2 = Standard Numbers and Computation The student uses numerical and computational concepts and procedures in a variety of situations.

Transcription

1 Standard Numbers and Computation The student uses numerical and computational concepts and procedures in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student MA550X.NC.NSP.1 explains and illustrates the relationship between the subsets of the complex number system [real numbers, imaginary numbers, natural (counting) numbers, whole numbers, integers, rational numbers, and irrational numbers] using mathematical models. MA550X.NC.NSP.2 identifies all the subsets of the complex number system [real numbers, imaginary numbers, natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] to which a given number belongs. MA550X.NC.NSP.3 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: = 0, Benchmark Number Systems and Their Properties The student demonstrates an understanding of the complex number system; recognizes, applies, and explains their properties, and extends these properties to algebraic expressions. Essential Concepts/Skills Implementation Assessment Examples What Students Need to Know 1. Operations of integers, real numbers. 2. How to make and use a Venn diagram or other structural diagrams or charts (check by having students make a diagram showing relationships between various automobiles). 3. Understanding of coordinate plane (axis, plotting). 4. Order of Operations. What Students Need to Do/Apply 1. Name, use, and explain number properties (closure, commutative, associative, identity, inverse, distributive). 2. Simplify and evaluate expressions using order of operation or properties of exponents. 3. Classify numbers (real, imaginary, complex, rational, irrational, whole, natural, integer). 4. Recognize the need for numbers beyond the real (imaginary, complex). 5. Simplify imaginary numbers. 6. Solve real world problems using very large and very small numbers. 7. Use applications from business, economics, chemistry, and physics. Strategies PEMDAS or other acronym or pneumonic devise for Scope and Sequence Geometry students explained and illustrated the relationship between the subsets of the real number system, as well as are able to identify all of the subsets. They recognized and used the properties of the real number system including the commutative and associative properties of addition and multiplication, distributive property, identity properties for addition and multiplication, as well as the properties of equality. Algebra 3 students will explore and illustrate the relationship between the subsets of the complex number system. They will be expected to recognize and use the properties of the complex number system. Pacing Considerations Operations with complex numbers 2 weeks 1. Use a Venn Diagram to show the relationships between the following numbers systems: real, imaginary, complex, integer, rational, irrational, whole, natural. Identify both the complex and imaginary parts of 7x-2iy. 2. Which of the following sets of numbers does not contain ? a. real b. rational c. irrational d. all of these Identify the property being illustrated in each of the following: 3a. 3x + 4x = 4x + 3x 3b. 5y+ 5y = = x 1= 6x 3c. Rewrite the following equation applying the symmetric property of equality: 3x + 5 = 12 3d. Use the addition property of equality to solve the equation, x 10 = 45. 1

2 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). MA550X.NC.NSP.4 uses and describes these properties with the real number system: a. transitive property (if a = b and b = c, then a = c). b. Reflexive property (a = a). Application Indicators None tested on the 9/10 Kansas Math Assessment. Vocabulary Students Know and Use addition property subtraction property multiplication property division property natural numbers whole numbers rational numbers irrational numbers real numbers integers imaginary numbers complex numbers complex plane imaginary axis closure property identify property inverse memorizing order of operation. Spreadsheets Have students create graphic organizers for visual learners to see relationships between number types. Use the coordinate plane, the horizontal axis and an imaginary axis to have students make the connection between the coordinate plane and the complex plane. 3e. If (x + 3) (x 5) = 0, what must be true about the values of x? 4a. Use the transitive property of equality to create a new equation if 5x+ 8= 3a 1 3a 1= 2x 4 and 4b. Write an equation which illustrates the reflexive property of equality. 2

3 distributive property zero product property conjugate transitive property reflexive property symmetric property 3

4 Standard Numbers and Computation The student uses numerical and computational concepts and procedures in a variety of situations. Benchmark Estimation The student uses computational estimation with complex numbers in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student MA550X.NC.E.1 estimates real number quantities using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology. MA550X.NC.E.2 uses various estimation strategies and explains how they were used to estimate real number quantities and algebraic expressions. MA550X.NC.E.3 knows and explains why a decimal representation of an irrational number is an approximate value. MA550X.NC.E.4 knows and explains between which two consecutive integers an irrational number lies. Application Indicators The student MA550X.NC.E.5 adjusts original rational number estimate of a real-world problem based on additional information (a frame of reference). Vocabulary Students Know and Use round truncate approximate integers Essential Concepts/Skills Implementation Assessment Examples What Students Need to Know 1. Operations of real numbers. 2. Order of operation. 3. Integers, scientific notation, absolute value, decimals, place value. 4. Convert between fractions, decimals, and percents. What Students Need to Do/Apply 1. Compare and order real numbers or algebraic expressions. 2. Know and use equivalent representations of the same real number and/or algebraic expressions. 3. Determine the reasonableness of solutions. 4. Use a variety of computational methods to estimate real number quantities involving rational numbers and pi. 5. Perform estimation, explain the method they chose, and use the estimated result to check the reasonableness of results and to make predictions. Strategies Use PEMDAS. Problems involving real numbers and algebraic expressions. Problem solving tool (3PR, UGLY, Polya s, etc.) Number lines or other visual Scope and Sequence Geometry students used the various estimation strategies and explained why a decimal representation of an irrational number is an approximate value. Algebra 3 students continue to use the various estimation strategies. Pacing Considerations Teach throughout coursework include estimation strategies as problems of the day. 1. Residential wastewater rates are based on a monthly customer charge of $4 plus $1.89 per 1,000 gallons of water used. Estimate the monthly cost of 8,400 gallons. 2. If you have a $4,000 debt on a credit card and the minimum of $30 is paid per month, is it reasonable to pay off the debt in 10 years? 3. Would it be better to use π or 3.14 when calculating the amount of outdoor carpet you will need to cover you patio? Explain your response. 4. Between what two integers does 51 lie? Approximate the value of 51 to the nearest tenth without using a calculator. 5. Mark's bowling scores last week were 151, 108, 151, and 152. Based on these scores he ESTIMATED his average score to be about 140. In the first two games Mark bowled this week he scored 165 and 162. Which statement best explains how Mark should adjust his ESTIMATE of his average score based on the additional information from his last two games? 4

5 aides. Mental math, Paper/pencil Estimating Techniques such as rounding, truncating, greatest integer, etc.). 5

6 Standard Numbers and Computation The student uses numerical and computational concepts and procedures in a variety of situations. Benchmark Computation The student models, performs, and explains computation with complex numbers and polynomials in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student MA550X.NC.C.1 computes with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology. MA550X.NC.C.2 performs and explains these computational procedures: a. N addition, subtraction, multiplication, and division using the order of operations. b. multiplication or division to find: i. a percent of a number. ii. percent of increase and decrease. iii. percent one number is of another number. iv. a number when a percent of the number is given. c. manipulation of variable quantities within an equation or inequality. d. simplification of radical expressions including square roots of perfect square monomials and cube roots of perfect cubic monomials. e. simplification or evaluation of real numbers and algebraic monomial expressions raised to a rational number power and algebraic binomial expressions squared or cubed. Essential Concepts/Skills Implementation Assessment Examples What Students Need to Know 1. How to enter exponential values into their calculator. 2. How to rename decimals, fractions, and percents. 3. Understanding of percent. What Students Need to Do/Apply 1. Explain and perform computational procedures emphasizing the order of operations (no calculator). 2. Simplify radical expressions. 3. Simplify and/or evaluate expressions with exponents (properties of exponents). 4. Simplify the products and quotients of real number and algebraic monomial expressions. 5. Calculate with percents and find percents when given data. 6. Manipulate variable quantities fluently and accurately. 7. Compute with matrices. 8. Understand and compute with imaginary numbers. 9. Use real number properties to perform various computational procedures and explain how they are used. Strategies Solve real-world problems using very large and very small numbers. Use a variety of computational Scope and Sequence Geometry students used the order of operations, simplified radical expressions, manipulated variable quantities within an equation or inequality, simplified or evaluated real numbers and algebraic expressions raised to a power and algebraic binomial expressions squared or cubed, and simplified products and quotients of real number and algebraic and algebraic monomial expressions using the properties of exponents. Algebra 3 students will simplify and evaluate radical expressions, rationalize denominators, simplify rational expressions including complex fractions. They will also simplify expressions using matrix operations. Evaluating natural exponential and logarithmic functions using a calculator as well as using the change-of-base formula to evaluate a logarithmic expression will also be covered. Pacing Considerations Properties of Exponents, Operations, Simplification throughout course Polynomials, operations, simplification 3 weeks Radical functions, operations, simplification, interest 3 weeks 1. A drop of water contains about 1.7 x water molecules. Find the number of molecules of water in a glass that has x 10 drops. How many millions of molecules are in the glass? Write your answer without using exponents. 2a. N In as many ways as possible, obtain a result of 10 by using each of the numbers 11, 1, 8, 3, and 14 only once and using any orders of operations. 2bii. Every year, more and more people in the United States become cellular phone subscribers. In 1990, there were 5.3 million subscribers; this number had increased to 16 million subscribers in Find the percent increase from 1990 to c. Solve: x = x + Solve for b 2 : A= 1 ( b ) 1+ b2 2 2d. Simplify: denominator. Simplify: 5 with a rational cd (6 cd ) Simplify: (-5-18 )-(6-50 ) 6

7 MA550X.NC.C.3 simplification of products and quotients of complex numbers and algebraic monomial expressions using the properties of exponents. Application Indicators The student MA550X.NC.C.4 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. b. volume and surface area given the measurement formulas of rectangular solids and probabilities. c. application of percents. Vocabulary Students Know and Use relative Magnitude evaluate simplify monomial polynomial radical root radicand binomial trinomial scientific notation exponents methods including mental mathematics, paper and pencil, manipulatives, and graphing. Use applications from business, economics, chemistry, and physics. Cognitive Tutor computerized software. Internet sites such as Scientific formulas PEMDAS or other pneumonic device for order of operation. Factor trees and other such concrete models for simplifying exponents. Fishing models for simplifying radicals. Area models for multiplying binomials. Balancing equations models for solving linear equations. FOIL or other acronym for multiplying binomials. 2e. Evaluate: Write the polynomial in standard 2 form: ( x + 2) 3. Simplify: ( 5 a b )( 2 a b ) a. Given the formula A = P(1+r) n, when A = amount, P= principal, r = interest rate for the compounded period, and n = number of times compounded, solve the following: If $1,000 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? 7

8 function power coefficient constant degree of monomial degree of polynomial rationalizing the denominator conjugate simple interest compound interest 8

9 Standard Algebra The student uses algebraic concepts and procedures in a variety of situations. Benchmark Patterns The student recognizes, describes, extends, develops, and explains the general rule of a pattern in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student. MA550X.A.P.1 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: a. arithmetic and geometric sequences using real numbers and/or exponents. b. patterns using geometric figures. c. algebraic patterns including consecutive number patterns or equations of functions. d. special patterns. MA550X.A.P.2 generates and explains a pattern. MA550X.A.P.3 classify sequences as arithmetic, geometric, or neither. MA550X.A.P.4 defines: a. a recursive or explicit formula for arithmetic sequences and finds any particular term. b. a recursive or explicit formula for geometric sequences and finds any particular term. Application Indicators None assessed on the 9/10 Kansas Math Assessment. Essential Concepts/Skills Implementation Assessment Examples What Students Need to Know 1. Write a linear equation given data. 2. How to enter data into tables in graphing calculator. 3. How to use regression tools in graphing calculator. 4. How to graph and use models to interpret types of functions. 5. A problem solving tool such as 3PR, UGLY, or Polya s. What Students Need to Do/Apply 1. Recognize the generalization of a pattern using symbolic notation to represent the nth term. 2. Recognize the same general pattern presented in different representations. 3. Identify and continue patterns presented in a variety of formats. Strategies Solve real world problems that suggest patterns. Find examples of real world data collection in newspapers or magazines. Probability experiments (i.e. handshake problem) Graphing Calculator-using quadratic and exponential regressions. Spreadsheets Use of tables Problem solving strategy templates. Scope and Sequence Geometry students identified, stated, and continued patterns for arithmetic and geometric sequences using real numbers and/or exponents. Also they recognized and identified patterns using geometric figures, algebraic patterns including consecutive number patterns or equations of functions and special patterns. Algebra 3 students find the explicit or recursive formula and indicated terms of an arithmetic and geometric sequence. They will evaluate given arithmetic and geometric series. Topics also include binomial expansions and Pascal s Triangle. Pacing Considerations Sequences, series, and summations 3 weeks 1, 2, 3. For each pattern a) Give the next three terms. b) Tell whether the pattern is arithmetic or geometric. c) Write an expression for the nth term. d) Find the 20 th term. 1. 6, 10, 14, 18, 22, 2. 2, 4, 8, 16, 3. 1, 4, 9, 16, 25, 4a. Write an expression for the nth term of the following sequence. 6, 10, 14, 18, 22 4b. Find the 8 th term for the sequence 2, 4, 8, 16, 9

10 Vocabulary Students Know and Use constant linear exponential quadratic function summation factorial sequence series rate of change slope lines of best fit predict 10

11 Standard Algebra The student uses algebraic concepts and procedures in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student MA550X.A.V.1 knows and explains the use of variables as parameters for a specific variable situation. MA550X.A.V.2 manipulates variable quantities within an equation or inequality. MA550X.A.V.3 solves: a. N linear equations and inequalities both analytically and graphically. b. quadratic equations with complex number solutions (may be solved by trial and error, graphing, quadratic formula, or factoring). c. N systems of linear equations with two unknowns using integer 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. Benchmark Variables, Equations, and Inequalities The student uses variables, symbols, complex numbers, and algebraic expressions to solve equations and inequalities. Essential Concepts/Skills Implementation Assessment Examples What Students Need to Know 1. Solve simple linear equations. What Students Need to Do/Apply 1. Set up and solve equations and inequalities. 2. Use a variety of ways to represent problem situations that involve variable quantities. 3. Formulate and solve problems involving symbols, percents, variables, expressions, inequalities, equations, and simple systems. Strategies Linear equations and inequalities: analytically and graphically both with and without graphing calculator (Linear forms of equations: standard, slope intercept, point slope, slope as a rate of change). Quadratic equations with rational, irrational, and imaginary solutions (solving in various forms and graphing). Systems of linear equations and inequalities (solving and graphing). Radical equations with and without extraneous roots (solving and graphing). Exponential equations (solving with same base). Scope and Sequence Geometry students had to know and explained the use of variables as parameters for a specific variable situation. They continued to solve linear equations and inequalities, quadratic equations, radical equations, and exponential equations. Algebra 3 students will solve quadratic equations by completing the square, quadratic formula, factoring, and taking the square root. Topics also include using the discriminant to classify the roots of a quadratic equation, solving radical equations, solving logarithmic and exponential equations, and solving rational equations. Pacing Considerations 4 weeks 1. Explain what b represents in y = mx+b. 2. Which of the following is equivalent to 5x 3y = 20? a y 5x = 20 5x b. y = 3 20 c. y = 5x 3y 20 d. 5 3y + = x 3. Find the total stopping distance of a car traveling at 60mph given d( x) = x+ x N Tickets for a school play are $5 for adults and $3 for students. You need to sell at least $65 in tickets. Give an inequality and a graph that represents this situation and three possible solutions. 11

12 Application Indicators The student MA550X.A.V.4 represents and/or solves real-world problems with: a. N linear equations and inequalities both analytically and graphically. Absolute value equations (solving and graphing). Manipulatives (Algeblocks, tiles, concrete models). Real-world applications from business, economics, chemistry, and physics. Vocabulary Students Know and Use Vocabulary includes vocabulary from entire course refer to textbook. 12

13 Standard Algebra The student uses algebraic concepts and procedures in a variety of situations. Benchmark Functions The student analyzes functions in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student MA550X.A.F.1 evaluates and analyzes functions using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology. MA550X.A.F.2 matches equations and graphs of constant and linear functions and quadratic functions limited to y = ax 2 + c. MA550X.A.F.3 determines whether a graph, list of ordered pairs, table of values, or rule represents a function. MA550X.A.F.4 determines x- and y- intercepts and maximum and minimum values of the portion of the graph that is shown on a coordinate plane. MA550X.A.F.5 identifies domain and range of: a. relationships given the graph or table. b. Linear, constant, and quadratic functions given the equation(s). MA550X.A.F.6 recognizes how changes in the constant and/or slope within a linear function change the appearance of a graph. MA550X.A.F.7 uses function notation. Essential Concepts/Skills Implementation Assessment Examples What Students Need to Know 1. How to graph. What Students Need to Do/Apply 1. Interpret, recognize and describe functions (Linear equations, constant, step, greatest integer, piecewise, Absolute values, Quadratic, Exponential, Radical, Logarithmic). 2. Recognize and perform various functional transformations. 3. Evaluate and graph functions and compositions of functions. 4. Interpret the meaning of points on a graph in the context of a real world situation. Strategies Cognitive Tutor Graph comparison math graphs to real world situations Transformation Lab discovery Scope and Sequence Geometry students evaluated and analyzed functions. They were expected to interpret the meaning of the x and y intercepts, slope, and/or points on and off the line of a graph in the context of a real world situation. Also they needed to recognize how changes in the constant and slope within a linear function changes the appearance of the graph. Algebra 3 students will identify and evaluate rational functions, find the inverse of a quadratic function, find the domain of a radical function both from a graph and algebraically. Topics also include finding all solutions of polynomial functions. Pacing Considerations 4 weeks 3. Which are functions? a. b. c. x y d Give the domain and range of y = x Consider y = 3x + 7. x y Describe how the line would change if the slop is increased. Describe how the line would change if the slope were negative. Describe how the line would change if the y-intercept were increased by 2. Describe how the line would change if the y-intercept were decreased by f(x) = 4x 2 + 7x + 2. Find f(x). 9. A tree service charges a basic fee of $50 to make a house call plus $20 per hour to trim trees. Write an equation that represents the situation. 13

14 MA550X.A.F.8 evaluates function(s) given a specific domain. MA550X.A.F.9 describes the difference between independent and dependent variables and identifies independent and dependent variables. Application Indicators The student MA550X.A.F.10 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. Vocabulary Students Know and Use linear constant step greatest integer piecewise absolute values quadratic exponential radical logarithmic scale function notation composite composition inverse domain range Identify the independent and dependent variables. What does the x-intercept represent? What does the y-intercept represent? 10. 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? The Water Tank Gallons Hours 14

15 Standard Geometry The student uses geometric concepts and procedures in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student MA550X.G.AP.1 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. MA550X.G.AP.2 determines if a given point lies on the graph of a given line or parabola without graphing and justifies the answer. MA550X.G.AP.3 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. MA550X.G.AP.4 finds and explains the relationship between the slopes of parallel and perpendicular lines. MA550X.G.AP.5 uses the Pythagorean Theorem to find distance (may use the distance formula). MA550X.G.AP.6 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. Benchmark Geometry from an Algebraic Perspective The student uses an algebraic perspective to analyze the geometry of two-dimensional and three -dimensional figures in a variety of situations. Essential Concepts/Skills Implementation Assessment Examples What Students Need to Know 1. Solve quadratic equations. What Students Need to Do/Apply 1. Recognize, classify, and discuss properties of conic sections. 2. Calculate the slope of a line from a list of ordered pairs and explain how the graph of the line is related to its slope (no calculator). 3. Find and explain the relationships between slopes of different lines. 4. Determine if a triangle is a right triangle and find the side lengths when the triangle is presented on the coordinate plane. 5. Recognize and describe single and multiple transformations on equations. 6. Explain how variations in constants, exponents, and/or coefficients within the equation change the appearance of the graph of the equation. Strategies Parabola, circle, ellipse, and hyperbola Lines of symmetry, foci, axis (major and minor), asymptotes Line of best fit. If Ax + By = C, then m = -a/b. Parallel and perpendicular lines. Distance formula. Slope Graphically and algebraically Scope and Sequence Geometry students represented a geometric situation in an algebraic manner in order to solve application problems. They used slope, distance, midpoint formulas, trigonometric functions, and Pythagorean theorem. Algebra 3 students will graph rational functions, write equations for the asymptotes, and identify any holes in the graph. Topics also include recognizing and graphing the equations for circle, parabola, ellipse, and hyperbola, finding the real roots/zeros of a quadratic function by locating the x-intercepts of the graph. Pacing Considerations 4 weeks 1. Identify the type of conic section, given 2 2 y = x Without graphing, determine if the coordinate (1, 2) lies on the line y=x Calculate the slope of a line given the following ordered pairs; (1, 0) (3, 1) and (5, 2). 4. The equation of a line 2x + 3y = 12. The slope of this line is. 2/3 What is the slope of a line perpendicular to this line? 5. Given the coordinates of the vertices for ABC, find the distance between point A and B. a. (0, 5) b. (-3, 1) c. (0, 1). 6. 4x + 3y = 12 Write the equation in slope-intercept form. Identify the slope. Identify the y-intercept. Graph the line. 7. Sketch a graph of y = -2x Does the graph have a maximum or minimum? 15

16 MA550X.G.AP.7 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. Application Indicators None tested on the 9/10 KS test. Vocabulary Students Know and Use parabola, circle, ellipse, and hyperbola parallel and perpendicular lines distance formula slope Conics and absolute values equations Line, absolute value equations, polynomial equations, and conics Graphing calculators Real world applications Find the maximum or minimum. Find the line of symmetry. Does the graph open up or down? What type of graph is this? a. line b. circle c. parabola d. hyperbola 16

17 Standard Data Analysis The student uses concepts and procedures of data analysis in a variety of situations. Benchmark Probability Apply probability theory to analyze the validity of arguments, draw conclusions, and make decisions in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student MA550X.A.P.1 finds the probability of two independent events in an experiment, simulation, or situation. MA550X.A.P.2 finds the conditional probability of two dependent events in an experiment, simulation, or situation. MA550X.A.P.3 explains the relationship between probability and odds and computes one given the other. Application Indicators None tested on the 9/10 Kansas Math Assessment. Vocabulary Students Know and Use probability odds combinations permutations compound event independent event Essential Concepts/Skills Implementation Assessment Examples What Students Need to Know 1. Find probability of a simple event. What Students Need to Do/Apply 1. Explain the relationship between probability and odds and compute one given the other. 2. Use theoretical or experimental probability to make predictions about real-world events such as work in economics, quality control, genetics, meteorology, and other areas of science, games, and situations involving geometric probability. Strategies For example: a 3 out of 4 probability is equivalent to 3 to 1 odds. Real life applications. Geometric probability Percentages derived from actual or simulated events. Determine the probability of a simple event or a compound event composed of 2 or more simple, independent events. Combinations and permutations Compare results to what is expected. Graphing calculators Scope and Sequence Geometry students found the probability of 2 independent events, conditional probability of 2 dependent events, explained relationship between probability and odds, and calculated the geometric probability of an event. Algebra 3 students will apply probability theory in the application of data analysis. Pacing Considerations 2 weeks 1. If two number cubes are rolled and the numbers that result are added, find the probability that the sum is an even number or a number less than 5. a. 5 b. 3 c. 2 d There are 38 members on the basketball teams and 54 members on the track teams at Central High. There are 13 people who are on both a basketball team and a track team. If one member of these teams is randomly selected, find the probability that the person is on a track team given that the person is on a basketball team. a. 13 b. 13 c. 54 d The weather forecast is for a 50% chance of rain today. According to the weather forecast, what are the odds it will rain today? 17

18 Standard Data Analysis The student uses concepts and procedures of data analysis in a variety of situations. Benchmark Statistics The student collects, organizes, displays, explains, and interprets numerical (rational) and non-numerical data sets in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student MA550X.A.S.1 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: a. frequency tables and line plots. 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. MA550X.A.S.2 explain how the reader s bias, measurement errors, and display distortions can affect the interpretation of data. MA550X.A.S.3 calculate and explains the meaning of range, quartiles and interquartile range for a real number data set. MA550X.A.S.4 explain the effects of outliers on the measures of central tendency (mean, median, mode) and range and interquartile range of a real number data set. Essential Concepts/Skills Implementation Assessment Examples What Students Need to Know 1. How to plot points. What Students Need to Do/Apply 1. Plot data points and use data analysis to determine relationships. Strategies Real life applications. Data displays: frequency distributions; box and whisker plots; stem and leaf plots; histograms; scatter plots/discrete graphs; bar, line, and circle graphs; Venn diagrams/pictorial displays; charts and tables. Statistical measures: measure of central tendency (mean, median, mode); range; quartiles; interquartile range; linear, quadratic, and exponential regression; correlation coefficient. Effects of outliers on the mean, median, and range of a real number set. Line of best fit given a scatter plot. Predictions using the equation of a line of best fit or the best fit model. Conduct experiments. Compare results to what is expected. Graphing calculators Read, create and compare/ Scope and Sequence Geometry students used data analysis to analyze decisions. They used and interpreted a variety of graphs/tables that represented statistical data. Algebra 3 students will analyze information generated by a table of data using graph, domain and range, regression equations, prediction, interpreting value, and maximum and minimum. Pacing Considerations 4 weeks 1. The percent of correct scores on a final exam for a class of 29 students are given in the stem and leaf plot below. Kiesha s score was above the median score for the class. Which of the following ranges would most likely include Kiesha s score? a. Score of 80 or higher b. Score in the 70s range c. Score in the 50s or 60s range d. Score of 50 or less 3. Find the mean, median, mode and range of the data. 13, 17, 31, 24, 46, 29, Describe what would happen to the mean if the maximum value (50) were removed from the data set above. How does this change the median? How does it change the range? 18

19 MA550X.A.S.5 approximate a line of best fit given a scatter plot and makes predictions using the graph or the equation of that line. MA550X.A.S.6 compare and contrast the dispersion of two given sets of data in terms of range and the shape of the distribution including: a. symmetrical (including normal). b. skew (left or right). c. bimodal, uniform (rectangular). Application Indicators MA550X.A.S.7 use 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: a. frequency tables and line plots. 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. Vocabulary Students Know and Use box and whiskers plot stem and leaf plot circle graphs Venn diagram mean median contrast various data displays (use test score data or postage stamp data). Create graphs from data. 5. The table below gives the price of a hamburger at a diner for selected years. Year Price 1950 $ $ $ $ $ $ $ $1.95 Create a scatter plot of the data. Label the y-axis from $0 to $2.00. Draw a line of best fit. Estimate the price of a hamburger in Redraw the graph with the y-axis labeled from $0 to $10 and intervals of $1. Compare and contrast this graph with the original. Which graph might the diner prefer to use for advertising purposes? 6. Create a scatter plot for the data and describe the correlation. x y

20 mode outlier scatter plot line of best fit linear regression quadratic regression exponential regression 20

21 Standard Problem Solving Indicator/Objective Critical Vocabulary Application Indicators The student MA550X.PS.1 uses appropriate representations of real numbers and algebraic expressions to formulate and solve real world problems. MA550X.PS.2 uses properties of the real number system to formulate and solve real world problems. MA550X.PS.3 uses arithmetic operations and inverse relationships to formulate and solve real world problems involving real numbers and algebraic expressions with special emphasis on topics such as: finding the volume and surface area when formulas are given; applications from business, economics, chemistry, and physics (avoiding logs); probabilities and exponential growth and decay. MA550X.PS.4 uses a variety of ways to represent problem situations that involve variable quantities. MA550X.PS.5 formulates and solves problems involving symbols, percents, variables, expressions, inequalities, equations, and simple systems. MA550X.PS.6 interprets the meaning of points on a graph in the context of a real world situation. Benchmark Apply algebraic content/objectives to solving problems in a variety of contexts. Essential Concepts/Skills Implementation Assessment Examples What Students Need to Know 1. Students must be able to master the skill indicators before they are ready to apply the skill to a real problem solving situation. 2. To be able to read on an 8 th grade reading level. 3. A problem solving plan and the strategies within problem solving. What Students Need to Do/Apply 1. Determine the appropriate problem solving strategy for a particular problem. Strategies Teach problem solving strategies. Teach problem solving tool (3PR, UGLY, Polya s, etc.). Use tables, graphs, and charts. Use spreadsheets, graphing calculators. Cognitive Tutor Keep problems authentic. Lots of practice. Scope and Sequence Occurs throughout the K-12 curriculum. Pacing Considerations Daily problem solving should occur on an on-going basis and continue throughout the school year. 1. 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 books and calculators for 9 math classrooms. 2. 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? 3a. 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? Solve this problem two ways. 3b. The purchase price (P) of a series EE Savings Bond is found by the formula ½ F = P where F is the face value of the bond. Use the formula to find the face value of a savings bond purchased for $500. 3c. 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. 21

22 MA550X.PS.7 uses the mathematical modeling process to make inferences about real world situations. MA550X.PS.8 uses theoretical or experimental probability to make predictions about real world events such as work in economics, quality control, genetics, meteorology, and other areas of science, games, and situations involving geometric probability. Application Indicators The student MA550X.PS.9 generates and/or solves real-world problems using equivalent representations of real numbers and algebraic expressions. MA550X.PS.10 determines whether or not solutions to real-world problems using real numbers and algebraic expressions are reasonable. MA550X.PS.11 generates and/or solves real-world problems with real numbers using the concepts of these properties to explain reasoning: a. commutative, associative, distributive, and substitution properties. b. identity and inverse properties of addition and multiplication. c. symmetric property of equality. d. addition and multiplication properties of equality. e. zero product property. MA550X.PS.12 analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions 3d. The total price for the purchase of three shirts in $62.54 including tax. If the tax is 3.89, what is the cost of one shirt? 3e. Jenny was thinking of two numbers. Jenny said that the product of the two numbers was 0. What could you deduce from this statement? Explain your reasoning. 4. A store sells CDs for $12.99 each. Knowing that the sales tax is 7%, Marie estimates the cost of a CD plus tax to be $ She selects nine CDs. The clerk tells Marie her bill is $ How can Marie explain to the clerk she has been overcharged? 5. Estimate how long it takes to walk from here to there; time how long it takes to take five steps and adjust your estimate. 6. If you have a $4,000 debt on a credit card and the minimum of $30 is paid per month, is it reasonable to pay off the debt in 10 years? 7. Do you need an exact or an approximate answer in calculating the area of the walls to determine the number of rolls of wallpaper needed to paper a room? What would you do if you were wallpapering 2 rooms? 22

23 (including mixed numbers), decimals or irrational numbers and their rational approximations in solving a given realworld problem. MA550X.PS.13 adjusts original rational number estimate of a real-world problem based on additional information (a frame of reference). MA550X.PS.14 estimates to check whether or not the result of a real-world problem using real numbers and/or algebraic expressions is reasonable and makes predictions based on the information. MA550X.PS.15 determines if a realworld problem calls for an exact or approximate answer and performs the appropriate computation using various computational strategies including mental math, paper and pencil, concrete objects, and/or appropriate technology. MA550X.PS.16 explains the impact of estimation on the result of a real-world problem (underestimate, overestimate, range of estimates). MA550X.PS.17 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, 8. If the weight of 25 pieces of paper was measured as grams, what would the weight of 2,000 pieces of paper equal to the nearest gram? If the student were to estimate the weight of one piece of paper as about 20 grams and then multiply this by 2,000 rather than multiply the weight of 25 pieces of paper by 80; the answer would differ by about 2,400 grams. In general, multiplying or dividing by a rounded number will cause greater discrepancies than rounding after multiplying or dividing. 9a. 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. 9b. A silo has a diameter of 8 feet and a height of 20 feet. How many cubic feet of grain can it store? 9c. 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? 9d. Given the formula A = P(1+r) n, when A = amount, P= principal, r = interest rate for the compounded period, and n = number of times compounded, solve the following: If $1,000 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? 23

24 division, squares, and square roots when the formulae are given as part of the problem and variables are defined. b. volume and surface area given the measurement formulas of rectangular solids and cylinders. c. probabilities. d. application of percents. e. simple exponential growth and decay (excluding logarithms) and economics. MA550X.PS.18 recognizes the same general pattern presented in different representations [numeric (list or table), visual (picture, table, or graph), and written]. MA550X.PS.19 solves real-world problems with arithmetic or geometric sequences by using the explicit equation of the sequence. MA550X.PS.20 represents real-world problems using variables, symbols, expressions, equations, inequalities, and simple systems of linear equations. MA550X.PS.21 represents and/or solves real-world problems with: a. N linear equations and inequalities both analytically and graphically. 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. d. radical equations with no more 9e. 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? 10. Have students identify and match equations to graphs to ordered pairs. 11a. An arithmetic sequence: A brick wall is 3 feet high and the owners want to build it higher. If the builders can lay 2 feet every hour, how long will it take to raise it to a height of 20 feet? 11b. A geometric sequence: Saving programs can double your money every 12 years. If you place $100 in the program, how many years will it take to have over $1,000? 12. Rebecca is the star forward on her high school broomball team. In one game, her field-goal total was 23 points, made up of 2-point and 3-point baskets. If she made 4 more 2-point baskets than 3-point baskets, how many of each type of basket did she make? 13a. Tickets for a school play are $5 for adults and $3 for students. You need to sell at least $65 in tickets. Give an inequality and a graph that represents this situation and three possible solutions. 24

25 than one inverse operation around the radical expression. e. a rational equation where the solution can be simplified as a linear equation with a nonzero denominator. MA550X.PS.22 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. MA550X.PS.23 translates between the numerical, graphical, and symbolic representations of functions. MA550X.PS.24 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. MA550X.PS.25 analyzes: a. the effects of parameter changes (scale changes or restricted domains) on the appearance of a function s graph. b. how changes in the constants and/or slope within a linear function affects the appearance of a graph. c. how changes in the constants and/or coefficients within a quadratic function in the form of y = ax 2 + c affects the appearance of a graph. MA550X.PS.26 recognizes that various mathematical models can be used to represent the same problem situation. 13b. A fence is to be built onto an existing fence. The three sides will be built with 2,000 meters of fencing. To maximize the rectangular area, what should be the dimensions of the fence? 13c. 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. 13d. A square rug with an area of 200 square feet is 4 feet shorter than a room. What is the length of the room? 13e. John is 2 feet taller than Fred. John s shadow is 6 feet in length and Fred s shadow is 4 feet in length. How tall is Fred? 14. Have students explain why an absolute value equation must have two possible solutions. 15. Have students identify which of the graphs represent f(x) = 2x² 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? (see graph on next page) 25