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SEVENTH GRADE The Seventh Grade mathematics curriculum features an in-depth, integrated preparation for algebra and geometry. Concepts include basic operations with fractions, decimals, number theory, measurement, data interpretation, geometry, integers, algebraic concepts, and percents. A variety of problem-solving techniques, real-world applications, and technology will be used when applying these concepts. This course is designed to prepare students for eighth grade mathematics or Pre-Algebra. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 1 SEVENTH GRADE CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 1. Apply concepts and perform the basic operations with decimals, fractions, and mixed numbers. (P, M, N) a. Compare, order, round, and estimate decimals. b. Add, subtract, multiply, and divide decimals in real-life situations with and without calculators. c. Use powers of ten to multiply and divide decimals. d. Convert among decimals, fractions, and mixed numbers. e. Express ratios as fractions. f. Add, subtract, multiply, and divide fractions and mixed numbers. g. Use estimation to add, subtract, multiply, and divide fractions. 2. Apply and use basic principles of number sense. (P, M, N) a. Use patterns to develop the concept of exponents. b. Write numbers in standard and exponential form. c. Convert between standard form and scientific notation. d. Find and use prime factorization with exponents to obtain the greatest common factor (GCF) and least common multiple (LCM). e. Describe and extend patterns in sequences. f. Identify and use the commutative, associative, distributive, and identity properties. g. Use patterns to develop the concepts of roots of perfect squares with and without calculators. 3. Use units of measurement with standard systems. (P, D, M, G, N) a. Convert within a standard measurement system (English and metric). b. Convert temperature using the Fahrenheit and Celsius formulas. c. Use standard units of measurement to solve application problems. 2 SEVENTH GRADE CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 4. Collect, organize, and summarize data and use simple probability. (P, D, M, G, N) a. Organize data in a frequency table. b. Interpret and construct histograms, line, and bar graphs. c. Interpret and construct circle graphs when given degrees. d. Interpret and construct stem and leaf plots and line plots from data. e. Estimate and compare data including mean, median, mode, and range of a set of data. f. Predict and recognize data from statistical graphs. g. Determine probability of a single event. h. Use simple permutations and combinations. 5. Use concepts of geometry in angles and polygons and extend the concepts of perimeter and area. (P, G, M, N) a. Identify polygons to twelve sides. b. Classify and compare the properties of quadrilaterals. c. Classify and measure angles of all types. d. Classify triangles by sides and angles. e. Find the perimeter of polygons. f. Find the area of triangles and quadrilaterals. g. Find the circumference and area of a circle. h. Identify congruent segments, angles, and polygons. i. Develop relationships of faces, vertices, and edges of three-dimensional figures. j. Perform transformations (rotations, reflections, translations) on plane figures using physical models and graph paper. k. Investigate symmetry of polygons. l. Develop and apply the Pythagorean Theorem to find missing sides of right triangles. 3 SEVENTH GRADE CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 6. Develop and apply the basic operations of integers. (P, D, M, G, N) a. Recognize and write integers including opposites and absolute value. b. Compare and order integers. c. Graph ordered pairs on a coordinate plane. d. Add, subtract, multiply, and divide integers with and without calculators. 7. Create and apply algebraic expressions and equations. (P, G, N) a. Translate between simple algebraic expressions and verbal phrases. b. Use the order of operations to simplify and/or evaluate numerical and algebraic expressions with and without calculators. c. Solve linear equations using the addition, subtraction, multiplication, and division properties of equality with integer solutions. d. Write and solve equations that represent problem-solving situations. e. Write a real-world situation from a given equation. 8. Survey and apply concepts of ratio, proportion, and percent. (P, D, M, G, N) a. Explore equivalent ratios and express them in simplest form. b. Solve problems involving proportions. c. Determine unit rates. d. Use models to illustrate the meaning of percent. e. Convert among decimals, fractions, mixed numbers, and percents. f. Determine the percent of a number. g. Estimate decimals, fractions, and percents. h. Use proportions and equations to solve problems with rate, base, and part with and without calculators. i. Find the percent of increase and decrease. j. Solve problems involving sales tax, discount, and simple interest with and without calculators. 4 Course: 7 Unit Theme: Concepts and Basic Operations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a, b Using grocery store and discount store ads, discuss Discussion; 8 c, j the cost of advertised items. Estimate and compare Student work sample total costs of items when given a certain amount of money to spend. Identify unit prices of several items and compare prices in order to find best buys. 1 a, b Using baseball batting averages from the newspaper, Student work sample 4 e round the averages to the nearest hundredth. Using calculators, estimate and calculate team averages. 1 a, b Using a menu from a local restaurant, calculate the Student work sample 8 j total cost of a meal, including tax and tip. 1 d, e, f, g Using recipes, double, triple and quadruple the Presentation; 3 c ingredients. Calculate ingredients needed to serve the Teacher observation 8 b class. Using several recipes, work in groups to plan a menu to serve at a party. Use addition, subtraction, multiplication, and division of fractions and mixed numbers, as well as whole numbers. Estimate these using calculators. 5 Course: 7 Unit Theme: Number Sense Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a Use a table to visualize the patterns in the concept of ● Teacher observation exponents. 1 b, c Research distances from each planet to the sun. ● Presentation; 2 Make a poster. Use findings to write standard form Student work sample and scientific notation. Use the calculator to develop this concept. 2 b, d Use a Venn diagram to find the GCF and LCM . ● Student work sample; In groups, find the prime numbers using the Sieve Teacher observation; of Eratosthenes. Written response; Divide the greater number by the lesser number to Presentation find GCF (Euclidean Algorithm). Research mathematicians—Eratosthenes and Euclid. Use this method of division of prime numbers to find LCM. 2 60, 12 2 30, 6 3 15, 3 5 5, 1 1, 1 2 2 3 5 60 Work in pairs to form fractions from statistics in a ● Student work sample 1 d, e 2 d school football game such as number of passes 8 a completed out of number thrown. Determine in simplest form. 2 e Discover the Fibonacci sequence in a pine cone or ● Presentation pineapple spiral. Make a bulletin board from facts about this sequence in nature. 2 e Derive a sequence of pay which would give the most ● Teacher observation money at the end of the month for each student such as the same amount paid each day or a small amount doubled everyday. 2 f Give everyday examples (putting on socks and shoes) ● Presentation using the properties—commutative, associative, distributive, and identity. 2 a, g Using graph paper, draw squares. Discuss the sides ● Presentation; and note representation of square root. These Teacher observation squares represent ―perfect squares.‖ 6 Course: 7 Unit Theme: Measurement Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a, b, c Collect data on high and low temperatures for one ● Student work sample; week. Calculate the Celsius temperature from a given Presentation; Fahrenheit temperature. Teacher-made test 1 c Convert metric units by multiplying and dividing by ● Student work sample; 3 a, c powers of 10. Teacher-made test 3 a, c Bring in grocery items and collect labels and note ● Student work sample weight, capacity, etc. Convert among units of standard and metric measurement. 7 Course: 7 Unit Theme: Data and Probability Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a, b, c, d, Collect and chart data on the height of students and ● Rubric e, f length of arms. Organize in a frequency table. (Note the mean, median, mode, and range using a line plot.) Organize data on a double bar graph. When given degrees, construct a circle graph of heights and lengths and compare data. 4 g In groups, flip coins and write expected outcomes. ● Student response; Rubric 4 g Use a spinner with letters. Find the probability that the ● Student response; pointer will stop on a certain letter. Rubric 4 h Introduce permutations and combinations using a ● Student response calculator (factorial key). 4 h Imagine a certain number of students eating at a ● Rubric restaurant. Calculate the different ways the group can be seated at a chosen number of tables. Choose three ingredients from a list of five at the salad bar (name the five ingredients). Ask ―In how many ways can three ingredients be chosen from the five?‖ 8 Course: 7 Unit Theme: Geometry Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a, b, c, d Use a Venn diagram to classify polygons. ● Teacher observation 5 a, b, c, d, Use quilt pattern books, measure angles of triangles ● Project; e, f, h and classify by sides and angles. Find the polygons Presentation and classify. Measure sides of polygons and find perimeter and area. Note congruence of angles and polygons. 1 b Verify the formula for circumference by measuring ● Demonstration; 5 g, f various sizes of cans or circular objects with string. Written response 5 i Recognize and identify faces and edges of objects ● Discussion; found in the classroom and on the campus. Written response 5 j Research M. C. Escher and model rotations, ● Student work sample; reflections, and translations using graph paper. Draw Presentation tessellations. 5 k Determine symmetry of capital letters of the alphabet. ● Presentation Draw the lines of symmetry and create a poster. 5 l Have the students draw a diagram of their room at ● Presentation; home. Place a chosen object in the corner of the Student work sample room. Determine the placement of other furniture, stereo system and speakers, etc. The meaning and application of the Pythagorean Theorem will be developed. 9 Course: 7 Unit Theme: Integers Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 b, c Research the high and low temperatures for five cities ● Presentation; 6 a, b, d in different regions of the United States for a week. Project; Find the difference in temperature among these cities. Student work sample Perform this activity in the winter and in the spring and order/compare. (Note: Below 0) Use calculators. 6 a, b Use number lines to compare and order integers and ● Student work sample; graph the integers. Teacher-made test 6 c Name coordinates and create patterns or figures (e.g., ● Teacher observation butterfly, umbrella, sailboat) by plotting points. 6 c Using a United States map, plot locations of chosen ● Student work samples cities, national parks, and other points of interests. Find latitude and longitude. 10 Course: 7 Unit Theme: Expressions and Equations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 7 a, b, c, d, e Read a sentence such as 2 x 10 as, 2 times what ● Student work sample; number is 10. A correct response will be 5. Write the Teacher made test related numerical expression and equation. Repeat this activity several times, and use the same procedure to write related multiplication and division sentences along with addition and subtraction. 7 a, b, c, d, e Use similarities and differences in appearance and ● Written response; dress of students. Write and solve equations about Teacher observation these criteria. Algebraic expressions may be written from phrases stated about appearance and dress. 7 c, d Use algebra tiles to model and solve equations. ● Teacher observation; Student work samples 11 Course: 7 Unit Theme: Ratio, Proportion, and Percent Suggested Suggested Comp. Obj. Teaching Strategies Assessment 8 a Play ―Concentration‖ to find matching pairs of ● Discussion; equivalent fractions. Teacher observation 8 b, f, h Compare the number of boys and girls in various ● Student work samples classrooms and convert boy/girl ratios to percent. Use proportions to convert percent and predict the number of boys or girls in other classes. 8 c Provide a price list from local grocery stores. Identify ● Presentation; unit prices of several items and compare prices in Student work samples order to find best buys. 1 a, b Show percent of change (increase or decrease) on ● Teacher observation; 8 d, i graph paper. Recognize and explain percent of Rubric change as shown on graph paper. 8 e, g Using a spinner, play a game by naming percents for ● Teacher observation; fractions, fractions for percents, percents for decimals, Student work samples and decimals for percents as the pointer lands on these sections. (Extension: Write estimations of the percent, decimal, fraction.) 8 h Assign each group a part, rate, or base problem. Write ● Teacher observation; problems from an ad in the newspaper of the type Student work samples problem using proportions and equations. Exchange among groups the assigned problems. 1 a, b Have students look at advertisements for discount ● Student work 8 j sales. Select one item and write the specific size, samples; brand, and other characteristics. From several stores Teacher-made test find the actual price of the same item. Compare prices to see if the sale is actually as good as the ad indicates. Calculate the discount and sale price to find the best store and best savings. Calculate the sales tax on the items. 12 EIGHTH GRADE The Eighth Grade mathematics curriculum will incorporate concepts which provide a smooth transition from concrete to abstract relationships in preparation for high school mathematics courses. Concepts include real numbers, algebraic concepts, geometric principles, ratio, proportion, percents, number theory, measurements, data analysis, and the coordinate system. A variety of problem-solving techniques and technology will be used when applying these concepts, which will enable students to solve real life problems. This course is designed to prepare students for Pre-Algebra. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 13 EIGHTH GRADE CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 1. Apply concepts and perform basic operations using real numbers. (P, D, G, N) a. Classify and give examples of real numbers such as natural, whole, integers, rational, and irrational. b. Identify, compare, and order fractions and decimals. c. Round and estimate fractions and decimals. d. Solve real-life problems involving addition, subtraction, multiplication, and division of fractions, decimals, and mixed numbers. e. Determine the absolute value and additive inverse of real numbers. f. Classify, compare, and order integers and rational numbers. g. Add, subtract, multiply, and divide integers and rational numbers with and without calculators. 2. Use basic concepts of number sense and perform operations involving order of operations, exponents, scientific notation. (P, M, N) a. Simplify expressions using order of operations. b. Use the rules of exponents when multiplying or dividing like bases, and when raising a power to a power. c. Multiply and divide numbers by powers of ten. d. Convert between standard form and scientific notation. e. Multiply and divide numbers written in scientific notation. f. Evaluate and estimate powers, squares, and square roots with and without calculators. 14 EIGHTH GRADE CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 3. Use properties to create and simplify algebraic expressions and solve linear equations and inequalities. (P, G, N) a. Identify and apply the commutative, associative, and distributive properties. b. Distinguish between numerical and algebraic expressions, equations, and inequalities. c. Convert between word phrases or sentences and algebraic expressions, equations, or inequalities. d. Simplify and evaluate numerical and algebraic expressions. e. Solve and check one and two-step linear equations and inequalities. f. Solve and check multi-step linear equations using the distributive property. g. Graph solutions to inequalities on a number line. h. Write a corresponding real-life situation from an algebraic expression. 4. Apply the concepts of ratio, proportion, and percent to solve real-life problems. (P, D, M, G, N) a. Write ratios comparing given data. b. Convert among ratios, decimals, and percents. c. Solve proportions. d. Solve for part, rate, or base. e. Find commissions and rates of commission, discounts, sale prices, sales tax, and simple interest. f. Find percent of increase and decrease. g. Write and solve real-life word problems using percents with and without calculators. 1. Convert and use standard units (English and metric) of measurement. (P, D, M, G, N) a. Convert, perform basic operations, and solve word problems using standard measurements. b. Measure line segments and find dimensions of given figures using standard measurements. c. Write and solve real-life problems involving standard measurements. d. Select appropriate units of measurement for real-life problems. 15 EIGHTH GRADE CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 6. Apply geometric principles to polygons, angles, and two and three- dimensional figures. (P, M, G, N) a. Identify parallel, perpendicular, intersecting, and skew lines. b. Identify and describe characteristics of polygons. c. Find the perimeter and area of polygons and circumference and area of circles. d. Classify, draw, and measure acute, obtuse, right, and straight angles. e. Identify and find the missing angle measure for adjacent, vertical, complementary, and supplementary angles. f. Locate and identify angles formed by parallel lines cut by a transversal (e.g., corresponding, alternate interior, and alternate exterior). g. Classify triangles by sides and angles and find the missing angle measure. h. Identify three-dimensional figures and describe their faces, vertices, and edges. i. Use the Pythagorean Theorem to solve problems, with and without a calculator. 7. Interpret, organize, and make predictions about a variety of data using concepts of probability and statistics. (P, D, M, G, N) a. Interpret and construct frequency tables and charts. b. Find mean, median, mode, and range of a given set of data. c. Interpret and construct bar, line, circle graphs, and pictographs from given data. d. Interpret and construct stem-and-leaf, box-and-whisker, and scatterplots from given data. e. Predict patterns or trends based on given data. f. Use combinations and permutations in application problems. g. Calculate and apply basic probability. 1. Apply the principles of graphing in the coordinate system. (P, D, M, G, N) a. Identify the x- and y-axis, the origin, and the quadrants of a coordinate plane. b. Plot ordered pairs. c. Label the x and y coordinates for a given point. d. Using tables, graph simple linear equations. 16 Course: 8 Unit Theme: Concepts and Basic Operations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Use yarn to create a Venn diagram of natural, whole, Student work sample; integers, rational, and irrational numbers. Choose an Observation index card with a number on it and place in the correct place. 1 b, e, f Use a number line to locate and compare numbers, Student work sample; including absolute values and additive inverses. Teacher-made test 1 f, g Using a weather map, compare temperatures around Discussion; 5 c the country. Find differences and average weekly Student presentation temperatures. 1 c, d, g Use maps, bus and plane schedules and fares, hotel Project; rates, etc., to plan a vacation. Estimate total Rubric; expenses. Teacher observation 17 Course: 8 Unit Theme: Number Sense Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a Divide the class into two groups. A student from each Student work sample; group goes to the board to work an order of operations Observation problem. The first correct answer wins and marks the tic-tac-toe board. Continue until one group wins. 2 b, c Distribute problems involving multiplying and dividing Constructed response; like bases or powers of ten. Work in groups looking for Discussion; a shortcut (a rule) for solving the problems. Discuss Teacher observation how rules can make solving problems easier. 2 d, e Use magazines and newspaper articles to find Discussion; examples of very large and very small numbers. Using Constructed response; the examples, write the numbers in scientific notation, Teacher-made test convert between standard form and scientific notation. Using different combinations, multiply and divide the numbers in scientific notation. Discuss the advantage of writing these numbers in scientific notation. 2 f Play Jeopardy with powers, squares, and square roots. Teacher observation; From an overhead transparency, select a category and Discussion; point value. Allow calculators on some categories and Performance-based point values. 2 Using grid paper, cut out squares presenting Rubric; f Teacher observation 1 6 2 through 2 powers. Discuss characteristics of the amount of squares and shapes that can be made from them. Each time the exponent is reduced by 1, the number of squares will be half. Emphasize 2° = 1. 18 Course: 8 Unit Theme: Expressions, Equations, and Inequalities Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a Divide the class into two teams. On an overhead, Teacher observation; write problems that can be solved easier by using Discussion properties. For example: 25 6 4 25 4 6 . One person from each team races to get the correct answer. (Explain the use of properties with each ` problem.) The team with the most number of correct answers wins. a, b, c, d, Play Algebraic Jeopardy. From an overhead Student work sample; 3 Teacher observation e, f, h transparency, choose a category and point value (e.g., expressions for 40). Categories include expressions, word phrases/sentences, properties, equations, inequalities, or algebraic phrases/sentences. Points range from 10 to 50 based on level of difficulty. Answers must be in the form of a question. Team with the most points when board is completed wins. 3 g Distribute a set of index cards containing inequalities. Teacher observation; The set should contain pairs of inequalities that have Discussion the same solution. Students solve and graph their inequalities on a number line, then search for the classmate with the same solution. Prizes or bonus points could be given for the first few pairs to match. 19 Course: 8 Unit Theme: Ratio, Proportion, and Percent Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a, b Count and record the number of boys and girls in the Student work sample; class. Write ratio of boys to girls, girls to boys, girls to Discussion ; total, etc. Convert the ratios to decimals and percents. Teacher-made test 4 a, b Play Percent Bingo. Make cards containing a column Discussion; for ratios, decimals (two columns), and percents (two Teacher observation columns). Draw a game piece and call it out. Players cover all spaces (except the one called out) that have 3 the same meaning (e.g., 75% = .75 = 4 ). First player to cover spaces vertically, horizontally, or diagonally wins. Discuss winner’s answers. 4 c, d Investigate the connections among test grades, total Presentation; problems, and number correct. Use proportions to find Discussion; how many problems would have to be correct on a 25 Observation; problem test to make an A, B, C, D, F. Find how many Student work sample; items were on the test if they made a grade of 80 and Rubric got 12 correct. Find the grade when given the number of problems on the test and the number correct. 4 e, f, g In groups, research local newspapers or businesses Student work sample; about percent topics (commissions, sales, percent Teacher-made test increase/decrease, interest) involving ways percent is used in business. Allow a specified amount of time, then have groups report findings to the class. Use calculators to convert among fractions, decimals, and percents involved in the groups findings. (Extend: Invite a guest speaker to discuss this topic.) 20 Course: 8 Unit Theme: Measurement Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a, c, d Design a deck to be added to a patio. Use basic ● Project; 6 c operations to find the perimeter and area and to find Rubric the amount of materials needed for the job. Select appropriate units of measurement. 5 b Given objects found in any classroom (e.g., books, ● Performance-based paper clips, desktops), measure and find the assessment dimensions of these objects. 21 Course: 8 Unit Theme: Geometry Suggested Suggested Comp. Obj. Teaching Strategies Assessment 6 a, b, c, d, Use manipulatives (e. g., D-Stix, plastic straws, flat Performance-based; g, h spaghetti, or connectors) to construct angles, Student work sample polygons, lines, triangles, and three-dimensional figures. Discuss the characteristics of each. 6 c Use geoboards or cm grid paper to make or draw Performance-based; shapes and find the perimeter and area (except Project; circles). Trace or draw a circle on the grid paper and Student work sample; use the above information to estimate the area and Observation circumference of the circle. Introduce formulas and have students calculate the area, perimeter, and circumference using formulas. 6 d Use a protractor to draw and measure angles. Classify Performance-based; each angle. Teacher-made test 6 e, f, g From given pictures, find the angle measure of Student work sample; adjacent, vertical, complementary, and supplementary Discussion; angles. Locate and identify corresponding and Presentation; alternate interior and exterior angles. Classify Rubric triangles and find missing angle measures. Draw one line on each of two transparencies. Arrange them so that they are parallel, then draw a transversal. Discuss the angles formed. Move transparencies to prove relationships among angles. 6 b, d, i Have students sketch right triangles on grid paper. Performance-based; Use the Pythagorean theorem to find the measure of Teacher observation; the hypotenuse. Verify the measure with a ruler or by Student work sample, counting the squares on the grid paper. Teacher-made test 22 Course: 8 Unit Theme: Probability and Statistics Suggested Suggested Comp. Obj. Teaching Strategies Assessment 7 a, c Interview classmates to determine their favorite foods. Project; Construct a frequency table and bar graph from the Student work sample data. 5 a, d For each family member record age, height in inches, Performance-based; 7 b, d, e and birth month. Find mean, median, mode, and Project; range of heights. Construct various plots from data Student work sample; (e.g., scatter plot from height and age, and from height Discussion; and month born, to determine if there is a correlation). Observation Predict from height/age plots. Use the graphing calculator to find the mean, median, and mode and to construct histograms, scatter plots, and box and whisker. 7 c Given a salary, plan a monthly budget, then construct Rubric; a circle graph. Student work sample 7 f, g Use coins, number cubes, menu items, and group Performance-based; memberships to calculate basic probability, Discussion; combinations, and permutations. Student work sample 23 Course: 8 Unit Theme: Coordinate System Suggested Suggested Comp. Obj. Teaching Strategies Assessment 8 a, b, c, d Write a linear equation on the overhead. Give each Teacher observation; row of the class different values to use in solving the Student work sample; equation. Let one row choose their own values. When Teacher-made test all have finished, have each row plot their points on a wall coordinate grid. Discuss the reasons that all points fall on the same line. If any points are not on the line, look for mistakes in calculations or have the class determine why the points are not on the line. Discuss the x-axis, the y-axis, the quadrants, and their characteristics. 8 a, b, c In pairs, listen to a selection of 20 song excerpts. ● Teacher observation; Each student rates the song on a scale from –5 to 5 Rubric based on whether or not they like the song. Using the ratings, the partners form an ordered pair. Plot the ordered pairs and discuss if partners are musically compatible and use quadrants in discussion. 8 b, c, d Graph linear equations on graph paper and check for ● Student work sample; accuracy using the graphing calculator. Teacher-made test 24 PRE-ALGEBRA The Pre-Algebra course is to serve as a bridge between elementary mathematics and Algebra. This course will build a foundation of algebraic concepts through the use of manipulatives and cooperative learning. Concepts include algebraic expressions, linear equations, polynomials, factoring, inequalities, geometry, statistics, and graphing. Students will learn to utilize the graphing calculator in appropriate situations. Problem solving, reasoning, estimation, and connections between math and everyday applications will be emphasized throughout Pre-Algebra. This course is designed to prepare students for Algebra I. This is a one credit course, if taken at the high school level. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 25 PRE-ALGEBRA CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 1. Explain, classify, and perform basic operations on the set of real numbers. (P, D, M, G, N) a. Classify numbers as natural, whole, integer, rational, irrational, and real. b. Identify and apply the properties of real numbers (include the use of mental mathematics and estimation methods). c. Model absolute value of real numbers as a measure of distance. d. Compare and order the real numbers and perform operations with rational numbers. e. Evaluate numerical and algebraic expressions using order of operations. f. Convert between repeating decimals and fractions. g. Recognize and evaluate perfect squares and approximate square roots. 2. Solve, check, and graph linear equations and inequalities in one variable. (P, G, N) a. Relate the language of mathematics to indicate mathematical operations. b. Translate between verbal expressions and algebraic expressions. c. Given an algebraic expression, write a corresponding real-life situation. d. Simplify algebraic expressions by combining like terms and using the distributive property. e. Solve, check, and graph one-step and two-step linear equations and inequalities. f. Solve and check multi-step linear equations and inequalities with variables on both sides involving the distributive property. 26 PRE-ALGEBRA CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 3. Recognize and perform basic operations on polynomials. (P, G, N) a. Classify types of polynomials. b. Determine the degree of polynomials. c. Simplify polynomials by combining like terms. d. Arrange polynomials in ascending or descending order of a variable. e. Use the rules of exponents to multiply and divide monomials. f. Use the rules of exponents to multiply monomials by polynomials. g. Model and use the distributive property and rules of exponents to multiply binomials by binomials. h. Multiply and divide numbers involving scientific notation. i. Use manipulative models to demonstrate operations of monomials and polynomials. 4. Use ratios, proportions, and percents to solve problems. (P, M, G, N) a. Represent, convert, and explain relationships among fractions, ratios, decimals, and percents in problem solving. b. Use proportions and equations to find part, rate, or base in real-world situations. c. Explain solutions and processes orally and in writing. 5. Use concepts of probability and statistics to interpret information. (P, D, G, N) a. Model the Fundamental Counting Principle to determine possible outcomes of an event. b. Use combinations and permutations in application problems. c. Calculate and apply basic probability. d. Collect, display, analyze, and draw appropriate conclusions from data. e. Interpret and construct stem-and-leaf, box-and-whisker, and scatter plots from data. 27 PRE-ALGEBRA CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 6. Solve, check, and graph solutions of equations and inequalities in two variables using the coordinate system. (P, D, M, G, N) a. Given a set of ordered pairs, draw a coordinate system using an appropriate scale. b. Create a table to graph equations and inequalities that are presented in slope intercept form. c. Use calculators/computers to check accuracy of tables and graphs as needed. d. Identify slope as positive, negative, zero, or undefined from a graph. e. Calculate slope from two points graphically and algebraically. f. Identify x- and y- intercepts from a graph. g. Identify the solution of a system of equations from a graph. 7. Use and apply properties and formulas to solve geometric problems. (P, D, M, G, N) a. Calculate perimeter, area, circumference, and volume using appropriate formulas. b. Recognize the irrational number pi (π) as the ratio of circumference to diameter of any given circle. c. Solve problems involving the use of the Pythagorean Theorem. d. Classify triangles by sides and angles. e. Use properties of similar triangles to solve problems. f. Recognize and determine degree measure of angles formed by parallel lines cut by a transversal. g. Develop, extend, and model the relationships of faces, vertices, and edges of three-dimensional figures. h. Perform transformations on plane figures. 28 Course: Pre-Algebra Unit Theme: Concepts and Basic Operations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Make a Venn diagram of the real number system. ● Rubric; Student work sample 1 b Given cards with variables, addition sign, multiplication ● Student response; sign, parentheses, and equal sign written on them, Teacher observation teams will arrange cards to create examples of properties. 1 c Create a number line on the floor. Model │3│ by ● Teacher observation walking from 0 to 3. Model │–3│ by walking from 0 to -3. Same number of steps were taken, but in opposite directions. Have students model other examples of absolute value. 1 d Choose four students as Olympic Athletes. Choose ● Teacher observation; seven students as judges. Students (athletes) Performance compete in categories such as high jump, toe-touch, assessment handstand, etc. Rank events according to difficulty and assign degree of difficulty to each. Judges score each athletes performance. Determine winners of gold, silver, and bronze by removing low and high score, total remaining five scores and multiply by degree of difficulty. 1 d, e, f, g To each group distribute the recipe for Butterfinger pie ● Performance in which the quantity of each ingredient is a numerical assessment or algebraic expression to be evaluated. Expressions should include decimals, fractions, perfect squares, and square roots. Once the group has determined the correct measurements make the recipe. Recipe: ● 12 oz. cream cheese ● 12 oz. Cool Whip ● 6 crushed Butterfingers Mix together in graham cracker pie crust. 29 ourse: Pre-Algebra Unit Theme: Equations and Inequalities Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a, b, c Brainstorm list of words that indicate math operations ● Student work sample; including real-life words such as deduction, raise, in Rubric addition. Encourage use of a thesaurus. Using the list of words, write an algebraic expression that corresponds to each word or combination of words. Emphasize that some words may change meaning depending upon context. 2 a, b, d Use algebra tiles to develop definitions of like terms. ● Discussion; 2 Explain that y and y are related because same color, Student work sample but the exponent creates a square with each side having a length of Y. . 1 y y y 2 e, f Progress from working problems using manipulatives ● Student work sample; to abstract. Teacher-made test 2 e, f Use graphing calculators to solve equations ● Student response graphically. Such as 2x 3 5x 4 . 2 e, f Play inequality BINGO Blackout. (see Glossary) Teacher observation Distribute to students a Bingo Card that contains fractions and decimals in each space. Give an inequality for the students to solve. The students with the correct answer on their card will cover the space. The first person to cover spaces vertically, horizontally, or diagonally will be declared the winner. 30 Course: Pre-Algebra Unit Theme: Polynomials Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a, d, i Give students a card with part of an algebraic ● Student response; expression written on it. Have students line up to Discussion create the given expression. Students will hold up the cards and explain what they are and what they do. 3 b Make a set of 24 cards. The set should contain 12 ● Teacher observation cards with polynomials written on them, all with different degrees. The other 12 cards should have the numerical degrees written on them. Turn all cards face down and play ―Concentration‖ to form pairs that match a polynomial to its corresponding degree. 3 c Use algebra tiles to illustrate combining like terms. ● Teacher observation; Performance assessment 3 e, f, g Use algebra tiles to model multiplication of monomials ● Performance and polynomials. Each factor represents either the assessment length or width of a rectangle. The area of the rectangle formed is the answer. . 1 d Use a graphing calculator to explore results of Rubric; 3 d,h multiplying and dividing numbers involving scientific Performance notation. Students should discover the result of raising assessment 10 to a positive or negative power. 31 Course: Pre-Algebra Unit Theme: Ratio, Proportion, and Percent Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a, b, c Use a real estate guide to choose a house to ● Student work sample; purchase. Given options such as 10% down payment, Rubric 1 20 year mortgage at a fixed rate of 7 2 , calculate down payment, principal, interest, principal plus interest, and monthly payment. 4 a, c Find examples of decimals, fractions, and percents in ● Rubric the newspaper. Convert each example. Discuss pros and cons for using each number form in its given context. 4 a Explore the ―golden ratio‖ and its influence on artists, ● Project; architects, and mathematicians throughout the years. Rubric Students work in cooperative groups to construct objects that use the ―golden ratio‖ in its design. 4 b, c Given a recipe that serves no more than six people, ● Rubric convert it to serve the entire class. Explain each step. 32 Course: Pre-Algebra Unit Theme: Probability and Statistics Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a Given a packet of cut-out doll clothes such as pants ● Presentation; and shirts in different colors, arrange clothes and Student work sample; determine possible number of outfits. Develop the Teacher observation; Fundamental Counting Principle. Discussion 5 b Have small groups determine the number of possible ● Presentation; order arrangements for their group and compare with Discussion; factorial. Extend to arrangement of students in a line Teacher observation for permutations and combinations. Use calculators as needed. 5 c, d Assign a number to each letter of the alphabet (some ● Discussion; 6 a positive, some negative, one zero). Write student Student work sample; initials on different colored sticky dots and create Teacher observation ordered pairs using the value given to the initials. Place dots on graph board and calculate probability of dots in each quadrant, colors in each quadrant, etc. 5 e Organize height of each student into stem- and leaf- ● Teacher observation plot. Extension: Use graphing calculators to analyze information by creating a box and whisker. 33 Course: Pre-Algebra Unit Theme: Coordinate System Suggested Suggested Comp. Obj. Teaching Strategies Assessment 6 a, c Create an Etch-A-Sketch style figure on graph paper. Rubric List the ordered pairs in the order necessary to connect each ordered pair if they were vertices of the figure. Enter the ordered pairs in the graphing calculator and view the figure. Adjust the window and scale appropriately to show entire figure. 6 b, c, f Using a graphing calculator, enter a linear equation. ● Teacher observation Use table function (if available) or build a table to view ordered pairs and determine intercepts. 6 d Indicate type of slope of the segments used in forming ● Discussion capital letters. 6 e Use a meter stick and level to determine slope of ● Student work sample; handicap ramp, stairs, and other structure examples Teacher observation found on campus. 6 g Use a graphing calculator to determine solution to ● Teacher observation system of equations. 34 Course: Pre-Algebra Unit Theme: Geometry Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Calculate surface area of desk and textbook. Measure ● Project 7 d and cut contact paper to cover. 1 a Make a poster showing ratio of circumference to ● Student work sample 7 b diameter for circles of varying size. 1 c Measure distance at the baseball field from home plate ● Teacher observation 7 g to first base and first base to second base. Use the Pythagorean Theorem to calculate distance from second base to home plate. Measure actual distance from second base to home plate and compare results to calculation. 7 d Go on a scavenger hunt around campus to find ● Presentation examples of different types of angles and triangles. Identify examples according to classification. 4 e Have students measure their height and the length of ● Teacher observation; 7 b their shadow. Measure the shadow of an object such Discussion; as a tree or flagpole. Use similar triangles to Student work sample approximate height of object. 7 f Use masking tape on the floor to create parallel lines ● Teacher observation cut by a transversal. Number the interior and exterior angles 1 to 8. Play ―twister‖ by placing hands and feet on indicated pairs of angles. 7 g Use tagboard and three-dimensional patterns to create ● Discussion; polyhedra. Use as classroom, library, or office Student work sample decorations. 7 h Use Miras to demonstrate symmetry, translations, ● Presentation; rotations, and reflections of figures. After using Miras Student work sample; to discover transformations, use centimeter grid paper Discussion; to complete transformations from given figures. Teacher observation (Extension: M. C. Escher video) 35 TRANSITION TO ALGEBRA Transition to Algebra is an elective course intended to be a bridge between the concrete concepts of Pre-Algebra and the abstract concepts of Algebra I and Geometry. This course will be activity-based, allowing students to explore and investigate algebraic and geometric concepts to build a stronger foundation of basic skills. Such explorations should emphasize physical models, data, graphs, and other mathematical representations in appropriate situations that facilitate the learning process. This course is designed for those students who have completed Pre-Algebra and desire an alternative before taking Algebra I. This is a one-credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 36 TRANSITION TO ALGEBRA CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 1. Recognize, classify, and model real numbers and their properties. (P, M, G, N) a. Identify the subsets of real numbers. b. Compare, order, and locate real numbers on a number line. c. Evaluate expressions with real numbers using order of operations emphasizing integers, rational numbers, and absolute value. d. Identify and demonstrate the properties of real numbers. e. Model real-life situations using real numbers. f. Evaluate powers, squares, square roots, and simplify non-perfect squares. g. Multiply and divide numbers involving scientific notation. 2. Demonstrate the connections between algebra and geometry. (P, D, M, G, N) a. Use formulas (e.g., perimeter, circumference, area, Pythagorean Theorem, distance, midpoint, slope) to solve problems. b. Reinforce formulas experimentally to verify solutions. c. Given a formula, solve for a specified variable of degree one. d. Apply ratios and proportions to solve problems. e. Using an appropriate scale, plot a set of ordered pairs and identify the domain and range. f. Calculate and apply concepts of probability. g. Explain and illustrate how changes in one variable may result in a change in another variable. 3. Explain and communicate the language of algebra. (P, D, M, N) a. Translate between verbal expressions and algebraic expressions. b. Use convincing arguments to justify solutions. c. Recognize and demonstrate the difference in ―evaluate,‖ ―simplify,‖ and ―solve.‖ 37 TRANSITION TO ALGEBRA CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 4. Solve and graph equations and inequalities in one or two variables. (P, D, G, N) a. Solve and check multi-step equations and inequalities, including distributive property, variables on both sides, and rational coefficients. b. Graph solutions to inequalities in one variable. c. Graph linear equations, and investigate the concepts of slope and y-intercept. d. Explore slope as a rate of change. e. Discuss the differences between the solutions of linear equations and inequalities. f. Use appropriate technology to explore and identify families of graphs (e.g., x is a line, x2 is a u shape, |x| is a v shape). 5. Model and simplify polynomials. (P, M, G, N) a. Use manipulatives to model operations of polynomials. b. Model polynomial operations to problems involving perimeter and area. c. Use exponent rules to multiply and divide monomials. d. Determine greatest common factor (GCF) and least common multiple (LCM) of polynomials. e. Arrange polynomials in descending or ascending order and determine the degree. 38 Course: Transition to Algebra Unit Theme: Real Numbers Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Show relationships using visual organizers among the Observation; subsets of the set of real numbers. Rubric 1 b Distribute cards containing a rational or irrational Observation number, arrange in order and justify placement. After ordering, place cards on a number line. 1 c Have teams comprised of four students form numerical Self-assessment using expressions to represent the numbers 1 to 26. Teams graphing calculator will use grouping symbols, the digits 1,2,3, and 5 only once, and the four basic operations to create the expression. 1 d From a list of equations, identify the property illustrated Teacher-made test by each. 1 e Prepare a poster illustrating the use of real numbers Rubric; from examples found in newspapers, magazines, and Checklist other resources. Write an explanation for each example. 1 f Create a table with three columns. Self-assessment using a calculator Numbers Square Square number 1 1 1 = 1 2 4 4 = 2 15 225 225 = 15 Use the table to estimate the square root of non-perfect squares. 1 g Use the graphing calculator in scientific model to Self-assessment using discover rules for multiplying and dividing numbers in graphing calculator scientific notation. 39 Course: Transition to Algebra Unit Theme: Connections Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a Given a formula, explain orally and in writing, Observation; representations of the variable and the process for Rubric applying the formula. 2 a, b Relate the midpoint formula to the average of two Observation grades by graphing the two grades and the average on a number line and emphasizing the location of the average. 2 a, b, g Using a compass, construct a circle with a given radius. Observation Use the formula to calculate area, and verify by estimating the area using the grid. Explain the increase to area if the radius is doubled or tripled. 2 a, b, e Plot two points on a coordinate plane. Use the ● Observation Pythagorean Theorem to find distance. Show how the distance formula is derived from the Pythagorean Theorem. 2 c Working in pairs, one student runs a specified distance Rubric while another uses a stopwatch to measure the time. Replace the distance run, and time in the formula to determine speed. 2 d Use scale drawings to determine actual Teacher test; measurements. Rubric 2 f Using a deck of cards, calculate the probability of Observation drawing a specific card from the deck. 40 Course: Transition to Algebra Unit Theme: Communication Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a Make charts of words that indicate various operations. Rubric Note difference among ―more than,‖ ―less than,‖ ―is more than,‖ and ―is less than.‖ 3 b Given a solved equation with mistakes, verify and Observation; 4 a explain why process is incorrect. Rubric 3 c Given several expressions and equations, sort and Observation classify according to the term ―evaluate,‖ ―simplify,‖ and ―solve.‖ 41 Course: Transition to Algebra Unit Theme: Graphing Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a Use manipulatives (e.g., algebra tiles or blocks) to Observation model processes used to solve equations. 4 a, b List ten solutions to an inequality and graph on a Observation number line. Show other possible solutions by shading. 4 c Graph linear equations on a graphing calculator to Self-assessment on explore slope and y-intercept. graphing calculator 4 d Using a graphing calculator, enter Observation; y x, y 2 x, y 1 x , and y 1 x , one at a time. Student response 2 4 Explore what happens with the steepness of each line. 4 c, e Graph an equation such as y 3 2 . Have Observation students choose solutions from a set of given ordered pairs (sticky notes on board work well) and place them in the correct place on the graph. Next, change the equation to an inequality and repeat procedure. Compare and contrast the equation and inequality. 4 f Explore graphs of simple linear, quadratic, and ● Observation absolute value equations on graphing calculators. Students will model the graphs represented on the calculator, using their arms. 42 Course: Transition to Algebra Unit Theme: Polynomials Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a Use algebra tiles to show differences among Observation " x x" and " x x", " x y" and " x y", "( y 1) x" and "( y 1) x" 2 a Given a rectangle of specific length and width, extend Observation; 5 b length and width by a variable and calculate new Constructed response perimeter and area in terms of the variable. 5 c Use expanded notation to multiply or divide monomials. Teacher test For example: x6 x x x x x x x x x x x4 x 2 x x 5 d Use factor trees and charts to determine GCF and Teacher test LCM. 5 e On index cards, write terms of a two or three variable Observation polynomial. Order terms in descending or ascending order and determine degree. 43 ALGEBRA I The Algebra I course will provide opportunities for students to develop and communicate an understanding of algebraic representation as a prerequisite to all higher mathematics courses. Concepts covered in this course include real numbers and their properties, functions, algebraic expressions, linear equations and inequalities, systems of equations and inequalities, graphing polynomials, formulas, slope, data analysis and probability. The use of graphing calculators will be an integral part of this course. This course is designed to prepare students for Geometry and/or Algebra II. This is a one-credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 44 ALGEBRA I CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 1. Recognize, classify, and use real numbers and their properties. (P, M, N) a. Describe the real number system using a diagram to show the relationships of component sets of numbers that compose the set of real numbers. b. Model properties and equivalence relationships of real numbers. c. Demonstrate and apply properties of real numbers to algebraic expressions. d. Perform basic operations on square roots excluding rationalizing denominators. 2. Recognize, create, extend, and apply patterns, relations, and functions and their applications. (P, D, G, N) a. Analyze relationships between two variables, identify domain and range, and determine whether a relation is a function. b. Explain and illustrate how change in one variable may result in a change in another variable. c. Determine the rule that describes a pattern and determine the pattern given the rule. d. Apply patterns to graphs and use appropriate technology. 3. Simplify algebraic expressions, solve and graph equations, inequalities and systems in one and two variables. (P, D, G, N) a. Solve, check, and graph linear equations and inequalities in one variable, including rational coefficients. b. Graph and check linear equations and inequalities in two variables. c. Solve and graph absolute value equations and inequalities in one variable. d. Use algebraic and graphical methods to solve systems of linear equations and inequalities. e. Translate problem-solving situations into algebraic sentences and determine solutions. 45 ALGEBRA I CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 4. Explore and communicate the characteristics and operations of polynomials. (P, M, G, N) a. Classify polynomials and determine the degree. b. Add, subtract, multiply, and divide polynomial expressions. c. Factor polynomials using algebraic methods and geometric models. d. Investigate and apply real-number solutions to quadratic equations algebraically and graphically. e. Use convincing arguments to justify unfactorable polynomials. f. Apply polynomial operations to problems involving perimeter and area. 5. Utilize various formulas in problem-solving situations. (P, D, M, G, N) a. Evaluate and apply formulas (e.g., circumference, perimeter, area, volume, Pythagorean Theorem, interest, distance, rate, and time). b. Reinforce formulas experimentally to verify solutions. c. Given a literal equation, solve for any variable of degree one. d. Using the appropriate formula, determine the length, midpoint, and slope of a segment in a coordinate plane. e. Use formulas (e.g., point-slope and slope-intercept) to write equations of lines. 6. Communicate using the language of algebra. (P, D, M, G, N) a. Recognize and demonstrate the appropriate use of terms, symbols, and notations. b. Distinguish between linear and non-linear equations. c. Translate between verbal expressions and algebraic expressions. d. Apply the operations of addition, subtraction, and scalar multiplication to matrices. e. Use scientific notation to solve problems. f. Use appropriate algebraic language to justify solutions and processes used in solving problems. 46 ALGEBRA I CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objectives: 7. Interpret and apply slope as a rate of change. (P, D, M, G, N) a. Define slope as a rate of change using algebraic and geometric representations. b. Interpret and apply slope as a rate of change in problem-solving situations. c. Use ratio and proportion to solve problems including direct variation b kxg y . d. Apply the concept of slope to parallel and perpendicular lines. 8. Analyze data and apply concepts of probability. (P, D, M, G, N) a. Collect, organize, graph, and interpret data sets, draw conclusions, and make predictions from the analysis of data. b. Define event and sample spaces and apply to simple probability problems. c. Use counting techniques, permutations, and combinations to solve probability problems. 47 Course: Algebra I Unit Theme: Real Numbers Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Write a journal, paragraph, or story to explain how the Rubric set of real numbers is like a family tree. 1 b Write each of the following on small individual paper Teacher observation squares: A, A, B, B, C, C, -, +, x, , IF, and THEN, =, ( ), and 0. Use these to model properties and equivalence relationships. 1 c Create foursomes such as: Teacher test; 6 f Constructed response 3 x 4 3x 12 3x 2 x 5x Distributive Property 5x 3 3 5x Which one does not belong? Explain. 1 d Find the perimeter and area of a rectangle with radical Teacher test terms as dimensions. 1 b, c, d Use and identify appropriate properties to explain a ● Teacher test; 6 f computational procedure. Extension: Given a real Constructed response world or mathematical problem identify the operational strategies involved and justify. 48 Course: Algebra I Unit Theme: Patterns Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a, b, d Using equations involving rational numbers, such as Rubric 7 a y .05x to represent the value of x nickels, explore how changes in x affect y. Identify domain as nickels and range as value. Use a T-chart to graph the relation and verify with graphing calculator. 2 c Use algebraic expressions to represent consecutive Rubric even or odd even integers that have a particular sum. Given a set of consecutive even or odd integers, write a verbal expression to represent the set. 49 Course: Algebra I Unit Theme: Graphing Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a Use manipulatives (e.g., algebra tiles or algeblocks) to Teacher test model the process of solving linear equations. Check solutions using the graphing calculator or substitution. 3 b Group students in pairs. Give each pair a set of linear Observation equations directing one student to graph using a graphing calculator and the other not using a calculator. Compare results and switch roles. 3 c Create a ―zero finder‖ as pictured to illustrate the Teacher test absolute value as a distance from the origin. For example: x2 5 5 4 3 2 1 0 1 2 3 4 5 3 2 1 0 1 2 3 4 5 6 7 Position with zero on the zero finder above the two on the number line because two makes the expression inside the absolute value zero. The solutions to the equation are five units from two on the number line. 3 d Use colored pencils to sketch and shade systems of Teacher test linear inequalities. 3 d Use Algebra Tiles and the graphing calculator to solve Rubric systems of equations. 3 d Compare solutions of systems of equations versus Observation inequalities. Use the graphing calculator to explore the different outcomes. 3 e Create constructed response items that involve Teacher test; translating problem-solving situations into algebraic Rubric sentences. Have students solve and exchange papers. - 50 Course: Algebra I Unit Theme: Polynomials Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a On each wall of the classsroom, put the classifications Observation 6 a of polynomials. Write assigned polynomials on index cards and place on the correct wall. In groups of four, assign a degree to each group and have them create a polynomial of that degree and present to large group. 4 b, c, f Given a rectangle of given length and width, extend the Teacher test length and width by a variable and find the perimeter and area. Given the area of a rectangle in one variable, find the length and width. 4 b Use the algebra tiles to model operations with Teacher test polynomial expressions. 4 c, d Use the quadratic formula to solve trinomial Teacher test equations, and use solutions to write binomial factors. 4 d Graph quadratic equations on a graphing calculator to Teacher test relate the x-intercepts to solutions. 4 e Use a graphing calculator to graph a quadratic Teacher test 6 f equation with no x-intercepts. Relate to the connections among x-intercepts, real solutions, and factors. 4 b, c, f Use algebra tiles to determine factors of a polynomial ● Observation expression. 4 b, c, f Use algebra tiles to create a rectangle of any area. ● Constructed response 5 a Determine the dimensions and perimeter of the sketched rectangle. 51 Course: Algebra I Unit Theme: Formulas Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a, b Given a cardboard box, measure the length, width and Rubric; height to determine perimeter of a side, area of a side, Teacher test and volume of the box. Find the diagonal of a side of the box. Extension: Determine the relationship between the dimensions of the box and the volume of the box. 5 a, b Determine and justify comparable pricing for different Presentation; 6 f size pizzas. Rubric 5 a, d Plot two points in a coordinate plane and use formulas Teacher test to calculate length, midpoint, and slope. Make comparisons among the formulas used for calculations. 5 c, e On index cards, write variables, symbols, operations, Observation and the equal sign, one per card. As formulas are given verbally, demonstrate by holding up appropriate index cards. EXTENSION: In pairs, demonstrate the ―Golden Rule of Algebra‖ to solve for lengths using the perimeter formula. 5 e Draw a line segment with endpoints in different ● Teacher test quadrants. Choose the appropriate formula to write the equation of the line formed using the line segment. Explain and show how standard form, point-slope formula, and slope-intercept formula are related. 52 Course: Algebra I Unit Theme: Communication Suggested Suggested Comp. Obj. Teaching Strategies Assessment 6 a, b Given several equations, classify as linear or non-linear Observation; and verify with a graphing calculator. Teacher test 6 c From two lists, match the algebraic expressions to their Rubric; corresponding verbal expressions. Extension: Create a Teacher test real-world problem using the corresponding matched algebraic and verbal expressions. 6 d Using two different brands of regular and diet soft Teacher test drinks arrange the price of each in matrix form and show the price doubling by using scalar multiplication. 6 a, e Using states that are rectangular in shape, estimate Presentation their actual area in square feet. Express the estimated area in scientific notation. 6 e, f Explore problems involving scientific notation using the ● Rubric graphing calculator. Explain the difference between multiplying by a positive power of ten and by a negative power of ten. 6 a, e, f Give examples of large numbers or small numbers ● Rubric containing more than three non-zero digits correctly represented in scientific notation. Explain and justify each example. 53 Course: Algebra I Unit Theme: Slope Suggested Suggested Comp. Obj. Teaching Strategies Assessment 7 a, b Relate income to the number of hours worked in Teacher test equations such as: y $5.25x and y $15.85x Use the graphing calculator to compare the change of income (y) as it relates to the change in hourly wage (slope). 7 a, b Place a yardstick across the incline of a set of steps. Rubric Measure the vertical change versus the horizontal change, then explore how changing these distances affect the steepness of the steps. 7 c Using a bicycle, demonstrate how the revolutions of the Teacher test pedal and the rear wheel illustrate the concept of direct variation. For example, y 3x . (In a particular gear perhaps the ratio is 3 to 1) x = number of revolutions of pedal y = number of revolutions of rear wheel Using a graphing calculator, graph a series of Observation 7 d equations to discover the relationship of slope to parallel and perpendicular lines. 54 Course: Algebra I Unit Theme: Probability Suggested Suggested Comp. Obj. Teaching Strategies Assessment 8 a In groups, assign each a topic from which to design Rubric and conduct a survey. Compile, graph, and interpret results and present to class. Extension: Use computer graphing software to organize collected data. 6 a Define ―events‖ and ―sample space‖ for experiments Observation 8 b involving number cubes, spinners, coin flipping, and cards. 6 f Determine how many handshakes there would be ● Rubric 8 c between five people if everyone had to shake hands with each person exactly once. Explain or sketch how the answer was determined. 55 GEOMETRY The Geometry course is the study of two and three-dimensional figures. This course will provide the opportunity for students to develop spatial sense and reasoning skills. Students will use the language of geometry to communicate an understanding of the properties and characteristics that encompass geometry. Students will also investigate patterns and relationships among geometric shapes. This course is designed for students who have successfully completed Algebra I. This is a one-credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 56 GEOMETRY CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 1. Communicate using the language of geometry. (P, M, G, N) a. Define and recognize terms and symbols of geometry and use them to communicate mathematical ideas. b. Differentiate between inductive and deductive reasoning. c. Use properties, theorems, postulates, and definitions to justify relationships involved with segment and angle congruence. d. Develop and evaluate mathematical arguments and proofs. 2. Identify, explore, discuss, and apply properties, theorems, postulates, and definitions related to angles, lines, and circles. (P, M, G, N) a. Identify and classify angles. b. Identify, explore, and apply angle relationships formed by parallel lines cut by a transversal. c. Explore, discuss, and apply the relationships among parts of a circle and between arcs and angles. d. Use angle and segment relationships to find unknown measures related to circles. 3. Identify, explore, discuss, and apply properties, theorems, postulates, and definitions related to polygons. (P, M, G, N) a. Identify and name different types of polygons and their subsets. b. Classify triangles and apply postulates and theorems to test for triangle congruence and triangle inequality. c. Identify altitude, median, angle bisectors, and perpendicular bisectors in a triangle. d. Apply definitions, postulates, and theorems to find angle measurements in polygons. 57 GEOMETRY CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 4. Explore and demonstrate the connections between algebra and geometry. (P, M, G, N) a. Apply ratios and proportions to solve for unknown measures in similar polygons. b. Solve for missing measurements in right triangles using the Pythagorean Theorem, special right triangle relationships, geometric mean, and trigonometric functions. c. Relate algebraic formulas to geometric properties to solve problems in the coordinate plane. d. Explore how change in perimeter results in a change in area. 5. Investigate, classify, compare, and contrast two and three-dimensional geometric figures. (P, M, G, N) a. Find the areas of triangles, quadrilaterals, and regular polygons. b. Find the area and circumference of a circle. c. Find the volumes of rectangular prisms, cylinders, pyramids, cones, and spheres. d. Use protractors, compasses, rulers, and/or technology to construct geometric figures and drawings. e. Compare, contrast, and classify two-dimensional figures and investigate their characteristics. f. Compare, contrast, and classify three-dimensional figures and investigate their characteristics. g. Use measurement to design and build a three-dimensional object. 6. Explore applications of patterns and transformational geometry. (P, D, M, G, N) a. Identify symmetry in common objects as examples of point, line, and rotational symmetry. b. Create designs using symmetry. c. Recognize and describe images of figures obtained by applying reflections, translations, rotations, and dilations. d. Create tessellations using translations and rotations. e. Determine the effect of scale factors on dilations. f. Use geometric probability to predict results. 58 Course: Geometry Unit Theme: Communication Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a As an on-going project, create a book consisting of Project; illustrations, real-life examples, and applications Rubric illustrating terms and symbols of geometry. 1 b Given situations that require logical thinking, classify Teacher test as inductive or deductive reasoning. 1 c, d On index cards, write statements and reasons to a two Rubric column proof (one per card). Shuffle, distribute, then have students put in logical order. 59 Course: Geometry Unit Theme: Segment and Angle Relationships Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Use a protractor to measure angles and classify Observation; 2 a according to definitions. Teacher test 5 d 1 a, c Construct a moveable model of parallel lines cut by a Observation 2 b transversal from three strips of tagboard fastened 5 d together with brads. Measure the various angles and show the relationship among the angles. 1 a Create a display illustrating parts of a circle, their Rubric 2 a, c definitions and properties. 5 d 1 a, c, d Construct a circle of any radius. Use a straight-edge Observation; 2 c, d to draw various angles formed by segments. Use a Teacher test 5 d protractor to measure and draw conclusions about formulas used to find these unknown measures. (Can be enhanced with appropriate technology.) 60 Course: Geometry Unit Theme: Polygons Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Design mobiles that illustrate the shape and Rubric 3 a characteristics of quadrilaterals. 5 e 1 a, c, d Given labeled sets of triangles, match to the Teacher test 3 b appropriate congruence postulate or theorem. 1 c, d Given three straws of different lengths, explore the Observation 3 b question: ―Is it always possible to form a triangle?‖ 3 c Fold different types of triangles to illustrate medians, Observation altitudes, and bisectors. 1 a, c Draw a polygon. Connect a vertex to the non-adjacent Teacher test 3 a, b, d vertices and form triangles. Discover the polygon interior angle theorem by counting the triangles and finding the sum of angles. 61 Course: Geometry Unit Theme: Connections Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a, d Given two similar polygons, use highlighters to color Teacher test 4 a code corresponding parts; set up ratios and proportions to find unknown measures. 4 a Use actual measures of a room in the school or home Rubric 5 d, g to make a scale drawing. Build a scale model of the room. 1 a, c Plot four vertices of a quadrilateral in a coordinate Rubric; 3 a plane. Use algebraic formulas to classify the Checklist 4 c quadrilateral and justify the conclusion. 5 e 1 a Form a square with string. Measure a side and Observation; 4 d calculate perimeter and area. Cut the string in half and Rubric 5 a repeat procedure. Record results and determine relationship between change in perimeter and resulting area. 62 Course: Geometry Unit Theme: Two and Three-Dimensional Figures Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a, e Given a variety of regular polygons, compare and ● Student work justify the relationship between the number of sides samples; and the number of diagonals. Rubric 1 a, c Construct a circle and inscribe a regular polygon of n Teacher test 5 a, b, d sides. Estimate area then calculate actual area by using the Area of Regular Polygon Theorem. 1 a Construct a single square with straightedge and Presentation; 5 a, d, e compass using least number of steps as possible. Project; Write instructions for the created construction. Rubric 5 b, c Measure and calculate the volume of cans of various Teacher test sizes in metric units. Test calculations by filling with water. (1cc = 1ml) 1 c, d Design and construct models of geometric solids and Project; Rubric 5 d, f create a table illustrating the relationship among faces, edges, and vertices of the solids. 63 Course: Geometry Unit Theme: Patterns or Transformations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Collect logos from newspapers and magazines and Rubric 6 a identify types of symmetry. 1 a Given an example of optical art: discuss symmetry Teacher test; 6 a, b involved to include features of the work and its relation Presentation; to symmetry groups. Extension: In groups, create a Rubric similar revision of an optical art. 6 b Draw half of a symmetrical design, exchange designs Observation and complete the drawing using vertical line symmetry. 1 a Design an original border on graph paper that Rubric 6 c incorporates reflections, translations, and rotations. 1 a Sketch a figure in the coordinate plane. Place the Observation 6 c, e vertices in a matrix . Apply scalar multiplication to obtain vertices of the dilated figure. 1 a Use pattern blocks to create tessellations. Investigate Project; 6 d works of M. C. Escher and use them as a model to Rubric create original tessellations. 6 f Divide a poster board into several rectangular regions. Observation Calculate the probability of a tossed penny landing in a particular region. 64 SURVEY OF MATHEMATICAL TOPICS Survey of Mathematical Topics is designed to provide students with the skills necessary in making wise financial decisions. The basic concepts of algebra will be reviewed and extended as students solve real-life problems which affect them and their families. This course will provide skills in probability and statistics, logic, linear programming, and regression analysis. Students are encouraged to use a variety of techniques and appropriate technology (calculators and/or computers) to solve problems. This course is designed for students who have successfully completed Algebra I, Geometry, and/or Algebra II. This is a one-credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 65 SURVEY OF MATHEMATICAL TOPICS CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 1. Demonstrate the skills necessary to manage personal finance. (P, D, M, N) a. Develop a household budget. b. Maintain and balance a checkbook. c. Investigate terminology and the process of filing personal income tax. d. Investigate and explore all the components necessary to own and operate a car. e. Analyze the options of housing alternatives. f. Connect and apply appropriate algebraic formulas to personal finance situations. 2. Compute, analyze, and develop a variety of personal and business investments. (P, D, M, N) a. Analyze information to make wise decisions regarding personal savings. b. Investigate life and health insurance. c. Study and investigate the economics of the stock market. d. Connect and apply appropriate algebraic formulas to personal and business investments. 3. Analyze and illustrate the practices that affect employer and employee decision-making. (P, D, M, G, N) a. Compute and compare various forms of earnings and calculate gross pay, deductions, and net pay. b. Analyze the relationships among cost, revenue, and profit. c. Apply linear programming to business decisions. d. Connect and apply appropriate algebraic formulas to employer and employee practices. 4. Demonstrate an understanding of the impact of consumer credit. (P, D, M, G, N) a. Compare and contrast the finances of credit cards. b. Explore the pros and cons of installment loans. c. Connect and apply appropriate algebraic formulas to consumer credit. 66 SURVEY OF MATHEMATICAL TOPICS CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 5. Collect and apply information in planning a trip. (P, D, M, G, N) a. Investigate and evaluate modes of transportation. b. Create a travel budget. c. Make travel plans based upon airline schedules. d. Utilize map-reading skills. e. Connect and apply appropriate algebraic formulas to planning a trip. 67 Course: Survey of Mathematical Topics Unit Theme: Personal Finance Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Create a budget for a family of four with a given yearly Student work sample income. 1 b Use simulated checks, check registers, and Portfolio reconciliation forms to maintain a checking account and to reconcile the checkbook with the bank statement. 1 c Obtain copies of 1040EZ and 1040A forms and Discussion instruction booklets from the IRS or local library. In groups, discuss the forms and provide sample information for students to complete both forms. 1 d Create a poster with the following headings for six Project; used cars cut out from newspaper advertisements: Rubric Sticker price Down payment (use 10% ) Loan amount Monthly payments (use current interest rate and three years for loan) Total payments Total amount including down payment Use a calculator and the monthly payment formula to complete the poster. Justify which car would be the best buy after verifying the condition of the car by visiting the dealership offering the car. 1 e Investigate the following for each of ten local Presentation apartments for rent: Square footage Monthly rent Number of bathrooms Number of bedrooms Using a graphing calculator, calculate linear regression and find the line of best fit to compare any two apartments. Use this information to make predictions. 1 f Use a calculator and the appropriate formula to Student work sample compute monthly payments when buying a car or house. 68 Course: Survey of Mathematical Topics Unit Theme: Personal and Business Investments Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a Visit local banks to gather information on savings Project accounts. Prepare a poster, which compares the data. 2 b Invite an actuary or local insurance agent to speak to Teacher observation the class concerning life and health insurance policies. 2 c Contact the Mississippi Economic Council (MEC) for Portfolio information on participating in the state Stock Market Game. 2 d Suppose that ancestors deposited $1 in a savings Discussion account 200 years ago. Using simple interest of 3% , calculate the value of that account today. Repeat using compound interest. Discuss the results. (Extend: Vary the amount originally deposited and/or the interest rate.) 2 d Use the Rule of 72 to estimate how long it would take Student work sample to become a millionaire with an initial deposit of $1000 with an interest rate of 10% . Repeat varying interest rates and initial deposit. 69 Course: Survey of Mathematical Topics Unit Theme: Employer/Employee Practices Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a Find gross pay based on commission sales and hourly Short answer rate. Use federal and/or state tax tables and FICA questions percentage rate to calculate deductions and net pay. 3 b Find the break even point given cost and revenue Constructed response equations. Analyze the regions between the two curves when graphed. 3 c Use the method of linear programming to maximize or Student work sample minimize certain factors in a business situation. 3 d Research different types and financial amounts of Checklist fringe benefits offered by local employers. Using this data, compute additional costs associated with employment. 70 Course: Survey of Mathematical Topics Unit Theme: Consumer Credit Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a Collect several credit card applications. Compare ● Discussion terms, finance charges, APR, etc. Determine which application is the most advantageous to the consumer 4 b Create an amortization schedule to illustrate the ● Student work sample concept of installment loans. 4 b Investigate car buying options involving rebates versus ● Discussion the offer of an extremely low interest rate. Discuss the advantages/disadvantages of each option for the dealer, loan institution, and buyer. 4 c Use the Rule of 78 to estimate the savings when a ● Short answer question loan of $1000 for 12 months at 7% is paid off after four months. 71 Course: Survey of Mathematical Topics Unit Theme: Travel Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a, b, c Plan a trip to a far away city within the 48 contiguous Presentation; United States. Decide on destination and length of Project; trip. Call a travel agent (or use the Internet) to Rubric compare various modes of transportation for cost and time constraints. Prepare a budget of anticipated expenses. 5 d, e Obtain state maps for each student. Given two Teacher observation; locations on the map, discuss the best route to travel Discussion from one location to another. Calculate the costs of driving a car to this destination. Discuss the pros and cons of driving versus other modes of transportation. 72 ALGEBRA II The Algebra II course is to serve as an extension of Algebra I with a variety of topics explored in greater depth. It will continue to provide opportunities for students to become mathematical problem solvers, to gain confidence in their ability to use mathematics, to learn to communicate and reason mathematically, to generalize when appropriate, and to make mathematical connections. Technology, especially graphing calculators, should be incorporated throughout this course. This course is designed for students who have successfully completed Algebra I and/or Geometry. This is a one- credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 73 ALGEBRA II CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 1. Explore the relationships among coefficients, exponents, degree and roots of equations. (P, M, G, N) a. Use acronyms such as SOPPS (Square, Opposite sign, Product, Plus, Square) to teach the sum/difference of cubes. b. Solve and explore equations using the quadratic formula, completing the square, synthetic division, graphing, and technology.+ c. Classify solutions of quadratic equations through observations of graphs and through use of the discriminant. d. Write a polynomial equation when given its roots. 2. Solve systems of equations and inequalities and interpret solutions. (P, D, M, G, N) a. Explore methods of solving systems of equations to include algebraic methods and matrices. b. Write a system of equations to solve a problem. c. Interpret by graphing, and solve systems of inequalities. d. Introduce linear programming as a method to solve problems. 3. Recognize, classify, and perform operations with irrational and complex numbers. (P, G, N) a. Explore and describe the complex number system. b. Explain and apply complex conjugate methods to simplify problems. c. Perform operations with complex numbers and review radicals. 74 ALGEBRA II CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 4. Identify and investigate relations and functions. (P, D, M, G, N) a. Determine the domain, range, roots, and inverse of a function. b. Recognize and determine graphs of linear, quadratic, absolute value, greatest integer, and piece-wise functions. c. Develop a complex coordinate plane for complex numbers (a + bi) where reals are represented on the x-axis and imaginary units are represented on the y-axis and model operations of complex numbers. d. Evaluate functions including composite functions. e. Explore and investigate solutions to compound and absolute value inequalities to include interval notation. f. Use scatter plots and apply regression analysis to data. 5. Investigate rational expressions and equations. (P, D, M, G, N) a. Perform basic operations and simplify rational expressions to include complex fractions. b. Solve and verify solutions to equations involving rational expressions. c. Analyze problems involving direct, inverse, joint, and combined variations. 6. Solve, graph, and apply the properties of exponential and logarithmic expressions and equations. (P, D, M, G, N) a. Illustrate and apply the relationships between exponential and logarithmic functions. b. Simplify radical, exponential, and logarithmic expressions. c. Solve equations involving radicals, exponents, and logarithms. d. Collect, organize, and interpret data from exponential, logarithmic, and power functions. 75 ALGEBRA II CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 7. Identify characteristics and extend operations and applications of matrices. (P, D, N) a. Explain dimensions of a matrix. b. Find the inverse and determinant of a matrix. c. Solve for unknown values in corresponding elements of equal matrices. d. Perform basic operations and apply to matrices. 76 Grade Level: Algebra II Unit Theme: Equations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Work problems and explain the process of factoring ● Small group sum/difference of cubes. observation 1 b Divide the class into groups. On individual cards, list Small group steps for deriving the quadratic formula by completing observation the square. Distribute one set each to be ordered in the proper sequence. 1 b In small groups, solve a given equation using at least Teacher observation four different methods. Have each student write a report explaining steps involved in each method. Designate and justify the preferred method. 1 c Provide to each student a list of quadratic equations. Self-evaluation using Using a graphing calculator, graph equations, observe graphing calculators the number of times the graph crosses the x-axis, then relate to roots and x-intercepts. 1 d Create a matching set of cards (one with equations Teacher observation and one with corresponding roots). Divide class into groups and provide each a set of cards to match each equation with its roots. 77 Grade Level: Algebra II Unit Theme: Systems of Equations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a In groups, solve systems of equations simultaneously Teacher observation using different methods. Compare and discuss solutions and the preferred process. Exchange methods and repeat until each student has used every method at least once. 2 b Fill a bag with two types of candy bars costing x Student work sample amount and y amount. On the outside of the bag, write the total number of candy bars and the dollar amount. Determine the number of each type. 2 c, d As an introduction to cost and profit linear Teacher observation programming problems, invite a businessman from industry to speak to the class. 78 Grade Level: Algebra II Unit Theme: Irrational and Complex Numbers Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a Given 1 sheet of cardboard, design and decorate a Rubric 4 math flag to represent the different sets of numbers to show how each set relates. Tape to the bottom of a wire hanger and display in the classroom. 3 b Given a problem that has been simplified incorrectly, ● Constructed response find the mistake and explain how to correct it. 3 c Discuss the history of complex numbers and their ● Teacher observation relationship to the Fundamental Theorem of Algebra. Extension: Discuss the relationship of complex numbers and fractals. 79 Grade Level: Algebra II Unit Theme: Relations and Functions Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a, b Given the transformation of parent graphs, match the Teacher observation; graphs with equations and word descriptions. Write Rubric observations and predictions based on the transformation. 4 a, b Give groups a function and its inverse. Have part of Presentation; the group algebraically justify the inverse of the Rubric function and have the remaining group justify graphically. 4 c Model the complex coordinate plane using a floor Teacher observation graph and students as coordinates. 4 d chchc h Explore the meaning of f 3 , f 1 , f x 1 as applied to a Teacher evaluation; function. Teacher test 4 e Write statements involving inequalities and absolute Rubric values that model finding the gas tank capacity, average city miles per gallon, and highway miles per gallon of a car. 4 f Collect data on any two situations related to the Teacher observation students in the class (e.g., education and salary, age and speeding tickets in a year). Graph the data and determine the line of best-fit. From the equation, make predictions based on this equation. 4 f Collect data on forearm length and height of students ● Teacher observation in the class. Use technology to draw a scatter plot and perform regression analysis. 80 Grade Level: Algebra II Unit Theme: Rational Expressions and Equations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a Make a set of cards with rational expressions and a Teacher observation second set of cards with the expression simplified; distribute cards and find the match. 5 b Explain why it is necessary to verify solutions to Teacher test; rational equations. Rubric 5 c Interview a science teacher on how the world of Presentation science uses variations. Present to the class the main ideas of the interview to include examples of how variations are used in science. 81 Grade Level: Algebra II Unit Theme: Exponential and Logarithmic Expressions and Equations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 6 a, c Use paper and pencil to draw graphs of exponentials Teacher observation and logarithms. Verify using a graphing calculator and compare and contrast the graphs. 6 b Show and explain the relationship between exponents Student evaluation and logarithms. 6 d Use technology to investigate the function which would Observation explain the process of cooling liquid in a cup. 82 Grade Level: Algebra II Unit Theme: Matrices Suggested Suggested Comp. Obj. Teaching Strategies Assessment 7 a, b Investigate the relationship among dimensions, Teacher test inverses, and determinants of matrices. 7 c List the necessary requirements for two matrices to be Checklist equal. 7 d Collect prices for individual orders of medium soda, Teacher test medium fries, and hamburgers from different fast food restaurants. Model through matrix multiplication the total cost for ordering 5 fries, 10 sodas, and 7 hamburgers. Determine the best deal. 83 ADVANCED ALGEBRA The Advanced Algebra course serves as an extension of algebraic concepts. Through a more in-depth study of algebra, students will further enhance their mathematical confidence and reasoning ability. This course will be an extension of Algebra II, and may be used as a prerequisite to Pre-Calculus. The use of graphing calculators and other appropriate tools of technology is strongly recommended. This is a one-half credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 84 ADVANCED ALGEBRA CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 1. Analyze and extend patterns of graphs in families of functions. (P, D, M, G, N) a. Determine domain and range. b. Relate symmetry to the behavior of even and odd functions. c. Use technology to analyze and sketch the graphs of polynomial, rational, exponential, and logarithmic functions. d. Explore properties of composites and inverses and their graphs as they relate to functions. e. Use linear programming to solve problems. 2. Investigate and apply the characteristics and operations connecting sequences and series. (P, D, G, N) a. Express sequences and series using recursive processes. b. Develop and use formulas for sequences. c. Evaluate and apply arithmetic and geometric series. d. Evaluate and apply infinite geometric series. e. Explore the relationships of Pascal’s triangle. 3. Explore and apply fundamental principles of probability and statistics. (P, D, G, N) a. Use summation () and factorial notation to solve problems. b. Expand and apply the Binomial Theorem to problem-solving situations. c. Draw inferences from and construct charts, tables, and/or graphs that summarize data. d. Use and apply the Fundamental Counting Principle, permutations, and combinations as a preface to probability. e. Use theoretical or experimental experiences to determine simple probability. f. Use curve-fitting to predict from data. 85 ADVANCED ALGEBRA CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 4. Identify, explore, and predict equations and graphs of conic sections. (P, M, G, N) a. Identify the parts essential to the graphs of the circle, parabola, ellipse, and hyperbola. b. Analyze and sketch the graphs of conics. c. Recognize conic sections by their graphs and equations. d. Apply algebraic techniques to write conics in standard form. e. Graph conic sections using translations. 5. Extend algebraic techniques to higher degree polynomial and complex rational problems. (P, D, N) a. Factor and find zeros of polynomial equations. b. Solve quadratic and simple polynomial inequalities. c. Solve inequalities containing simple rational expressions. 6. Explore and extend properties and applications of exponential and logarithmic equations. (P, D, M, G, N) a. Explore and simplify exponential expressions and solve exponential equations. b. Evaluate logarithmic expressions and solve logarithmic equations. c. Explore applications of logarithms. 86 Course: Advanced Algebra Unit Theme: Functions Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a, c Using a graphing calculator or computer simulation, Teacher evaluation investigate and discuss the domain and range of families of functions by comparing the equation, graph, and table of values. 1 b Using a 6" x 6" graph grid and pipe cleaner, model Teacher observation even and odd functions. 1 d Using a graphing calculator or computer simulation, Self evaluation compare the graphs of two functions to their composite function. Predict the graph and verify using technology. 1 e Create, construct, and solve a linear programming Peer evaluation problem with at least four equations. 87 Course: Advanced Algebra Unit Theme: Sequences and Series Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a, b, d Using a calculator, take the square root of a positive Student evaluation integer. Continue to take the square root of the answer. Discuss the results and model the pattern. 2 c Explain the difference between a geometric and Constructed response arithmetic series and give an example of each. 2 e Using the Internet, explore and investigate the patterns Student work sample of Pascal’s Triangle. 88 Course: Advanced Algebra Unit Theme: Probability and Statistics Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a Given a pattern of one whole, a half, one-sixth, and Teacher evaluation one twenty-fourth, investigate and relate to n. 3 b Work problems involving batting averages and coin Teacher evaluation tossing using the Binomial Theorem or Pascal’s Triangle. 3 c, f Students will plot their shoe size and wrist Peer evaluation measurement on a large graph. After drawing the line of best fit, predict a professional athlete’s wrist size based on a given shoe size. 3 d Given the school lunch menu for the day, determine Teacher evaluation the number of possible combinations of meals. 3 e Flip coins repeatedly or draw objects from a sack to Peer evaluation compare the outcomes to the expected probability. 3 f Using a graphing calculator, use curve fitting to find the Self-evaluation equation of the curve of best fit containing three or more non-linear points. Make predictions using the equation and the graph. 89 Course: Advanced Algebra Unit Theme: Conic Sections Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a, b, c, d Given a list of quadratic equations, determine the type Teacher evaluation of conic section. Write each equation in standard form and identify specific characteristics. 4 b, c, e Graph parent conic sections and predict translations. Self-evaluation Verify using a graphing calculator or computer simulation. 90 Course: Advanced Algebra Unit Theme: Polynomial Equations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a Create an equation given the zeros to see the Self-evaluation relationship between the zeros and the equation. Use technology to verify the zeros. 5 b, c Given a polynomial inequality or a rational inequality, Teacher evaluation find and verify the values of x and express the solution in inequality notation, interval notation, and graph form. 91 Course: Advanced Algebra Unit Theme: Exponential and Logarithmic Equations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 6 a, c Using pb to represent the beginning population, pe to Teacher evaluation represent the ending population, and t to represent growth time intervals, use the following formula to aft determine bacteria growth: p e p b 2 . Given values for any two unknowns, solve for the third. Construct and complete a two-day chart logging total bacteria at specific time intervals for growth. 6 b, c Given logarithmic and exponential expressions, Teacher evaluation explain the process of converting from one form to the other and make connections for solving exponential and logarithmic equations such as log2 x 4 or 5 x 71 . Work application examples that include growth and decay problems involving half-life. 92 PRE-CALCULUS The Pre-Calculus course serves as a bridge between Algebra II or Advanced Algebra and Calculus. It will extend students’ knowledge of concepts mastered in Algebra II or Advanced Algebra. This course will increase analysis skills and enhance students’ reasoning ability and mathematical confidence. The use of technology, especially graphing calculators, should be an integral part of this course. This course is designed to prepare students for Calculus/Advanced Placement Calculus. This is a one-half credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 93 PRE-CALCULUS CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 1. Investigate, predict, and extend patterns of graphs in families of functions. (P, D, M, G, N) a. Demonstrate proficiency in determining domain and range. b. Relate powers and coefficients to the end behavior of graphs of functions. c. Relate symmetry to the behavior of even and odd functions. d. Analyze and sketch the graphs of polynomials, rational, piece-wise, greatest integer, exponential, and logarithmic functions, and verify using technology. e. Explore properties of composites and inverses and their graphs as they relate to functions. 2. Illustrate and explore the characteristics and operations connecting sequences and series. (P, D, G, N) a. Express sequences and series using recursive processes. b. Develop and use formulas for sequences. c. Evaluate and apply arithmetic and geometric series. d. Evaluate and apply infinite geometric series. e. Use the Principle of Mathematical Induction as a form of mathematical proof. 3. Explore and apply fundamental principles of probability. (P, D, G, N) a. Use summation () and factorial notations to solve problems. b. Expand and apply the Binomial Theorem to problem-solving situations. c. Use and apply the fundamental counting principle, permutations, and combinations as a preface to probability. d. Use theoretical or experimental experiences to determine simple probability. 94 PRE-CALCULUS CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 4. Extend algebraic problem-solving techniques to higher degree polynomial and complex rational equations. (P, D, G, N) a. Factor and find zeros of polynomial equations. b. Graph and write equations using the behavior of linear, even, and odd factors. c. Solve simple polynomial inequalities to include quadratic inequalities. d. Solve inequalities containing simple rational expressions. e. Investigate optimization problems. 5. Extend operations and applications of matrices. (P, N) a. Calculate determinants of matrices to include expansion of minors. b. Solve systems of n equations and explain the solutions. 6. Extend properties and applications of exponential and logarithmic equations. (P, D, M, G, N) a. Explore and simplify exponential expressions and solve exponential equations. b. Evaluate logarithmic expressions and solve logarithmic equations. c. Explore the application of logarithms to problem-solving situations. 95 Course: Pre-Calculus Unit Theme: Families of Functions Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Given a function, predict the domain and range. Enter Teacher evaluation the function on the graphing calculator. Use a piece of spaghetti to find the domain and range. Compare the prediction to the spaghetti results. 1 b Using the graphing calculator, determine the end Rubric; behavior of a function and write a paragraph explaining Teacher evaluation the results. Discuss how the degree of the function affected the end behaviors. Create a spreadsheet of values to determine the end behavior of a graph. 1 c Discuss the properties of linear, even and odd factors. Self-evaluation 4 b In small groups, discuss the following situation: If f x is an even function and g x is an odd function, is f x g x an even function, an odd function, or neither. Draw the graph of an even function and of an odd function. Demonstrate symmetry with respect to the x-axis, the y-axis, the line y x and the line y x . Verify using the graphing calculator. 1 e In small groups, determine if families of functions have Teacher observation the same symmetry as the parent function and justify the answer. 1 e Fold graph paper about the line y x . Draw the Self-evaluation graph of any function. Trace the graph on the other side of the fold to reveal the inverse. 1 e Create and graph a function. Find and graph the Report inverse. Write a paragraph describing the relationship between a relation and its inverse. Include how to determine if a relation is a function and whether the function has an inverse. 96 Course: Pre-Calculus Unit Theme: Sequences and Series Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a, b, c On the first day of January, Bob ate one candy bar. Class discussion Each day thereafter he ate one more candy bar than the previous day. Determine the number of candy bars he ate during the month of January. 2 a, b, c Find the number of calories in a candy bar of your ● Class discussion choice. Calculate the caloric intake for that month. Estimate the possible weight gain by the end of the month. 2 c Tear a square piece of paper with an area of one, in Demonstration half. Tear it in half again. Predict the area of one of the resulting rectangles after six tears. 2 d Divide the class into groups and provide each group Teacher observation with a ball. As the ball is thrown or dropped, use technology to record the path of the ball. 2 e Use mathematical induction to prove that a formula is Teacher evaluation valid for all positive integral values of n. 97 Course: Pre-Calculus Unit Theme: Probability Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 b In small groups, create a geometric series which can Student evaluation 3 a be expressed in sigma notation. Exchange papers and write the series in sigma notation form. 3 b Give an example involving a baseball player’s batting Teacher evaluation average. In small groups, use the Binomial Theorem to determine the probability of getting at least three hits in the next five times at bat. 3 c Write a paragraph explaining the difference between Report; permutations and combinations. Create problems Rubric involving each. 3 d In small groups, provide each group with a different Demonstration type of manipulative (e.g., cards, number cubes, coins, spinners, and slips of paper with numbers). Given a probability problem, determine the theoretical probability. Perform the experiment to compare theoretical prediction to the experimental result. 98 Course: Pre-Calculus Unit Theme: Polynomial Equations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a, b Create and graph an equation of a polynomial function. Teacher evaluation Write a paragraph explaining the zeros of a function and how to determine where they are located on the graph. 4 b Provide small groups or individuals with pictures of 10 Rubric functions which have varying degrees, but all of which c c ch c h h h are factorable. For example, f x x 1 x 2 x 3 . Have each group come up with the equation for the function, using the smallest possible degree. Students can expand the factors or leave them in factored form. Have them justify why they chose the degree they did for each factor. 4 c, d, e In small groups, create, solve, and graph rational and Student evaluation polynomial inequalities. Extend to graphing systems of inequalities by shading solutions with colored pencils. Given a function, find the maximum and minimum of the shaded region. 99 Course: Pre-Calculus Unit Theme: Matrices Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a Write a paragraph discussing the process to find the Rubric determinant of a 3 x 3 matrix. Include both the lattice method and expansion by minors. 5 b In small groups, create a word problem involving a Class discussion 2 x 2 matrix. Solve the system of equations by matrices and explain the results. 100 Course: Pre-Calculus Unit Theme: Exponential and Logarithmic Equations Suggested Suggested Comp. Obj. Teaching Strategies Assessment 6 a In small groups, create, explain, and verify specific Student evaluation examples for each of the properties of exponents. 6 b Compare the relationships of a logarithmic function Student evaluation and the inverse of an exponential function. Write an equation in one form, exchange papers, and write the inverse form. 6 c Solve growth and decay problems involving half-life Teacher evaluation using logarithms. 101 TRIGONOMETRY The Trigonometry course forms a foundation for later development of Calculus concepts. This course is a comprehensive study of trigonometric functions with emphasis on applications. The study of trigonometry extends algebraic concepts to the exploration of circular and triangular functions with their properties and graphs. The use of graphing calculators is an essential part of this course. This course is designed for students who have successfully completed Algebra I, Geometry, and Algebra II, and is a prerequisite for Calculus/Advanced Placement Calculus. This is a one-half credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 102 TRIGONOMETRY CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 1. Identify, locate, and apply trigonometric functions to the unit circle. (P, M, G, N) a. Identify and locate angles in radians and degrees based on the unit circle. b. Convert between degree and radian measurements of angles. c. Use the definition of the six trigonometric functions to find missing parts of a triangle. d. Determine the values of inverse trigonometric functions. e. Utilize special right triangle relationships and symmetry as they apply to the unit circle. f. Relate the unit circle to the right triangle. 2. Explore, communicate, and apply the connections between the patterns of trigonometric functions and graphing with and without appropriate technology. (P, D, M, G, N) a. Recognize, sketch, and interpret the graphs of the six basic trigonometric functions and their inverses to include restrictions on the domain. b. Recognize, sketch, and interpret graphs of the trigonometric functions using all transformations. 3. Utilize and extend algebraic and geometric techniques to trigonometric equations and applications. (P, D, M, G, N) a. Solve for unknown parts of triangles to include Law of Sines and Law of Cosines. b. State, verify, and utilize trigonometric identities. c. Find arc length and area of a sector of a circle. d. Find the area of a triangle using Heron’s Formula and/or 2 bc sin A . 1 e. Solve trigonometric equations, using both radians and degrees. f. Model and apply right triangle formulas, Law of Sines, and Law of Cosines to problem-solving situations. 103 TRIGONOMETRY CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 4. Introduce and investigate basic concepts of vectors and operations with vectors. (P, M, G, N) a. Recognize different notations for vectors. b. Apply addition to vector sums and resultants. c. Determine the norm (magnitude) of a vector. d. Create a unit vector in the same and in the opposite direction of a given vector. e. Draw a vector to represent a quantity. 104 Course: Trigonometry Unit Theme: Circle Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a, b, e, f Using a protractor and a paper plate, show the Project; multiples of 30, 45 , and 60 and quadrantals in all four Rubric quadrants labeling each in degrees and radians. 1 b, d Cards marked with two sides in degree measure and Demonstration two sides in radian measure are dealt with the last card placed face up in the middle of the table. Playing left to right, match cards with degree/radian equivalents. Design a similar game using inverses. 1 c, d, e With a protractor and string, use angle of elevation and Demonstration distance from the tree to compute the height of the tree. 1 d, e Use bow tie visuals with 30 , 45 , and 60 angles Student work sample marked to determine the values of inverse trigonometric functions. 1 2 2 1 3 3 0 30 -1 2 2 -1 1 f Use the acronym, All Students Take Calculus, for Demonstration finding the sign of the six trigonometric functions in all four quadrants. Begin with ―A” in the first quadrant, all functions are positive. With ―S‖ in the second quadrant, only sin x and its reciprocal are positive. With ―T‖ in the third quadrant, only tan x and its reciprocal are positive. With ―C‖ in the fourth quadrant, only cos x and its reciprocal are positive. 105 Course: Trigonometry Unit Theme: Graphs Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a Review inverses of algebraic equations and discuss Self-evaluation reflections about the line y = x. Predict and sketch the inverse of the six basic trigonometric functions. Verify using the graphing calculator. 2 a Using the definitions of cos 0 and sin 0 (x and y Self-evaluation coordinate of corresponding point on the unit circle) determine the values of the six trigonometric functions for the quadrantal angles. (0, 1) ● (1, 0) ● ● (-1, 0) ● (0, -1) 2 b Using a graphing calculator or computer simulation ● Discussion; program, investigate the phase shift, amplitude, and Self-evaluation period changes of trigonometric graphs. 106 Course: Trigonometry Unit Theme: Identities Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a Given a set of equally spaced points on a circle of a Self-evaluation given radius, use the Law of Sines and Law of Cosines to find horizontal and vertical distances from point to point to the nearest thousandth. Use a computer program to verify solutions. 3 b Write each trigonometric function in terms of all the Teacher evaluation other trigonometric functions. For example: sin x = 1 cos2 x 3 b The hexagon demonstrates the following relationships: Demonstration Functions across the heavy lines are reciprocals. Functions across horizontal lines (heavy and light lines) are co-functions. Going around the hexagon, choose any three consecutive functions. The product of the outer two functions results in the middle function. Within a shaded triangle, begin at the left vertex, move right, then down. The Pythagorean identities are formed. 3 c Using several different size balls, determine the radius Student evaluation of each. Find the arc length of a cross-section of each ball with a central angle of 30, 90 , 135 . Find the area of each cross-section. 3 c, d, f Find the area of a piece of irregularly shaped land Self-evaluation given the legal land description, and compare to the area listed in the deed. 3 e Starting with a degree or radian measure, write a Student evaluation trigonometric equation with that solution (e.g., given x 45 , an equation would be tan tan x = 1 ) Justify the value for x. 107 Course: Trigonometry Unit Theme: Vectors Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a Research different notations for vectors using the Rubric Internet or the library. Compare and contrast the different notations. 4 b Using necessary directional tools, locate selected Self-evaluation items on the school campus by finding the resultant of two given vectors from a given point. 4 c Use the Pythagorean Theorem to calculate the norm Teacher observation (magnitude) of a vector. 4 d Draw examples of equal, opposite, parallel, and Short answer perpendicular vectors on an overhead transparency and investigate their relationships. 4 e Using a protractor and ruler, construct vectors given Student work sample the magnitude and direction. 108 CALCULUS ADVANCED PLACEMENT CALCULUS AB ADVANCED PLACEMENT CALCULUS BC Calculus is the study of the mathematics of change. The major focus is on differential and integral calculus. The Calculus course provides a survey of calculus without the theory and rigor necessary to receive advanced placement credit. The Advanced Placement Calculus courses are intended for those students who wish to seek college credit and/or placement from institutions of higher learning. Topics marked by an asterisk (*) are for the additional topics to be taught in Advanced Placement Calculus BC. The use of graphing calculators and other technologies are integral parts of each calculus course. These courses are designed for the student who has a thorough knowledge of college preparatory mathematics. Calculus, Advanced Placement Calculus AB and Advanced Placement Calculus BC are each one-credit courses. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. Please adjust the course content and kind and use of the calculator as outlined in the latest version of AP Course Description, published by the College Board each year. For more information contact: The College Board, Advanced Placement Program, P. O. Box 6670, Princeton, New Jersey, 08541-6670. 109 CALCULUS ADVANCED PLACEMENT CALCULUS AB ADVANCED PLACEMENT CALCULUS BC CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 1. Demonstrate basic knowledge of functions, their behavior, and characteristics. (P, D, G, N) a. Predict and explain the characteristics and behavior of functions and their graphs. b. Investigate, describe, and determine asymptotic behavior. c. Discuss and determine continuity and discontinuity of functions. d. *Analyze parametric, polar, and vector functions. 2. Evaluate limits and communicate an understanding of the limiting process. (P, D, G, N) a. State and apply properties of limits. b. Calculate limits using algebra. c. Estimate limits from graphs or tables of data. d. Verify the behavior and direction of non-determinable limits. e. Use L'Hopital's Rule to evaluate simple indeterminate forms. f. *Apply L'Hopital's Rule to determine convergence of improper integrals and series. 3. Use the definition and formal rules of differentiation to compute derivatives. (P, G, N) a. State and apply the formal definition of a derivative. b. Apply differentiation rules to sums, products, quotients, and powers of functions. c. Discuss and demonstrate the differences between average and instantaneous rates of change. d. Use the chain rule and implicit differentiation. e. Extend knowledge of derivatives to include exponential, logarithmic, trigonometric and inverse trigonometric functions. f. *Calculate derivatives of parametric, polar and vector functions. 110 CALCULUS ADVANCED PLACEMENT CALCULUS AB ADVANCED PLACEMENT CALCULUS BC CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 4. Apply derivatives to find solutions in a variety of situations. (P, D, M, G, N) a. Interpret and communicate the purposes of the derivatives. b. Interpret the derivative as a rate of change in varied applied contexts, including velocity, speed and acceleration. c. Apply the derivative to find tangent lines and normal lines to given curves at given points. d. Apply Rolle’s Theorem and the Mean Value Theorem and their geometric consequences. e. Apply differentiation techniques to curve sketching. f. Explain and predict the relationships between functions and their derivatives. g. Model rates of change to solve related rate problems. h. Solve optimization problems. i. Determine an understanding of Newton’s Method to approximate roots. j. Investigate local linear approximations. k. *Interpret differential equations using slope fields. l. *Solve differential equations by Euler’s Method. m. *Analyze planar curves given in parametric, polar and vector form including velocity and acceleration vectors. 5. Employ various integration properties and techniques to evaluate integrals. (P, D, M, G, N) a. Demonstrate the concept of the integral as an accumulator. b. Use Reimann’s Sum and the Trapezoidal Rule to approximate definite integrals. c. State and apply the First and Second Fundamental Theorem of Calculus. d. Evaluate the average value of a function on an interval. e. Apply the power rule and u-substitution to evaluate indefinite integrals. f. *Extend techniques of integration to include integration by parts and simple partial fractions. 111 CALCULUS ADVANCED PLACEMENT CALCULUS AB ADVANCED PLACEMENT CALCULUS BC CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 6. Adapt integration methods to model solutions to problems. (P, D, M, G, N) a. Investigate and apply integration to solve problems including area, volume, and cross sections. b. Employ integration to compute distance traveled by a particle along a line. c. Solve differential equations using integration and separation of variables. d. Utilize integrals to model solutions to real-world problems. e. *Solve logistic differential equations and use them in modeling. f. *Apply integration to find length of a curve. 7. *Explore the concepts affecting relationships among different kinds of series. (P, D, G, N) a. *Identify different types of series and their characteristics. b. *Apply different types of tests to create valid arguments to determine convergence or divergence of series. c. *Use Lagrange’s Method for computing errors of Taylor polynomials. d. *Formulate new series from known series to include Maclaurin and Taylor series. * Topics marked by an asterisk (*) are for the additional topics to be taught in Advanced Placement Calculus BC. 112 Course: Calculus/Advanced Placement Calculus AB/ Advanced Placement Calculus BC Unit Theme: Functions Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a, b, c Distribute examples of graphed functions. For each Short answer example: a. Use the graph to identify intervals where the function is continuous. b. Discuss and identify the values of the function where failure occurs for each of the three tests of continuity. 1 c Explore Layman’s version of continuity: A function is Teacher observation continuous if you can draw it without ever lifting your pencil. 1 d Use technology to model parametrics by revisiting an Peer evaluation old algebra problem of two trains traveling on the same track. 113 Course: Calculus/Advanced Placement Calculus AB/ Placement Advanced Calculus BC Unit Theme: Functions Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a, b, c Divide the class into groups. Each group will Group work; investigate the function: Class discussion 3 ch xx 11 f x Group assignments: 1) Have one group create a table of 10 to 20 function values for 1, 2 2) Create table values for 0, 1 . 3) Graph function using a decimal (friendly) calculator window. List five (5) observations about what happens to y values as x gets closer to 1. 4) Predict what graph will look like and list at least five characteristics. 5) Algebraically explore the function: ―Can it be factored? ― 2 c, d Compare the graphs of several rational functions to Rubric table values for behavior at points near where the denominator is undefined. 2 d Compare a list of indeterminate forms and discuss why Student work sample they are indeterminant. 2 d Use x b g lim 1 1 x x to show/explore why 1 is an Short answer indeterminant form. 114 Course: Calculus/Advanced Placement Calculus AB/ Advanced Placement Calculus BC Unit Theme: Derivatives Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a Using an overhead graphing calculator to create Presentation overheads of different functions, create two bugs (from hole punched dots) to travel along the overhead functions. Get students to predict what will happen as both bugs walk along the curve toward each other and the two bugs are connected by a string—one bug stays still and the other approaches the first bug. 3 b Quotient Rule Hi = Numerator Demonstration Lo = Denominator Lo de Hi – Hi de Lo And down below the denominator squared must go. 3 c Provide students with a table of values of time and Discussion speed. Have them calculate the average speed. What method(s) were used? Compare to instantaneous rates. 3 b, d, e After basic differentation rules have been introduced, Test provide memory tools. For example, PI (Power then do the Inside), and PTA (Power, Trig, Angle). 3 f Make a set of match cards with derivatives, graphs, Free response and different forms (parametrics, polar, and vector) and have groups match and sort. 4 a, b Given the graph of a function draw the tangent line at a Student work sample variety of points on the function. Estimate the slope and analyze in terms of rate of change. 4 c Determine the tangents to the curve Short answer 4x2 9y2 36 at the ends of each axis. Describe the relationship between the two sets of tangents. 4 d Explain the similarities and differences between Rolle’s Essay Theorem and the Mean Value Theorem. 115 Course: Calculus/Advanced Placement Calculus AB/ Advanced Placement Calculus BC Unit Theme: Derivatives Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 e, f Give students a function like f bg x5 3x4 4x3 12x2 x Group work a) Where are the zeros for f’(x)? b) Identify intervals where graph is increasing/decreasing. c) Have students compute derivative and graph the ch derivative. Where is f x above the x-axis; Below the x-axis? d) State x coordinates of max/min points for f x . ch 4 e, f Make a set of match-cards to include f x, f ' , x, f " f Group investigation ’(x), f― (x) for each group of students. (Extend: Critical number cards) Have groups match all the parts, then present one complete solution to the class. 4 g Use a table approach to organizing student work in Student work sample solving related rates. Know What to Find (lots of things) (only one here) 4 h Find examples of real-world situations that involve Project solving optimization problems. Follow-up with a class discussion. 4 h Investigate why a soda can is the shape and size it is? Project 4 i Use technology to demonstrate finding roots using Demonstration Newton’s Method. 4 j Justify how linear approximations are used to model Short answer local linearity of different functions. 4 k Create an overhead with families of curves that are Small groups solutions to a particular differential equation. Give each group a copy of an extra transparency. Have groups draw tangent lines at given points for different curves. Bring all group transparencies and place on overhead. Discuss the meaning of the slope field. 4 l Get a copy of an Euler method program or use a Discussion spreadsheet. Investigate what happens for different functions and different step sizes when using Euler’s method. 4 m Model tossing a baseball to a person sitting on a ferris Self assessment using wheel using parameter equations and/or vectors. graphing calculator 116 Course: Calculus/Advanced Placement Calculus AB/ Advanced Placement Calculus BC Unit Theme: Integrals Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a Provide a data set where an over-estimate and an Constructed response under-estimate of an integral could be computed. Relate to an example of velocity data and estimate distance traveled. 5 b Use technology to investigate numerical methods such Teacher observation as the Trapezoidal Rule. 5 c Use the Fundamental Theorem of Calculus to explain Constructed response the difference between definite and indefinite integrals. 5 d Create a graph that would model the average value Short answer formula. 5 e, f Divide the class into two teams. Use a football field to Constructed response score points. Team 1 has four chances to move +0 yards (correct answer = 10 yards). The team quarterback will designate a player to answer a question. All class members will work on the problem. If the designated player misses the question, the side of the room that has the most correct answers either wins the play or blocks the play. 117 Course: Calculus/Advanced Placement Calculus AB/ Advanced Placement Calculus BC Unit Theme: Integrals Suggested Suggested Comp. Obj. Teaching Strategies Assessment 6 a Compute the area between a curve and the x-axis Group investigation using geometric shapes and rectangular areas (from grid). 6 a Use playdough to create solids formed by revolving a Teacher observation region about an axis. Slice into discs to demonstrate where the disc formula for volumes is derived. 6 b Use a graph to explain how an integral would model Class discussion distance traveled. 6 c Explain the process for solving differential equations Essay by separation of variables. 6 d, e Investigate exponential decay and/or logistic functions Test as they apply to integrals. 6 f Derive the formula for arc length. Demonstration 118 Course: Calculus/Advanced Placement Calculus AB/ Advanced Placement Calculus BC Unit Theme: Series Suggested Suggested Comp. Obj. Teaching Strategies Assessment 7 a, b Create a set of matching cards with all the tests for Student work sample convergence, sample series, and blank index cards. Students will match tests with examples, then use index cards to write an appropriate argument proving convergence or divergence. 7 c Discuss how to find a value for c on a specific interval Class discussion as it relates to errors of Taylor polynomials. 7 d Obtain either a computer or calculator program that will Constructed response compute the Taylor polynomial. Explain the computer/calculator results for the examples given. 119 DISCRETE MATHEMATICS Discrete Mathematics is the study of mathematics as it applies to systems that have a finite number of elements. A few of the topics that will be explored are set and binary systems, logic, graph theory, simple games, and the geometry of fractals. Technology will be used when appropriate throughout the course. Discrete Mathematics is usually considered important for potential application to computer science, but is not limited to that area. This course is designed for students who have successfully completed Algebra II. It may be an alternative to pre-calculus, trigonometry, or calculus. This is a one-half credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 120 DISCRETE MATHEMATICS CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 1. Perform operations on sets and investigate properties of fields. (P, G, N) a. Define and recognize binary operations. b. Perform operations on a set. c. Identify properties of fields. d. Identify simple operations using set theory to include Venn diagrams. 2. Apply the rules of logic to discuss the validity of arguments. (P, G, N) a. Investigate and apply rules of logic to include negations, connectives, conditionals, inverses, and patterns of inference. b. Construct truth tables. c. Apply the principles of logic to determine the validity of arguments. d. Use basic Boolean Algebra to create elementary logic circuits. 3. Explore and investigate graph theory and its applications. (P, M, G) a. Define and identify the basic terminology of graph theory. b. Recognize properties of graphs having Eulerian and Hamiltonian paths and circuits. c. Construct and use tree diagrams to solve graph theory problems. d. Apply graph theory techniques to determine shortest paths and scheduling situations. 4. Investigate and explain strategies for solving simple games. (P, D, N) a. Determine the characteristics that result in a fair game. b. Identify winning strategies for basic games. 121 DISCRETE MATHEMATICS CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 5. Apply and compare approaches to problem-solving situations. (P, D, M, G, N) a. Perform basic operations with matrices. b. Explain and represent a relation (on a finite set) by a digraph or by a matrix. c. Apply matrices to solving problems. d. Use difference equations to model real-life problems. e. Investigate and apply fair division concepts to problem-solving situations. f. Use and compare recursive approaches to problem-solving and identifying numerical patterns. g. Apply algorithms to solving problems. h. Analyze networks and their applications including roads and airline routes. 6. Investigate the geometry of fractals. (P, D, G, N) a. Identify fundamental characteristics of fractals. b. Explain the outcomes of the Chaos game. c. Determine patterns in area and perimeter of simple fractal patterns. d. Explore and determine the concepts of fractal dimension. 122 Course: Discrete Mathematics Unit Theme: Operations of Sets Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a, b, c Create an operation rule: a # b 3a 2b . Investigate Teacher observation the characteristics, properties, and which set of numbers work with the rule. 1 d Locate examples of Lewis Carroll puzzles that use Project Venn diagram solutions (Web investigation). Discuss the characteristics of the examples. 123 Course: Discrete Mathematics Unit Theme: Rules of Logic Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a, b, c, d Gather materials to build a simple circuit (battery, Small groups switch, light bulb, alligator clips). Create situations like ―The seat belt must be secure before the car will start,‖ and model with the circuit and logic. 124 Course: Discrete Mathematics Unit Theme: Graph Theory Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a, b, d Use the game ―Instant Insanity‖ to show how graph Class discussion; theory makes solutions easy. Teacher observation 3 c Obtain a map of a five-block downtown area or within a Project five-block radius of the school. Design a graph of all possible paths from a designated starting point to a specific location (school). Display options using tree diagrams. 125 Course: Discrete Mathematics Unit Theme: Strategies and Simple Games Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a, b Use ―Master Mind‖ to teach terminology and basic Rubric game strategies. 126 Course: Discrete Mathematics Unit Theme: Problem Solving Suggested Suggested Comp. Obj. Teaching Strategies Assessment 5 a Record scores for foul shots and goals for five Student work sample basketball players during one game. Model each player’s total scores by matrix multiplication. 5 b, c Research different mathematical methods that have Report been used throughout history to code message, specifically role of matrices. 5 d Consider a circular shaped pizza. If size or shape do Constructed response not matter, what is the pattern to the number of pieces produced by cutting once, twice, three times, etc.? 5 e Divide the class into groups of three to four students. Discussion Give each group a circle with 10" diameter that represents a cake. Have a group develop a method of cutting the cake for class members that would be ―fair.‖ 5 f Take the square root of a positive number on the Short answer calculator, then take square root of answer . . . ENTER, ENTER . . . What happens? Why? 5 g Divide class into groups. Devise a plan for dividing a Constructed response cake among 3, 4, or more people. Solutions should be in the form of algorithms. 5 h Find a copy of course offerings for the freshman class. Project Design a network that would model possible schedules. 127 Course: Discrete Mathematics Unit Theme: Fractals Suggested Suggested Comp. Obj. Teaching Strategies Assessment 6 a, c, d Enlarge a Mississippi map of the coastline. Apply Student work sample techniques for evaluating fractal dimensions to the Mississippi map. 6 b Form groups. Play the Chaos game with equal Class discussion probabilities of one-third. Change probabilities and discuss similarities and differences of the outcomes. 128 PROBABILITY AND STATISTICS The Probability and Statistics course is intended for those students who would like to explore more closely the topics of probability and statistics. Probability provides concepts and methods for dealing with uncertainty and for interpreting predictions based on uncertainty. Statistics deepens and builds understanding of the methods of data analysis. Use of appropriate tools of technology should be an integral part of this course. This course is designed for students who have successfully completed Algebra II. This is a one-half credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 129 PROBABILITY AND STATISTICS CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 1. Collect, read, interpret, and analyze data as it relates to the real world. (P, D, M, G, N) a. Draw inferences from charts, tables, and graphs that summarize data. b. Find mean, median, mode, and percentile information from a given set of data. c. Use curve-fitting to predict from collected data. d. Explain and defend regression models using correlation coefficients and residuals. e. Use an understanding of algebraic concepts to determine mathematical models of best fit. 2. Collect and decide on the most appropriate form of displaying data and be able to create tables and different kinds of graphs to represent data. (D, M, G) a. Collect and organize data using frequency distributions, stem-and-leaf plots, and histograms. b. Choose the graph type, such as bar, circle, pictograph, line, or x-y, that best represents a given set of data. c. Create graphs with scales which fairly display the data. 3. Demonstrate how patterns can be used to explain probability. (P, D, M, G) a. Represent probability as a rational number. b. Explain the relationship between theoretical and experimental probability. c. Apply the counting principles, including permutations and combinations. d. Construct and interpret sample spaces, events, and tree diagrams. e. Identify types of events, including mutually exclusive, independent, and complementary. f. Calculate geometric probability using two-dimensional models, and explain the processes used. g. Create simulations and experiments that correlate to theoretical probability. h. Use Markov Chains to calculate probability by constructing matrix models. i. Apply the concept of a random variable to generate and interpret probability distributions. 130 PROBABILITY AND STATISTICS CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 4. Investigate algebraic concepts as they apply to one and two variable data. (P, D, M, G, N) a. Describe the sampling process and effects of sampling on outcomes of statistical processes. b. Calculate mean, median, mode, standard deviation, z-scores, t-test, t-scores, quartiles, and ranges, and explain their applications. c. Apply statistics in decision-making and hypothesis testing. d. Design, execute, make conclusions, and communicate the results of a statistical experiment. 131 Course: Probability and Statistics Unit Theme: Data Analysis Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a, b, c, d, e Gather information on closing prices of selected stocks Rubric; for a one-year period. In small groups and using Teacher observation; different companies: Class discussion; Examine differences in percentile growth from Report month to month. Interpret and analyze data using the necessary formulas. Communicate results in written and oral form to the class. After discussion, make conclusions about which stock would be the best investment based upon one year’s growth. 1 b Explore to find the possible differences between the Short answer largest and smallest of five integers whose mean is 5, median is 5, and whose mode is 8. 1 a, c, d, e Time 30 periods of a pendulum swing for different Project string lengths. Analyze results. Predict how tall a pendulum is in a science museum. 132 Course: Probability and Statistics Unit Theme: Representing Data Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a, b, c Analyze monthly income/expenses using current Student graphs; market values, which are independently and Rubric realistically determined. Use the following categories of expenses: Taxes: federal income tax, state income tax, FICA Housing: mortgage or rent, insurance, taxes Groceries Utilities: water, electric, gas, phone, sanitation fee, cable Automobile: payment, insurance, tag, gas Entertainment Savings Charitable contributions Insurance: medical, life Clothing Collect and organize data, then choose the graph type that represents the data and construct this graph. Analyze results to see if future adjustment should be made in expense patterns. 2 a Gather nutritional data about favorite cereals. Decide Presentation on best means to organize information; frequency, stem-leaf plots, and/or histograms. 2 b, c Provide each group with a different data set. Each Peer evaluation group decides on best type of graph to display data. Groups share graphs and justification to the class. 133 Course: Probability and Statistics Unit Theme: Probability Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a, b Discuss the probability of tossing a coin. Conduct Teacher observation experiments varying the number of tosses. Compare and contrast theoretical and experimental probability. 3 a, b, g Reasearch the Buffon Needle Problem and perform Small groups; the classic experiment by dropping pipe cleanerss on a Class discussion tiled floor. Use data to compare with actual formulas involving n. 3 c Investigate how a state, like Mississippi, determines Report the sequence patterns of numbers and letters for license plates or how the phone company decides to issue new area codes. 3 d Use the school lunch menu and construct a tree Portfolio diagram to determine the number of possible meals. 3 e Discuss whether the following example is a mutually Teacher observation; exclusive event. Discussion; Given a standard deck of 52 cards, find the probability Student response of drawing a card that is a red card or a face card. Validate by randomly pulling the red card and the face cards and count the total number. Then change the situation to drawing two cards from the deck that are red cards or face cards and illustrate differences with cards and explain use of combination formula for this example. 3 e Discuss the differences between independent and Rubric dependent events. Present the class with a bag of marbles consisting of 5 red, 6 blue, and 4 green marbles. Ask students to determine the probability of drawing out a blue, a red, and another blue marble in that order without replacement. Then, perform the experiment again with replacement. Divide the class into groups and discuss whether ―with replacement‖ or ―without replacement‖ has the greatest probability of success. The large group will then discuss results of the experiment and will explain their conclusions. 3 f Design a target with five sections so that the Project probability of hitting only one particular section is 25%. 134 Course: Probability and Statistics Unit Theme: Probability Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 g Using dice and decks of cards, work in small groups to Teacher observation create a theoretical/experimental probability simulation for one of the other groups to carry out. 3 h Suppose a presidential election has just taken place. Teacher-made test A large sample of voters were interviewed on whether item or not they switched party affiliations. The following contains the probability data resulting from this survey. Democrat Republican L0.8 M.6 0.2 O P N0 0.4 Q Given that a voter is a Democrat at this election, what is the probability that party affiliation will be switched in the election after the next two transitions? According to statistics, at the time the survey was taken, 60% of the voters were Democrat and 40% were Republicans. Based on the survey results, what percent of the population will be Democrats in the election after two transitions? 3 i Repeatedly toss four coins and record the number of Class activity and heads obtained on each trial. Find the mean number discussion of heads in 5, 10, 25, 50, and 100 trials of the experiment. For each number of trials, find the probability distribution for the number of heads obtained. (The mean of the random variable is 2.) The mean number of heads observed when four coins are tossed many times approaches the population mean of the probability distribution. 135 Course: Probability and Statistics Unit Theme: Inferential Statistics Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a Design a method for obtaining a simple random Teacher Critique sample of students. Sample to determine the typical number of hours studied each week-night by students in grades 11 and 12 at your school. 4 a Design a method for obtaining a stratified Teacher Critique sample to determine who among three hypothetical candidates will be elected Homecoming Queen at your school. 4 b The teacher writes down all scores on the last Class discussion major test. Each student will standardize his/her score. Students will discuss measures of center for the test scores and also measures of spread. 4 c, d Design an experiment to compare the means of Teacher grades two samples. Write hypotheses, collect and project analyze data, draw appropriate conclusions, and communicate the results 136 ADVANCED PLACEMENT STATISTICS The Advanced Placement Statistics course introduces students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Four major areas of concentration include data explorations, design of experiments, production of models using probability and simulation and statistical inference. The use of technology will be an integral part of the course. This course is designed for students who have successfully completed Algebra II. This is a one-credit course. The competencies are printed in bold face type and are required to be taught. The competencies combine the content strands: patterns/algebraic thinking, data analysis/prediction, measurement, geometric concepts, and number sense, and the process strands: problem solving/reasoning, estimating, incorporating technology, communicating, and making connections/applications. The competencies may relate to one, many, or all of the mathematics curriculum strands and may be combined and taught with other competencies throughout the school year. Competencies are not listed in order of importance; rather the sequence of competencies relates to the broader K-12 framework. Competencies provide a general guideline of on-going instruction, not isolated units, activities, or skills. The suggested teaching objectives are optional. Objectives indicate concepts that enable fulfillment of competencies, describe competencies in further detail, or show the progression of concepts throughout the grades. School districts may adopt the objectives, modify them, and are encouraged to write their own objectives to meet the needs of students in their school district. 137 ADVANCED PLACEMENT STATISTICS CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 1. Use graphical and numerical techniques to study patterns and to explore, describe, and interpret data. (P, D, M, G, N) a. Interpret graphical displays of distributions of univariate data (dot plots, stem plots, histograms, box plots). b. Summarize distribution of univariate data and correctly find and use measures of center (mean, median, mode); measures of spread (range, interquartile range, standard deviation); and measures of position (quartiles, percentiles, standardized scores). c. Explore bivariate data by analyzing patterns in scatterplots and residual plots, performing logarithmic and power transformations to achieve linearity, finding least squares regression lines, and finding correlation coefficients. d. Explore categorical data, construct, and interpret frequency tables. 2. Plan a study by clarifying a question and deciding upon a method of data collection and analysis. (P, D, N) a. Know the characteristics of a well-designed and well-conducted study and be able to distinguish between observational studies, surveys, and experiments. b. Design a method for obtaining a simple random sample for a population of interest and for obtaining a stratified sample when appropriate. c. Identify sources of bias and discuss the concept of sampling error in studies. d. Design experiments, to include the concepts of confounding variables, control groups, placebo effects, blinding, randomization, replication, blocking, and generalizability of results. 138 ADVANCED PLACEMENT STATISTICS CONTENT STRANDS: Patterns/Algebraic Thinking (P) Geometric Concepts (G) Data Analysis/Prediction (D) Number Sense (N) Measurement (M) COMPETENCIES and Suggested Teaching Objective(s): 3. Use probability to predict what the distribution of data should look like under a given method. (P, D, M, G, N) a. Use concepts of independent and mutually exclusive events, and apply the addition, multiplication, and conditional probability rules to find the probability of events. b. Produce models using probability and simulation, and explain the ―law of large numbers.‖ c. Find the mean and standard deviation of a random variable and the mean and standard deviation for the sums and differences of independent random variables. d. Know properties of the normal distribution, use normal distribution tables, and make inferences from these tables. e. Simulate sampling distributions (distributions of a sample proportion, distribution of a sample mean, distribution of a difference between two independent sampling proportions, distribution of a difference between two independent sample means). f. Discuss and illustrate the Central Limit Theorem. 4. Use statistical inference to analyze data, draw appropriate conclusions, and effectively communicate those conclusions. (P, D, G, N) a. Find and interpret large sample confidence intervals for a proportion, a mean, a difference between two proportions, and a difference between two means. b. Appropriately use the following tests of significance: large sample tests for a proportion, a mean, a difference between two proportions, and a difference between two means (unpaired and paired); Chi-square test for goodness of fit, homogeneity of proportions, and independence; single sample and two sample t-procedures; and inference for slope of least squares line. c. Write null and alternate hypotheses for studies, distinguish between one and two- sided tests, calculate appropriate test statistics, find p-values, arrive at appropriate conclusions, and communicate those conclusions effectively. 139 Grade Level: Advanced Placement Statistics Unit Theme: Patterns and Data Interpretation Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 a Open a magazine arbitrarily and record the lengths of Check students’ all words in the first complete paragraph on the page. graphs and written Create a dot plot of the lengths (number of letters) of description words that were recorded. Write a few sentences describing this distribution of word lengths. (Students may choose various magazines and compare results.) 1 a, b Reconsider the data collected with word lengths. Check students’ graph Calculate the five number summary of this distribution and related comments and draw a boxplot. Comment on what the boxplot reveals about the distribution of word lengths. Are there outliers? 1 a Consult the Farmer’s Almanac or U. S. Census Report Check students’ to find a data set of interest. The Internet is also a graphs and analysis of source for interesting data sets. Choose a one- graphs variable data set such as percentage of residents 65 years of age or older in each of the fifty states. Draw a histogram for the data. Make a stem plot for this data. Describe the main features of the distribution. Is it symmetric, right skewed, or left skewed? Single or double peaked? Are there gaps or outliers? 1 b Write down all scores on the last major test. Each Class discussion; student will standardize his/her score. Discuss Teacher observation measures of center for the test scores and also measures of spread. 1 c Collect data for number of students’ siblings, and Check scatter plots number of students’ mothers’ siblings. Draw a scatter plot of students’ siblings versus mothers’ siblings. Analyze patterns found in the scatter plot. 1 c Obtain from a favorite fast food restaurant nutritional Check scatter plot, information about their sandwiches. List all regression line, sandwiches, serving size (in ounces) of each residual plot, and sandwich, and calories for each sandwich. Draw a analysis scatter plot and reveal an association between a sandwich’s serving size and its calories? Determine the least squares regression line for predicting calories from serving size. Find the correlation coefficient. Sketch a plot of residuals. How well does the least- squares regression line fit the data? 140 Grade Level: Advanced Placement Statistics Unit Theme: Patterns and Data Interpretation Suggested Suggested Comp. Obj. Teaching Strategies Assessment 1 c A courtier was offered a reward by an ancient king of Self-check Persia. He asked for a grain of rice on the first square of a chessboard, two grains on the second square, then 4, 8, 16, etc. Plot the number of grains on each square against the number of the square for squares 1 to 10 and connect the points with a smooth curve (exponential curve). Take the logarithm of each of the numbers of grains. Plot these logarithms against the numbers of squares from 1 to 10. (straight line) Find the least squares regression line for the logarithms of the number of grains versus the number of squares. Use this equation to predict the number of grains for the 64th square. 1 d Classify each member of Congress according to Whole class his/her gender and political party. Construct a assessment; frequency table with row headings of Republican, Peer assessment Democrat or other. Use column heading of male or female. Interpret the frequency table. 141 Grade Level: Advanced Placement Statistics Unit Theme: Sampling and Experimental Design Suggested Suggested Comp. Obj. Teaching Strategies Assessment 2 a, c Consult a scientific journal. Find an example of an Class discussion; observational study, a survey, and an experiment. Peer assessment Critique each study to determine if it is a well-designed and well-conducted study. Identify any sources of bias. 2 b Design a method of obtaining a simple random sample Teacher observation to determine the typical number of hours studied each and critique week night by students in grades 11 and 12 at your school. 2 b Design a method for obtaining a stratified sample to Teacher critique determine who among three hypothetical candidates will be elected Homecoming Queen at your school. 2 d Divide class into groups of three. Each group will Teacher and peer design an experiment, keeping in mind the concepts of critique of confounding variables, control groups, placebo effects, experimental design blinding, randomization, and replication. 142 Grade Level: Advanced Placement Statistics Unit Theme: Probability and Data Distributions Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 a Using M&Ms, obtain probabilities for various colors. ● Class activity and Apply the addition principle to compute the probability discussion of choosing a red or blue M&M, when selecting one at random. 3 a Use the multiplication and conditional probability rules ● Class activity and to find the probability of selecting at random two male discussion members of the class. (Assuming all names of class members were put in a hat and two names were drawn without replacement.) Find the conditional probability of selecting a male member of the class, given the student chosen has blonde hair. 3 b, c Repeatedly toss four coins and record the number of ● Class activity and heads obtained on each trial. Find the mean number discussion of heads in 5, 10, 25, 50, and 100 trials of the experiment. (The mean number of heads x observed when four coins are tossed many times approaches the population mean of the probability distribution.) The mean of the random variable = 2.) An illustration of the ―Law of Large Numbers‖ follows. x will approach 2 more closely as the number of trials grow. 3 d Each student should calculate the ratio of his height ● Student and whole and his arm span (e.g., height divided by arm span). class activity; Produce a dotplot of the distribution of these ratios (for Teacher critique of all students in class). Does the distribution appear to work be roughly normal? Calculate the mean and standard deviation of these ratios. Suppose that these ratios in the population of all college students do in fact follow a normal distribution with mean and standard deviation equal to those found in your classroom sample. Under this assumption, calculate the proportion of all students who have a ratio greater than one (height greater than arm span). 3 e Consider the population of the Reese’s Pieces candies ● Individual and whole made by Hershey. Suppose you want to learn about class activity the distribution of colors of these candies but you can only afford to take a sample of 25 candies. Record the number and proportion of each color in your sample. Each student should calculate the proportion of orange candies obtained by the students in the class. If every student estimated the population proportion of orange candies by the proportion of orange candies in his sample, would everyone arrive at the same conclusion? Observing the sample results from the entire class, estimate the population proportion of orange candies. Observe the variation of the sample proportions from sample to sample—the sampling distribution of the sample proportion. 143 Grade Level: Advanced Placement Statistics Unit Theme: Probability and Data Distributions Suggested Suggested Comp. Obj. Teaching Strategies Assessment 3 f Suppose a population consists of five employees for a ● Teacher critique of firm. The number of years of employment are 5, 3, 6, answers 2, 4. Compute the mean length of employment for the c h population 4 . Select all possible samples of size two from the population. Compute the mean of each sample. Does the mean of the sample means equal the population mean? Give the sampling distribution of the means. Plot the probability distribution of the sample means and the population. Is the population normally or non-normally distributed? Does the sampling distribution tend to approximate a normal distribution? (Central Limit Theorem) 144 Grade Level: Advanced Placement Statistics Unit Theme: Statistical Inference Suggested Suggested Comp. Obj. Teaching Strategies Assessment 4 a Have students think of a real situation in which they Teacher critique would be interested in producing a confidence interval to estimate a population proportion. Have them describe how they would compute a 95% confidence interval. 4 a, b, c Select one page from the white pages of a telephone Teacher grades book. Disregard all listing of businesses, which project provide only initials, and listing with first names that are not gender-specific (like Pat or Chris). For the listings, which can be identified as male or female, count how many are male and how many are female. What is the sample proportion of females in the sample? Use the sample data to form a 95% confidence interval for the actual proportion of all humans who are female. Does the confidence interval provide a reasonable estimate of the actual proportion of all humans who are female? (No) Explain. Using your sample data, perform a test of significance to address whether the sample data support the theory that less than half of all of the telephone books’ individual listings carry female names. Write null and alternate hypotheses. Calculate appropriate test statistics, find p-value, and write a paragraph describing your findings and explain how conclusions follow from the test results. 145 146

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