator 3
(K) The student recognizes and describes the symmetries (point, line, plane) that exist in three-dimensional figures.
Indicator 4
(K) The student recognizes that similar figures have congruent angles, and their corresponding sides are proportional.
Indicator 5
(K) The student uses the Pythagorean Theorem to (2.4.K1g):
a. determine if a triangle is a right triangle,
b. find a missing side of a right triangle.
Indicator 6
(K) The student recognizes and describes (2.4.K1h):
a. congruence of triangles using: Side-Side-Side (SSS), Angle-Side Angle (ASA) Side-Angle-Side (SAS), and Angle-Angle-Side (AAS);
b. the ratios of the sides in special right triangles: 30°-60°-90° and 45°-45°-90°.
Indicator 7
(K) The student recognizes, describes, and compares the relationships of the angles formed when parallel lines are cut by a transversal.
Indicator 8
(K) The student recognizes and identifies parts of a circle: arcs, chords, sectors of circles, secant and tangent lines, central and inscribed angles.
Indicator 1
(A) The student solves real-world problems by (2.4.A1a,f):
a. using the properties of corresponding parts of similar and congruent figures, e.g., scale drawings, map reading, or proportions;
b. applying the Pythagorean Theorem, e.g., When checking for square corners on concrete forms for a foundation, determine if a right angle is formed by using the Pythagorean Theorem.;
c. using properties of parallel lines, e.g., street intersections.
Indicator 2
(A) The student uses deductive reasoning to justify the relationships between the sides of 30°-60°-90° and 45°-45°-90° triangles using the ratios of sides of similar triangles.
Indicator 3
(A) The student understands the concepts of and develops a formal or informal proof through understanding of the difference between a statement verified by proof (theorem) and a statement supported by examples.
Benchmark 2
Measurement and Estimation - The student estimates, measures and uses geometric formulas in a variety of situations.
Indicator 1
(K) The student determines and uses real number approximations (estimations) for length, width, weight, volume, temperature, time, distance, perimeter, area, surface area, and angle measurement using standard and nonstandard units of measure (2.4.K1a) $.
Indicator 2
(K) The student 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, distance, area, surface area, mass, midpoint, and angle measurements $.
Indicator 3
(K) The student approximates conversions between customary and metric systems given the conversion unit or formula.
Indicator 4
(K) The student states, recognizes, and applies formulas for (2.4.K1h) $:
a. perimeter and area of squares, rectangle, and triangles;
b. circumference and area of circles;
c. volume of rectangular solids.
Indicator 5
(K) The student uses given measurement formulas to find perimeter, area, volume, and surface area of two- and three-dimensional figures (regular and irregular).
Indicator 6
(K) The student recognizes and applies properties of corresponding parts of similar and congruent figures to find measurements of missing sides.
Indicator 7
(K) The student knows, explains, and uses ratios and proportions to describe rates of change $, e.g., miles per gallon, meters per second, calories per ounce, or rise over run.
Indicator 1
(A) The student solves real-world problems by (2.4.A1a) $:
a. converting within the customary and the metric systems, e.g., Marti and Ginger are making a huge batch of cookies and so they are multiplying their favorite recipe quite a few times. They find that they need 45 tablespoons of liquid. To the nearest ¼ of a cup, how many cups would be needed?
b. finding the perimeter and the area of circles, squares, rectangles, triangles, parallelograms, and trapezoids, e.g., a track is made up of a rectangle with dimensions 100 meters by 50 meters with semicircles at each end (having a diameter of 50 meters). What is the distance of one lap around the inside lane of the track?
c. finding the volume and the surface area of rectangular solids and cylinders, e.g., if a car engine has 6 cylinders and each cylinder has a height of 8 4 cm and a diameter of 8.8 cm, then what is the total volume of the cylinders?
d. using the Pythagorean theorem, e.g., a baseball diamond is a square with 90 feet between each base. What is the approximate distance from home plate to second base?
e. using rates of change, e.g., the equation w = -52 + 1.6t can be used to approximate the wind chill temperatures for a wind speed of 40 mph. Find the wind chill temperature (w) when the actual temperature (t) is 32 degrees. What part of the equation represents the rate of change?
Indicator 2
(A) The student estimates to check whether or not measurements or calculations for length, weight, volume, temperature, time, distance, perimeter, area, surface area, and angle measurement in real-world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference) (2.4.A1a) $.
Indicator 3
(A) The student uses indirect measurements to measure inaccessible objects, e g., You are standing next to the railroad tracks and a train passes. The number of cars in the train can be determined if you know how long it takes for one car to pass and the length of time the whole train takes to pass you.
Benchmark 3
Geometry from an Algebraic Perspective - The student uses an algebraic perspective to analyze the geometry of two- and three-dimensional figures in a variety of situations.
Indicator 1
(K) The student recognizes and examines two- and three-dimensional figures and their attributes including the graphs of functions on a coordinate plane using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology (2.4.K1I).
Indicator 2
(K) The student determines if a given point lies on the graph of a given line or parabola without graphing and justifies the answer (2.4.K1i).
Indicator 3
(K) The student calculates the slope of a line from a list of ordered pairs on the line and explains how the graph of the line is related to its slope (2.4.K1i).
Indicator 4
(K) The student finds and explains the relationship between the slopes of parallel and perpendicular lines.
Indicator 5
(K) The student uses the Pythagorean Theorem to find distance (may use the distance formula) (2.4.K1i).
Indicator 6
(K) The student recognizes the equation of a line and transforms the equation into slope-intercept form in order to identify the slope and y intercept and uses this information to graph the line (2.4.K1i).
Indicator 7
(K) The student recognizes the equation y = ax^2 + c as a parabola; represents and identifies characteristics of the parabola including opens upward or opens downward, steepness (wide/narrow), the vertex, maximum and minimum values, and line of symmetry; and sketches the graph of the parabola (2.4.K1i).
Indicator 8
(K) The student explains the relationship between the solution(s) to systems of equations and systems of inequalities in two unknowns and their corresponding graphs (2.4.K1i), e.g., for equations, the lines intersect in either one point, no points, or infinite points; and for inequalities, all points in double-shaded areas are solutions for both inequalities.
Indicator 1
(A) The student represents, generates, and/or solves real-world problems that involve distance and two-dimensional geometric figures including parabolas in the form ax^2 + c (2.4.A1g,i), e.g., ex.
Indicator 2
(A) The student translates between the written, numeric, algebraic, and geometric representations of a real-world problem (2.4.A1f-g) $, e.g., given a situation (ex), write a function rule, make a T-table of the algebraic relationship, graph the order pairs.
Indicator 3
(A) The student recognizes and explains the effects of scale changes on the appearance of the graph of an equation involving a line or parabola (2.4.A1g).
Indicator 4
(A) The student analyzes how changes in the constants and/or leading coefficients within the equation of a line or parabola affects the appearance of the graph of the equation (2.4.A1g).
The student uses concepts and procedures of data analysis in a variety of situations.
Benchmark 1
Probability - The student applies probability theory to draw conclusions, generate convincing arguments, make predictions and decisions, and analyze decisions including the use of concrete objects in a variety of situations.
Indicator 1
(K) The student finds the probability of two independent events in an experiment simulation, or situation.
Indicator 2
(K) The student finds the conditional probability of two dependent events in an experiment, simulation, or situation.
Indicator 3
(K) The student explains the relationship between probability and odds and computes one given the other.
Indicator 1
(A) The student conducts an experiment or simulation with two dependent events; records the results in charts, tables, or graphs; and uses the results to generate convincing arguments, draw conclusions and make predictions.
Indicator 2
(A) The student uses theoretical or empirical probability of a simple or compound event composed of two or more simple, independent events to make predictions and analyze decisions about real-world situations including:
a. work in economics, quality control, genetics, meteorology, and other areas of science;
b. games;
c. situations involving geometric models, e.g., spinners or dartboards (2.4.A1j).
Indicator 3
(A) The student compares theoretical probability (expected results) with empirical probability (experimental results) of two independent and/or dependent events and understands that the larger the sample size, the greater the likelihood that experimental results will match theoretical probability.
Indicator 4
(A) The student uses conditional probabilities of two dependent events in an experiment, simulation, or situation to make predictions and analyze decisions.
Benchmark 2
Statistics - The student collects, organizes, displays, explains, and interprets numerical (rational) and non-numerical data sets in a variety of situations.
Indicator 1
(K) The student organizes, displays, and reads quantitative (numerical) and qualitative (non-numerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays (2.4.K1l).
a. frequency tables;
b. bar, line, and circle graphs;
c. Venn diagrams or other pictorial displays;
d. charts and tables;
e. stem-and-leaf plots (single and double);
f. scatter plots;
g. box-and-whiskers plots;
h. histograms.
Indicator 2
(K) The student explains how the reader's bias, measurement errors, and display distortions can affect the interpretation of data.
Indicator 3
(K) The student calculates and explains the meaning of range, quartiles and interquartile range for a real number data set.
Indicator 4
(K) The student explains the effects of outliers on the measures of central tendency (mean, median, mode) and range and interquartile range of a real number data set.
Indicator 5
(K) The student approximates a line of best fit given a scatter plot, and makes predictions using the equation of that line.
Indicator 6
(K) The student compares and contrasts the dispersion of two given sets of data in terms of range and the shape of the display including:
a. symmetrical (including normal),
b. skew-left or skew-right,
c. bimodal,
d. uniform (rectangular).
Indicator 1
(A) The student uses data analysis (mean, median, mode, range, quartile, interquartile range) in real-world 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. stem-and-leaf plots (single and double);
f. scatter plots
g. box-and-whiskers plots;
h. histograms.
Indicator 2
(A) The student determines and describes appropriate data collection techniques observations, surveys, or interviews) and sampling techniques (random sampling, samples of convenience, biased sampling, census of total population, or purposeful sampling) in a given situation.
Indicator 3
(A) The student uses changes in scales, intervals, and categories to help support a particular interpretation of the data.
Indicator 4
(A) The student determines and explains the advantages and disadvantages of using each measure of central tendency and the range to describe a data set (2.4.K1i).
Indicator 5
(A) The student analyzes the effects of:
a. outliers on the mean, median, and range of a real number data set;
b. changes within a real number data set on mean, median, mode, range, quartiles, and interquartile range.
Indicator 6
(A) The student approximates a line of best fit given a scatter plot, makes predictions, and analyzes decisions using the equation of that line.
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