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# Real Numbers Properties Worksheet by sqo14436

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Real Numbers Properties Worksheet document sample

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```									                                   Math-Related Credit Crosswalk
for
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
REASONING ABOUT NUMBERS, SYSTEMS AND QUANTITATIVE LITERACY

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
sign.
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
situations.
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
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
integer division as the inverse of integer
multiplication.
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
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
numbers.
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
denominators).
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
averages.
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
tables.
Compute with Rational Numbers

N.FL.07.07          Solve problems involving operations with        All worksheets
integers.
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,
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
correctly.
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

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
converse).
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
satisfied.

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Convert within Measurement Systems

M.UN.06.01          Convert between basic units of                    Torque wrenches
measurement within a single measurement
system.
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
trigonometric)
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,
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
graph.
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
tables.

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
equations.
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
inverses commutativity, associativity, and
the distributive property of multiplication
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.

G1.4.1              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.

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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