**Power Standards (most important concepts), for each Unit are Highlighted in Yellow.

Unit 1

KEY STANDARDS -Understand congruence and similarity using physical models, transparencies, or geometry software. MCC8.G.1 Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. MCC8.G.2 Understand 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. MCC8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two‐dimensional figures using coordinates. MCC8.G.4 Understand 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. MCC8.G.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 three angles appear to form a line, and give an argument in terms of transversals why this is so.

ENDURING UNDERSTANDINGS

Coordinate geometry can be a useful tool for understanding geometric shapes and transformations.

Reflections, translations, and rotations are actionsthat produce congruent geometric objects.

A dilation is a transformation that changes the size of a figure, but not the shape.

The notation used to describe a dilation includes a scale factor and a center of dilation. A dilation of scale factor k with the center of dilation at the origin may be described by the notation (kx, ky).

If the scale factor of a dilation is greater than 1, the image resulting from the dilation is an enlargement. If the scale factor is less than 1, the image is a reduction.

Two shapes are similar if the lengths of all the corresponding sides are proportional and all the corresponding angles are congruent.

Two similar figures are related by a scale factor, which is the ratio of the lengths of the corresponding sides.

Congruent figures have the same size and shape. If the scale factor of a dilation is equal to one, the image resulting from the dilation is congruent to the original figure.

When parallel lines are cut by a transversal, corresponding, alternate interior and alternate exterior angles are congruent.

ESSENTIAL QUESTIONS:

How can the coordinate plane help me understand properties of reflections, translations, and rotations?

What is the relationship between reflections, translations, and rotations?

What is a dilation and how does this transformation affect a figure in the coordinate plane?

How can I tell if two figures are similar?

In what ways can I represent the relationships that exist between similar figures using the scale factors, length ratios, and area ratios?

What strategies can I use to determine missing side lengths and areas of similar figures?

Under what conditions are similar figures congruent?

When I draw a transversal through parallel lines, what are the special angle and segment relationships that occur?

What information is necessary before I can conclude two figures are congruent?

Unit 2

Exponents

KEY COMMON CORE GEORGIA PERFORMANCE STANDARDS:
Work with radicals and integer exponents. MCC8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3(–5) = 3(–3) = = .MCC8.EE.2 Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
MCC8.EE.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 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
MCC8.EE.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. MCC8.EE.7 Solve linear equations in one variable. a. 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 = b, or a = a results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
MCC8.NS.1 Know that numbers that are not rational are called irrational. Understand 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.
MCC8.NS.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 (square root 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.

ENDURING UNDERSTANDINGS:

Square roots can be rational or irrational.
An irrational number is a real number that cannot be written as a ratio of two integers.
Every number has a decimal expansion, for rational numbers it repeats eventually, and can be converted into a rational number.
All real numbers can be plotted on a number line. approximations of irrational numbers can be used to compare the size or irrational numbers, locate them approximately on a number line, and estimate the value of expressions. √2 is irrational.
Exponents are useful for representing very large or very small numbers.
Properties of integer exponents can be use to generate equivalent numerical expressions.
Scientific notation can be used to estimate very large or very small quantities and to compare quantities.
Linear equations in one variable can have one solution, infinitely many solutions, or no solutions.

ESSENTIAL QUESTIONS:

When are exponents used and why are they important?
How can I apply the properties of integer exponents to generate equivalent numerical expressions?
How can I represent very small and large numbers using integer exponents and scientific notation?
How can I perform operations with numbers expressed in scientific notation?
How can I interpret scientific notation that has been generated by technology?
Why is it useful for me to know the square root of a number?
How do I simplify and evaluate numeric expressions involving integer exponents?
What is the difference between rational and irrational numbers? hen are rational approximations appropriate?
Why do we approximate irrational numbers?
What strategies can I use to create and solve linear equations with one solution, infinitely many solutions, or no solutions?

Unit 3

Geometric Applications of Exponents

KEY GEORGIA PERFORMANCE STANDARDS:
Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.

Understand and apply the Pythagorean Theorem.

MCC8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

MCC8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

MCC8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Solve real‐world and mathematical problems involving volume of cylinders, cones, and spheres.

MCC8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Work with radicals and integer exponents.

MCC8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cubed roots of small perfect cubes. Know that √2 is irrational.

RELATED STANDARDS

MCC8.EE.7 Solve linear equations in one variable.

a. 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 = b, or a = a results (where a and b are different numbers).

b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

ENDURING UNDERSTANDINGS

• The Pythagorean Theorem can be used both algebraically and geometrically to solve problems involving right triangles

• There is a relationship between the Pythagorean Theorem and the distance formula.

• Both the Pythagorean Theorem and distance formula can be used to find missing side lengths in a coordinate plane and real-world situation.

• How to solve simple and complex linear and literal equations with one solution.

• Finding the square root of a number is the inverse operation of squaring that number.

• Finding the cube root of a number is the inverse operation of cubing that number.

• Right triangles have a special relationship among the side lengths which can be represented by a model and a formula

• Pythagorean Triples can be used to construct right triangles.

• How to simplify radicals and solve quadratic equations

• Attributes of geometric figures can be used to identify figures and find their measures.

• Relationships between change in length of radius or diameter, height, and volume exist for cylinders, cones and spheres.

ESSENTIAL QUESTIONS:

• What method is used to determine the missing length of a line segment given two polygons?

• What is the length of the side of a square of a certain area?

• What is the relationship among the lengths of the sides of a right triangle?

• How can the Pythagorean Theorem be used to solve problems?

• What is the correlation between the Pythagorean Theorem and the distance formula?

• How can I use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle?

• How do I use the Pythagorean Theorem to find the length of the legs of a right triangle?

• How do I know that I have a convincing argument to informally prove Pythagorean Theorem?

• What is Pythagorean Theorem and when does it apply?

• How can I determine the length of a diagonal?

• How can I find the altitude of an equilateral triangle?

• How could I find the shortest distance from one point to another if there was an obstacle in the way?

• Where can I find examples of two and three-dimensional objects in the real-world?

• How does the change in radius affect the volume of a cylinder, cone, or sphere?

• How does the change in height affect the volume of a cylinder, cone, or sphere?

• How does the volume of a cylinder, cone, and sphere with the same radius change if it is doubled?

• How do I simplify and evaluate algebraic equations involving integer exponents, square and cubed root?

• How do I know when an estimate, approximation, or exact answer is the desired solution?

Unit 4

Functions

KEY STANDARDS

Define, evaluate, and compare functions.
MCC8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. MCC8.F.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.

ENDURING UNDERSTANDINGS

• A function is a specific type of relationship in which each input has a unique output.
• A function can be represented in an input-output table.
• A function can be represented graphically using ordered pairs that consist of the input and the output of the function in the form (input, output).
• A function can be represented with an algebraic rule.

ESSENTIAL QUESTIONS

• What is a function?
• What are the characteristics of a function?
• How do you determine if relations are functions?
• How is a function different from a relation?
• Why is it important to know which variable is the independent variable?
• How can a function be recognized in any form?
• What is the best way to represent a function?
• How do you represent relations and functions using tables, graphs, words, and algebraic equations?
• What strategies can I use to identify patterns?
• How does looking at patterns relate to functions?
• How are sets of numbers related to each other?
• How can you use functions to model real-world situations?
• How can graphs and equations of functions help us to interpret real-world problems?

EVIDENCE OF LEARNING

By the conclusion of this unit, students should be able to demonstrate the following competencies:
• recognize functions in all various forms;
• determine independent and dependent variables; and
• compare functions represented in any form.

Unit 5

Linear Functions

STANDARDS ADDRESSED IN THIS UNIT

Understand the connections between proportional relationships, lines, and linear equations. MCC8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
MCC8.EE.6 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.
Define, evaluate, and compare functions.
MCC8.F.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.

ENDURING UNDERSTANDINGS

• Patterns and relationships can be represented graphically, numerically, and symbolically.
• Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportion problems.

ESSENTIAL QUESTIONS

• How can patterns, relations, and functions be used as tools to best describe and help explain real-life relationships?
• How can the same mathematical idea be represented in a different way? Why would that be useful?
• What is the significance of the patterns that exist between the triangles created on the graph of a linear function?
• When two functions share the same rate of change, what might be different about their tables, graphs and equations? What might be the same?
• What does the slope of the function line tell me about the unit rate?
• What does the unit rate tell me about the slope of the function line?

EVIDENCE OF LEARNING

By the conclusion of this unit, students should be able to demonstrate the following competencies:
• choose any two points on a line and find the slope;
• determine the unit rate and interpret it as the slope of the function;
• use the slope and y intercept to write an equation in slope-intercept form;
• choose two different points on the same line and find the slope; and
• draw two similar triangles ΔABC and ΔDEF on a coordinate plane. Name each point. Use sides AB and DE to compare the slope.

Unit 6

Linear Models and Tables

KEY STANDARDS

Use functions to model relationships between quantities. MCC8.F.4 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 value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
MCC8.F.5 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.
Investigate patterns of association in bivariate data.
MCC8.SP.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.
MCC8.SP.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.
MCC8.SP.4 Understand 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.

ENDURING UNDERSTANDINGS

• Collecting and examining data can sometimes help one discover patterns in the way in which two quantities vary.
• Changes in varying quantities are often related by patterns which, once discovered, can be used to predict outcomes and solve problems.
• Written descriptions, tables, graphs, and equations are useful in representing and investigating relationships between varying quantities.
• Different representations (written descriptions, tables, graphs, and equations) of the relationships between varying quantities may have different strengths and weaknesses.
• Linear functions may be used to represent and generalize real situations.
• Slope and y-intercept are keys to solving real problems involving linear relationships.

ESSENTIAL QUESTIONS

• How can I find the rate of change from a table, graph, equation, or verbal description?
• How can I find the initial value from a table, graph, equations, or verbal description?
• How can I write a function to model a linear relationship?
• How can I sketch a graph given a verbal description?
• How can I describe a situation given a graph?
• How can I analyze a scatter plot?
• How can I create a linear model given a scatter plot?
• How can I use a linear model to solve problems?
• How can I use bivariate data to solve problems?
• What strategies can I use to help me understand and represent real situations involvinglinear relationships
• How can the properties of lines help me to understand graphing linear functions?
• What can I infer from the data?
• How can functions be used to model real-world situations?
• How does a change in one variable affect the other variable in a given situation?
• Which tells me more about the relationship I am investigating – a table, a graph or an equation? Why?

EVIDENCE OF LEARNING

• find the rate of change from a table, graph, equation, or verbal description
;• find the initial value from a table, graph, equations, or verbal description;
• write a function to model a linear relationship;
• sketch a graph given a verbal description;
• describe a situation given a graph;
• analyze a scatter plot;
• create a linear model given a scatter plot;
• create a line of best fit given a scatter plot;
• use a linear model to solve problems; and
• use bivariate data to solve problems.

Unit 7

KEY STANDARDS

Analyze and solve linear equations and pairs of simultaneous linear equations. MCC8.EE.8. a,b,c Analyze and solve pairs of simultaneous linear equations. a. Understand 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. b. 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. c. 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. RELATED STANDARDS Analyze and solve linear equations and pairs of simultaneous linear equations. MCC8.EE.7 Solve linear equations in one variable.

ENDURING UNDERSTANDINGS

There are situations that require two or more equations to be satisfied simultaneously.

There are several methods for solving systems of equations.

Solutions to systems can be interpreted algebraically, geometrically, and in terms of problem contexts.

The number of solutions to a system of equations or inequalities can vary from no solution to an infinite number of solutions.

ESSENTIAL QUESTIONS

How do I solve pairs of simultaneous linear equations?

How can I translate a problem situation into a system of equations?

What does the solution to a system tell me about the answer to a problem situation?

How can I interpret the meaning of a “system of equations” algebraically and geometrically?

What does the geometrical solution of a system mean?

How can I translate a problem situation into a system of equations?

What does the solution to a system tell me about the answer to a problem situation?

EVIDENCE OF LEARNING

analyze and solve systems of equations;

understand solutions of systems of two linear equations is represented by the intersection point of the two lines when graphed;

solve systems of two linear equations algebraically;

estimate solutions of systems of two linear equations when graphed; and apply the use of systems of equations in real-world situations.

Unit CRCT Review

Cumulative Review of all 8th grade CCGPS:

MCC8.NS.1-2:

*Know that there are numbers that are not rational, and approximate themby rational numbers.

MCC8.EE.1-4:

*Work with radicals and integer exponents.

MCC8.EE.5-6:

*Understand the connections between proportional relationships, lines,and linear equations.

MCC8.EE.7-8:

*Analyze and solve linear equations and pairs of simultaneous linearequations.

MCC8.F.1-3:

*Define, evaluate, and compare functions.

MCC8.F.4-5:

*Use functions to model relationships between quantities.

MCC8.G.1-5:

*Understand congruence and similarity using physical models,transparencies, or geometry software.

MCC8.G.6-8:

*Understand and apply the Pythagorean Theorem.

MCC8.G.9:

*Solve real-world and mathematical problems involving volume ofcylinders, cones, and spheres.

MCC8.SP.1-4:

*Investigate patterns of association in bivariate data.

## The l

ink below is for the 8th GradeCommon Core Georgia Performance Standards:## https://www.georgiastandards.org/Pages/default.aspx

## **Power Standards (most important concepts), for each Unit are Highlighted in Yellow.

Unit 1KEY STANDARDS -Understand congruence and similarity using physical models, transparencies, or geometry software.MCC8.G.1Verify experimentally the properties of rotations, reflections, and translations:a.Lines are taken to lines, and line segments to line segments of the same length.b.Angles are taken to angles of the same measure.c.Parallel lines are taken to parallel lines.MCC8.G.2Understand 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.MCC8.G.3Describe the effect of dilations, translations, rotations, and reflections on two‐dimensional figures using coordinates.MCC8.G.4 Understand 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.

MCC8.G.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 three angles appear to form a line, and give an argument in terms of transversals why this is so

.ENDURING UNDERSTANDINGSactionsthat produce congruent geometric objects.kwith the center of dilation at the origin may be described by the notation (kx,ky).ESSENTIAL QUESTIONS:Unit 2ExponentsKEY COMMON CORE GEORGIA PERFORMANCE STANDARDS:Work with radicals and integer exponents.

MCC8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3(–5) = 3(–3) = = .MCC8.EE.2 Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

MCC8.EE.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 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.

MCC8.EE.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.

MCC8.EE.7 Solve linear equations in one variable.

a. 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 = b, or a = a results (where a and b are different numbers).

b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

MCC8.NS.1 Know that numbers that are not rational are called irrational. Understand 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.

MCC8.NS.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 (square root 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.

Square roots can be rational or irrational.ENDURING UNDERSTANDINGS:An irrational number is a real number that cannot be written as a ratio of two integers.

Every number has a decimal expansion, for rational numbers it repeats eventually, and can be converted into a rational number.

All real numbers can be plotted on a number line. approximations of irrational numbers can be used to compare the size or irrational numbers, locate them approximately on a number line, and estimate the value of expressions. √2 is irrational.

Exponents are useful for representing very large or very small numbers.

Properties of integer exponents can be use to generate equivalent numerical expressions.

Scientific notation can be used to estimate very large or very small quantities and to compare quantities.

Linear equations in one variable can have one solution, infinitely many solutions, or no solutions.

When are exponents used and why are they important?ESSENTIAL QUESTIONS:How can I apply the properties of integer exponents to generate equivalent numerical expressions?

How can I represent very small and large numbers using integer exponents and scientific notation?

How can I perform operations with numbers expressed in scientific notation?

How can I interpret scientific notation that has been generated by technology?

Why is it useful for me to know the square root of a number?

How do I simplify and evaluate numeric expressions involving integer exponents?

What is the difference between rational and irrational numbers? hen are rational approximations appropriate?

Why do we approximate irrational numbers?

What strategies can I use to create and solve linear equations with one solution, infinitely many solutions, or no solutions?

Unit 3Geometric Applications of ExponentsKEY GEORGIA PERFORMANCE STANDARDS:Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.

## Understand and apply the Pythagorean Theorem.

## MCC8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

## MCC8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

## MCC8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

## Solve real‐world and mathematical problems involving volume of cylinders, cones, and spheres.

## MCC8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

## Work with radicals and integer exponents.

## MCC8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cubed roots of small perfect cubes. Know that √2 is irrational.

## RELATED STANDARDS

## MCC8.EE.7 Solve linear equations in one variable.

## a. 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 = b, or a = a results (where a and b are different numbers).

## b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

ENDURING UNDERSTANDINGS• The Pythagorean Theorem can be used both algebraically and geometrically to solve problems involving right triangles• There is a relationship between the Pythagorean Theorem and the distance formula.• Both the Pythagorean Theorem and distance formula can be used to find missing side lengths in a coordinate plane and real-world situation.• How to solve simple and complex linear and literal equations with one solution.• Finding the square root of a number is the inverse operation of squaring that number.• Finding the cube root of a number is the inverse operation of cubing that number.• Right triangles have a special relationship among the side lengths which can be represented by a model and a formula• Pythagorean Triples can be used to construct right triangles.• How to simplify radicals and solve quadratic equations• Attributes of geometric figures can be used to identify figures and find their measures.• Relationships between change in length of radius or diameter, height, and volume exist for cylinders, cones and spheres.ESSENTIAL QUESTIONS:## • What method is used to determine the missing length of a line segment given two polygons?

## • What is the length of the side of a square of a certain area?

## • What is the relationship among the lengths of the sides of a right triangle?

## • How can the Pythagorean Theorem be used to solve problems?

## • What is the correlation between the Pythagorean Theorem and the distance formula?

## • How can I use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle?

## • How do I use the Pythagorean Theorem to find the length of the legs of a right triangle?

## • How do I know that I have a convincing argument to informally prove Pythagorean Theorem?

## • What is Pythagorean Theorem and when does it apply?

## • How can I determine the length of a diagonal?

## • How can I find the altitude of an equilateral triangle?

## • How could I find the shortest distance from one point to another if there was an obstacle in the way?

## • Where can I find examples of two and three-dimensional objects in the real-world?

## • How does the change in radius affect the volume of a cylinder, cone, or sphere?

## • How does the change in height affect the volume of a cylinder, cone, or sphere?

## • How does the volume of a cylinder, cone, and sphere with the same radius change if it is doubled?

## • How do I simplify and evaluate algebraic equations involving integer exponents, square and cubed root?

## • How do I know when an estimate, approximation, or exact answer is the desired solution?

Unit 4Functions

Define, evaluate, and compare functions.KEY STANDARDSMCC8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

MCC8.F.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.

• A function is a specific type of relationship in which each input has a unique output.ENDURING UNDERSTANDINGS• A function can be represented in an input-output table.

• A function can be represented graphically using ordered pairs that consist of the input and the output of the function in the form (input, output).

• A function can be represented with an algebraic rule.

• What is a function?ESSENTIAL QUESTIONS• What are the characteristics of a function?

• How do you determine if relations are functions?

• How is a function different from a relation?

• Why is it important to know which variable is the independent variable?

• How can a function be recognized in any form?

• What is the best way to represent a function?

• How do you represent relations and functions using tables, graphs, words, and algebraic equations?

• What strategies can I use to identify patterns?

• How does looking at patterns relate to functions?

• How are sets of numbers related to each other?

• How can you use functions to model real-world situations?

• How can graphs and equations of functions help us to interpret real-world problems?

By the conclusion of this unit, students should be able to demonstrate the following competencies:EVIDENCE OF LEARNING• recognize functions in all various forms;

• determine independent and dependent variables; and

• compare functions represented in any form.

Unit 5Linear Functions

Understand the connections between proportional relationships, lines, and linear equations.STANDARDS ADDRESSED IN THIS UNITMCC8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

MCC8.EE.6 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.

Define, evaluate, and compare functions.

MCC8.F.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.

• Patterns and relationships can be represented graphically, numerically, and symbolically.ENDURING UNDERSTANDINGS• Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportion problems.

• How can patterns, relations, and functions be used as tools to best describe and help explain real-life relationships?ESSENTIAL QUESTIONS• How can the same mathematical idea be represented in a different way? Why would that be useful?

• What is the significance of the patterns that exist between the triangles created on the graph of a linear function?

• When two functions share the same rate of change, what might be different about their tables, graphs and equations? What might be the same?

• What does the slope of the function line tell me about the unit rate?

• What does the unit rate tell me about the slope of the function line?

By the conclusion of this unit, students should be able to demonstrate the following competencies:EVIDENCE OF LEARNING• choose any two points on a line and find the slope;

• determine the unit rate and interpret it as the slope of the function;

• use the slope and y intercept to write an equation in slope-intercept form;

• choose two different points on the same line and find the slope; and

• draw two similar triangles ΔABC and ΔDEF on a coordinate plane. Name each point. Use sides AB and DE to compare the slope.

Unit 6Linear Models and Tables

Use functions to model relationships between quantities.KEY STANDARDSMCC8.F.4 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 value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

MCC8.F.5 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.

Investigate patterns of association in bivariate data.

MCC8.SP.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.

MCC8.SP.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.

MCC8.SP.4 Understand 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.

• Collecting and examining data can sometimes help one discover patterns in the way in which two quantities vary.ENDURING UNDERSTANDINGS• Changes in varying quantities are often related by patterns which, once discovered, can be used to predict outcomes and solve problems.

• Written descriptions, tables, graphs, and equations are useful in representing and investigating relationships between varying quantities.

• Different representations (written descriptions, tables, graphs, and equations) of the relationships between varying quantities may have different strengths and weaknesses.

• Linear functions may be used to represent and generalize real situations.

• Slope and y-intercept are keys to solving real problems involving linear relationships.

• How can I find the rate of change from a table, graph, equation, or verbal description?ESSENTIAL QUESTIONS• How can I find the initial value from a table, graph, equations, or verbal description?

• How can I write a function to model a linear relationship?

• How can I sketch a graph given a verbal description?

• How can I describe a situation given a graph?

• How can I analyze a scatter plot?

• How can I create a linear model given a scatter plot?

• How can I use a linear model to solve problems?

• How can I use bivariate data to solve problems?

• What strategies can I use to help me understand and represent real situations involvinglinear relationships

• How can the properties of lines help me to understand graphing linear functions?

• What can I infer from the data?

• How can functions be used to model real-world situations?

• How does a change in one variable affect the other variable in a given situation?

• Which tells me more about the relationship I am investigating – a table, a graph or an equation? Why?

• find the rate of change from a table, graph, equation, or verbal descriptionEVIDENCE OF LEARNING;• find the initial value from a table, graph, equations, or verbal description;

• write a function to model a linear relationship;

• sketch a graph given a verbal description;

• describe a situation given a graph;

• analyze a scatter plot;

• create a linear model given a scatter plot;

• create a line of best fit given a scatter plot;

• use a linear model to solve problems; and

• use bivariate data to solve problems.

Unit 7KEY STANDARDSAnalyze and solve linear equations and pairs of simultaneous linear equations.MCC8.EE.8. a,b,c Analyze and solve pairs of simultaneous linear equations.

a. Understand 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.

b. 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.

c. 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

.RELATED STANDARDSAnalyze and solve linear equations and pairs of simultaneous linear equations.MCC8.EE.7Solve linear equations in one variable.ENDURING UNDERSTANDINGSESSENTIAL QUESTIONSEVIDENCE OF LEARNINGUnit CRCT ReviewCumulative Review of all 8th grade CCGPS:MCC8.NS.1-2:*Know that there are numbers that are not rational, and approximate themby rational numbers.MCC8.EE.1-4:*Work with radicals and integer exponents.MCC8.EE.5-6:*Understand the connections between proportional relationships, lines,and linear equations.MCC8.EE.7-8:*Analyze and solve linear equations and pairs of simultaneous linearequations.MCC8.F.1-3:*Define, evaluate, and compare functions.MCC8.F.4-5:*Use functions to model relationships between quantities.MCC8.G.1-5:*Understand congruence and similarity using physical models,transparencies, or geometry software.MCC8.G.6-8:*Understand and apply the Pythagorean Theorem.MCC8.G.9:*Solve real-world and mathematical problems involving volume ofcylinders, cones, and spheres.MCC8.SP.1-4:*Investigate patterns of association in bivariate data.