Evidence Outcomes:

1. Demonstrate informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Define irrational numbers as numbers that are not rational. (CCSS: 8.NS.A.1)

2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. (CCSS: 8.NS.A.2)

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

• Use appropriate tools to deepen understanding of mathematical concepts

• Articulate how mathematical concepts relate to one another in the context of a problem or abstract relationships

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

• Investigate rational and irrational numbers and their relative approximate positions on a number line

• Use technology to understand repetition in decimal expansions

• Apply understanding of rational and irrational numbers to work within the structure of the real number system

• Recognize when a decimal expansion can and cannot be represented by a rational number

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

• That there are numbers that are not rational

• Rational numbers have terminating or repeating decimals

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

• Approximate irrational numbers using rational numbers

• Demonstrate informally that every number has a decimal expansion

• Convert a decimal into a rational number

• Locate irrational numbers approximately on a number line

Evidence Outcomes:

1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^2×3^(-5)=3^(-3)=1/3^3 =1/27 (see image at bottom of page for example). (CCSS: 8.EE.A.1)

2. Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares (up to 100) and cube roots of small perfect cubes (up to 64). Know that √2 is irrational. (CCSS: 8.EE.A.2)

3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 10^8 and the population of the world as 7 times 10^9, and determine that the world population is more than 20 times larger. (CCSS: 8.EE.A.3)

4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. (CCSS: 8.EE.A.4)

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

• Use appropriate tools to deepen understanding of mathematical concepts

• Articulate how mathematical concepts relate to one another in the context of a problem or abstract relationship

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

• Reason about unusually large or small quantities

• Use technology to compute with and approximate radicals and roots and understand how such tools represent scientific notation

• Explore the structure of numerical expressions with integer exponents

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

• Properties of integer exponents

• Positive integer exponents are equivalent to repeated multiplication

• Negative integer exponents are equivalent to repeated division

• What an irrational number is

• The Real Number System

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

• Apply properties of integer exponents to generate equivalent expressions

• Use square roots and cube roots to solve equations

  • Perfect squares up to 100

  • Perfect cubes up to 64

• Compare large numbers represented by a single digit times an integer power of 10

• Perform operations with numbers expressed in scientific notation

  • Choose units of appropriate size for measurements of very large or very small quantities

  • Interpret scientific notation

Evidence Outcomes:

1. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. (CCSS: 8.EE.B.5)

2. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. (CCSS: 8.EE.B.6)

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

• Fluidly move between contextualizing and decontextualizing during the problem solving process

• Use appropriate tools to deepen understanding of mathematical concepts

• Articulate how mathematical concepts relate to one another in the context of a problem or abstract relationships

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

• Make connections between representations of linear growth

• Transfer knowledge of constant rates of change to graphs and equations

• Make sense of and compare proportional relationships represented in different forms

• Compare, contrast, and make claims with proportional relationships based on properties of equations, tables, and/or graphs

• Understand the structure of proportional relationships in both equations and graphs

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

• The formula for equation of a line

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

• Graph proportional relationships

• Interpret the unit rate as the slope of the graph

  • Find slope algebraically using yx

  • Find slope using a graph

• Compare two different relationships represented in different ways (ex. Equation and a graph)

• Use similar triangles to explain why the slope is the same between any two point on a non-vertical line

• Derive the y = mx and y = mx + b equations (Why not y = b + mx?)

Evidence Outcomes:

1. Solve linear equations in one variable. (CCSS: 8.EE.C.7)Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). (CCSS: 8.EE.C.7.a) Solve linear equations with rational number coefficients, including equations with variables on both sides and whose solutions require expanding expressions using the distributive property and collecting like terms. (CCSS: 8.EE.C.7.b)

2. Analyze and solve pairs of simultaneous linear equations. (CCSS: 8.EE.C.8) Explain that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. (CCSS: 8.EE.C.8.a) Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. (CCSS: 8.EE.C.8.b) Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. (CCSS:8.EE.C.8.c)

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

• Fluidly move between contextualizing and decontextualizing during the problem solving process

• Based on an understanding of any problem, initiate a plan, execute it, and evaluate the reasonableness of the solution

• Examine and apply a variety of methods to accurately and efficiently solve problems

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

• Solve problems involving linear equations and systems of linear equations with accuracy that makes sense in the real-world context being modeled

• Solve problems that require a system of linear equations in two variables

• Model real-world problems with linear equations and systems of linear equations, with variables defined in their real-world contexts

• Recognize the structure of equations and systems of equations that produce one, infinitely many, or no solution

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

• that solutions to systems are intersection points

• What equations look like that have 0, 1, or infinitely many solutions

  • x = a, a = a, a = b; a and b are different numbers

• Standard form of a linear equation

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

• Solve linear equations in one variable

  • 0, 1, and infinitely many solutions

  • Rational number coefficients

  • Variables on both sides

  • Distribution

• Solve pairs of simultaneous linear equations

  • Explain that solutions are intersection points

  • Two linear equations, algebraically (written in standard form)

  • Solve simple cases by inspection

  • Real-world systems and other mathematical problems

Evidence Outcomes:

1. Define a function as a rule that assigns to each input exactly one output. Show that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required for Grade 8.) (CCSS: 8.F.A.1)

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 linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. (CCSS: 8.F.A.2)

3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4) and (3, 9), which are not on a straight line. (CCSS: 8.F.A.3)

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

• Use appropriate tools to deepen understanding of mathematical concepts

• Articulate how mathematical concepts relate to one another in the context of a problem or abstract relationships

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

• Make connections between the information gathered through tables, equations, graphs, and verbal descriptions of functions

• Define variables as quantities and interpret ordered pairs in terms of those variables

• With and without technology, analyze and describe non-linear functions using equations, graphs and tables

• Recognize patterns of linear growth

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

• That a function is a rule that assigns to each input exactly one output

• That a function can be represented in multiple ways

  • Algebraically, graphically, tabular, description

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

• Show that the graph of a function is the set of ordered pairs consisting of an input and output

• Compare different representations of two functions

  • Algebraically, graphically, tabular, verbal descriptions

• Interpret the equation y = mx + b as defining a linear function whose graph is a line

• Understand examples of functions that are not linear (ex. Area of a square)

Evidence Outcomes:

1. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. (CCSS: 8.F.B.4)

2. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. (CCSS: 8.F.B.5)

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

• Communicate effectively based on purpose, task, and audience

• Fluidly move between contextualizing and decontextualizing during the problem solving process

• Examine and apply a variety of methods to accurately and efficiently solve problems

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

• Describe the qualitative features of linear or nonlinear functions

• Model real-world situations with linear functions

• Use strategies to calculate the rate of change in a linear function

• Use properties of linear functions to create equations

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

• The form of a linear equation

  • What values are needed to write a linear equation

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

• Construct a linear function

  • Determine the rate of change and initial value from a description or given two (x, y) values

    • (x,y) values can be read from a table or graph

  • Interpret a rate of change and initial value within the context of a situation, and in terms of its graph or a table of values

• Describe qualitatively the relationship between quantities by analyzing a graph

  • Where a function is increasing, decreasing, linear or nonlinear

  • Sketch a graph of a function described verbally

Evidence Outcomes:

1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. (CCSS: 8.SP.A.1)

2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. (CCSS: 8.SP.A.2)

3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr. as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. (CCSS: 8.SP.A.3)

4. Explain that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

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

• Fluidly move between contextualizing and decontextualizing during the problem solving process

• Use appropriate tools to deepen understanding of mathematical concepts

• Articulate how mathematical concepts relate to one another in the context of a problem or abstract relationships

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

• Recognize and describe patterns in bivariate data

• Interpret the contextual meaning of slope and y-intercept where applicable

• Build a statistical model to explore, describe, and generalize relationships between two variables

• Use scatter plots and two-way tables to describe associations in data

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

• That straight lines are widely used to model relationships between two quantitative variables

• Vocabulary: scatter plot, bivariate, clustering, outliers, positive or negative association, linear association, nonlinear association, categorical data, frequencies, two-way table

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

• Construct and interpret scatter plots

  • Describe patterns such as clustering, outliers, positive and negative association, linear and nonlinear associations

  • Informally fit a straight line and informally assess the fit of the model by judging the closeness of the line to the points

• Use the linear model, fit to a scatter plot, to solve problems

  • Interpret the slope and intercept within the context of the situation

• Construct and interpret a two-way table (involving categorical variables)

  • Explain patterns

  • Use relative frequencies to describe possible association between the variables

Evidence Outcomes:

1. Verify experimentally the properties of rotations, reflections, and translations: (CCSS: 8.G.A.1) Lines are taken to lines, and line segments to line segments of the same length. (CCSS: 8.G.A.1.a) Angles are taken to angles of the same measure. (CCSS: 8.G.A.1.b) Parallel lines are taken to parallel lines. (CCSS: 8.G.A.1.c)

2. Demonstrate that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. (CCSS: 8.G.A.2)

3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (CCSS: 8.G.A.3)

4. Demonstrate that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. (CCSS: 8.G.A.4)

5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. (CCSS: 8.G.A.5)

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

• Use appropriate tools to deepen understanding of mathematical concepts

• Articulate how mathematical concepts relate to one another in the context of a problem or abstract relationships

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

• Make connections between the perseverance of congruence in geometry and the perseverance of equivalence of arithmetic and algebraic expression

• Use physical models, transparencies, geometric software, or other appropriate tools to explore the relationships between transformations and congruence and similarity

• Use the structure of the coordinate system to describe the locations of figures obtained with rotations, reflections, and translations

• Reason that since any one rotation, reflection or translation of a figure preserves congruence, then any sequence of those transformations must also preserve congruence

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

• Vocabulary: Translations, rotations, reflections, transformations, lines, line segment, congruence, angles, parallel lines, similarity

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

• Verify experimentally the properties of rotations, reflections, and translations

  • Lines are taken to lines, line segments to line segments of the same length

  • Angles are taken to angle of the same measure

  • Parallel lines are taken to parallel lines

• Demonstrate that a 2D figure is congruent to another if the second can be obtained by a sequence of rotations, reflections, and translations

  • Describe a sequence of transformations that show two figures are congruent

• Describe the effect of dilations, translations, rotations, and reflections using coordinates

• Demonstrate that a 2D figure is similar to another if the second can be obtained by a sequence of rotations, reflections, dilations, and translations

  • Describe a sequence of transformations that show two figures are similar

• Use informal arguments to establish facts about

  • the angle sum and exterior angles of triangles

  • Angles created when parallel lines are cut by a transversal

  • AA criterion for similarity in triangles

Evidence Outcomes:

1. Explain a proof of the Pythagorean Theorem and its converse. (CCSS: 8.G.B.6)

2. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (CCSS: 8.G.B.7)

3. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. (CCSS: 8.G.B.8)

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

• Use appropriate tools to deepen understanding of mathematical concepts

• Articulate how mathematical concepts relate to one another in the context of a problem or abstract relationships

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

• Think of the Pythagorean Theorem not just as a formula that only holds true under certain conditions

• Test to see if a triangle is a right triangle by applying the Pythagorean Theorem

• Use patterns to recognize and generate Pythagorean triples

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

• The formula for the Pythagorean Theorem

• Some pythagorean triples

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

• Explain a proof of the Pythagorean Theorem and its converse

• Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and other mathematical problems in 2D and 3D

• Use the Pythagorean Theorem to find the distance between two points

Evidence Outcomes:

1. State the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (CCSS: 8.G.C.9)

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

• Fluidly move between contextualizing and decontextualizing during the problem solving process

• Based on an understanding of any problem, initiate a plan, execute it, and evaluate the reasonableness of the solution

• Examine and apply a variety of methods to accurately and efficiently solve problems

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

• Efficiently solve problems using established volume formulas

• Describe how the formulas for volumes of cones, cylinders, and spheres relate to one another and to the volume formulas for solids with rectangular bases

• Use appropriate precision when solving problems involving measurement and volume formulas that describe real-world shapes

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

• Formulas for volumes of cones, cylinders, and spheres

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

• Solve real-world and mathematical problems using the formulas for the volumes of cones, cylinders, and spheres