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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:

                                                                x2 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.




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