Evidence Outcomes:

1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. (CCSS: HS.N-Q.A.1)

2. Define appropriate quantities for the purpose of descriptive modeling. (CCSS: HS.N-Q.A.2)

3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. (CCSS: HS.N-Q.A.3))

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

• Reason abstractly and quantitatively

• Attend to precision

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

• Reason quantitatively and use units to solve problems

  • Use units to understand problems and guide solutions

  • Use units to determine the scale and origin in graph

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

• Units can be used to better understand scenarios

• The meaning of unit quantities and not just how to compute them

• How to specify appropriate units when labeling axes (domain/range)

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

• Solve problems with appropriate units

• Create representations of problems with consideration to units

• Choose and interpret units consistently in formulas

• Discern the difference between: “Let x = number of gallons of gas” and “Let x = gas”

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

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

• Slope-intercept form, Point Slope form, Standard form of linear equations

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

• 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 1 - Linear

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

• Properties of operations can be used to change expressions on either side of the equation to equivalent expressions

  • Use reverse order of operations to solve equations

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

Evidence Outcomes:

1. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (CCSS: HS.A-REI.B.3)

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

• 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...

• Solve equations and inequalities in one variable

Math 1 - Linear/Exponential

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

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

• Solve linear equations and inequalities in one variable

• Including equations with coefficients represented by letters

Evidence Outcomes:

1. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. (CCSS: HS.A-REI.C.5)

2. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (CCSS: HS.A-REI.C.6)

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

• Reason abstractly and quantitatively

• Model with mathematics

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

• Solve systems of equations

• Solve systems of linear equations algebraically and graphically

Math 1 - Linear Systems

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

• When and how to use the elimination method to solve a system of equations

• When and how to use substitution to solve a system of equations

• How to use graphs to approximate solutions of systems of equations

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

• Solve systems of linear equations exactly and approximately using graphs and algebraic methods

Evidence Outcomes:

1. Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (CCSS: HS.A-REI.D.10)

2. 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)

3. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. (CCSS: HS.A-REI.D.12)

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 1 - Linear/Exponential

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

• Graphs of equations are different from graphs of inequalities.

• The graph of an equation is the set of all solutions to the equation

• The graph of a linear inequality in 2 variables is a half-plane and the graph of a system of inequalities is the intersection of corresponding half-planes

• How to make sense of correspondences between equations, verbal descriptions, tables, and graphs

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

• Use technology to reason about and solve systems of equations and inequalities

• Analyze and use the information presented in equations and visually in graphs

• 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

  • Linear

• Graph the solutions to a linear inequality in two variables

• Graph the solution set to a system of linear inequalities in two variables

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

Evidence Outcomes:

1. Explain that a function is a correspondence from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. If f  is a function and x is an element of its domain, then f (x) denotes the output of f  corresponding to the input x. The graph of f  is the graph of the equation y=f (x0. (CCSS: HS.F-IF.A.1)

2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (CCSS: HS.F-IF.A.2)

3. Demonstrate that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0)=f(1)=1, f(n+1)=f(n)+f(n-1) for n≥1 . (CCSS: HS.F-IF.A.3)

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

• Reason abstractly and quantitatively

• 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...

• Understand the concept of a function and use function notation

  • Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of context

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

• If f is a function and x is an element of its domain, then f(x) represents the output of f corresponding to the input x.

• One-to-one correspondence for functions

• How to connect function notation with real-world scenarios

• What makes a function a function

• Difference between a function and an equation

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

• Describe sequences as functions, including recursive

• Use accurate terms and symbols when describing functions and using function notation

• Evaluate functions for input in their domains

• Interpret statements involving function notation in terms of 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 1 - Linear/ Exponential

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, and exponential

• Definition of domain

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

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

  • Intercepts

  • Intervals of inc, dec, positive, negative

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

• Use functions and their graphs to model, interpret, and generalize about real-world situations

• Calculate and interpret the average rate of change presented symbolically or as a table

  • Estimate the rate of change from a graph

• Use technology as a tool to visualize and understand how various functions behave

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)

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 1 - Linear/ Exponential

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

• Vocabulary: intercepts, max/min, zeros, end behavior

• 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

  • Linear and intercepts

• 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)

   • Determine an explicit expression, a recursive process, or steps for calculation from a context. (CCSS: HS.F-BF.A.1.a)

   • 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)

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.* (CCSS: HS.F-BF.A.2)

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 1 - Linear/ Exponential

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

• Vocabulary: explicit expressions, recursive process, arithmetic sequences, geometric sequences

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

• Write an explicit expression, recursive process, or steps for calculation from a context

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

• Write arithmetic and geometric sequences both recursively and with an explicit formula

• Apply understanding of functions to real-world contexts

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)

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

Math 1 - Linear/Exponential

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

• Vocabulary: function family, and 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 graphs 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

Evidence Outcomes:

1. Distinguish between situations that can be modeled with linear functions and with exponential functions. (CCSS: HS.F-LE.A.1)

   • Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. (CCSS: HS.F-LE.A.1.a)

   • Identify situations in which one quantity changes at a constant rate per unit interval relative to another. (CCSS: HS.F-LE.A.1.b)

   • Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. (CCSS: HS.F-LE.A.1.c)

2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (CCSS: HS.F-LE.A.2)

3. Use graphs and tables to describe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (CCSS: HS.F-LE.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

• 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 1 - Linear/ Exponential

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

• The difference between situations that can be modeled by linear and exponential functions

• The connection between table, graph, and function notation to better understand the function

• Exponential functions will always surpass all other functions in the long run

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

• Prove that linear functions change by equal differences over equal intervals and exponentials grow by equal factors

• Identify situations where one quantity changes at a constant rate per unit interval

• Identify situations where one quantity changes at a constant percent rate per unit interval

• Construct linear and exponential functions given any form from the rule of 4 including arithmetic and geometric sequences

• Use graphs and tables to describe that an exponential function surpasses linear and polynomial functions in the long run

• Compare and contrast any two functions

Evidence Outcomes:

1. Interpret the parameters in a linear or exponential function in terms of a context. (CCSS: HS.F-LE.B.5)

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

• Reason abstractly and quantitatively

• Model with mathematics

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

• Interpret expressions for functions in terms of the situation they model (Modeling)

  • Use mathematics to model, interpret, and reason about real-world contexts

Math 1 - Linear and exponential

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

• What each component/constant of a function represents

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

• Communicate the meaning of a mathematical model of a real-world situation both mathematically and contextually

• Represent a situation mathematically using symbols

• Translate symbols and mathematical representations into a meaningful context

Evidence Outcomes:

1. Model data in context with plots on the real number line (dot plots, histograms, and box plots). (CCSS: HS.S-ID.A.1)

2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. (CCSS: HS.S-ID.A.2)

3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (CCSS: HS.S-ID.A.3)

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, shape, center, spread, outlier

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

• Model data with dot plots, histograms, and box plots

• Compare the appropriate measures of center and spread for two or more data sets

• Interpret differences in shape, center, and spread of data sets

  • Account for possible effects of outliers

Evidence Outcomes:

1. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. (CCSS: HS.S-ID.B.5)

2. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (CCSS: HS.S-ID.B.6)

   • Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. (CCSS: HS.S-ID.B.6.a)

   • Informally assess the fit of a function by plotting and analyzing residuals. (CCSS: HS.S-ID.B.6.b)

   • Fit a linear function for a scatter plot that suggests a linear association. (CCSS: HS.S-ID.B.6.c)

3. Distinguish between correlation and causation. (CCSS: HS.S-ID.C.9)

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

• 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...

• Summarize, represent, and interpret data on two categorical and quantitative variables

  • Analyze, synthesize, and interpret information from scatter plots, residual plots, and two-way tables

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

• Vocabulary: two-way frequency table, relative frequencies, joint, marginal, conditional relative frequencies, scatter plot, residuals, linear association

• Understand the difference between correlation and causation

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

• Create two-way tables

• Create scatter plots

  • Fit a function to the scatter plot (linear, quadratic, and exponential models)

  • Graph a residual plot and use it to determine the appropriateness of the model

• Use technology to compute with large data sets then interpret and make sense of the results

Evidence Outcomes:

1. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. (CCSS: HS.S-ID.C.7)

2. Using technology, compute and interpret the correlation coefficient of a linear fit. (CCSS: HS.S-ID.C.8)

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

• Reason abstractly and quantitatively

• Use appropriate tools strategically

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

• Interpret linear models

  • Interpret slope, y-intercept, and correlation coefficient of a linear model in the context of the data

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

• Vocabulary: Correlation coefficient, slope, intercept

• When the slope and intercept have no meaning within the context of the situation (oftentimes because of extrapolation)

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

• Use technology to interpret the correlation coefficient

• Make meaning of statistics in their lives outside of school

Evidence Outcomes:

1. State precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. (CCSS: HS.G-CO.A.1)

2. Represent transformations in the plane using e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). (CCSS: HS.G-CO.A.2)

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (CCSS: HS.G-CO.A.3)

4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (CCSS: HS.G-CO.A.4)

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools (e.g., graph paper, tracing paper, or geometry software). Specify a sequence of transformations that will carry a given figure onto another. (CCSS: HS.G-CO.A.5)

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

• 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...

• Experiment with transformations in the plane

  • Understand and represent transformations using a variety of tools

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

• Vocabulary: angle, circle, perpendicular line, parallel line, line segment, based on constructions

• The meaning of rotations, reflections, and translations based on angles, circles, perpendicular lines, parallel lines and line segments

• Rigid transformations vs fluid transformations

  • Shift, Rotate, Reflect vs Stretch/Compression

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

• Represent transformations in the plane using tools and technology

• Describe transformations using points, one as the input and one as the output

• Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself

• Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments

• Perform transformations of a given figure using appropriate tools (graph/tracing paper or technology)

• Determine a sequence of transformations to move a given figure onto another

Evidence Outcomes:

1. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. (CCSS: HS.G-CO.B.6)

2. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. (CCSS: HS.G-CO.B.7)

3. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (CCSS: HS.G-CO.B.8)

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 congruence in terms of rigid motions

  • Triangle Congruence Theorems

  • Corresponding Parts of Congruent Figures are Congruent and its converse

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

• Rigid motions create congruent figures

• Two figures are congruent if you can use rigid motions to move one onto another

• The minimum amount of information you need to know about two triangles in order to determine if they are congruent

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

• Use congruent parts of two triangles to show congruence

• Use the definition of congruence to show parts of triangles are congruent

• Use Triangle Congruence Theorems to determine congruent triangles (ASA, SSS, SAS)

Evidence Outcomes:

1. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. (CCSS: HS.G-CO.D.12)

2. (+) Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (CCSS: HS.G-CO.D.13)

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...

• Make geometric constructions

  • Using a variety of tools

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

• Vocabulary: Segment, angle, bisecting, perpendicular lines, perpendicular bisector, parallel, (+) equilateral triangle, (+)square, (+)regular hexagon, (+)inscribed

• The steps needed to create constructions

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

• Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, technology)

• Copy a segment and an angle

• Bisect a segment and an angle,

• Construct perpendicular lines (including perpendicular bisector), parallel lines, lines of reflection

• (+) Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle

Evidence Outcomes:

1. Use coordinates to prove simple geometric theorems algebraically. For example, 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). (CCSS: HS.G-GPE.B.4)

2. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). (CCSS: HS.G-GPE.B.5)

3. Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles.* (CCSS: HS.G-GPE.B.7)

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

• Reason abstractly and quantitatively

• 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...

• Use coordinates to prove simple geometric theorems algebraically (Modeling)

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

• Distance formula

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

• Use coordinates to prove theorems (ex. prove or disprove that a figure in the coordinate plane is a rectangle)

• Prove and use the slope criteria for parallel and perpendicular lines

• Use the distance formula (pythagorean theorem) to compute side lengths to find perimeter and area of triangles and rectangles.