SM2 Curriculum Notebook

SM2 Curriculum Notebook
2017


Curriculum Notebook Table of Contents
Standards indicate the broad goals for a student to master in a course. Standards are typically set by a state or district school board.
Mathematics Practice Standards........................................................................................................... Page 4 Secondary Math 2 Content Standards .................................................................................................. Page 8
Essential Learning Standards
Particular standards/objectives/indicators that a school/district defines as critical for student learning. In fact, they are so critical that students will receive intervention if they are not learned. Essentials are chosen because they 1. have endurance, 2. have leverage, and 3. are important for future learning. ....................................................................................................................................................................... Page 15
Curriculum Resources
The materials teachers use to plan, prepare, and deliver instruction, including materials students use to learn about the subject. Such materials include texts, textbooks, tasks, tools, and media. Sometimes organized into a comprehensive program format, they often provide the standards, units, pacing guides, assessments, supplemental resources, interventions, and student materials for a course.
USBE Core Curriculum Guides ....................................................................................................................... Page 16
Pacing Guide
The order and timeline of the instruction of standards, objectives, indicators, and Essentials over the span of a course (semester or year). ....................................................................................................................................................................... Page 100
Units
A plan for several weeks of instruction, usually based on a theme, that includes individual lesson plans. Units often also include: Standards, learning targets/goals, skills, formative and summative assessment, student materials, essential questions, big ideas, vocabulary, questions, and instructional methods.
Understanding by Design .............................................................................................................................. Page 101 Secondary Math 2 Units ................................................................................................................................ Page 102
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Assessment Standards
A set of criteria to guide the assessment of student learning in a course that is based on Standards/Essentials of the course; this might include formative assessment practices, summative assessments/practices, common assessment plans, feedback practices, and a schedule for testing. ............................................................................................................................................................... Page Ethics ..................................................................................................................................................... Page 104
Intervention Standards
A set of criteria to guide teachers to provide additional instruction to students who did not master the content in Tier 1 instruction. This might include: commercial intervention programs, teacher-developed intervention materials, diagnostic testing, RTI/MTSS processes, and a list of essential knowledge/skills that will prompt intervention if the student does not demonstrate mastery.
RTI ......................................................................................................................................................... Page 106 MTSS...................................................................................................................................................... Page 108
Supplemental Resources
Instructional materials, beyond the main curricular materials, used to strategically fill gaps/weaknesses of the core program materials.
Provo Way Instructional Model ............................................................................................................ Page 110 ............................................................................................................................................................... Page
Evidence-based Pedagogical Practices
A list of teaching strategies that are supported by adequate, empirical research as being highly effective.
John Hattie ............................................................................................................................................ Page 115
Glossary
Terms and acronyms used in this document ........................................................................................ Page 116
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Secondary Math 2 Standards
Standards indicate the broad goals for a student to master in a course. Standards are typically set by a state or district school board.
Mathematics
Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solutions pathway rather than simply jumping into a solution attempt. The consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and the continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
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2 Reason abstractly and quantitatively
Mathematically proficient students make a sense of the quantities and their relationships in problem situations. Students bring two complimentary abilities to bear on problems involving quantitative relationships: the ability to decontextualize–to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents–and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
3 Construct viable arguments and critique the reasoning of others
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of tow plausible arguments, distinguish correct logic or reasoning form that which is flawed and–if there is a flaw in an argument–explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
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4 Model with mathematics
Mathematically proficient students can apply mathematics the know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity on interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possible improving the model if it has not served its purpose.
5 Use appropriate tools strategically
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
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6 Attend to precision
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
7 Look for and make use of structure
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y )2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
8 Look for and express regularity in repeated reasoning
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding(x–1)(x +1),(x –1)(x2 +x+1),and(x–1)(x3 +x2 +x +1)might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
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Content Standards for Secondary Mathematical 2 Numbers – The Real Number System NRN
Extend the properties of exponents to rational exponents
1. Explainhowthedefinitionofthemeaningofrationalexponentsfollows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
2. Rewriteexpressionsinvolvingradicalsandrationalexponentsusingthe properties of exponents.
Use properties of rational and irrational numbers.
3. Explainwhythesumorproductoftworationalnumbersisrational;that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Numbers – The Complex Number System NCN Perform arithmetic operations with complex numbers.
1. Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
2. Usetherelationi2=–1andthecommutative,associative,anddistributive properties to add, subtract, and multiply complex numbers.
Use complex numbers in polynomial identities and equations.
7. Solve quadratic equations with real coefficients that have complex solutions.
Algebra – Seeing Structure in Expressions ASSE Interpret the structure of expressions.
1. Interpretexpressionsthatrepresentaquantityintermsofitscontext.
a. Interpretpartsofanexpression,suchasterms,factors,andcoefficients. b. Interpretcomplicatedexpressionsbyviewingoneormoreoftheirparts
as a single entity.
2. Usethestructureofanexpressiontoidentifywaystorewriteit
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Write expressions in equivalent forms to solve problems.
3. Chooseandproduceanequivalentformofanexpressiontorevealand explain properties of the quantity represented by the expression.
a. Factoraquadraticexpressiontorevealthezerosofthefunctionit defines.
b. Completethesquareinaquadraticexpressiontorevealthe maximum or minimum value of the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions
Arithmetic with polynomials and rational expressions AAPR Perform arithmetic operations on polynomials.
1. Understandthatpolynomialsformasystemanalogoustotheintegers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Creating Equations ACED Create equations that describe numbers or relationships.
1. Createequationsandinequalitiesinonevariableandusethemtosolve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
2. Createequationsintwoormorevariablestorepresentrelationships between quantities; graph equations on coordinate axes with labels and scales.
4. Rearrangeformulastohighlightaquantityofinterest,usingthesame reasoning as in solving equations.
Reasoning with equations and inequalities AREI Understand solving equations as a process of reasoning and explain the reasoning.
4. Solve quadratic equations in one variable.
a. Usethemethodofcompletingthesquaretotransformanyquadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
b. Solvequadraticequationsbyinspection(e.g.,forx2=49),takingsquare
roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
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Solve systems of equations.
7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically
Functions – Interpreting Functions FIF Understand the concept of a function and use function notation.
4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
5. Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothe quantitative relationship it describes.
6. Calculateandinterprettheaveragerateofchangeofafunction(presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Analyze functions using different representations
7. Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthe graph, by hand in simple cases and using technology for more complicated cases.
8.
a. Graphlinearandquadraticfunctionsandshowintercepts,maxima,and minima.
b. Graphsquareroot,cuberoot,andpiecewise-definedfunctions, including step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
d. (+)Graphrationalfunctions,identifyingzerosandasymptoteswhen suitable factorizations are available, and showing end behavior.
e. Graphexponentialandlogarithmicfunctions,showinginterceptsand end behavior, and trigonometric functions, showing period, midline, and amplitude.
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Usetheprocessoffactoringandcompletingthesquareinaquadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
b. Usethepropertiesofexponentstointerpretexpressionsfor exponential functions.
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9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Functions – Building Functions FBF Build a function that models a relationship between two quantities.
1. Writeafunctionthatdescribesarelationshipbetweentwoquantities. a. Determineanexplicitexpression,arecursiveprocess,orstepsfor
calculation from a context.
b. Combinestandardfunctiontypesusingarithmeticoperations.
Build new functions from existing functions
3. Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an
inverse and write an expression for the inverse.
Functions – Linear, Quadratic, and Exponential Models FLE
3. Observeusinggraphsandtablesthataquantityincreasingexponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Functions – Trigonometric Functions FTF Prove and apply trigonometric identities
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
Construct and compare linear, quadratic, and exponential models and solve
problems
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Geometry – Congruence GCO Prove geometric theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Geometry – Similarity, Right Triangles, and Trigonometry GSRT Understand similarity in terms of similarity transformations
1. Verifyexperimentallythepropertiesofdilationsgivenbyacenteranda scale factor
a. Adilationtakesalinenotpassingthroughthecenterofthedilation to a parallel line, and leaves a line passing through the center unchanged
b. Thedilationofalinesegmentislongerorshorterintheratiogiven by the scale factor
2. Giventwofigures,usethedefinitionofsimilarityintermsofsimilarity transformations to decide if they are similar, explain, using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides
3. UsethepropertiesofsimilaritytransformationstoestablishtheAA criterion for two triangles to be similar
Prove theorems involving similarity
4. Provetheoremsabouttriangles(Theoremsinclude:alineparalleltoone side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity)
5. Usecongruenceandsimilaritycriteriafortrianglestosolveproblemsandto prove relationships in geometric figures
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Define trigonometric ratios and solve problems involving right triangles
6. Understandthatbysimilarity,sideratiosofrighttrianglesarepropertiesof the angles in the triangle leading to definitions of trigonometric ratios for acute angles
7. Explainandusetherelationshipbetweenthesineandcosineof complementary angles
8. UsetrigonometricratiosandthePythagoreanTheoremtosolveright triangles in applied problems
Geometry – Circles GC Understand and apply theorems about circles
1. Provethatallcirclesaresimilar
2. Identifyanddescriberelationshipsamonginscribedangles,radii,and
chords. (Including the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle)
3. Constructtheinscribedandcircumscribedcirclesofatriangle,andprove properties of angles for a quadrilateral inscribed in a circle
Geometry – Expressing Geometric Properties with
Equations GGPE Understand and apply theorems about circles
1. Derivetheequationofacircleofgivencenterandradiususingthe Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation
2. Derivetheequationofparabolagivenafocusanddirectrix
4. Usecoordinatestoprovesimpletheoremsalgebraically(forexample,prove
or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0,2))
Use coordinates to prove simple geometric theorems algebraically
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio
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Geometry – Modeling with Geometry GGMD Explain volume formulas and use them to solve problems
1. Giveaninformalargumentfortheformulasforthecircumferenceofa circle, area of a circle, volume of a cylinder, pyramid, and cone (Use dissection arguments, Cavalieri’s principle, and informal limit arguments)
3. Usevolumeformulasforcylinders,pyramids,cones,andspherestosolve problems
Statistics – Conditional Probability and the Rules
of Probability SCP
1. Describeeventsassubsetsofasamplespace(thesetofoutcomes)using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or”, “and”, “not”)
2. UnderstandthattwoeventsAandBareindependentiftheprobabilityofA and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent
3. UnderstandtheconditionalprobabilityofAgivenBasP(AandB)/P(B),and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B
4. Constructandinterprettwo-wayfrequencytablesofdatawhentwo categories are associated with each object being classified. Use the two- way table as a sample space to decide if events are independent and to approximate conditional probabilities
5. Recognizeandexplaintheconceptsofconditionalprobabilityand independence in everyday language and everyday situations
6. FindtheconditionalprobabilityofAgivenBasthefractionofB’soutcomes that also belong to A, and interpret the answer in terms of the model
7. Applytheadditionrule,P(AorB)=P(a)+P(B)–P(AandB),andinterpret the answer in terms of the model
Understand independence and conditional probability and use them to interpret
data
Use the rules of probability to compute probabilities of compound events in a
uniform probability model
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Course Essential Learning Standards
Particular standards/objectives/indicators that a school/district defines as critical for student learning. In fact, they are so critical that students will receive intervention if they are not learned. Essentials are chosen because they 1. have endurance, 2. have leverage, and 3. are important for future learning.
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Curriculum Resources
The materials teachers use to plan, prepare, and deliver instruction, including materials students use to learn about the subject. Such materials include texts, textbooks, tasks, tools, and media. Sometimes organized into a comprehensive program format, they often provide the standards, units, pacing guides, assessments, supplemental resources, interventions, and student materials for a course.
USBE Core Curriculum Guides
Click on a standard to go to the USBE Core Curriculum Guide
NRN1 Explain how definition of rational exponents follows from integer exponents
NRN2 Rewrite expressions involving radicals and rational exponents
NRN3 Explain why sums & products are rational or irrational
NCN1 Know there is a complex number i such that i2 = -1
NCN2 Use properties to add, subtract, and multiply complex numbers
ASSE1a Interpret parts of an expression, such as terms, factors, and coefficients
ASSE1b Interpret complicated expressions
ASSE2 Use the structure of an expression to identify ways to rewrite it
ASSE3 Produce equivalent form of an expression to explain properties of the expression
AAPR1 Understand that polynomials form a system analogous to the integers
ACED1 Create equations and inequalities in one variable & use them to solve problems
ACED2 Create equations in two or more variables to represent relationships
ACED4 Rearrange formulas to highlight a quantity of interest
AREI4 Solve quadratic equations in one variable
AREI7 Solve a simple system of a linear equation & a quadratic equation in 2 variables
FIF4 Interpret key features of graphs and tables in terms of quantities of a function
FIF5 Relate the domain of a function to its graph
FIF6 Calculate and interpret the average rate of change of a function
FIF7a Graph linear functions and show intercepts
FIF7e Graph exponential functions, showing intercepts and end behavior
FIF8 Write a function defined by an expression in different but equivalent forms
FIF9 Compare properties of two functions each represented in a different way
FBF1a Determine an explicit expression or a recursive process from a context
FBF1b Write a function that describes a relationship between two quantities
FBF3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k)
FBF4 Find inverse functions
FLE3 Observe relationships of linear and exponential functions using graphs & tables FTF8 Prove the Pythagorean Identity sinø2 + cosø2 = 1 & use it to find sinø, cosø & tanø
GCO9 Prove theorems about lines and angles
GCO10 Prove theorems about triangles
GCO11 Prove theorems about parallelograms
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GSRT1 GSRT2 GSRT3 GSRT4 GSRT5 GSRT6 GSRT7 GSRT8
GC1 GC2 GC3
GGPE1 GGPE2 GGPE4 GGPE6
GGMD1 GGMD3
SCP1 SCP2 SCP3 SCP4 SCP5 SCP6 SCP7
Verify experimentally the properties of dilations
Given two figures, use the definition of similarity to decide if they are similar Use the properties of similarity transformations to establish the AA criterion Prove theorems about triangles
Use congruence and similarity criteria for triangles
Understand that by similarity, side ratios of right triangles are proportional Explain & use relationships between the sine & cosine of complementary angles Use trigonometric ratios & the Pythagorean Theorem to solve problems
Prove that all circles are similar
Identify and describe relationships among inscribed angles, radii, and chords Construct the inscribed and circumscribed circles of a triangle
Derive the equation of a circle
Derive the equation of parabola
Use coordinates to prove simple geometric theorems algebraically
Find the point on a line segment that partitions the segment in a given ratio
Informally argue formulas of circles, area, volume of a cylinder, pyramid, & cone Use volume formulas
Describe events as subsets of a sample space
Understand independent events
Understand conditional probability
Construct and interpret two-way frequency tables of data
Recognize and explain the concepts of conditional probability and independence Find the conditional probability of A given B
Apply the addition rule, P(A or B) = P(a) + P(B) – P(A and B)
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Number–The Real Number System NRN1
Core Content
Cluster Title: Extend the properties of exponents to rational exponents
Standard: N.RN.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allow for a notation for radicals in terms of rational exponents. (For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so 5(1/3)3=5.
Concepts and skills to master
Conceptual:
• Understand the meaning of a rational exponent
Procedural:
• Calculate expressions that use rational exponents
Representational:
• Represent the reciprocal values of rational exponents such as
• Model the properties of rational exponents as multiple multiplications
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know the properties of integer exponents
• Understand that all non-perfect square roots and cube roots are irrational
Procedural:
• Apply the properties of integer exponents to simplify and evaluate numerical expressions
• Evaluate the square roots of small perfect squares and cube roots of small perfect cubes
Representational:
• Model the properties of integer exponents as multiple multiplications
• Represent the solutions to equations using square root and cube root symbols
Academic Vocabulary and notation
Rational exponent, radical, radicand, index, nth root, exponential notation
Suggested Instructional Strategies
• Relate rational exponents to integer and whole number exponents
• Teach with N.RN.2
Sample Formative Assessment Tasks Skill-based Task:
What is 8(1/3)?
What is the area of a rectangle with a length squarerootofsevenandawidthofcuberootof seven?
Resources
Problem Task:
What is the square root of the cubed root of x?
Whatisthenthrootofthemrootofthep root of x?
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Number–The Real Number System NRN2
Core Content
Cluster Title: Extend the properties of exponents to rational exponents
Standard: N.RN.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents
Concepts and skills to master
Conceptual:
• Extend the properties of integer exponents to rational exponents Procedural:
• Use the properties of integer exponents to simplify expressions
• Convert radical notation to rational exponent notation
• Convert rational exponent notation to radical notation
Representational:
• Model exponential notation
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know the relationship between radical notation and rational exponent notation
• Become familiar with radical notation
Procedural:
• Use properties of exponents to simplify expressions
Representational:
• Model exponential notation
Academic Vocabulary and Notation
Rational exponent, radical, radicand, index, nth root, exponential notation, radical notation
Suggested Instructional Strategies
• Compare contexts where radical form is preferable to rational exponents and vice versa
Sample Formative Assessment Tasks
Skill-based Task:
Compute the cubed root of 25 times the square root of five to the third power
What is the area of a rectangle with a length square root of seven and a width the cubed root of seven?
Resources
Problem Task:
What is the square root of the cubed root of x?
What is the nth root of the m root of the p root of x?
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Number–The Real Number System NRN3
Core Content
Cluster Title: Use properties of rational and irrational numbers
Standard: N.RN.3: Explain why sums and products of rational numbers are rational, why the sum of a rational number and an irrational number is irrational, and why the product of a nonzero rational number and an irrational number is irrational
Concepts and skills to master
Conceptual:
• Understand why adding and multiplying two rational numbers results in a rational
number
• Understand why adding a rational number to an irrational number results in an irrational
number
• Understand why multiplying a nonzero number to an irrational number results in an
irrational number Procedural:
• Add, subtract and multiply real numbers
• Simplify radical expressions Representational:
• Model addition, subtraction and multiplication of real numbers Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know that the square root of 2 is irrational
• Understand that rational numbers can be written as a terminating or repeating decimal
• Understand that irrational numbers are non-terminating, non-repeating decimals
Procedural:
• Identify rational and irrational numbers by examining the decimal traits
Representational:
• Model the square roots of irrational numbers such as diagonals of squares
Academic Vocabulary and Notation
Rational, irrational, radical notation
Suggested Instructional Strategies
• Teach computation by using formal definitions
• Explore sums and products of rational and irrational numbers to discover patterns where the results are either rational or irrational
Sample Formative Assessment Tasks
Skill-based Task:
Simplify the square root of 3 times the quantity two plus the square root of 6
What type of number is the product of 3 and the square root of 3
Resources
Problem Task:
Given a right triangle whose hypotenuse is irrational, find measures for legs where:
• Both legs are rational
• Both legs are irrational
• One leg is rational and the other one is
irrational
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Number–The Complex Number System NCN1
Core Content
Cluster Title: Perform arithmetic operations with complex numbers
Standard: N.CN.1: Know there is a complex number i such that i2 = -1, and every complex number hastheforma+biwithaandbreal
Concepts and skills to master
Conceptual:
• Understand that the set of complex numbers includes the set of all real numbers and the set of imaginary number
Procedural:
• Express numbers in the form a + bi
Representational:
•
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the Fundamental Theorem of Algebra
• Understand square roots
Procedural:
• Use the definition of square root and the Fundamental Theorem of Algebra to show the
need for the square root of negative 1
• Solvex2 –a2=0andx2 +a2=0whereaisaninteger
Representational:
• Connect imaginary solutions to the graphs of quadratic functions
Academic Vocabulary and Notation
Real numbers, complex numbers, imaginary numbers, i, a + bi Suggested Instructional Strategies Resources
• Use the definition of square root Use geometer’s sketchpad to model the roots of andtheFundamentalTheoremof quadraticsandanticipatetheimaginaryroots.
Algebra to show the need for the square root of negative
one. Consider this in a historical context
• Solvex2 –a2=0andx2 +a2=0 where a is an integer
• Connect imaginary solutions to the graphs of quadratic functions
Sample Formative Assessment Tasks
Skill-based Task:
Write the square root of negative twenty- five plus the square root of nine as a complex number in the form a + bi
http://www.learnzillion.com/lessons/225-determine- whether-a-square-root-is-real-or-imaginary
http://www.learnzillion.com/lessons/226-write-the- square-root-of-a-negative-number-as-imaginary
http://www.learnzillion.com/lessons/227-classify- complex-numbers-as-real-or-imaginary
Problem Task:
Given px2 + q = 0 find values of p and q that result in:
• A real number solution
• An imaginary number solution
Generalize the relationship between p and q that
would result in each type of solution
21


Number–The Complex Number System NCN2
Core Content
Cluster Title: Perform arithmetic operations with complex numbers
Standard: N.CN.2: Use the relation i 2 = -1 and the commutative, associative, and distribution properties to add, subtract, and multiply complex numbers
Concepts and skills to master
Conceptual:
• Understand how the commutative, associative and distribution properties work with complex numbers
Procedural:
• Add, subtract, and multiply complex numbers
Representational:
•
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the definition of i
Procedural:
• Combine like terms in polynomials
Representational:
• Model the commutative, associative and distributive properties with real numbers
Academic Vocabulary and Notation
Complex numbers, i
Suggested Instructional Strategies Resources
• Relate operations with complex Add complex numbers: combining like terms numbers to familiar operations with http://learnzillion.com/lessons/232-add-complex- numbers or polynomial expressions numbers-combining-like-terms
Subtract complex numbers:combining like terms
http://learnzillion.com/lessons/233-subtract- complex-numbers-combining-like-terms
Multiplying 1-term complex numbers
http://learnzillion.com/lessons/234-multiply-1term- complex-numbers
Multiplying complex numbers
http://learnzillion.com/lessons/235-multiply- complex-numbers
Simplifying complex numbers
http://learnzillion.com/lessons/236-simplify-positive- integer-powers-of-i
22


Sample Formative Assessment Tasks
Skill-based Task:
Perform the following operations and simplify the solutions (2+3i)+(5–7i)
(3 – 5i)(2 + 4i)
Problem Task:
Under what circumstances does (a + bi)(c + di) result in the following:
• Arealnumber
• An imaginary number
• A non-real complex number
23


Algebra–Seeing Structure in Expressions ASSE1a
Core Content
Cluster Title: Interpret the structure of expressions.
Standard: A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients.
MASTERY Patterns of Reasoning:
Conceptual:
• Recognize terms, bases, exponents, coefficients, and factors.
• Determine real world context of the variables in an expression.
Procedural:
• Identify the individual factors of a given term within an expression.
Representational:
• Model real world context using variables in an expression.
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the meaning of symbols indicating mathematical operations.
• Understand the meaning of implied operations (e.g. 2x).
• Understand the meaning of exponents.
• Understand grouping symbols.
Procedural:
• Use order of operations including grouping symbols.
Representational:
• Model mathematical operations.
Academic Vocabulary and Notation
2x, x^2 ,(), { }, [ ], Bases, Coefficients, Exponents, Expression, Factors, Grouping symbols,Terms,
Instructional Strategies Used
Resources Used
Unit 3 from PHS algebra book, 5 section 3.1 expressions and equations Alg1_VariablesAndExpressions (ASSE1a).pdf Unit 2-1 from CAS:
sec 2-1n variables and expressions.doc lesson plan 2-1.doc
Worksheet 2-1.doc
Problem Task Interprettheexpression:5-3(x-y)^2.Explainthe output values possible.
• •
Design a game around identifying
terms, bases, exponents, coefficients, and factors
Create formulas based on context
Assessment Tasks Used
Skill-based Task ConsidertheformulaSurfaceArea=2B+Ph
• What are the terms of this formula?
• What are the coefficients?
24


Algebra–Seeing Structure in Expressions ASSE1b
Core Content
Cluster Title: Interpret the structure of expressions.
Standard: A.SSE.1: Interpret expressions that represent a quantity in terms of its context. b. Interpret complicated expressions by viewing one or more of their parts as a single
entity. For example, interpret
P(1+r)^n as the product of P and a factor not depending on P.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that the terms within grouping symbols represent a single value e.g., in
the expression (x + 4)
• x+4isasinglevalueforeveryx.
• Recognize that the expression (x + 3)/3 cannot be simplified by simply canceling
the threes. Procedural:
• Be able to calculate the expression (x+6)/3 correctly for every value of x. Representational:
• Be able to represent real world contexts in expressions requiring more than one operation.
Supports for Teachers
Critical Background Knowledge
Conceptual:
•
•
Procedural:
•
• Be able to create expressions using variables for real world contexts Academic Vocabulary and Notation
2x, ,(), { }, [ ], Bases, Coefficients, Exponents, Expression, Factors, Grouping symbols,Terms,
Understand the importance of grouping symbols within the order of operations Understand the use of variables
Be able to fluently compute with the order of operations Representational:
Instructional Strategies Used
• Given a word problem and a formula, have students examine the structure and explain the context of different parts of the formula.
Assessment Tasks Used
Skill-based Task
Consider the Area formula for a trapezoid
Resources Used
Problem Task:
Interpret P(1+r)^n as the product of P and a factor not depending on P.
• • •
What are the terms of this formula? What are the coefficients?
What is the relationship between (b1 + b2) and h?
25


Algebra–Seeing Structure in Expressions ASSE2
Core Content
Cluster Title: Build new functions from existing functions
Standard: A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2)
Concepts and skills to master
Conceptual:
• Understand that an expression has different forms.
Procedural:
• Justify the different forms of an expression based on mathematical properties
• Interpret different symbolic notation
Representational:
• Model the difference of two squares
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the distributive property in simplifying and expanding expressions
Procedural:
• Use various types of factoring skills
Representational:
• Model factoring
Academic Vocabulary and Notation
Factors, coefficients, terms, exponent, base, constant, variable, binomial, monomial, and
polynomial
Suggested Instructional Strategies
• This standard should be taught in conjunction with standard A.SSE.3 with a heavy emphasis on justification
Sample Formative Assessment Tasks
Skill-based Task:
Explain how you can use the quadratic formula to solve x4 – 2x2 + 35 = 0
Resources
Distributing and factoring using area: http:illuminations.nctm.org/LessonDetail.aspx?id=L744
Difference of squares: http:illuminations.nctm.org/LessonDetail.aspx?id=L276
Problem Task:
Factor x6 – y6 as the difference of two squares and as the difference of cubes. Justify that the resulting expressions are equivalent
26


Algebra–Seeing Structure in Expressions ASSE3
Core Content
Cluster Title: Write expressions in equivalent forms to solve problems
Standard: A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression
A. Factor a quadratic expression to reveal the zeros of the function it defines
B. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines
C. Use the properties of exponents to transform expressions for exponential functions (For example the expression 1.15t can be written as (1.15 1/12)12t – 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.)
Concepts and skills to master
Conceptual:
• Understand the quantity represented by equivalent expressions Procedural:
• Rewrite expressions in different forms using mathematical properties
• Given a context, determine the best form of an expression
• Factor a quadratic expression to reveal the zeros of the function it defines
• Complete the square in a quadratic expression to reveal the maximum or minimum value
of the function it defines Representational:
• Model the properties of exponents to transform expressions for exponential functions Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the distributive property in simplifying and expanding expressions
Procedural:
• Use various types of factoring skills
Representational:
• Model factoring
Academic Vocabulary and Notation
Factors, coefficients, terms, exponent, base, constant, variable, binomial, monomial, and polynomial
27


Suggested Instructional Strategies
Resources
Identifying the zeros of a quadratic function by factoring
http://learnzillion.com/lessons/802-identifying-the-zeros- of-a-quadratic-function-by-factoring-out-a-monomial- factor
Identifying zeros of a perfect square quadratic function by factoring http://learnzillion.com/lessons/803-identifying-zeros-of-a- perfect-square-quadratic-function-by-factoring
Identifying zeros of a difference of squares function by factoring http://learnzillion.com/lessons/804-identifying-zeros-of-a- difference-of-squares-function-by-factoring
Factor a quadratic function in standard form to reveal zeros part 1 http://learnzillion.com/lessons/1124-factor-a-quadratic- function-to-reveal-zeros
Factor a quadratic function in standard form to reveal zeros part 2 http://learnzillion.com/lessons/1120-factor-a-quadratic- function-in-standard-form-2-to-reveal-zeros
Factor a quadratic function in standard form to reveal zeros part 3 http://learnzillion.com/lessons/1123-factor-a-quadratic- function-in-standard-form-3-to-reveal-zeros
Factor special products to reveal zeros of a quadratic function http://learnzillion.com/lessons/1121-factor-special- products-1-to-reveal-zeros-of-a-quadratic-function
Identify the zeros of a quadratic function in standard form by factoring http://learnzillion.com/lessons/800-identify-the-zeros-of- a-quadratic-function-in-standard-form-by-factoring
Identify the zeros of a quadratic function in standard form by factoring when the leading coefficient is greater than one http://learnzillion.com/lessons/801-identifying-the-zeros- of-a-quadratic-function-in-standard-form-by-factoring- when-leading-coefficient-is-greater-than-one
• •
Connect point-slope form to transformation of a line
Connect to the forms of a quadratic function
28


Sample Formative Assessment Tasks
Skill-based Task:
Problem Task:
NCTM horseshoes in flight task:
http://nctm.org/standards/content.aspx?id=23749
•
•
• •
Given a quadratic in standard form, rewrite in vertex form and list the properties used in each step
One of the factors of 0.2x3 – 1.2x2 – 0.6x is (x - 2). Find the other factors
Find multiple ways to rewrite x6 – y6
Rewrite x 2/3 in radical form
29


Algebra–Arithmetic with Polynomials and Rational Expressions AAPR
Core Content
Cluster Title: Perform arithmetic operations on polynomials
Standard: A.APR.1: Understand that polynomials form a system analogous to the integers – namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials
Concepts and skills to master
Conceptual:
• Understand closure of polynomials for addition, subtraction and multiplication
Procedural:
• Add and subtract polynomials
• Multiply polynomials using the distributive property, and then simplify
Representational:
• Model the addition, subtraction and multiplication of polynomials
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand operations and properties of integers, including closure
• Understand the distributive property
Procedural:
• Add and subtract like terms
Representational:
• Model operations and properties of integers
Academic Vocabulary and Notation
Like terms, binomial, trinomial, polynomial, closure
Suggested Instructional Strategies
Resources
Polynomial puzzler:
http://illuminations.nctm.org/Lessons.aspx
Algebra tiles, polyominoes
http://nlvm.usu.edu/
Problem Task:
Jeff owns three rectangular pieces of land as shown in the diagram below. Find two representations for the area of the land.
•
•
Use algebra tiles or other manipulatives for addition, subtraction and multiplication of polynomials
Try to find two polynomials whose sum/product is not a polynomial
Sample Formative Assessment Tasks
Skill-based Task:
Multiply (x2 +3x – 5)(x + 4) and determine if the result is a polynomial
30


Algebra–Creating Equations ACED1
Core Content
Cluster Title: Create equations that describe numbers or relationships.
Standard: A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
MASTERY Patterns of Reasoning:
Conceptual:
• Recognize one-variable linear equations and inequalities within contextual situations. (stories)
• Know properties of exponents needed to solve and interpret solutions to exponential equations and inequalities.
Procedural:
• Create one-variable linear equations and inequalities from contextual situations.
• Create one-variable simple rational and simple exponential equations and inequalities
from contextual situations.
• Create one-variable quadratic equations and inequalities from contextual situations.
Representational:
• Model constant change.
• Model change that is not constant.
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand inverse operations to isolate variables and solve equations.
• Understand notation for inequalities.
• Understand the properties of exponents.
Procedural:
• Use inverse operations to isolate variables and solve equations.
• Use the properties of exponents.
• Efficiently use order of operations.
Representational:
• Model linear functions.
Academic Vocabulary and Notation
=, >, <, At least, At most, Greater than, Less than, No less than, No more than
31


Instructional Strategies Used
Guide students in the conversion of contextual information into mathematical notation.
Use story contexts to create linear and exponential equations and inequalities. Have students anticipate when the change is constant and when the change is not constant.
Assessment Tasks Used
Skill-based Task
Juan pays $52.35 a month for his cable bill andJuan pays $52.35 a month for his cable bill and an
Resources Used
Making it Happen (NCTM)
Sections from CAS algebra book:
Sec 5-4n equations of lines.doc lesson plan 5-4.doc
WS 5–4.doc
an additional $.199 for each streamed movie. If his last cable bill was $68.27, how many movies did Juan watch?
additional $.199 for each streamed movie. Gail pays $40.32 a month for her cable bill and an additional $2.49 for each streamed movie. Who has the better deal? Justify your choice.
Problem Task:
32


Algebra–Creating Equations ACED2
Core Content
Cluster Title: Create equations that describe numbers or relationships.
Standard: A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. MASTERY Patterns of Reasoning:
Conceptual:
• Choose the best form of an equation to model linear and exponential functions. Procedural:
• Write an equation to represent a linear equation.
• Write an equation to represent an exponential relationship. Representational:
• Represent a linear relationship with a graph.
• Represent an exponential relationship with a graph.
• Model a data set using an equation.
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand how to read a graph.
• Understand slope as a rate of change of one quantity in relation to another quantity.
Procedural:
• Choose appropriate scales and labels for graphs.
Representational:
• Model location of a point with coordinates.
Academic Vocabulary and Notation
Dependent variable, Domain, Independent variable, Range, Scale, Variable
Instructional Strategies Used
Use technology to explore linear and exponential graphs.
Use story contexts to create linear and exponential graphs. Have students anticipate when the change is constant and when the change is not constant.
Use data sets to generate linear and exponential graphs and equations.
Assessment Tasks Used
Skill-based Task Writeandgraphanequationthatmodels the cost of buying and running an air conditioner with a purchase price of $250 and costs $0.38/hour to run.
Resources Used
Compound interest simulator at: www.illuminatinos.NCTM.org (NCTM)
Problem Task: Jeannettecaninvest$2000at3%interest compounded annually or she can invest $1500 at 3.2% interest compounded annually. Which is the better investment and why?
33


Algebra–Creating Equations ACED4
Core Content
Cluster Title: Create equations that describe numbers or relationships.
Standard: A.CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand the rules for rearranging formulas needed to isolate a particular variable.
Procedural:
• Employ the rules of arithmetic to isolate a particular variable.
Representational:
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the rules of arithmetic used to find the value of an expression or equation
when given all but one value. Procedural:
• Use the order of operations efficiently. Representational:
• Use formulas to represent familiar calculations such as area, volume or constant rates. Academic Vocabulary and Notation
Constant, Formula, Literal equation, Variable
Use formulas from a www.iofm.net/community/kidscorner/maths/common_formulas.htm variety of disciplines such Sections from CAS algebra book:
Instructional Strategies Resources Used Used
asphysics,chemistry,or sports to explore the advantages of different formats of the same formula.
Assessment Tasks Used
Skill-based Task I=Prtsolveforr.
sec2-3nformulasp1.doc lesson plan 2-3.doc Worksheet 2-3.doc
sec 2-4n formulas p2.doc lesson plan 2-4.doc Worksheet 2-2.doc Worksheet 2-3.doc
sec 2-5n formulas p3.doc lesson plan 2-5.doc
Problem Task:
Paul just arrived in England and heard the temperature in degrees Celsius. He remembers that. How will Paul use this formula find the temperature in Farenheit?
34


Algebra–Reasoning with Equations and Inequalities AREI4
Core Content
Cluster Title: Solve equations and inequalities in one variable
Standard: A.REI.4: Solve quadratic equations in one variable
A. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form
B. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives complex solutions and write them as a plus or minus bi for real numbers a and b
Concepts and skills to master
Conceptual:
• •
Understand the zero product property and use it to solve a factorable quadratic equation Know when one method of finding the roots of a quadratic are more efficient than the other
Interpret the discriminant Procedural:
•
• •
•
Complete the square
Solve quadratic equations, including complex solutions, using completing the square,
quadratic formula, factoring and by taking the square root Derive the quadratic formula from completing the square
Representational:
• Use technology to represent the roots of a quadratic equation
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand real numbers and the complex number system • Understand complex numbers
Procedural:
• Simplify radicals
• Factor
Representational:
• Use area models to represent the factors of a quadratic equation
Academic Vocabulary and Notation
Radicals, complex numbers, solve, factor, discriminant
35


Suggested Instructional Strategies
Resources
Web links for
Illuminations: Proof without words and Completing thesquare
Understand the discriminant and nature of the roots
http://learnzillion.com/lessons/275-understand-the- discriminant-and-nature-of-the-roots
Find the vertex of a graph and determine whether it is a maximum or minimum http://learnzillion.com/lessons/269-find-the-vertex- of-a-graph-and-determine-whether-it-is-a-maximum- or-minimum
Factor an expression: taking out the greatest common factor http://learnzillion.com/lessons/271-factor-an- expression-taking-out-the-greatest-common-factor
Find the axis of symmetry of a quadratic equation
http://learnzillion.com/lessons/276-find-the-axis-of- symmetry-of-a-quadratic-equation
Factor a quadratic equation: difference between two squares http://learnzillion.com/lessons/272-factor-a- quadratic-equation-difference-between-2-squares
Draw the graph of a quadratic equation
http://learnzillion.com/lessons/277-draw-the-graph- of-a-quadratic-equation
Factor perfect square trinomials
http://learnzillion.com/lessons/273-factor-perfect- square-trinomials
Determine the domain and range of a parabola: looking at the graph http://learnzillion.com/lessons/278-determine-the- domain-and-range-of-a-parabola-looking-at-the- graph
Factor: using factor sums
http://learnzillion.com/lessons/274-factor-using- factor-sums
•
Use algebra tiles to demonstrate completing the square problems (SeeNCTMMTMS,March2007,p. 403)
36


Sample Formative Assessment Tasks
Skill-based Task:
Solve the equation 6x2 – x – 15 = 0 by factoring and by completing the square. Justify each method using mathematical properties
Transform a quadratic equation by completing the square http://learnzillion.com/lessons/1240-transform-a- quadratic-equation-by-completing-the-square
Transform a quadratic equation by completing the square, a = 1 http://learnzillion.com/lessons/1239-transform-a- quadratic-equation-by-completing-the-square-a1
Problem Task:
Solve the quadratic equation 49x2 – 70x + 37 = 0 using two methods. Describe the advantages of each method
37


Algebra–Reasoning with Equations and Inequalities AREI7
Core Content
Cluster Title: Solve systems of equations
Standard: A.REI.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically (for example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3)
Concepts and skills to master
Conceptual:
• Recognize that solutions of a system that includes a unit circle centered at the origin and
a line with a y-intercept of 0 are points on the unit circle Procedural:
• Solve a simple system consisting of a linear equation and a quadratic equation in two variables graphically
• Solve a simple system consisting of a linear equation and quadratic equation in two variables algebraically
Representational:
• Model the solution of a simple system consisting of a linear equation and a quadratic in
two variables Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand what a system is and the nature of the solutions
• Know that a quadratic function is a vertical parabola
• Know that a quadratic equation can be a parabola or any conic section
Procedural:
• Solve systems of equations
Representational:
• Model the solutions of systems of equations
Academic Vocabulary and Notation
Quadratic function, unit circle, system of equations, conic sections
Suggested Instructional Strategies Resources
• Find the intersection of a line with http://www.learnzillion.com/lessons/228-determine- a y-intercept of 0 and a unit circle whether-a-number-is-real-or-imaginary-isolating-the- centered at the origin with a quadratic-term
solution that is a fraction. The
solution will be a point on the unithttp://www.learnzillion.com/lessons/230-solve-
circle that corresponds to a right triangle and the trigonometric ratios
quadratic-equations-with-real-coefficients-using-the- quadratic-formula
http://www.learnzillion.com/lessons/229-determine- whether-a-number-is-real-or-imaginary-calculating- the-value-of-the-discriminant
http://www.learnzillion.com/lessons/231-solve- equations-completing-the-square
38


Sample Formative Assessment Tasks
Skill-based Task: Problem Task:
Find the intersection of the circle with a For a system consisting of a linear equation and a radius of 1 centered at the origin and the quadratic equation, how many possible solutions are line y = -3(x – 2). Show your work both there? Give an example for each possibility and graphically and algebraically include the graph and system
39


Functions–Interpreting Functions FIF4
Core Content
Cluster Title: Understand the concept of a function and use function notation.
Standard F.IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
Concepts and Skills to Master
Conceptual
• Know what intercepts are.
• Know what increasing, decreasing, positive and negative intervals are.
Procedural
• Given a graph, identify key features such as x- and y-intercepts; intervals where the
function is increasing, decreasing,
• positive, or negative.
• Given a table, identify key features such as x- and y-intercepts; intervals where the
function is increasing, decreasing,
• positive, or negative.
• Find key features of a function.
Representational
• Graph functions using key features of the function.
• Represent the key features of graphs with interval notation and symbols of inequality.
Supports for Teachers
Critical Background Knowledge
Conceptual
• Recognize linear and exponential functions.
Procedural
• Read tables and equations.
Representational
• Graph linear and exponential functions.
Academic Vocabulary and Notation
Decreasing, Increasing, Intercepts, Intervals, Interval notation, Negative, Positive
Suggested Instructional Strategies
• Use graphing technology to explore and identify key features of a function.
• Use key features of a function to graph functions by hand.
Resources
Online graphing calculator at:
http://rentcalculators.org/stheli.html
40


Sample Formative Assessment Tasks
Skill-based Task
Identify the intervals where the function is increasing and decreasing.
Problem Task
Create a story that would generate a linear or exponential function and describe the meaning of key features of the graph as they relate to the story.
41


Functions–Interpreting Functions FIF5
Core Content
Cluster Title: Understand the concept of a function and use function notation.
Standard F.IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relations example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then t be an appropriate domain for the function.
Concepts and Skills to Master
Conceptual
• Know what the domain of a function is.
Procedural
• Identify domains of functions given a graph.
• Identify reasonability of a domain in a particular context.
Representational
• Graph a function, given a restricted domain.
Supports for Teachers
Critical Background Knowledge
Conceptual
• Familiarity with function notation and domain.
• Understand what dependent and independent variables are.
Procedural
• Identify dependent and independent variables.
Representational
• Express functions with function notation.
Academic Vocabulary and Notation
Domain, Dependent variable, Function, Independent variable, Integers,
Suggested Instructional Strategies Resources
•
•
Discuss contexts where the domain Domain representations at:
of a function should be limited to a subset of integers, positive or negative values, or some other restriction to the real numbers.
Find examples of functions with limited domains from other curricular areas (science, physical education, social studies, consumer science).
www.illuminations.NCTM.org
42
h


Sample Formative Assessment Tasks
Skill-based Task Problem Task
You are hoping to make a profit on Create a function in context where the domain would be:
the school play and have determined the function describing the profit to be f(t) = 8t -2654 where t is the number of tickets sold. What is a reasonable domain for this function? Explain.
• All real numbers
• Integers
• Negative numbers • Rational numbers • (10, 40).
43


Functions–Interpreting Functions FIF6
Core Content
Cluster Title: Understand the concept of a function and use function notation.
Standard F.IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a special interval. Estimate the rate of change from a graph. Concepts and Skills to Master
Conceptual
• Understand rate of change for linear and exponential functions.
• Understand interval requirements for functions. Procedural
• Calculate the rate of change given a linear function from the equation or a table.
• Calculate the rate of change over a given interval in an exponential function from an
equation or a table where the
• domain is a subset of the integers.
Representational
• Use a graph to estimate the rate of change over an interval in a linear or exponential
function. Supports for Teachers
Critical Background Knowledge
Conceptual
• Understand definition of slope.
Procedural
• Calculate slope.
Representational
• Represent slope on a graph.
Academic Vocabulary and Notation
Average, Decreasing, Function, Increasing, Interval, Rate of change
Suggested Instructional Strategies Resources
• Use graphical data from birth rates, BMI Drug Filtering
in growing children, electricity rates, population growth or other linear or exponential data to explore and discuss the meaning of rate of change
•
Growth rate At:
www.illuminations.NCTM.org
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Sample Formative Assessment Tasks
Skill-based Task
Find the average rate of change on the interval [-3, 1] Table 1
y 8 3 -2 -7 -12 -17 -22 Table 2
y 6 12 24 48 96 192 384 Table 3
y 7 2 -1 0 2 4 6
Problem Task
The graph models the speed of a
car. Tell a story using the graph to describe what is happening in various intervals.
x
-3
-2
-1
0
1
2
3
x
-3
-2
-1
0
1
2
3
x
-3
-2
-1
0
1
2
3
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Functions–Interpreting Functions FIF7a
Core Content
Cluster Title: Understand the concept of a function and use function notation.
Standard F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
a. Graph linear functions and show intercepts.
Concepts and Skills to Master
Conceptual
• Understand what intercepts are.
Procedural
• Write linear functions in slope-intercept form.
Representational
• Graph lines expressed in slope-intercept form or standard form by hand.
Supports for Teachers
Critical Background Knowledge
Conceptual
• Understand coordinate system including ordered pairs and quadrants..
Procedural
• Find corresponding dependent values given independent values.
Representational
• Graph points on the coordinate plane.
Academic Vocabulary and Notation
Intercept, Linear, Slope,
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Suggested Instructional Strategies
• Allow students to develop graphs from tables and use those graphs to generalize graphing strategies.
• Graph equations generated from real-life contexts.
Resources
Kuta Software Worksheets at:
http://www.kutasoftware.com/free.html
Geogebra
http://www.geogebra.org/cms/
Sections from CAS Algebra book:
Sec 5-3n Intercepts of lines.doc lesson plan 5-3.doc
WS 5–3.doc
Sec 5-4n equations of lines.doc lesson plan 5-4.doc
WS 5–4.doc
Sec 5-5n graphing linear equations.doc lesson plan 5-5.doc
WS 5–5.doc
Sec 7-1n graphing quadratics.doc lesson plan 7-1 parts of parabola.doc ws7-1.doc
ws7-1A.doc
Sec 8-5n exponential functions.doc lesson plan 8 - 5A.doc ws8-5A.doc
lesson plan 8 - 5B.doc ws8-5B.doc
Problem Task
An athlete running the 100 m dash averages 1.02 seconds every 10 meters. Model the situation graphically. If the athlete comes in second by only 0.22 sec. What was the average for the
champion? Model the champion’s race graphically.
Sample Formative Assessment Tasks
Skill-based Task
• Graph the function f(x) = 2x – 3
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Functions–Interpreting Functions FIF7e
Core Content
Cluster Title: Understand the concept of a function and use function notation.
Standard F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
e. Graph exponential functions, showing intercepts and end behavior.
Concepts and Skills to Master
Conceptual
• Understand what intercepts are.
Procedural
• Calculate values for y given values for x in exponential functions.
• Use technology to model complex exponential functions.
• Identify intercepts in graphs of exponential functions.
Representational
• Graph exponential functions by hand.
Supports for Teachers
Critical Background Knowledge
Conceptual
• Understand meaning of exponential notation.
Procedural
• Calculate expressions involving exponents.
Representational
• Graph points on the coordinate plane.
Academic Vocabulary and Notation
End behavior, Exponential, Intercept,
Suggested Instructional Strategies
• Allow students to develop graphs from tables and use those graphs to generalize graphing strategies.
• Graph exponential equations generated from real-life contexts.
Sample Formative Assessment Tasks
Resources
Kuta Software Worksheets at:
http://www.kutasoftware.com/free.html
Geogebra
Skill-based Task
• Graph the function f(x) = 2x
http://www.geogebra.org/cms/
The population of salmon in a lake triples each year. The current population is 472. Model the situation graphically. Include the last three years and the next two. Model the situation with a function.
Problem Task
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Functions–Interpreting Functions FIF8
Core Content
Cluster Title: Interpret functions that arise in applications in terms of a context
Standard: F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of a graph, and interpret these in terms of a context
b. Use the properties of exponents to interpret expressions for exponential functions. (For example, identify percent rate of change in functions such as , and classify them as representing exponential growth or decay)
Concepts and skills to master
Conceptual:
• Understand different properties of functions
• Understand different but equivalent expressions of a function Procedural:
• Factor quadratic functions to show zeros, extreme values and symmetry of the graph
• Complete the square to show zeros, extreme values and symmetry of the graph of a
quadratic function
• Transition between different forms of quadratic functions and identify the advantages of
each Representational:
• Model functions with technology
• Model factors of trinomials Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know key features of quadratic functions
Procedural:
• Graph and show key features of quadratic functions
• Identify transformations of functions
• Interpret key features of a graph
• Multiply binomials
Representational:
• Model linear and exponential functions
• Model the multiplication of binomials
Academic Vocabulary and Notation
Binomial, trinomial, perfect square trinomial, completing the square, zero, extreme values (maximum and minimum), vertex, axis of symmetry
Suggested Instructional Strategies Resources
• Use manipulatives for multiplying, Library of virtual manipulatives, and algebra tiles
factoring, and completing the
square
• Create a graphic organizer
comparing all three forms of quadratic functions
http://nlvm.usu.edu/en/nav/vlibrary.html
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Sample Formative Assessment Tasks
Skill-based Task:
Factor the expression : 2x2 – 9x – 5 Transform f(x) = x2 + x – 12 into another form to identify the zeros
Transorm f(x) = x2 + x – 12 into another form to identify the vertex
Problem Task:
You are the head of the marketing department at Harmonix, and have just determined your revenue can be modeled by the function f(x) = -10x2 + 100x – 210, where x is the amount spend on advertising in thousands of dollars. Determine an advertising budget
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