Embedded Files
Key Areas of Focus for Grade 8: Linear algebra
Eighth grade mathematics is about
- Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations
- Grasping the concept of a function and using functions to describe quantitative relationships
- Analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
Module 1: Integer Exponents and Scientific Notation
Module 1: Integer Exponents and Scientific Notation
(13 Hours)
(13 Hours)
Overview
In Grade 8 Module 1, students expand their basic knowledge of positive integer exponents and prove the Laws of Exponents for any integer exponent. Next, students work with numbers in the form of an integer multiplied by a power of 10 to express how many times as much one is than the other. This leads into an explanation of scientific notation and continued work performing operations on numbers written in this form.
Priority Standards
8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.
8.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.
8.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.
Unit Vocab
Scientific Notation (The scientific notation for a finite decimal is the representation of that decimal as the product of a decimal π and a power of 10, where π satisfies the property that it is at least 1, but smaller than 10, or in symbolic notation, 1 β€ π < 10. For example, the scientific notation for 192.7 is 1.927 Γ 102 .)
Order of Magnitude (The order of magnitude of a finite decimal is the exponent in the power of 10 when that decimal is expressed in scientific notation. For example, the order of magnitude of 192.7 is 2 because when 192.7 is expressed in scientific notation as 1.927 Γ 102 , 2 is the exponent of 102 . Sometimes we also include the number 10 in the definition of order of magnitude and say that the order of magnitude of 192.7 is 102 .)
Topic A: Exponential Notations and Properties of Integer Exponents
Topic A: Exponential Notations and Properties of Integer Exponents
Standard
8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 x 3-5 = 3-3 = 1/33 = 1/27.
Overview
Students build upon their foundation with exponents as they make conjectures about how zero and negative exponents of a number should be defined and prove the properties of integer exponents. These properties are codified into three laws of exponents. They make sense out of very large and very small numbers, using the number line model to guide their understanding of the relationship of those numbers to each other
Lessons
Lesson 1: Exponential Notation - Priority Lesson-
Lesson 2: Multiplication of Numbers in Exponential Form β- Priority Lesson-
Lesson 4: Applying Properties of Exponents to Generate Equivalent Expressions β- Priority Lesson-
Lesson 5: Negative Exponents and the Laws of Exponents β-- Priority Lesson-
Topic B: Magnitude and Scientific Notation
Topic B: Magnitude and Scientific Notation
Standards
8.EE.A.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.
8.EE.A.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.
Overview
Having established the properties of integer exponents, students learn to express the magnitude of a positive number through the use of scientific notation and to compare the relative size of two numbers written in scientific notation.
Students explore the use of scientific notation and choose appropriately sized units as they represent, compare, and make calculations with very large quantities (e.g., the U.S. national debt, the number of stars in the universe, and the mass of planets) and very small quantities, such as the mass of subatomic particles.
Lessons
Lesson 7: Magnitude ****Priority Lesson****
Lesson 8: Estimating Quantities ****Priority Lesson****
Lesson 9: Scientific Notation ****Priority Lesson****
Lesson 12: Choice of Unit ****Priority Lesson****
Lesson 13: Comparing and Interpreting Scientific Notation Using Technology ****Priority Lesson****
Module 2: The Concept of Congruence
Module 2: The Concept of Congruence
(16 Hours)
(16 Hours)
Overview
Students learn about translations, reflections, and rotations in the plane and, more importantly, how to use them to precisely define the concept of congruence.
Priority Standards
8.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.
8.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.
Unit Vocab
Transformation (A transformation is a rule, to be denoted by πΉ, that assigns each point π of the plane a unique point which is denoted by πΉ(π).)
Basic Rigid Motion (A basic rigid motion is a rotation, reflection, or translation of the plane. Basic rigid motions are examples of transformations. Given a transformation, the image of a point π΄ is the point the transformation maps the point π΄ to in the plane.) Translation (A translation is a basic rigid motion that moves a figure along a given vector.)
Rotation (A rotation is a basic rigid motion that moves a figure around a point, π degrees.)
Reflection (A reflection is a basic rigid motion that moves a figure across a line.) Image of a point, image of a figure (Image refers to the location of a point or figure after it has been transformed.)
Sequence (Composition) of Transformations (A sequence of transformations is more than one transformation. Given transformations πΊ and πΉ, πΊ Β° πΉ is called the composition of πΉ and πΊ.)
Vector (A Euclidean vector (or directed segment) π΄π΅βββββ is the line segment π΄π΅ together with a direction given by connecting an initial point π΄ to a terminal point π΅.)
Congruence (A congruence is a sequence of basic rigid motions (rotations, reflections, translations) of the plane.)
Transversal (Given a pair of lines πΏ and π in a plane, a third line π is a transversal if it intersects πΏ at a single point and intersects π at a single but different point.)
Topic A: Definitions and properties of the Basic Rigid Motions
Topic A: Definitions and properties of the Basic Rigid Motions
Standard
8.G.A.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
Overview
Throughout Topic A, on the definitions and properties of the basic rigid motions, students verify experimentally their basic properties and, when feasible, deepen their understanding of these properties using reasoning. In particular, what students learned in Grade 4 about angles and angle measurement is put to good use here. They learn that the basic rigid motions preserve angle measurements as well as segment lengths.
Lessons
Lesson 1: Why Move Things Around? **Priority Lesson***
Lesson 2: Definition of Translation and Three Basic Properties
Lesson 4: Definition of Reflection and Basic Properties **Priority Lesson***
Lesson 6: Rotations of 180 Degrees **Priority Lesson***
Topic B: Sequencing the Basic Rigid Motions
Topic B: Sequencing the Basic Rigid Motions
Standards
8.G.A.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.
Overview
Topic B focuses on the first part of 8.G.A.2 in the respect that students learn how to sequence rigid motions.
Lessons
Lesson 7: Sequencing Translations **Priority Lesson***
Lesson 10: Sequences of Rigid Motions **Priority Lesson***
Topic C: Congruence and Angle Relationships
Topic C: Congruence and Angle Relationships
Standards
8.G.A.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.
8.G.A.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.
Overview
In Topic C, students learn that congruence is just a sequence of basic rigid motions. The fundamental properties shared by all the basic rigid motions are then inherited by congruence: Congruence moves lines to lines and angles to angles, and it is both distance and angle-preserving. In Grade 7, students used facts about supplementary, complementary, vertical, and adjacent angles to find the measures of unknown angles. This module extends that knowledge to angle relationships that are formed when two parallel lines are cut by a transversal.
In Topic C, on angle relationships related to parallel lines, students learn that pairs of angles are congruent because they are angles that have been translated along a transversal, rotated around a point, or reflected across a line.
Standards
Lesson 11: Definition of Congruence and Some Basic Properties *Priority Lesson*
Lesson 13: Angle Sum of a Triangle *Priority Lesson*
Topic D: The Pythagorean Theorem (OPTIONAL in this unit)
Topic D: The Pythagorean Theorem (OPTIONAL in this unit)
Optional Topic D introduces the Pythagorean theorem. Students are shown the βsquare within a squareβ proof of the Pythagorean theorem. The proof uses concepts learned in previous topics of the module, that is, the concept of congruence and concepts related to degrees of angles. Students begin the work of finding the length of a leg or hypotenuse of a right triangle using π 2 + π 2 = π 2 . Note that this topic is not assessed until Module 7.
Lessons
Module 3: Similarity: Geometry
Module 3: Similarity: Geometry
(14 Hours)
(14 Hours)
Overview
In Module 3, students learn about dilation and similarity and apply that knowledge to a proof of the Pythagorean Theorem based on the Angle-Angle criterion for similar triangles. The module begins with the definition of dilation, properties of dilations, and compositions of dilations. One overarching goal of this module is to replace the common idea of βsame shape, different sizesβ with a definition of similarity that can be applied to geometric shapes that are not polygons, such as ellipses and circles.
Priority Standards
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.A.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.
8.G.A.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.
8.g.B.6 Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions
Unit Vocab
Dilation (Dilation, π·, is a transformation of the plane with center π and scale factor π (π > 0). If π·(π) = π and if π β π, then the point π·(π), to be denoted by π, is the point on the ray ππβββββ so that |ππ| = π|ππ|. If the scale factor π β 1, then a dilation in the coordinate plane is a transformation that shrinks or magnifies a figure by multiplying each coordinate of the figure by the scale factor.)
Congruence (A finite composition of basic rigid motionsβreflections, rotations, translationsβof the plane. Two figures in a plane are congruent if there is a congruence that maps one figure onto the other figure.) Similar (Two figures in the plane are similar if there exists a similarity transformation taking one figure to the other.)
Similarity Transformation (A similarity transformation, or similarity, is a composition of a finite number of basic rigid motions or dilations. The scale factor of a similarity transformation is the product of the scale factors of the dilations in the composition; if there are no dilations in the composition, the scale factor is defined to be 1.)
Similarity (A similarity is an example of a transformation.)
Topic A: Dilation
Topic A: Dilation
Focus Standard
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Overview
Topic A begins by demonstrating the need for a precise definition of dilation instead of βsame shape, different sizeβ because dilation will be applied to geometric shapes that are not polygons. Students begin their work with dilations off the coordinate plane by experimenting with dilations using a compass and straightedge to develop conceptual understanding. It is vital that students have access to these tools in order to develop an intuitive sense of dilation and to prepare for further work in high school Geometry.
Lessons
Lesson 1: βWhat Lies Behind Same Shapeβ? *Priority Lesson*
Lesson 2: Properties of Dilations *Priority Lesson*
Lesson 3: Examples of Dilations *Priority Lesson*
Lesson 4: Fundamental Theorem of Similarity (FTS) *Priority Lesson*
Lesson 6: Dilations on the Coordinate Plane *Priority Lesson*
Topic B: Similar Figures
Topic B: Similar Figures
Focus Standards
8.G.A.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.
8.G.A.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
Overview
Students learn definition of similarity and the properties of similarities. This Topic demonstrates that a two-dimensional figure is similar to another if the second can be obtained from a dilation followed by congruence. Knowledge of basic rigid motions is reinforced throughout the module, specifically when students describe the sequence that exhibits a similarity between two given figures. Previously, students used vectors to describe the translation of the plane. Module 3 begins in the same way, but once figures are bound to the coordinate plane, students describe translations in terms of units left or right and units up or down. When figures on the coordinate plane are rotated, the center of rotation is the origin of the graph. In most cases, students describe the rotation as having center π and degree π unless the rotation can be easily identified (e.g., a rotation of 90Β° or 180Β°). Reflections remain reflections across a line, but when possible, students should identify the line of reflection as the π₯-axis or π¦-axis
Lessons
Lesson 8: Similarity *Priority Lesson*
Lesson 10: Informal Proof of AA Criterion for Similarity *Priority Lesson*
Lesson 11: More About Similar Triangles *Priority Lesson*
Topic C: The Pythagorean Theorem
Topic C: The Pythagorean Theorem
Focus Standards
8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Overview
It is recommended that students have some experience with the lessons in Topic D from Module 2 before beginning these lessons. In Lesson 13 of Topic C, students are presented with a general proof that uses the Angle-Angle criterion. In Lesson 14, students are presented with a proof of the converse of the Pythagorean Theorem.
Lessons
Module 4: Linear Equations
Module 4: Linear Equations
(31 Hours)
(31 Hours)
Focus Standards
8.EE.B.5 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.
8.EE.B.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 π¦ = ππ₯ + π for a line intercepting the vertical axis at π.
8.EE.C.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 π₯ = π, π = π, or π = π results (where π and π 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.
8.EE.C.8 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.
Overview
In Module 4, students extend what they already know about unit rates and proportional relationships to linear equations and their graphs. Students understand the connections between proportional relationships, lines, and linear equations in this module. Students learn to apply the skills they acquired in Grades 6 and 7, with respect to symbolic notation and properties of equality to transcribe and solve equations in one variable and then in two variables. Students learn the connection between proportional relationships, lines, and linear equations as they develop ways to represent a line by different equations (y = mx + b, y β y1 = m (x β x1), etc.). They analyze and solve linear equations and pairs of simultaneous linear equations. The equation of a line provides a natural transition into the idea of a function explored in the next two modules.
Unit Vocab
Slope (Slope is a number that describes the βsteepnessβ or βslantβ of a line. It is the constant rate of change. Example: The slope, π, of the graph of line π to the right is π = 3 7 .)
Solution to a System of Linear Equations (The solution to a system of linear equations is a pair of numbers from the domain of the variables that, when each number from the pair is substituted into all instances of its corresponding variable, makes the equation a true number sentence. Example: The solution to the system of linear equations { π₯ + π¦ = 15 3π₯ β 7π¦ = β2 is the ordered pair ( 103 10 , 47 10) because the ordered pair is a solution to each linear equation of the system, and it is the point on the plane where the graphs of the two equations intersect.)
System of Linear Equations (A system of linear equations, also referred to as simultaneous linear equations, is the set of at least two linear equations. Example: { π₯ + π¦ = 15 3π₯ β 7π¦ = β2 is a system of linear equations.)
Topic A: Writing and Solving Linear Equations
Topic A: Writing and Solving Linear Equations
Standards
8.EE.C.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 π₯ = π, π = π, or π = π result (where π and π 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
Overview
In Topic A, students begin by transcribing written statements using symbolic notation. Then, students write linear and non-linear expressions leading to linear equations, which are solved using properties of equality (8.EE.C.7b). Students learn that not every linear equation has a solution. In doing so, students learn how to transform given equations into simpler forms until an equivalent equation results in a unique solution, no solution, or infinitely many solutions (8.EE.C.7a). Throughout Topic A students must write and solve linear equations in real-world and mathematical situations.
Lessons
Lesson 1: Writing Equations Using Symbols *Priority Lesson*
Lesson 2: Linear and Nonlinear Expressions in x *Priority Lesson*
Lesson 3: Linear Equations in x *Priority Lesson*
Lesson 4: Solving a Linear Equation *Priority Lesson*
Lesson 6: Solutions of a Linear Equation *Priority Lesson*
Lesson 7: Classification of Solutions *Priority Lesson*
Topic B: Linear Equations in Two Variables and Their Graphs
Topic B: Linear Equations in Two Variables and Their Graphs
Focus Standard
8.EE.B.5 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.
Overview
In Topic B, students work with constant speed, a concept learned in Grade 6, but this time with proportional relationships related to average speed and constant speed. These relationships are expressed as linear equations in two variables. Students find solutions to linear equations in two variables, organize them in a table, and plot the solutions on a coordinate plane.
It is in Topic B that students begin to investigate the shape of a graph of a linear equation. Students predict that the graph of a linear equation is a line and select points on and off the line to verify their claim. Also in this topic is the standard form of a linear equation, ax + by = c, and when a, b β 0, a non-vertical line is produced. Further, when a or b = 0, then a vertical or horizontal line is produced.
Lessons
Lesson 10: A Critical Look at Proportional Relationships *Priority Lesson*
Lesson 11: Constant Rate *Priority Lesson*
Lesson 12: Linear Equations in Two Variables *Priority Lesson*
Lesson 13: The Graph of a Linear Equation in Two Variables *Priority Lesson*
Lesson 14: The Graph of a Linear EquationβHorizontal and Vertical Lines
Topic C: Slope and Equations of Lines
Topic C: Slope and Equations of Lines
Focus Standards
8.EE.B.5 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.
8.EE.B.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 π¦ = ππ₯ for a line through the origin and the equation π¦ = ππ₯ + π for a line intercepting the vertical axis at π.
Overview
In Topic C, students know that the slope of a line describes the rate of change of a line. Students first encounter slope by interpreting the unit rate of a graph. In general, students learn that slope can be determined using any two distinct points on a line. Students verify this fact by checking the slope using several pairs of points and comparing their answers. In this topic, students derive y = mx and y = mx + b for linear equations by examining similar triangles.
Students generate graphs of linear equations in two variables first by completing a table of solutions, then using information about slope and y-intercept. Students graph equations using information about x- and y-intercepts. Next, students learn some basic facts about lines and equations, such as why two lines with the same slope and a common point are the same line, how to write equations of lines given slope and a point, and how to write an equation given two points.
Students compare two different proportional relationships represented by graphs, tables, equations, or descriptions. Finally, students learn that multiple forms of an equation can define the same line.
Lessons
Lesson 15: The Slope of a Non-Vertical Line *Priority lesson*
Lesson 16: The Computation of the Slope of a Non-Vertical Line *Priority lesson*
Lesson 17: The Line Joining Two Distinct Points of the Graph *Priority lesson*
Lesson 18: Only One Line Passing Through a Point with a Slope
Lesson 19: Graph of Linear Equation in Two Variables Is a Line *Priority lesson*
Lesson 21: Some Facts About Graphs of Linear Equations in Two Variables *Priority lesson*
Lesson 22: Constant Rates Revisited *Priority lesson*
Topic D: Systems of Linear Equations and Their Solutions
Topic D: Systems of Linear Equations and Their Solutions
Standard
8. EE.B.5 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.
8.EE.C.8 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, 3π₯ + 2π¦ = 5 and 3π₯ + 2π¦ = 6 have no solution because 3π₯ + 2π¦ 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.
Overview
Students begin by comparing the constant speed of two individuals to determine which has greater speed. Students graph simultaneous linear equations to find the point of intersection and then verify that the point of intersection is in fact a solution to each equation in the system. To motivate the need to solve systems algebraically, students graph systems of linear equations whose solutions do not have integer coordinates. Students use an estimation of the solution from the graph to verify their algebraic solution is correct.
Students learn to solve systems of linear equations by substitution and elimination. Students understand that a system can have a unique solution, no solution, or infinitely many solutions, as they did with linear equations in one variable. Finally, students apply their knowledge of systems to solve problems in real-world contexts, including converting temperatures from Celsius to Fahrenheit.
Lessons
Lesson 24: Introduction to Simultaneous Equations *Priority Lesson*
Lesson 25: Geometric Interpretation of the Solutions of a Linear System
Lesson 27: Nature of Solutions of a System of Linear Equations *Priority Lesson*
Lesson 28: Another Computational Method of Solving a Linear System *Priority Lesson*
Lesson 29: Word Problems *Priority Lesson*
Topic E: The Pythagorean Theorem (OPTIONAL in this Unit)
Topic E: The Pythagorean Theorem (OPTIONAL in this Unit)
Standard
8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.EE.C.8 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, 3π₯ + 2π¦ = 5 and 3π₯ + 2π¦ = 6 have no solution because 3π₯ + 2π¦ 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
Overview
Optional Topic E is an application of systems of linear equations. Specifically, a system that generates Pythagorean triples. First, students learn that a Pythagorean triple can be obtained by multiplying any known triple by a positive integer. Then, students are shown the Babylonian method for finding a triple that requires the understanding and use of a system of linear equations.
Lessons
Lesson 31: System of Equations Leading to Pythagorean Triples
Module 5: Examples of Functions from Geometry
Module 5: Examples of Functions from Geometry
(11 Hours)
(11 Hours)
Focus Standards
8.F.A.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.
8.F.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 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.
8.F.A.3 Interpret the equation π¦ = ππ₯ + π as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function π΄ = π 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.
8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems
Overview
Students understand why functions are necessary for describing geometric concepts and occurrences. Students are introduced to functions in the context of linear equations and area/volume formulas in Module 5. They define, evaluate, and compare functions using equations of lines as a source of linear functions and area and volume formulas as a source of non-linear functions.
Unit Vocab
Function (A function is a rule that assigns to each input exactly one output.)
Input (The number or piece of data that is put into a function is the input.)
Output (The number or piece of data that is the result of an input of a function is the output.)
Topic A: Functions
Topic A: Functions
Standards
8.F.A.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.
8.F.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 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.
8.F.A.3 Interpret the equation π¦ = ππ₯ + π as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function π΄ = π 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.
Overview
A function is defined as an assignment to each input, exactly one output. Students learn that the assignment of some functions can be described by a mathematical rule or formula. Students consider functions of discrete and continuous rates and understand the difference between the two.
Students apply their knowledge of linear equations and their graphs to graphs of linear functions. Students relate a function to an input/output machine: a number or piece of data, known as the input, goes into the machine, and a number or piece of data, known as the output, comes out of the machine. Students inspect the rate of change of linear functions and conclude that the rate of change is the slope of the graph of a line. They learn to interpret the equation π¦ = ππ₯ + π as defining a linear function whose graph is a line.
Lessons
Lesson 1: The Concept of a Function *Priority Lesson*
Lesson 2: Formal Definition of a Function *Priority Lesson*
Lesson 5: Graphs of Functions and Equations *Priority Lesson*
Lesson 6: Graphs of Linear Functions and Rate of Change *Priority Lesson*
Lesson 7: Comparing Linear Functions and Graphs *Priority Lesson*
Topic B: Volume
Topic B: Volume
Standards
8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Overview
In Topic B, students apply their knowledge of volume from previous grade levels to the learning of the volume formulas for cones, cylinders, and spheres. First, students are reminded of what they already know about volume. Next, students use what they learned about the area of circles to determine the volume formulas of cones and cylinders. In each case, physical models will be used to explain the formulas, first with a cylinder seen as a stack of circular disks that provide the height of the cylinder.
Students compare the volume of a sphere to its circumscribing cylinder (i.e., the cylinder of dimensions that touches the sphere at points, but does not cut off any part of it). Students learn that the formula for the volume of a sphere is two-thirds the volume of the cylinder that fits tightly around it.
Lessons
Lesson 9: Examples of Functions from Geometry *Priority Lesson*
Lesson 10: Volumes of Familiar SolidsβCones and Cylinders *Priority Lesson*
Lesson 11: Volume of a Sphere *Priority Lesson*
Module 6: Linear Functions
Module 6: Linear Functions
(14 Hours)
(14 Hours)
Focus Standards
8.F.B.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 (π₯π₯,π¦π¦) 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.
8.F.B.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.
8.SP.A.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. 8.SP.A.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. 8.SP.A.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.
8.SP.A.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. 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?
Overview
In Grades 6 and 7, students worked with data involving a single variable. Module 6 introduces students to bivariate data. Students are introduced to a function as a rule that assigns exactly one value to each input. In this module, students use their understanding of functions to model the possible relationships of bivariate data. This module is important in setting a foundation for studentsβ work in algebra in Grade 9.
Unit Vocab
Association (An association is a relationship between two variables. The tendency for two variables to vary together in a predictable way.)
Column relative frequency (In a two-way table, a column relative frequency is a cell frequency divided by the column total for that cell.)
Row relative frequency (In a two-way table, a row relative frequency is a cell frequency divided by the row total for that cell.)
Two-way table (A two-way table is a table used to summarize data on two categorical variables. The rows of the table correspond to the possible categories for one of the variables, and the columns of the table correspond to the possible categories for the other variable. Entries in the cells of the table indicate the number of times that a particular category combination occurs in the data set or the frequency for that combination.)
Topic A: Linear Functions
Topic A: Linear Functions
Focus Standards
8.F.B.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 (π₯π₯,π¦π¦) 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.
8.F.B.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
Overview
Topic A examines the relationship between two variables using linear functions. Linear functions are connected to a context using the initial value and slope as a rate of change to interpret the context. Students represent linear functions by using tables and graphs and by specifying rate of change and initial value. Slope is also interpreted as an indication of whether the function is increasing or decreasing and as an indication of the steepness of the graph of the linear function. Nonlinear functions are explored by examining nonlinear graphs and verbal descriptions of nonlinear behavior.
Lessons
Lesson 1: Modeling Linear Relationships *Priority Lessons*
Lesson 2: Interpreting Rate of Change and Initial Value *Priority Lessons*
Lesson 3: Representations of a Line *Priority Lessons*
Lesson 4: Increasing and Decreasing Functions *Priority Lessons*
Topic B: Bivariate Numerical Data
Topic B: Bivariate Numerical Data
Focus Standards
8.SP.A.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.
8.SP.A.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.
Overview
Topic B uses linear functions to model the relationship between two quantitative variables as students move to the domain of Statistics and Probability. Students make scatter plots based on data. They also examine the patterns of their scatter plots or given scatter plots. Students assess the fit of a linear model by judging the closeness of the data points to the line.
Lessons
Lesson 6: Scatter Plots *Priority Lesson*
Lesson 8: Informally Fitting a Line *Priority Lesson*
Topic C: Linear and Nonlinear Models
Topic C: Linear and Nonlinear Models
Focus Standards
8.SP.A.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.
8.SP.A.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.
8.SP.A.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.
Overview
In Topic C, students use linear and nonlinear models to answer questions in context. They interpret the rate of change and the initial value in context. They use the equation of a linear function and its graph to make predictions. Students also examine graphs of nonlinear functions and use nonlinear functions to model relationships that are nonlinear. Students gain experience with the mathematical practice of βmodeling with mathematicsβ
Lessons
Lesson 9: Determining the Equation of a Line Fit to Data *Priority Lesson*
Lesson 10: Linear Models *Priority Lesson*
Topic D: Bivariate Categorical Data
Topic D: Bivariate Categorical Data
Focus Standards
8.SP.A.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. 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?
Overview
In Topic D, students examine bivariate categorical data by using two-way tables to determine relative frequencies. They use the relative frequencies calculated from tables to informally assess possible associations between two categorical variables.
Lessons
Lesson 13: Summarizing Bivariate Categorical Data in a Two-Way Table *Priority Lesson*
Module 7: Introduction to Irrational Numbers Using Geometry
Module 7: Introduction to Irrational Numbers Using Geometry
(23 Hours)
(23 Hours)
Focus Standards
8.NS.A.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.
8.NS.A.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., π 2 ). For example, by truncating the decimal expansion of β2, show that β2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get a better approximation.
8.EE.A.2 Use square root and cube root symbols to represent solutions to the equations of the form π₯ 2 = π and π₯ 3 = π, where π is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that β2 is irrational.
8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Overview
Module 7 begins with work related to the Pythagorean Theorem and right triangles. Before the lessons of this module are presented to students, it is important that the lessons in Modules 2 and 3 related to the Pythagorean Theorem are taught. In Modules 2 and 3, students used the Pythagorean Theorem to determine the unknown length of a right triangle. In cases where the side length was an integer, students computed the length. When the side length was not an integer, students left the answer in the form of x2=c, where c was not a perfect square number. Those solutions are revisited and are the motivation for learning about square roots and irrational numbers in general.
Unit Vocab
Perfect Square (A perfect square is the square of an integer.)
Square Root (The square root of a number π is equal to π if π2 = π. It is denoted by βπ.)
Cube Root (The cube root of a number π is equal to π if π3 = π. It is denoted by βπ 3 .)
Irrational Number (Irrational numbers are numbers that are not rational.) Infinite Decimals (Infinite decimals are decimals that do not repeat nor terminate.)
Rational Approximation (Rational approximation is the method for determining the approximated rational form of an irrational number).
Truncated Cone (A truncated cone is a solid obtained from a cone by removing the top portion above a plane parallel to the base.)
Topic A: Square and Cube Roots
Topic A: Square and Cube Roots
Focus Standards
8.NS.A.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.
8.NS.A.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., ). For example, by truncating the decimal expansion of show that is between 1 and 2, then between 1.4 and 1.5, and explain how to continue to get better approximations.
8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the form and where is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational.
Overview
The definition for irrational numbers relies on studentsβ understanding of rational numbers, that is, students know that rational numbers are points on a number line and that every quotient of integers (with a non-zero divisor) is a rational number. Then irrational numbers are numbers that can be placed in their approximate positions on a number line and not expressed as a quotient of integers.
Though the term βirrationalβ is not introduced until Topic B, students learn that irrational numbers exist and are different from rational numbers. Students learn to find positive square roots and cube roots of expressions and know that there is only one such number. Topic A includes some extension work on simplifying perfect square factors of radicals in preparation for Algebra I.
Lessons
Lesson 1: The Pythagorean Theorem *Priority Lesson*
Lesson 2: Square Roots *Priority Lesson*
Lesson 3: Existence and Uniqueness of Square and Cube Roots *Priority Lesson*
Lesson 5: Solving Radical Equations *Priority Lesson*
Topic B: Decimal Expansions of Numbers
Topic B: Decimal Expansions of Numbers
Focus Standards
8.NS.A.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.
8.NS.A.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., ). For example, by truncating the decimal expansion of show that is between 1 and 2, then between 1.4 and 1.5, and explain how to continue to get better approximations.
8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the form and where is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational.
Overview
In Topic B, students learn that to get the decimal expansion of a number, they must develop a deeper understanding of the long division algorithm learned in Grades 6 and 7. Students learn a procedure to get the approximate decimal expansion of. At this point, students learn that the definition of an irrational number is a number that is not equal to a rational number.
Defining irrational numbers as those that are not equal to rational numbers provides an important guidepost for studentsβ knowledge of numbers. Students learn that an irrational number is something quite different than other numbers they have studied before. They are infinite decimals that can only be expressed by a decimal approximation. Now that students know that irrational numbers can be approximated, they extend their knowledge of the number line gained in Grade 6 to include being able to position irrational numbers on a line diagram in their approximate locations.
Lessons
Lesson 6: Finite and Infinite Decimals *Priority Lesson*
Lesson 8: The Long Division Algorithm *Priority Lesson*
Lesson 9: Decimal Expansions of Fractions, Part 1 *Priority Lesson*
Lesson 10: Converting Repeating Decimals to Fractions *Priority Lesson*
Lesson 11: The Decimal Expansion of Some Irrational Numbers *Priority Lesson*
Lesson 14: The Decimal Expansion of Ο *Priority Lesson*
Topic C: The Pythagorean Theorem
Topic C: The Pythagorean Theorem
Focus Standards
8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Overview
Topic C revisits the Pythagorean Theorem and its applications, now in a context that includes the use of square roots and irrational numbers. Students learn another proof of the Pythagorean Theorem involving areas of squares off of each side of a right triangle. Another proof of the converse of the Pythagorean Theorem is presented to students, which requires an understanding of congruent triangles. With the concept of square roots firmly in place, students apply the Pythagorean Theorem to solve real-world and mathematical problems to determine an unknown side length of a right triangle and the distance between two points on the coordinate plane
Lessons
Lesson 15: The Pythagorean Theorem, Revisited *Priority Lesson*
Lesson 17: Distance on the Coordinate Plane *Priority Lesson*
Lesson 18: Applications of the Pythagorean Theorem *Priority Lesson*
Topic D: Applications of Radicals and Roots
Topic D: Applications of Radicals and Roots
Focus Standards
8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems
Overview
The Pythagorean Theorem is applied to three-dimensional figures in Topic D as students learn some geometric applications of radicals and roots. In order for students to determine the volume of a cone or sphere, they must first apply the Pythagorean Theorem to determine the height of the cone or the radius of the sphere.
Students learn that truncated cones are solids obtained by removing the top portion above a plane parallel to the base. Students know that to find the volume of a truncated cone they must access and apply their knowledge of similar figures learned in Module 3. Their work with truncated cones is an exploration of solids that is not formally assessed. In general, students solve real-world and mathematical problems in three dimensions in Topic D.
Lessons
Lesson 19: Cones and Spheres *Priority Lesson*
Lesson 21: Volume of Composite Solids *Priority Lesson*
Lesson 22: Average Rate of Change *Priority Lesson*
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