Evidence Outcomes:

1. (+) Extend polynomial identities to the complex numbers. For example, rewrite as x^2+4 as (x+2i)(x-2i). (CCSS: HS.N-CN.C.8)

2. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (CCSS: HS.N-CN.C.9

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Part major and part ADVANCED

• Make sense of problems and persevere in solving them

• Use appropriate tools strategically

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Use complex numbers in polynomial identities and equations

  • Complex numbers arise when working with quadratic equations

In order to meet these essential understandings, students must know...

• (+)That Polynomial identities extend to complex numbers

• (+)The Fundamental Theorem of Algebra

In order to meet these essential understandings, students must be able to...

• (+) Rewrite quadratics as the product of two complex numbers x^2 + 4 = (x + 2i)(x - 2i)

Evidence Outcomes:

1. Use the formula for the sum of a finite geometric series (when the common ratio is not ) to solve problems. For example, calculate mortgage payments.★ (CCSS: HS.A-SSE.B.4)

   • (+) Derive the formula for the sum of a finite geometric series (when the common ratio is not ). (CCSS: HS.A-SSE.B.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Look for and make use of structure

• Look for and express regularity in repeated reasoning

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Write expressions in equivalent forms to solve problems (Modeling)

  • Move between multiple forms of functions and using them to find key characteristics of the function and to solve

  • Recognize the difference in the structure of linear, quadratic, and other equations, and apply strategies to solve

    • This standard is the procedural piece of this concept - the previous standard is the conceptual piece.

• Math 3 - Polynomial/Rational

In order to meet these essential understandings, students must know...

In order to meet these essential understandings, students must be able to...

• Use the formula for the sum of a finite geometric series

  • (+)Derive the formula when the ratio is not 1

Evidence Outcomes:

1. Explain 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. (CCSS: HS.A-APR.A.1)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Look for and make use of structure

• Look for and express regularity in repeated reasoning

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Perform arithmetic operations with polynomials

  • Understand that polynomials form a system analogous to the integers (closed under addition, subtraction, and multiplication)

In order to meet these essential understandings, students must know...

• Understand how types of numbers and operations form a closed system

• How operations on polynomials yield polynomials

In order to meet these essential understandings, students must be able to...

• Explain how polynomials are closed under operations of +, -, x

• Add, subtract, multiply polynomials

• Make hypotheses and draw conclusions about polynomials and operations on them

Evidence Outcomes:

1. Know and apply the Remainder Theorem. For a polynomial p(x) and a number a, the remainder on division by x-a is p(a), so p(a)=0 if and only if (x-a) is a factor of p(x). (Students need not apply the Remainder Theorem to polynomials of degree greater than 4.) (CCSS: HS.A-APR.B.2)

2. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. (CCSS: HS.A-APR.B.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Make sense of problems and persevere in solving them

• Reason abstractly and quantitatively

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Understand the relationship between zeros and factors of polynomials

  • Identify zeros of polynomials and use them to construct a rough graph of the function

In order to meet these essential understandings, students must know...

• The Remainder Theorem up to degree 4

• The relationship between zeros and a polynomial in factored form and how they connect to real-world context

• The Zero Product Property of Polynomials

In order to meet these essential understandings, students must be able to...

• Use appropriate methods to reveal zeros of polynomials

• Make sense of zeros within the context of graphs, tables, real-world context, and equations in factored forms

• Use zeros to construct a polynomial function

• Use the Remainder Theorem to find zeros/factors of polynomials

Evidence Outcomes:

1. (+) Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2+y^2 )^2=(x^2-y^2)^2+(2xy)^2 can be used to generate Pythagorean triples. (CCSS: HS.A-APR.C.4)

2. (+) Know and apply the Binomial Theorem for the expansion of in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) (CCSS: HS.A-APR.C.5)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Look for and express regularity in repeated reasoning

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• (+) Use polynomial identities to solve problems

In order to meet these essential understandings, students must know...

• (+) The connection between the Binomial Theorem and coefficients of Pascal’s Triangle

In order to meet these essential understandings, students must be able to...

• (+) Prove polynomial identities (ex. Generating pythagorean triples)

• (+) Use the Binomial Theorem to expand expressions

Evidence Outcomes:

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. (CCSS: HS.A-CED.A.1)

2. Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales. (CCSS: HS.A-CED.A.2)

3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. (CCSS: HS.A-CED.A.3)

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. (CCSS: HS.A-CED.A.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Reason abstractly and quantitatively

• Model with mathematics

• Use appropriate tools strategically

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Create equations that describe numbers or relationships (Modeling)

  • Model and solve problems arising in the real-world

Math 1 - Linear/ Exponential

Math 2 - Quadratic/Exponential

Math 3 - Polynomial/Rational

In order to meet these essential understandings, students must know...

• When to use one-variable equations vs two-variable equations

• Solutions can be nonviable depending on the real-life context

In order to meet these essential understandings, students must be able to...

• Create equations and inequalities in one variable and use them to solve problems

  • Linear, quadratic, and simple rational and exponentials

• Create equations in two or more variables and graph (label scales and axes)

• Represent constraints using equations or inequalities, including systems of equations/inequalities

• Use structure and operations to rearrange formulas to isolate a variable (ex. solve for ‘a’ given the formula: F = ma)

• Interpret mathematical results in the context of the situation and reflect on whether the results make sense and serve a purpose

• Use pencil and paper and technology to make sense of and solve mathematical equations

Evidence Outcomes:

1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. (CCSS: HS.A-REI.A.1)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Construct viable arguments and critique the reasoning of others

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Understand solving equations as a process of reasoning and explain the reasoning

Math 3 - Radical/Rational

In order to meet these essential understandings, students must know...

• What an extraneous solution is and why it occurs

In order to meet these essential understandings, students must be able to...

• Explain each step in solving a simple equation, starting from the assumption that the original equation has a solution using written communication skills

• Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise

Evidence Outcomes:

1. Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. (CCSS: HS.A-REI.D.11)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Make sense of problems and persevere in solving them

• Use appropriate tools strategically

• Attend to precision

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Represent and solve equations and inequalities graphically

  • Use characteristics and structures of function families to understand and generalize about solutions

Math 3 - Combine polynomial, rational, radical, absolute value, exponential

In order to meet these essential understandings, students must know...

In order to meet these essential understandings, students must be able to...

• Explain why the x-coordinate of an intersection point of two graphs, f(x) and g(x), is the solution to the equation f(x) = g(x)

  • Find solutions approximately by using technology, table of value, or successive approximations

  • Polynomial, rational, absolute value, exponential, and logarithmic

• Label graphs and specify units of measure with a degree of precision appropriate for the problem context.

Evidence Outcomes:

1. 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (CCSS: HS.F-IF.B.4)

2. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.* (CCSS: HS.F-IF.B.5)

3. Calculate and interpret the average rate of change presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph.* (CCSS: HS.F-IF.B.6)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Model with mathematics

• Use appropriate tools strategically

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Interpret functions that arise in applications in terms of the context (Modeling)

  • Explain how real-world context influences the domain of the function

Math 3 - Rational/Square Root/Cube Root

In order to meet these essential understandings, students must know...

• Key features and understand how various functions behave in different representations

• Similarities and differences between linear, exponential, quadratic, polynomial, rational, radical, and trigonometric functions

In order to meet these essential understandings, students must be able to...

• Interpret key features of graphs and tables and sketch graph showing key features

  • Intercepts

  • Intervals of inc, dec, positive, negative

  • Relative mins and maxs

  • Symmetries

  • End behavior

  • Periodicity

• Describe an appropriate domain for a function, especially functions that model real-world situations

Evidence Outcomes:

1. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* (CCSS: HS.F-IF.C.7)

   • Graph linear and quadratic functions and show intercepts, maxima, and minima. (CCSS: HS.F-IF.C.7.a)

   • Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (CCSS: HS.F-IF.C.7.b)

   • Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (CCSS: HS.F-IF.C.7.c)

   • Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. (CCSS: HS.F-IF.C.7.e)

2. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (CCSS: HS.F-IF.C.8)

1. • Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (CCSS: HS.F-IF.C.8.a)

   • Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (​0.97)^t, y = (1.01)12^t, and y = (1.2)^t /10, and classify them as representing exponential growth or decay. (CCSS: HS.F-IF.C.8.b)

2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (CCSS: HS.F-IF.C.9)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Reason abstractly and quantitatively

• Use appropriate tools strategically

• Attend to precision

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Analyze functions using different representations (Modeling)

  • Reason abstractly and understand the connections between the symbolic representation, the table of values, and the key features of the graph of a function (Rule of 4)

  • Compare properties of functions written in different forms

Math 3 - Rational/Radical

In order to meet these essential understandings, students must know...

• Vocabulary: intercepts, max/min, zeros, (+)asymptotes, end behavior, period, midline, amplitude

• Which forms of functions are used to find specific features of that function

In order to meet these essential understandings, students must be able to...

• Graph functions and show key features both by hand and using technology

  • Square root, cube root, piece-wise (step) and absolute value

  • Polynomial functions: zeros and end behavior

  • Exponential and logarithmic functions: intercepts, end behavior

  • Trigonometric functions: period, midline, amplitude

• Write and compare functions in equivalent forms to reveal properties

• Use models of functions (equations, graphs, tables, scenarios) to compare key features of multiple, different functions

Evidence Outcomes:

1. Write a function that describes a relationship between two quantities.* (CCSS: HS.F-BF.A.1)

   • Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (CCSS: HS.F-BF.A.1.b)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Model with mathematics

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Build a function that models a relationship between two quantities (Modeling)

  • Use both explicit and recursive formulas

Math 3 - All

In order to meet these essential understandings, students must know...

In order to meet these essential understandings, students must be able to...

• Combine function types (subtract a constant function from an exponential function)

• (+)Compose functions

Evidence Outcomes:

1. Identify the effect on the graph of replacing f(x) by f(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. (CCSS: HS.F-BF.B.3)

2. Find inverse functions. (CCSS: HS.F-BF.B.4)

   • 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. For example, f(x)=2x^3 or f(x)=(x+1)/(x-1) for x≠1. (CCSS: HS.F-BF.B.4.a)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Construct viable arguments and critique the reasons of others

• Attend to precision

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Build new functions from existing functions

  • Use function notation to write transformations

  • Finding inverse functions

Math 1 - Linear/Exponential

Math 2 - Quadratic/Exponential/Absolute Value

Math 3 - above and include radical, rationals

In order to meet these essential understandings, students must know...

• Vocabulary: even/odd functions, function family, inverse functions, transformation

• How transformations in function notation affect graphs

  • f(x)+k, f(x+k), kf(x), f(kx)

In order to meet these essential understandings, students must be able to...

• Show how transformations by k appear on graphs of functions

• Find transformations from graph and write in function notation

• Use technology to create, describe and analyze related functions

• Communicate explanations of generalities across function families using accurate terms, definitions and mathematical symbols

• Find the inverse function for a simple function f that has an inverse

Evidence Outcomes:

1. For exponential models, express as a logarithm the solution to ab^ct=d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. (CCSS: HS.F-LE.A.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Make sense of problems and persevere in solving them

• Model with mathematics

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Construct and compare linear, quadratic, and exponential models and solve problems (Modeling)

  • Apply models to real-world scenarios

Math 3 - Logarithms as solutions to Exponentials

In order to meet these essential understandings, students must know...

In order to meet these essential understandings, students must be able to...

• For exponential models, express as a logarithm and evaluate using technology (the base is 2, 10, or e)

Evidence Outcomes:

1. (+) Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle. (CCSS: HS.F-TF.A.1)

2. (+) Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. (CCSS: HS.F-TF.A.2)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

ADVANCED

• Look for and make use of structure

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Extend the domain of trigonometric functions using the unit circle

  • Using radian measures of angles

In order to meet these essential understandings, students must know...

• Radians measure distance around a circle in terms of the radius of one unit

• The radian measure stays the same for all equivalent angles, regardless of the circle’s size.

• Radian measure of an angle can be defined as the quotient of the arc length to radius- making it dimensionless-explaining why the “unit” is often omitted when measuring angles in radians

In order to meet these essential understandings, students must be able to...

• Explain the connection between the unit circle and graphs of trigonometric functions

Evidence Outcomes:

1. (+) Model periodic phenomena with trigonometric functions with specified amplitude, frequency, and midline.* (CCSS: HS.F-TF.B.5)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

ADVANCED

• Reason abstractly and quantitatively

• Model with mathematics

• Look for and express regularity in repeated reasoning

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• (+)Model periodic phenomena with trigonometric functions (Modeling)

  • (+)Graph trigonometric functions on a coordinate plane, where the x-axis is the radian angle measures

In order to meet these essential understandings, students must know...

• (+)Which real world scenarios are periodic

• (+)The relationship between the unit circle and the graphs of the trig functions

In order to meet these essential understandings, students must be able to...

• (+) Model periodic functions using trigonometric functions, using specified amplitude, frequency, and midline

• (+) Use properties of periodic functions to gain deeper understanding of real world scenarios

Evidence Outcomes:

1. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages and identify data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. (CCSS: HS.S-ID.A.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Reason abstractly and quantitatively

• Model with mathematics

• Use appropriate tools strategically

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Summarize, represent, and interpret data on a single count or measurement variable

  • Use statistics and statistical reasoning to make sense of, interpret, and generalize about real-world situations

In order to meet these essential understandings, students must know...

• Vocabulary: dot plot, histogram, box plot, median, mean, interquartile range, standard deviation, shape, center, spread, outlier, normal distribution

In order to meet these essential understandings, students must be able to...

• Create normal curves from mean and standard deviation to estimate percentages (68-95-99.7%)

  • Identify data sets for which such a procedure is not appropriate

  • Use calculators, spreadsheets, and tables to estimate areas under the normal curve

Evidence Outcomes:

1. Describe statistics as a process for making inferences about population parameters based on a random sample from that population. (CCSS: HS.S-IC.A.1)

2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? (CCSS: HS.S-IC.A.2)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Construct viable arguments and critique the reasoning of others

• Attend to precision

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Understand and evaluate random processes underlying statistical experiments

  • Use a variety of statistical tools to construct and defend logical arguments based on data

In order to meet these essential understandings, students must know...

• Statistics is used to make inferences about population parameters based on a random sample from the population

• The difference between a statistic and a parameter

In order to meet these essential understandings, students must be able to...

• Decide if a specified model is consistent with results from a data generating process (simulation)

Evidence Outcomes:

1. Identify the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. (CCSS: HS.S-IC.B.3)

2. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. (CCSS: HS.S-IC.B.4)

3. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. (CCSS: HS.S-IC.B.5)

4. Evaluate reports based on data. Define and explain the meaning of significance, both statistical (using p-values) and practical (using effect size). (CCSS: HS.S-IC.B.6)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Construct viable arguments and critique the reasoning of others

• Model with mathematics

• Look for and express regularity in repeated reasoning

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Make inferences and justify conclusions from sample surveys, experiments, and observational studies

  • Apply statistical methods to interpret information and draw conclusions in real-world context

In order to meet these essential understandings, students must know...

• Vocabulary: sample surveys, experiments, observational studies, treatments, margin of error, p-values, effect size

• What reasonable means in terms of statistics

In order to meet these essential understandings, students must be able to...

• Identify the purposes of and differences between sample surveys, experiments, and observational studies

  • How does randomization relate to each

• Use data to estimate a population mean or proportion

• Develop a margin of error

• Use data from a randomized experiment to compare two treatments and decide if differences between parameters are significant

• Define and explain the meaning of significance, statistical (p-value) and practical (effect size)

• Evaluate reports based on data

Evidence Outcomes:

1. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). (CCSS: HS.S-MD.B.6)

2. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). (CCSS: HS.S-MD.B.7)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

ADVANCED

• Model with mathematics

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• (+)Use probability to evaluate outcomes of decisions

  • (+))Analyze decisions and strategies using probability concepts

In order to meet these essential understandings, students must know...

• (+)What “fair” means in the context of statistics

In order to meet these essential understandings, students must be able to...

• (+)Use a random process to make decisions (RNG, Random number tables, big ol’ hat and mix it up)

Evidence Outcomes:

1. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. (CCSS: HS.G-GMD.B.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Make sense of problems and persevere in solving them

• Reason abstractly and quantitatively

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Visualize relationships between two-dimensional and three-dimensional objects

In order to meet these essential understandings, students must know...

• Vocabulary: cross-section

In order to meet these essential understandings, students must be able to...

• Identify shapes of cross sections of 3D objects

• Identify 3D objects formed by rotations of 2D objects

Evidence Outcomes:

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* (CCSS: HS.G-MG.A.1)

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* (CCSS: HS.G-MG.A.2)

3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* (CCSS: HS.G-MG.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Make sense of problems and persevere in solving them

• Model with mathematics

• Use appropriate tools strategically

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Apply geometric concepts in modeling situations (Modeling)

  • Use geometric shapes, their measures, and their properties to describe objects and solve problems in the real-world

In order to meet these essential understandings, students must know...

• Vocabulary: Density

• Common area and volume formulas

In order to meet these essential understandings, students must be able to...

• Use geometric shapes, their measures, and their properties to describe objects

• Apply concepts of density based on area and volume (ex. population density)

• Apply geometric methods to solve design problems (ex. Minimize cost)