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Real Numbers Properties Worksheet by sqo14436


Real Numbers Properties Worksheet document sample

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									                                   Math-Related Credit Crosswalk
                                  Career Technical Education Classes
                                         in Macomb County

                                         Program Information
                                 District:   Chippewa Valley Schools
                         Program Name: Small Engines
                      CIP Code Number:       47.0606
                         Career Pathway: EMIT
                        Instructor Name:     Rex Phelps
                                    Date: 12/03/08
Strand              STANDARDS                                                   CTE APPLICATION and PRACTICE

L1.1                                             Number Systems and Number Sense

L1.1.1              Know the different properties that hold in          Two Stroke engine measurements.
                    different number systems and recognize              Shop math worksheet
                    that the applicable properties change in the        Grade sheets
                    transition from the positive integers to all
                    integers, to the rational numbers, and to the
                    real numbers.
L1.1.2              Explain why the multiplicative inverse of a         Two Stroke piston/cylinder measurements
                    number has the same sign as the number,
                    while the additive inverse has the opposite
L1.1.3              Explain how the properties of associativity,        Two Stroke worksheet.
                    commutativity, and distributivity, as well as       Oversize service limits
                    identity and inverse elements, are used in          Shop math worksheet
                    arithmetic and algebraic calculations.
L1.1.4              Describe the reasons for the different              Shop math worksheet
                    effects of multiplication by, or
                    exponentiation of, a positive number by a
                    number less than 0, a number between 0
                    and 1, and a number greater than 1.
L1.1.6              Explain the importance of the irrational            Cylinder bore area
                    numbers √2 and √3 in basic right triangle           Engine displacement calculations
                    trigonometry, the importance of ╥ because
                    of its role in circle relationships, and the role
                    of e in applications such as continuously
                    compounded interest.
L1.2                                              Representations and Relationships

L1.2.1              Use mathematical symbols (e.g., interval            Cylinder bore area calculations

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                    notation, set notation, summation notation)     Shop math worksheet
                    to represent quantitative relationships and
L1.2.2              Interpret representations that reflect          Service manual reading
                    absolute value relationships (e.g.,│x-a│< b,    Service limit ranges
                    or a± b) in such contexts as error tolerance.
L1.2.4              Organize and summarize a data set in a          Horsepower vs. rpm charts
                    table, plot, chart, or spreadsheet; find        Shop math worksheet
                    patterns in a display of data; understand       Tool crib organization
                    and critique data displays in the media.
L1.3                                            Counting and Probabilistic Reasoning

L1.3.1              Describe, explain, and apply various            Piston/cylinder wear predictions
                    counting techniques (e.g., finding the
                    number of different 4-letter passwords;
                    permutations; and combinations); relate
                    combinations to Pascal’s triangle; know
                    when to use each technique.
L1.3.2              Define and interpret commonly used              Career search / motivational survey
                    expressions of probability (e.g., chances of
                    an event, likelihood, odds).
L1.3.3              Recognize and explain common probability        Career search / job trends
                    misconceptions such as “hot streaks” and
                    “being due.”
                                                       Multiply and Divide Fractions

N.MR.06.01          Understand division of fractions as the         Grade sheets
                    inverse of multiplication.
N.FL.06.02          Given an applied situation involving dividing   Shop math worksheet
                    fractions, write a mathematical statement to
                    represent the situation.
N.MR.06.03          Solve for the unknown.                          Shop math worksheet
                                                                    Two stroke worksheet
N.FL.06.04          Multiply and divide any two fractions,          Shop math worksheet
                    including mixed numbers, fluently.
                                     Represent Rational Numbers as Fractions or Decimals

N.ME.06.05          Order rational numbers and place them on        Shop math worksheet
                    the number line.                                Measuring instruments
N.ME.06.06          Represent rational numbers as fractions or      Shop math worksheet
                    terminating decimals when possible and          Measuring instruments
                    translate between these representations.
N.ME.06.07          Understand that a fraction or a negative        Shop math worksheet
                    fraction is a quotient of two integers.         Measuring instruments
                    Add and Subtract Integers and Rational Numbers

N.ME.06.08          Understand integer subtraction as the           Two stroke worksheet
                    inverse of integer addition. Understand
                    integer division as the inverse of integer
N.FL.06.09          Add and multiply integers between -10 and       Two stroke power curve charts
                    10; subtract and divide integers using the
                    related facts. Use the number line and chip
                    models for addition and subtraction.
N.FL.06.10          Add, subtract, multiply and divide positive     Grade sheets
                    rational numbers fluently.                      Two stroke worksheet.
                                                           Find Equivalent Ratios
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N.ME.06.11          Find equivalent ratios by scaling up or         Electrical anolog multi-meter
                    scaling down.
                                  Solve Decimal, Percentage and Rational Number Problems

N.FL.06.12          Calculate part of a number given the            Grade sheets
                    percentage and the number.                      Gas/oil mixtures or two-stroke engines
N.MR.06.13          Solve contextual problems involving             Customer work orders parts/labor
                    percentages such as sales taxes and tips.
N.FL.06.14          For applied situations, estimate the answers    Customer work order estimates
                    to calculations involving operations with       Shop math worksheet
                    rational numbers
N.FL.06.15          Solve applied problems that use the four        Four stoke valve adjustment using shims
                    operations with appropriate decimal
                         Understand Rational Numbers and Their Location on the Number Line

N.ME.06.17          Locate negative rational numbers (including     Micrometer reading using various adaptors and
                    integers) on the number line. Know that         standards
                    numbers and their negatives add to 0 and
                    are on opposite sides and at equal distance
                    from 0 on a number line.
N.ME.06.18          Understand that rational numbers are            Shop math worksheet
                    quotients of integers (non zero
N.ME.06.19          Understand that 0 is an integer that is         Micrometer reading using various adaptors and
                    neither negative nor positive.                  standards.

N.ME.06.20          Know that the absolute value of a number is     Two stroke worksheet
                    the value of the number ignoring the sign; or   Cylinder taper and out of round
                    is the distance of the number from 0.
                                                      Understand Derived Quantities

N.MR.07.02          Solve problems involving derived quantities     Syllabus /grading procedure
                    such as density, velocity and weighted
                       Understand and Solve Problems Involving Rates, Ratios, and Proportions

N.FL.07.03          Calculate rates of change including speed.      Horsepower vs. rpm
                                                                    Power curves
                                                                    Engine design… bore vs. stroke, camshaft design
N.MR.07.04          Convert ratio quantities between different      Torque wrench inch lbs./foot pounds
                    systems of units, such as feet per second to    Metric/inch
                    miles per hour.
N.FL.07.05          Solve proportion problems using such            Shop math worksheet
                    methods as unit rate, scaling, finding          Service manual worksheets
                    equivalent fractions, and solving the
                    proportion equation a/b = c/d; know how to
                    see patterns about proportional situations in
                                                    Compute with Rational Numbers

N.FL.07.07          Solve problems involving operations with        All worksheets
N.FL.07.08          Add, subtract, multiply and divide positive     Most worksheets
                    and negative rational numbers fluently.

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N.FL.07.09          Estimate results of computations with              Customer work orders
                    rational numbers.

                                                  Understand Real Number Concepts
N.ME.08.02          Understand meanings for zero and negative          Measuring tools and instruments
                    integer exponents.
N.ME.08.03          Understand that in decimal form, rational          Grade sheets
                    numbers either terminate or eventually             Shop math worksheet
                    repeat, and that calculators truncate or
                    round repeating decimals; locate rational
                    numbers on the number line; know fraction
                    forms of common repeating decimals.
N.ME.08.04          Understand that irrational numbers are             Grade sheets
                    those that cannot be expressed as the              Four stroke cylinder bore and engine displacement
                    quotient of two integers, and cannot be            calculations
                    represented by terminating or repeating
                    decimals; approximate the position of
                    familiar irrational numbers.
                                                                   Solve Problems

N.MR.08.07          Understand percent increase and percent            Customer purchase order worksheets
                    decrease in both sum and product form.
N.MR.08.08          Solve problems involving percent increases         Customer purchase order worksheets
                    and decreases.
N.FL.08.09          Solve problems involving compounded
                    interest or multiple discounts.
N.MR.08.10          Calculate weighted averages such as                Syllabus and grade sheets
                    course grades, consumer price indices and
                    sports ratings.
N.FL.08.11          Solve problems involving ratio units, such         Four stroke basic physics
                    as miles per hour, dollars per pound or            Two stroke fuel/oil mixtures
                    persons per square mile.
L2                  STANDARDS                                                CTE APPLICATION and PRACTICE

                                        CALCULATION, ALGORITHMS, AND ESTIMATION

L2.1                                      Calculation Using Real and Complex Numbers
L2.1.1              Explain the meaning and uses of weighted           Syllabus and grade work sheets
                    averages (e.g., GNP, consumer price index,
                    grade point average).
L2.1.2              Calculate fluently with numerical                  Four stroke engine displacement calculations
                    expressions involving exponents. Use the
                    rules of exponents, and evaluate numerical
                    expressions involving rational and negative
                    exponents, and transition easily between
                    roots and exponents.
L2.1.6              Recognize when exact answers aren’t                Compression and leakdown tests
                    always possible or practical. Use
                    appropriate algorithms to approximate
                    solutions to equations (e.g., to approximate
                    square roots).
L2.2                                                      Sequences and Iteration

L2.2.3              Use iterative processes in such examples           Compression and leakdown estimates
                    as computing compound interest or
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                    applying approximation procedures.
L3                  STANDARDS                                                CTE APPLICATION and PRACTICE

                                               MEASUREMENT AND PRECISION
L3.1                                       Measurement Units, Calculations, and Scales

L3.1.1              Convert units of measurement within and            Shop math worksheet
                    between systems; explain how arithmetic            Service manual usage
                    operations on measurements affect units,           Two stroke worksheet
                    and carry units through calculations
L3.2                                                         Understanding Error

L3.2.1              Determine what degree of accuracy is               Four stroke valve adjustment and shim calculations
                    reasonable for measurements in a given
                    situation; express accuracy through use of
                    significant digits, error tolerance, or percent
                    of error; describe how errors in
                    measurements are magnified by
                    computation; recognize accumulated error
                    in applied situations.
L3.2.2              Describe and explain round-off error,              Four stroke valve adjustment and shim calculations
                    rounding, and truncating.                          Grade sheets

L3.2.3              Know the meaning of and interpret                  Valve adjustment, flywheel / ignition coil clearances
                    statistical significance, margin of error, and
                    confidence level.
L4.1                                                      Mathematical Reasoning

L4.1.1              Distinguish between inductive and                  Troubleshooting engine problems
                    deductive reasoning, identifying and               Group work for cooperative units
                    providing examples of each.
L4.1.2              Differentiate between statistical arguments        Troubleshooting engine problems
                    (statements verified empirically using
                    examples or data) and logical arguments
                    based on the rules of logic.
L4.1.3              Define and explain the roles of axioms             Troubleshooting engine problems
                    (postulates), definitions, theorems,
                    counterexamples, and proofs in the logical
                    structure of mathematics. Identify and give
                    examples of each.
L4.2                                                   Language and Laws of Logic

L4.2.1              Know and use the terms of basic logic (e.g.,       Syllabus and safety units
                    proposition, negation, truth and falsity,
                    implication, if and only if, contrapositive, and
L4.3                                                                   Proof
L4.3.3              Explain the difference between a necessary         Syllabus/teacher lesson
                    and a sufficient condition within the
                    statement of a theorem. Determine the
                    correct conclusions based on interpreting a
                    theorem in which necessary or sufficient
                    conditions in the theorem or hypotheses are

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                                                Convert within Measurement Systems

M.UN.06.01          Convert between basic units of                    Torque wrenches
                    measurement within a single measurement
A1                  STANDARDS                                               CTE APPLICATION and PRACTICE

                                        EXPRESSIONS, EQUATIONS, AND INEQUALITIES

A1.1                      Construction, Interpretation, and Manipulation of Expressions (linear,
                          quadratic, polynomial, rational, power, exponential, logarithmic, and
A1.1.1              Give a verbal description of an expression        Shop math worksheet
                    that is presented in symbolic form, write an
                    algebraic expression from a verbal
                    description, and evaluate expressions given
                    values of the variables.
A1.1.2              Know the definitions and properties of            Engine displacement calculations
                    exponents and roots and apply them in
                    algebraic expressions.
A1.1.3              Factor algebraic expressions using, for           Shop math worksheet
                    example, greatest common factor, grouping,
                    and the special product identities (e.g.,
                    differences of squares and cubes).
A1.2                    Solutions of Equations and Inequalities (linear, exponential, logarithmic,
                                      quadratic, power, polynomial, and rational)
A1.2.1              Write and solve equations and inequalities        Shop math worksheet
                    with one or two variables to represent
                    mathematical or applied situations.
A1.2.9              Know common formulas (e.g., slope,                Shop math worksheet
                    distance between two points, quadratic            Power curves for horsepower vs. rpm
                    formula, compound interest, distance = rate
                    · time), and apply appropriately in
                    contextual situations.
A2.4                                                    Lines and Linear Functions

A2.4.2              Graph lines (including those of the form x =      Power curve charts
                    h and y = k) given appropriate information.
A2.4.3              Relate the coefficients in a linear function to   Power curve charts, usable horsepower
                    the slope and x- and y-intercepts of its
                                                          Calculate Rates - Algebra

A.PA.06.01          Solve applied problems involving rates,           Power curve charts
                    including speed.
                      Use Variables, Write Expressions and Equations, and Combine Like Terms

A.FO.06.06          Represent information given in words using        Troubleshooting unit
                    algebraic expressions and equations.

                             Represent Linear Functions Using Tables, Equations, and Graphs

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A.RP.06.08          Understand that relationships between             Power curve charts
                    quantities can be suggested by graphs and

A.PA.06.09          Solve problems involving linear functions         Power curve charts
                    whose input values are integers; write the
                    equation; graph the resulting ordered pairs
                    of integers.
A.RP.06.10          Represent simple relationships between            Power curve charts
                    quantities using verbal descriptions,
                    formulas or equations, tables and graphs.
                        Understand and Apply Directly Proportional Relationships and Relate to
                                          Linear Relationships - Algebra

A.AP.07.01          Recognize when information given in a             Power curve charts
                    table, graph or formula suggests a directly
                    proportional or linear relationship.
A.PA.07.04          For directly proportional or linear situations,   Calculating oil / fuel ratios
                    solve applied problems using graphs and           Compression ratios
                     Understand and Solve Problems about Inversely Proportional Relationships

A.PA.07.09          Recognize inversely proportional                  Oil / fuel / air ratios
                    relationships in contextual situations; know
                    that quantities are inversely proportional if
                    their product is constant.
                               Apply Basic Properties of Real Numbers in Algebraic Contexts

A.PA.07.11          Understand and use basic properties of real       Grade worksheets
                    numbers: additive and multiplicative
                    identities, additive and multiplicative
                    inverses commutativity, associativity, and
                    the distributive property of multiplication
                    over addition.
G1                     STANDARDS                                            CTE APPLICATION and PRACTICE

                                                  FIGURES AND THEIR PROPERTIES

G1.1                            Lines and Angles; Basic Euclidean and Coordinate Geometry

G1.1.5              Given a line segment in terms of its              Chain adjustment procedures on motorcycles. Both
                    endpoints in the coordinate plane,                sides of the swing arm must come out equal lengths
                    determine its length and midpoint.                and the chain must have proper slack.
G1.4                                              Quadrilaterals and Their Properties

G1.4.1              Solve multi-step problems and construct           Riding mower, go- kart wheel alignment and steering
                    proofs involving angle measure, side length,      adjustments.
                    diagonal length, perimeter, and area of
                    squares, rectangles, parallelograms, kites,
                    and trapezoids.
G1.6                                                    Circles and Their Properties

G1.6.1              Solve multi-step problems involving               Wheel lacing and different spoke patterns.
                    circumference and area of circles.

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G1.6.3              Solve problems and justify arguments about       Wheel lacing and different spoke patterns.
                    central angles, inscribed angles, and
                    triangles in circles.
G2                     STANDARDS                                          CTE APPLICATION and PRACTICE

                                             RELATIONSHIPS BETWEEN FIGURES
G2.1                                   Relationships Between Area and Volume Formulas

G2.1.3              Know and use the relationship between the        Combustion chamber shapes, compression ratios,
                    volumes of pyramids and prisms (of equal         bore versus stroke dimensions.
                    base and height) and cones and cylinders
                    (of equal base and height).
G3                  STANDARDS                                           CTE APPLICATION and PRACTICES

                                        TRANSFORMATIONS OF FIGURES IN THE PLANE

G3.1                                    Distance-preserving Transformations: Isometries

G3.1.1              Define reflection, rotation, translation, and    Making gaskets for multiple engines.
                    glide reflection and find the image of a
                    figure under a given isometry.
G3.1.2              Given two figures that are images of each        Gasket making
                    other under an isometry, find the isometry
                    and describe it completely.
G3.1.3              Find the image of a figure under the             Gasket making
                    composition of two or more isometries and
                    determine whether the resulting figure is a
                    reflection, rotation, translation, or glide
                    reflection image of the original figure.
                                       Understand and Apply Basic Properties – Geometry

G.GS.06.01          Understand and apply basic properties of         Wheel alignments, gasket making, chain tension
                    lines, angles and triangles, including:          adjustments, engine design of bore vs. stroke, etc
                    triangle inequality; relationships of vertical
                    angles, complementary angles,
                    supplementary angles; congruence of
                    corresponding & alternate interior angles
                    when parallel lines are cut by a transversal,
                    and that such congruencies imply parallel
                    lines; locate interior and exterior angles of
                    any triangle, and use the property that an
                    exterior angle of a triangle is equal to the
                    sum of the remote (opposite) interior
                    angles; know that the sum of the exterior
                    angles of a convex polygon is 360 degrees.

                                              Solve Problems about Geometric Figures

G.SR.08.03          Understand the definition of a circle; know      Four stroke engine design…..determining engine
                    and use the formulas for circumference and       displacement by bore area X stroke X number of
                    area of a circle to solve problems.              cylinders.
                        Understand Concepts of Volume and Surface Area, and Apply Formulas

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G.SR.08.06          Know the volume formulas for generalized       Calculations of engine displacement. Uses both
                    cylinders ((area of base) x height),           metric and inch systems. Also used with combustion
                    generalized cones and pyramids (1/3 (area      chamber shapes.
                    of base) x height), and spheres (4/3‫תּ‬
                    (radius) ) and apply them to solve problems.
S1                     STANDARDS                                         CTE APPLICATION and PRACTICE

                                        UNIVARIATE DATA - EXAMINING DISTRIBUTIONS

S1.1                                                Producing and Interpreting Plots

S1.1.1              Construct and interpret dot plots,             Examining power charts of engines using
                    histograms, relative frequency histograms,     horsepower on the vertical and engine speed on the
                    bar graphs, basic control charts, and box      horizontal lines. Charts are used to compare engine
                    plots with appropriate labels and scales;      to engine, four-stroke to two-stroke, and
                    determine which kinds of plots are             modifications to any particular engine.
                    appropriate for different types of data;
                    compare data sets and interpret differences
                    based on graphs and summary statistics.
S3                     STANDARDS                                          CTE APPLICATION and PRACTICE

                                             SAMPLES, SURVEYS, AND EXPERIMENTS

S3.1                                                   Data Collection and Analysis

S3.1.3              Distinguish between an observational study     Used during engine troubleshooting unit and for lab
                    and an experimental study, and identify, in    projects throughout open lab. Theories and
                    context, the conclusions that can be drawn     observations are compared and added to test results
                    from each.                                     using instrumentation.

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