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Real Numbers Properties Worksheet document sample
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 C:\Docstoc\Working\pdf\cfb7be8d-0403-4975-a76c-433971d8c135.doc 1 08/01/2011 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 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 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 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 C:\Docstoc\Working\pdf\cfb7be8d-0403-4975-a76c-433971d8c135.doc 2 08/01/2011 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. C:\Docstoc\Working\pdf\cfb7be8d-0403-4975-a76c-433971d8c135.doc 3 08/01/2011 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 C:\Docstoc\Working\pdf\cfb7be8d-0403-4975-a76c-433971d8c135.doc 4 08/01/2011 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 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 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. C:\Docstoc\Working\pdf\cfb7be8d-0403-4975-a76c-433971d8c135.doc 5 08/01/2011 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, 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 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 C:\Docstoc\Working\pdf\cfb7be8d-0403-4975-a76c-433971d8c135.doc 6 08/01/2011 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 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. C:\Docstoc\Working\pdf\cfb7be8d-0403-4975-a76c-433971d8c135.doc 7 08/01/2011 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 C:\Docstoc\Working\pdf\cfb7be8d-0403-4975-a76c-433971d8c135.doc 8 08/01/2011 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. C:\Docstoc\Working\pdf\cfb7be8d-0403-4975-a76c-433971d8c135.doc 9 08/01/2011