West Windsor-Plainsboro Regional School District Math Resource Center Grade 8

West Windsor-Plainsboro Regional School District Math Resource Center Grade 8

Content Area: Mathematics Course & Grade Level: Math 8 Unit 1 - Foundations of Algebra Summary and Rationale This unit involves the study of numbers that are rational and irrational including but not limited to perfect and non-perfect squares and cube roots. Students will expand upon the fundamental rules of simplifying algebraic expressions. They will build upon their current knowledge of one step equations to solve more complex linear equations. As students master these concepts, they will have the building blocks for further study of mathematics. 42 days Standard 8.NS.1 8.NS.2 Recommended Pacing State Standards Cumulative Progress Indicator (CPI) 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. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. 8.EE.2 Use square root and cube root symbols to represent solutions to equations of the formx2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. 8.EE.7a 8.EE.7b 7.NS.A.1d 7.NS.A.2c 6.NS.C.7c Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Apply properties of operations as strategies to add and subtract rational numbers. Apply properties of operations as strategies to multiply and divide rational numbers. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation. For example, for an account balance of -30 dollars, write -30 = 30 to describe the size of the debt in dollars.

Instructional Focus Unit Enduring Understandings Numbers can be represented in a variety of forms that do not change the value of the number. All numbers belong to the set of complex numbers. Within that set are many subsets that help us describe and characterize numbers by their properties. Unit Essential Questions How is the cube root of a number different from the square root of a number? How can you solve a multi-step equation? How can you check the reasonableness of your solution? How do you determine whether a number is rational or irrational? Objectives Students will know: Vocabulary: Square Root, Cube Root, Rational number, Irrational number, Natural number, Whole number, Integers Students will be able to: Classify and understand numbers as rational and irrational. Evaluate square roots and cube roots and use them to make rational approximations of irrational numbers. Simplify multi-step algebraic expressions using order of operations, distributive property, combining like-terms, and integer rules. Solve multi-step linear equations with rational number coefficients. Solve and identify linear equations in one variable that have one solution, infinitely many solutions, or no solution.* Rewrite equations and formulas to solve for one variable in terms of the other variable(s). Resources Core Text: Big Ideas Math 8 Textbook - (Chapter 7 Sections 1, 2, 4), (Big Ideas Math 8 Skills Review and Basic Skills Handbook), (Chapter 1 sections 1-4) Suggested Resources: Big Ideas Math 8 Supplemental Materials * Optional, extension topics for RC students as dictated by IEP

Content Area: Mathematics Course & Grade Level: Math 8 8.F.2 Unit 2 - Linear Functions Summary and Rationale Functions are all around us, though students do not always realize this. Algebraic tools allow us to express these functional relationships very efficiently; find the value of one thing (such as the gas price) when we know the value of the other (the number of gallons); and display a relationship visually in a way that allows us to quickly grasp the direction, magnitude, and rate of change in one variable over a range of values of the other. Being able to describe a function as a line, equation or table of values allows us to use the function to interpret the relationship between the variables. Recommended Pacing 32 days Standard 8.F.4 8.EE.7a 8.EE.7b 8.EE.8a 8.EE.8b 8.EE.8c 8.F.1 State Standards Cumulative Progress Indicator (CPI) Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 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. 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. 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. 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. 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.3 8.F.4 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2giving 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. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.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. Instructional Focus Unit Enduring Understandings Linear functions are used to model, understand and interpret real-life data. We describe linear functions in a variety of ways slope, intercepts, form etc. Unit Essential Questions What does the slope of a line tell you about the line? What are the different ways to describe a line? How can you use different representations of functions to show the relationship between two data sets? How can you solve a system of linear equations? What are the types of possible numbers of solutions to a system of equations? Objectives Students will know: Vocabulary: Linear Equation, Slope, Rise, Run, x-intercept, y-intercept, Slope-Intercept Form, Function, Linear Function, System of Equations, Input, Output Students will be able to: Understand the behavior of a line by interpreting its slope Determine the slope of a line by analyzing the graph and/or applying the slope formula. Understand the components of slope-intercept form of a line and apply them to create its graph. Write an equation of the line using the slope and a point on the line. * Apply the slope of a line to determine if two lines are parallel, or perpendicular. Determine if a relation is a function. Compare and write functions represented in different ways (words, tables, and graphs). Understand that y= mx +b is a linear function and recognize nonlinear functions. Solve a system of equations through graphing, substitution, and elimination. Solve system of equations with no solution or infinitely many solutions.

Resources Core Text: Big Ideas Math 8 Textbook - (Chapter 4 Sections 2, 4, 6), (Chapter 6 Sections 1-5), (Chapter 5 Sections 1-4) Suggested Resources: Big Ideas Math 8 Supplemental Materials * Optional, extension topics for RC students as dictated by IEP

Content Area: Mathematics Course & Grade Level: Math 8 Unit 3 - Exponents and Scientific Notation Summary and Rationale This unit involves the study of integer exponents and scientific notation. Scientific Notation is a method for writing very large and very small numbers efficiently. It allows mathematicians to calculate using these large and small numbers in an effective way. The laws of exponents help us work with numbers in scientific notation and can be expanded to simplifying many algebraic expressions. This simplification often makes the expressions easier to work with. In this unit, students will explore the different exponent rules and their applications. 11 days Standard 8.EE.1 8.EE.3 Cumulative Progress Indicator (CPI) Recommended Pacing State Standards Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 3-5 = 3-3 = 1/33 = 1/27. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger. Instructional Focus Unit Enduring Understandings Scientific notation is used to represent very large and very small numbers. Understanding the rules of exponents helps us to compare and calculate with very large and very small numbers Unit Essential Questions How can you use inductive reasoning to observe patterns and write general rules involving properties of exponents? How can you evaluate a nonzero number with an exponent of zero? With a negative integer? How can you read and write numbers that are written in scientific notation? Objectives Students will know: Vocabulary: Power, Base, Exponent, Scientific Notation Students will be able to: Apply the properties of integer exponents (power of a power, power of a product, power of a quotient, zero, negative) to generate equivalent expressions Use scientific notation to estimate very large or small quantities. Interpret scientific notation generated by technology. Resources Core Text: Big Ideas Math 8 Textbook - (Chapter 10 Sections 1-6) Suggested Resources: Big Ideas Math 8 Supplemental Materials

Content Area: Mathematics Course & Grade Level: Math 8 Unit 4 - Geometry Summary and Rationale Geometry is the study of the size, shape and position of 2 dimensional shapes and 3 dimensional figures. In geometry, one explores spatial sense and geometric reasoning. This unit involves the study of congruence and similarity of figures. The students will understand and apply the Pythagorean Theorem and solve real world and mathematical problems involving volume of cylinders, cones, and spheres. 42 days Standard 8.G.1a Cumulative Progress Indicator (CPI) Recommended Pacing State Standards Verify experimentally the properties of rotations, reflections, and translations: Lines are taken to lines, and line segments to line segments of the same length. 8.G.1b 8.G.1c 8.G.2 8.G.3 8.G.4 8.G.5 8.G.6 8.G.7 Verify experimentally the properties of rotations, reflections, and translations: Angles are taken to angles of the same measure. Verify experimentally the properties of rotations, reflections, and translations: Parallel lines are taken to parallel lines. 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. Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. 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. 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. Explain a proof of the Pythagorean Theorem and its converse. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Instructional Focus Unit Enduring Understandings Geometry is the mathematics we use to describe our physical world. Geometry is the mathematics of measurement. Unit Essential Questions How can you identify congruent triangles? How can you manipulate an object on a plane? How do changes in dimensions of similar geometric figures affect the perimeters and areas of the figures? How can you describe angles formed by parallel lines and transversals? How are the lengths of the sides of a right triangle related? How can you find the volume of 3-dimensional figures? Objectives Students will know: Vocabulary: Congruency, Transformation, Translation, Reflection, Rotation, Dilation, Scale Factor, Transversal, Interior Angles, Exterior Angles, Corresponding Angles, Regular Polygon, Legs, Hypotenuse, Pythagorean Theorem, Cylinder, Cone, Sphere Students will be able to: Use corresponding angles and sides to identify congruent figures Identify translations, reflections, rotations and dilations* of polygons in the coordinate plane Understand the relationships between corresponding sides and angles of similar figures Identify and find measures of angles formed when parallel lines are cut by a transversal Understand and find the measures of interior and exterior angles of triangles Understand and find the measures of interior and exterior angles of polygons* Apply the Pythagorean Theorem to find missing side lengths of right triangles Apply the Pythagorean Theorem to find the distance between points on a coordinate plane* Know and apply the formula for the volume of cylinders to solve real world problems Know and apply the formula for the volume of cones to solve real world problems Know and apply the formula for the volume of spheres to solve real world problems Resources Core Text: Big Ideas Math 8 Textbook - (Chapter 2 Sections 1-7), (Chapter 3 Sections 1-4), (Chapter 7 Sections 3 and 5), Chapter 8 Sections 1-3) Suggested Resources: Big Ideas Math 8 Supplemental Materials * Optional, extension topics for RC students as dictated by IEP

Content Area: Mathematics Course & Grade Level: Math 8 Unit 5 - Statistics and Probability Summary and Rationale Statistics is the science of learning from data. As a discipline, it is concerned with the collection, analysis and interpretation of data as well as the effective communication and presentation of results relying on data. In this unit, students will use a line of best fit as a statistical method to make predictions 5 days Standard 8.SP.1 8.SP.2 Cumulative Progress Indicator (CPI) Recommended Pacing State Standards 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. 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. Instructional Focus Unit Enduring Understandings Statistics helps us understand data and use it to make decisions. Probability helps us understand what is likely to happen and we can use that information to make decisions. Unit Essential Questions How does a scatter plot help you understand a set of data? How can you use data/line of best fit to predict an event? Objectives Students will know: Positive correlation, negative correlation, line of best fit, outliers, clusters, scatter plot Students will be able to: Interpret scatter plots. Make predictions based on data and/or line of best fit. Conduct an experiment and display their data on a scatterplot Present their findings to the class Resources Core Text: Big Ideas Math 8 Textbook - (Chapter 9 Sections 1-2) Suggested Resources: Big Ideas Math 8 Supplemental Materials