Mathematics Standards Clarification For Number & Quantity .
MathematicsStandardsClarification forNumber & QuantityConceptual CategoryHigh SchoolDesigned for teachers by teachers!2019
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ContentsThe Real Number System(M). 4Quantities(Q) . 9The Complex Number System (CN) .11Vector and Matrix Quantities (VM) .20Acknowledgements .36References .373
The Real Number SystemCluster:Extend the properties of exponents to rational exponents.NVACS HSN.RN.A.1Explain how the definition of the meaning of rational exponents follows from extending the properties ofinteger exponents to those values, allowing for a notation for radicals in terms of rational exponents. Forexample, we define 51/3 to be the cube root of 5 because we want (51/3)3 5(1/3)3 to hold, so (51/3)3 must equal 5.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 1: Students will make sense of problems and persevere insolving problems by rewriting exponential and rational expressions. MP 7: Students will look for and make use of structure by applyingproperties of exponents and radicals to simplify expressions. Use problems such as ( 3)2 and ((32)) to consider exponents. Provides problems that give a context for using rational exponents. Presses students to explain the meaning of a fractional in differentcontexts. Provides problems that enable students to determine the differencebetween computing a value such as 4 and finding the solution setof a related equation, such as x2 4. Know perfect squares and cubes. Know properties of exponents. Know the definition of a rational exponent. Properties of exponents. Graphing radical equations. Simplifying radicals. Solving equations using radicals and logarithms. Achieve the Core (Achieve the Core) MVP Math Lesson 2.4, 2.6, 2.7 (MVP Math) Illustrative Math 3 Tasks (Illustrative Math) Big Ideas Math Chapter 5 Test page 289, exercises 5–8 (Big Ideas)4
The Real Number SystemCluster:Extend the properties of exponents to rational exponents.NVACS HSN.RN.A.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 1: Students will make sense of problems and persevere insolving problems by rewriting radicals as expressions with rationalexponents. MP 7: Students will look for and make use of structure by applyingproperties of exponents and radicals to simplify expressions. Use problems that allow students to use either radical or exponentialforms and requires them to explain their reasoning for their choice. Know perfect squares and cubes. Know properties of exponents. Know the definition of a rational exponent. Properties of exponents. Graphing radical equations. Simplifying radicals. Solving equations using radicals and logarithms. Achieve the Core (Achieve the Core) MVP Math Lesson 2.4, 2.6 (MVP Math) Shmoop Click Sample Assessments for Drills (Schmoop)5
The Real Number SystemCluster:Use properties of rational and irrational numbers.NVACS HSN.RN.B.3Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and anirrational number is irrational; and that the product of a nonzero rational number and an irrational number isirrational.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 3: Students will construct viable arguments and critique thereasoning of others as they use results about the sums and productsof rational and irrational numbers to determine whether a sum orproduct is rational, and to justify their reasoning. Have students explore computations with only rationals to get a feelfor what it means for a computation to have a value that is or is notpart of the set being used. Have students explore different computations with irrationals. Have students explore different computations with rational andirrational numbers. Have students explain and justify their answers. Classify numbers as rational or irrational. Know and apply the properties of equality. Closure property. Achieve the Core (Achieve the Core) Shmoop Click Sample Assessments for Drills (Schmoop)6
The Real Number SystemCluster:Reason quantitatively and use units to solve problems.NVACS HSN.Q.A.1 (Major Supporting Work)Use units as a way to understand problems and to guide the solution of multi-step problems; choose andinterpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsExemplars MP 1: Make sense of the quantities and their relationship within thecontext of the problem. MP 4: Model the quantities within the context of the problem toapply to solve everyday situations MP 7: Look for repeated reasoning in calculations for shortcuts.(For example, some students will convert yards straight to inchesinstead of converting yards to feet and then feet to inches.) This standard also includes converting measurements anddimensional analysis. Using, choosing, and interpreting units should occur in the contextof applications which contain them, especially mathematicalmodeling and formulas.For example: When applying the formula distance (speed)(time),students should recognize that when distance is measured in km,and speed is measured in km/hr, then time must be measured inhours. Attention should be made to the scale of each axis from the originof graphs in all graphing situations, especially those where data andapplication are present. This includes choosing appropriate viewingwindows on graphing calculators and software.For example: When graphing an exponential function, thehorizontal scale may be in ones but the vertical scale in tens,hundreds, thousands, or more. Also, the graph may not need toshow negative values of the dependent variable, depending on thecontext or function.For example: When fitting a line to data, a graph may not includethe origin to best display the data. Understand reciprocals. Understand unit rates, rate of change, and proportions. Unit conversions of measurement. Interpreting graphs. Breaking down modeling problems.7
ElementConnections Within andBeyond High t ExamplesExemplars Link to their science courses (The science teachers can be veryhelpful with this topic.) Working with quantities and the relationships between themprovides grounding for work with expressions, equations, andfunctions. Slope and average rate of change. Modeling. Blast Module (Blast) Illustrative Mathematics 10 Tasks (Illustrative Math ) RPDP Math, High School, Unit 2 Relationships Between QuantitiesNotes, Pages 9–11 (RPDP)8
QuantitiesCluster:Reason quantitatively and use units to solve problems.NVACS HSN.Q.A.2 (Major Supporting Work)Define appropriate quantities for the purpose of descriptive modeling.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 1: Make sense and explain the meaning of the problem, as wellas the quantities and their relationships in the problem. MP 3: Attend to precision when reporting solutions to problemsrequiring discrete responses. Mathematical models require specifically defined inputs andoutputs, measured in appropriate units. There should be attention tothis whenever applications are employed, whether linear, quadratic,exponential, or other type of function in this course.For example: Let T total cost of a taxi ride in dollars, let d distance of the taxi in miles, where each 1/7 mile costs 15 cents, andlet f the flat fee for a taxi ride in dollars. A model for this wouldbe T f (0.15)(7d). Create discourse with students to check that an answer isreasonable. Understand reciprocals. Understand unit rates, rate of change, and proportions. Unit conversions of measurement. Interpreting graphs. Breaking down modeling problems. Working with quantities and the relationships between themprovides grounding for work with expressions, equations, andfunctions. Modeling. Blast Module (Blast) Shmoop Click Samples Assessment for Drills (Schmoop) Spark 101 NASA Bone Density in Space (Spark) Illustrative Mathematics 3 Tasks (Illustrative Math)9
QuantitiesCluster:Reason quantitatively and use units to solve problems.NVACS HSN.Q.A.3 (Major Supporting Work)Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 1: Make sense of the meaning of the problem in context. MP 3: Attend to precision by applying rules of significant digits. MP 7: Look for repeated reasoning in calculations for shortcuts.(For example, some students will convert yards straight to inchesinstead of converting yards to feet and then feet to inches.) Intermediate calculations should not be rounded; rounding to theappropriate degree of precision occurs only at the end of a string ofcomputations. Solutions should be addressed both from a conceptual / commonsense level and from a formal perspective.For example: Conceptually, if the sides of a rectangle are measuredto tenths of centimeters, then the calculated area should not bereported to a greater precision than tenths of square centimeters.For example: Formally, if the sides of a rectangle are measured suchthat the lengths have two and three significant digits respectively,then the area should be reported to two significant digits. Rounding rules Working with quantities and the relationships between themprovides grounding for work with expressions, equations, andfunctions. Link to their science courses (The science teachers can be veryhelpful with this topic). Checking for a reasonable solution. Modeling. Blast Module (Blast) Illustrative Mathematics 7 Tasks (Illustrative Math) Math is Fun Help to clarify accuracy versus precision (MathisFun) MathBits Notebook Working with units (MathBits) RPDP Math, High School, Unit 2 Relationships Between QuantitiesNotes, Page 6 (RPDP)10
The Complex Number SystemCluster:Perform arithmetic operations with complex numbers.NVACS HSN.CN.A.1 (Major Supporting Work)Know there is a complex number i such that i2 –1, and every complex number has the form a bi with a andb real.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 2: Students will reason abstractly to identify imaginarynumbers. MP 3: Construct viable arguments and critique the reasoning ofothers. Example: Defend or analyze the work of others. Finding patterns for the powers of i using multiplication, rules ofexponents, and the definition of i. Identifying and categorizing real and complex numbers. Definition of like terms. Definition of a coefficient. Knowledge of number sets. Solving Quadratic Equations. Restrictions on the Domain on Radical Functions. Polar and Cartesian Forms of Complex Numbers. Modular Arithmetic. Illustrative Math Complex Numbers and Patterns (Powers of i)(Illustrative Math) RPDP Identify the discriminant of a quadratic and describe the roots(RPDP) RPDP Unit Test (RPDP)11
The Complex Number SystemCluster:Perform arithmetic operations with complex numbers.NVACS HSN.CN.A.2 (Major Supporting Work)Use the relation i2 –1 and the commutative, associative, and distributive properties to add, subtract, andmultiply complex numbers.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond Grade LevelInstructionalExamples/Lessons/TasksAssessment ExamplesExemplars MP 2: Students will use abstract reasoning to understand the valueof squaring an imaginary number. MP 8: Students will develop an understanding of what happenswhen operations are performed on imaginary numbers through theuse of repeated reasoning. Connect to operations on scientific notation as a review ofproperties. Connect to operations on binomials as a review of properties. Powers of i. Distributive Property, especially for two binomials. Commutative Property for regrouping like terms. Associative Property. Solving quadratic equations. Graphing on the Complex plane. Rectangles, Squares, Cubes Vectors and Polar Graphs Complex Number Products Greatest Value (Open Middle) Complex Number Products (Open Middle) Multiply Complex Numbers (Open Middle) Factoring Complex Numbers (Open Middle) Engage NY Lesson 37 (EngageNY) RPDP Unit Test (RPDP)12
The Complex Number SystemCluster:Perform arithmetic operations with complex numbers.NVACS HSN.CN.A.3 (Major Supporting Work)Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.ElementStandards forMathematical PracticeInstructional StrategiesExemplars MP 2: Students will use abstract reasoning to understand the valueof squaring and imaginary number. MP 8: Students will develop an understanding of what happenswhen operations are performed on imaginary numbers through theuse of repeated reasoning. Open withhave students determinevalues of a and b using the numbers between –9 and 9 such that theproduct is a real number. Give each table group a different example eg:and have each groupdetermine which values of a and b that yield a real product. Students then share out their results and identify the patternto determine the conjugate.Define moduli as the plural of modulus, which is the absolute valueof a complex number.Operations on complex numbers.Distance formula.Pythagorean Theorem.Rationalizing a denominator. pp 18–21 (RPDP)Complex Numbers Worksheet (RPDP)CPalmsRPDP pp 21–22 (RPDP)Shmoop Click Samples Assessments for Drills (Schmoop) Prerequisite SkillsConnections Within andBeyond High t Examples13
The Complex Number SystemCluster:Represent complex numbers and their operations on the complex plane.NVACS HSN.CN.B.4 (Major Supporting Work)( ) Represent complex numbers on the complex plane in rectangular and polar form (including real andimaginary numbers), and explain why the rectangular and polar forms of a given complex number represent thesame number.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 2: Students will use abstract reasoning to understand the valueof squaring and imaginary number. MP 8: Students will develop an understanding of what happenswhen operations are performed on imaginary numbers through theuse of repeated reasoning. Use scaffolding to build the concept as follows: A single real number on the rectangular plane to the polarplane. An imaginary unit on the rectangular plane to the polarplane. Finally, a complex number on the rectangular plane to thepolar plane. Graphing on a Cartesian Plane. Distance formula and Pythagorean Theorem. Right triangle trigonometry. Degree to radian conversions. Transformations. (Dilations & Rotations) Engage NY Lesson 13 (Engage NY)CPalms Virtual Manipulative and Video (cpalms)TI Calculator Activities Texas Instruments)Exit Ticket Lesson 13 (Engage NY)14
The Complex Number SystemCluster:Represent complex numbers and their operations on the complex plane.NVACS HSN.CN.B.5 (Major Supporting Work)( ) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on thecomplex plane; use properties of this representation for computation. For example, (-1 3 i)3 8 because (–1 3 i) has modulus 2 and argument 120.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 2: Students will use abstract reasoning to understand the valueof squaring an imaginary number. MP 8: Students will develop an understanding of what happenswhen operations are performed on imaginary numbers through theuse of repeated reasoning. Introduce using powers of i graphed on the complex plane. Draw a vector diagram of each complex number to modeloperations geometrically. Operations with like terms, binomials, and complex numbers. Pythagorean Theorem. Right triangle trigonometry. Transformations. Graphing complex numbers on the complex plane. Vector operations. Geometric representation of Complex Number operations usingVectors (Math Facutly) TI Calculator Activities (Texas Instruments) Addition & Subtraction (Engage NY) Multiplication (Engage NY) Illustrative Math 1 Task (Illustrative Math) Exit Ticket pp 7–9 (Engage NY) Exit Ticket p 121 (Engage NY)15
The Complex Number SystemCluster:Represent complex numbers and their operations on the complex plane.NVACS HSN.CN.B.6 (Major Supporting Work)( ) Calculate the distance between numbers in the complex plane as the modulus of the difference, and themidpoint of a segment as the average of the numbers at its endpoints.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 2: Students will use abstract reasoning to understand the valueof squaring and imaginary number. MP 8: Students will develop an understanding of what happenswhen operations are performed on imaginary numbers through theuse of repeated reasoning. Help the students to understand this complex topic by relating it tofinding the distance between any two points in the Cartesian Plane. Graphing in the Cartesian Plane. Distance Formula. Pythagorean Theorem. Finding the distance between points. Connects to work that will be done with vectors. Modulus of the Difference & Midpoint (Engage NY) Midpoint of a Segment (Engage NY) RPDP Notes pp 20–22 (RPDP) Illustrative Math Complex Distance (Illustrative Math) Exit Ticket pp 7–10 (Engage NY)16
The Complex Number SystemCluster:Use complex numbers in polynomial identities and equations.NVACS HSN.CN.C.7 (Major Supporting Work)Solve quadratic equations with real coefficients that have complex solutions.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 2: Students will use abstract reasoning to understand how aproblem with real numbers can generate a solution that includesimaginary numbers. MP 6: Students will need to be precise in dealing with negativequantities as these will be source for their complex solutions. Graph the parent function y x2. Ask what are the solutions to thisgraph? What would be the equation if we shift this function 4 unitsdown? What are the solutions to this graph? What if we shift theparent function up 4 units? What is the equation? What are itssolutions? Quadratic formula Simplifying radicals Completing the square (including area modeling) Solving quadratic equations. Solving polynomial equations. Illustrative Mathematics 1 Task (Illustrative Math) Engage NY Lesson 4 (Engage NY) Interactive Maths: Solving Quadratic Equations (select Complexoption) (Interactive Maths) RPDP pp 21–30 (RPDP) Engage NY Exit Ticket p 54 (Engage NY) RPDP Multiple Choice Questions pp 13–14, 29-30 (RPDP)17
The Complex Number SystemCluster:Use complex numbers in polynomial identities and equations.NVACS HSN.CN.C.8 (Major Supporting Work)( ) Extend polynomial identities to the complex numbers. For example, rewrite x2 4 as (x 2i)(x – 2i).ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 7: Students will use the structural similarity betweenpolynomial expressions and complex numbers to apply the rules foroperations from the polynomials to the complex. Start with a review of factoring polynomials with real solutions,then extend it to the complex numbers. Factoring polynomials. Polynomial operations. Factoring Polynomials Properties of Real Numbers such as Associative, Commutative,Distributive & Identity Properties. Illustrative Mathematics 1 Task (Illustrative Math) Engage NY Lesson 3 (Engage NY) Exit Ticket pp 11 (Engage NY)18
The Complex Number SystemCluster:Use complex numbers in polynomial identities and equations.NVACS HSN.CN.C.9 (Major Supporting Work)( ) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 8: Look for and express regularity in repeated reasoning todiscover the nature of the Fundamental Theorem of Algebra. Connect the roots of a quadratic graph to the solutions of anequation to the linear factors of the equation. Then extend tocomplex solutions and linear factors. Then extend to polynomialfunctions. Create discourse among students regarding the use of the wordsroots, zeros, and solutions. These words, although notinterchangeable, have very similar meanings. Definition of linear factors. Degree of a polynomial. Remainder Theorem. Repeated Zeros. End behavior of graphs. x-intercepts of Polynomial Functions Odd/Even Roots Engage NY Lesson 40 (Engage NY) TI Calculator Activities (Texas Instruments) Exit Ticket p 477 (Engage NY)19
Vector and Matrix QuantitiesCluster:Represent and model with vector quantities.NVACS HSN.VM.A.1 (Major Supporting Work)( ) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directedline segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, v , v , v).ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP2: When students understand that vectors are quantities theyextend their ability to use quantitative reasoning. Introduce vectors by looking at them in a real world context. Forexample, discuss the difference between velocity (vector) and speed(magnitude). Relate vector magnitude to the distance formula. Take time to expose students to the notations used to describevectors. Pythagorean Theorem. Connect vectors to polar coordinates. Vectors are used in many higher-level science courses to describeand explain physical movements and forces. RPDP pp 1–5 (RPDP) Collaborate with science instructors (physics) for examples ofprojects utilizing vectors. RPDP (RPDP)20
Vector and Matrix QuantitiesCluster:Represent and model with vector quantities.NVACS HSN.VM.A.2 (Major Supporting Work)( ) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of aterminal point.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 1: Students will make sense of vector problems by looking atthe components of the vectors as coordinate changes. Create right triangles on the coordinate plane where the givenvector is the hypotenuse and the legs become the components of thevectors. Move vectors to different locations and with different directions toexplore how these relate to the components of the vector. Treat the starting point of a vector as a translation from the origin. Coordinate geometry. Pythagorean Theorem. Connect vectors to polar coordinates. Vectors are used in many higher-level science courses to describeand explain physical movements and forces. RPDP pp 1–5 (RPD) TI Calculator Activity (Texas Instruments) Collaborate with science instructors (physics) for examples ofprojects utilizing vectors. RPDP (RPDP)21
Vector and Matrix QuantitiesCluster:Represent and model with vector quantities.NVACS HSN.VM.A.3 (Major Supporting Work)( ) Solve problems involving velocity and other quantities that can be represented by vectors.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 3: Students will justify their representations of vector quantitiesand critique the representations of other students. MP 4: Students will be able to use vectors to model real worldsituations. Provide examples that relate to force, momentum, velocity, andwork. Connect with the Physics teacher for some examples that will buildupon the work students do in that class. The concepts of units and rates. Right triangle trigonometry. Vectors are used if you study polar coordinates. Vectors are used in many higher-level science courses to describeand explain physical movements and forces. RPDP pp 6–9 (RPDP) Vectors & Sports (Prezi) RPDP (RPDP) Shmoop Click sample assessments for drills. (Schmoop)22
Vector and Matrix QuantitiesCluster:Perform operations on vectors.NVACS HSN.VM.B.4 (Major Supporting Work)( ) Add and subtract vectors.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 2: Students reason abstractly and quantitatively by adding andsubtracting vectors algebraically and graphically, using differentrepresentations to find solutions. Add and subtract vectors algebraically and graphically. Simplifying expressions by combining like terms. Basic coordinate geometry, plotting points. Scalar multiplication and vector multiplication. Sum of magnitude/direction. Connections to right triangle trigonometry. Polar and complex numbers. Geometric representation of vector operations using parallelogramson the coordinate plane. Graphing Adding and Subtracting Vectors (Khan Academy) Sum of Two Vectors (Wolfram Demonstrations Project) Open Middle (Open Middle) Shmoop Click sample assignments for drills (Schmoop) Algebraically Add and Subtract Vectors Practice (Khan Academy) Graphically Add and Subtract Vectors Practice (Khan Academy) RPDP (RPDP)23
Vector and Matrix QuantitiesCluster:Perform operations on vectors.Represent complex numbers on the complex plane in rectangular and polar form(including real and imaginary numbers), and explain why the rectangular and polar forms of a given complexnumber represent the same number.NVACS HSN.VM.B.4.A (Major Supporting Work)Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of asum of two vectors is typically not the sum of the magnitudes.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 2: The parallelogram rule allows students to use quantitativereasoning of vector components to add vectors. MP 4: Students will model vector addition using the parallelogramrule. Placing the vectors on a coordinate grid allows students to see howthe components of the original vectors relate to the resultingaddition. Help students to understand how the parallelogram rule relate tobasic geometry work students have done in the past. Parallelogram rule is Pythagorean Theorem.Basic trigonometry.Law of Cosines.Vectors are used in many higher-level science courses to describeand explain physical movements and forces.RPDP pp 1–5 (RPDP)TI Calculator Activity (Texas Instruments)Engage NY Lesson 19 (Engage NY)RPDP (RPDP)24
Vector and Matrix QuantitiesCluster:Perform operations on vectors.NVACS HSN.VM.B.4.B (Major Supporting Work)Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.ElementStandards forMathematical PracticeInstructional StrategiesPrerequisite SkillsConnections Within andBeyond High t ExamplesExemplars MP 7: Using their knowledge of the structure of a vector studentswill be able to add them when given only a direction and magnitudeof the original vectors. Have students practice moving between different representations ofvectors so that th
Extend the properties of exponents to rational exponents. NVACS HSN.RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3
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