Unit Title: Expressions and Equations Standard 8.EE Expressions and Equations ROCKAWAY TOWNSHIP PUBLIC SCHOOLS MATHEMATICS UNIT GUIDE GRADE 8 MATH Time Frame: First Marking Period 21 st Century Theme 9.1.A Critical Thinking and Problem Solving 9.1.B Creativity and Innovation Enduring Understandings: Exponents can be used to represent numbers of varying magnitude. Slope of a line is a constant rate of change. One form of an equation for a line is y = mx+b, where m is the slope and b is the y-intercept. Inverse operations are used to solve equations. Linear equations in one variable can have one solution, infinitely many solutions, or no solution. Solutions to systems of equations in two variables are either one, infinitely many, or no solutions. Essential Questions: How are exponents and roots helpful in real-world data? How is scientific notation helpful in representing very large or very small numbers? How can the rate of change be found in various representations of linear data? How can multiple equations be used together to help solve real-world situations? How is thinking algebraically different from thinking arithmetically? Cumulative Progress Indicator Number(s): 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. 8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. 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. 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. 8.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 + b for a line intercepting the vertical axis at b. 8.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 = a, or a = b 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. 8.EE.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 real-world and mathematical problems leading to two linear equations in two variables. Unit Learning Targets: Suggested Activities: Vocabulary The student will be able to. Use powers and exponents to write large and small numbers Simplify real number expressions by multiplying and dividing monomials Use laws of exponents to find powers of monomials Write and evaluate expressions using negative exponents Use scientific notation to write large and small numbers Compute with numbers written in scientific notation Find square roots and cube roots Estimate square roots and cube roots Compare mathematical expressions involving real numbers Identify proportional and nonproportional linear relationships by finding a constant rate of change Find rates of change Find slope of a line Compare and contrast proportional and non-proportional linear relationships Use direct variation to solve problems Graph linear equations using the x- and y- intercept Including Differentiated Strategies (DI) Students will work with a partner to explore multiplying with the same base (Starting with a salary of $.01 and doubling it each day for a month-30 days) Students can use colored pencils to color each base a different color (DI) Use algebra tiles to model the Quotient of Powers Property (DI) Students will use a table showing the density of one atom of various elements and list the elements in order from least to greatest Use algebra tiles to model and solve multistep equations Equation System of equations Slope Rise Run y-intercept x-intercept Slope-intercept form Standard form Solution Linear Elimination Linear combination Coordinate Inverse operations Distributive property Constant rate of change Direct variation Constant of variation Unit rate Origin Radical Square root
Write and solve algebraic equations from verbal sentences and problem situations Solve equations using addition, subtraction, multiplication and division properties of equality Write and solve multi-step equations Write, solve, and graph multi-step inequalities in one variable Graph equations written in standard form Find one solution for a set of two equations Solve systems of equations by graphing, substitution and elimination (linear combination) Explore special systems of linear equations that have no solution or an infinite number of solutions Resources and Technology Integration Current textbook and textbook resources Graph paper Illuminations.nctm.org Mathwire.com www.brainpop Easiteach Interactive whiteboard Kuta software www.shodor.org/interactive Edhelper.com Eduplace.com/kids/mw Geometer s Sketchpad MathSnacks.com Quia.com nlvm.usu.edu Algebra tiles Cube root Scientific notation Integer exponents Perfect squares Perfect cubes Power Base Exponent Monomial Binomial Trinomial Assessments Quizzes Formal assessments Participation Homework Rubric specific activities or projects Teacher created assessments Open-ended assessments Math journal Teacher observation during cooperative learning groups
Unit Title: The Number System Standard 8.NS The Number System ROCKAWAY TOWNSHIP PUBLIC SCHOOLS MATHEMATICS UNIT GUIDE GRADE 8 MATH Time Frame: Second Marking Period 21 st Century Theme 9.1.A Critical Thinking and Problem Solving 9.1.B Creativity and Innovation Enduring Understandings: Real numbers consist of both rational and irrational numbers Essential Questions: How is an understanding of rational and irrational numbers and relationships useful in problem solving? What is the difference between a rational and irrational number? When is it appropriate to use estimation and /or approximation? Cumulative Progress Indicator Number(s): 8.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. 8.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., π 2 ). Unit Learning Targets: Suggested Activities: Vocabulary The student will be able to. Identify rational and irrational numbers Approximate square roots as being between two integers Find cube roots Use properties of square roots to simplify expressions Including Differentiated Strategies (DI) Have students work in groups to find decimal equivalents of rational numbers Use graphic organizers to depict rational and irrational numbers Use Geo boards to estimate roots of nonperfect squares Use uni-cubes to calculate perfect cubes and approximate Real number system Rational Irrational Square root Repeating decimal Terminating decimal Radical Non-repeating Non-terminating Integers
Resources and Technology Integration Current textbook and textbook resources Graph paper Illuminations.nctm.org Mathwire.com www.brainpop Easiteach Interactive whiteboard Kutasoftware www.shodor.org/interactive Edhelper.com MathSnacks.com Quia.com nlvm.usu.edu Geo boards Assessments Quizzes Formal assessments Participation Homework Rubric specific activities or projects Teacher created assessments Open-ended assessments Math journal Teacher observation during cooperative learning groups
Unit Title: Functions Standard ROCKAWAY TOWNSHIP PUBLIC SCHOOLS MATHEMATICS UNIT GUIDE GRADE 8 MATH Time Frame: Second Marking Period 21 st Century Theme 8.F Functions 9.1.A Critical Thinking and Problem Solving 9.1.B Creativity and Innovation Enduring Understandings: Functions can be represented algebraically, graphically, numerically in tables (ordered pairs), or by verbal descriptions. Functions are used to model real-world phenomena. Essential Questions: What are the characteristics of a function? What is the difference between an input and an output of an equation? What values in the coordinate plane can be represented by the domain and/or range? What kind of relationships can be found when comparing two sets of data? Cumulative Progress Indicator Number(s): 8.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. 1 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 8.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. 8.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 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. 1 function notation is not required in Grade 8.
Unit Learning Targets: The student will be able to. Identify the domain and range of a graph and table of values Graph functions and determine if the domain is discrete or continuous Explore linear patterns in tables and graphs to write linear equations Compare linear and non-linear functions Compare proportional relationships and functions Write the equation of a line given the slope and a point Assume that the rate of change (slope) remains constant for a period of time in real-life situations Suggested Activities: Including Differentiated Strategies (DI) Use an input/output table to show a function Use the vertical line test to determine whether it is a function or not Describe what is meant by domain and range of a function Look for patterns and sequences in given tables to find functions and be able to write equations (function rules) Graph and see that speed is a function Vocabulary Functions Linear function Nonlinear Domain Range Resources and Technology Integration Current textbook and textbook resources Graph paper Illuminations.nctm.org Mathwire.com www.brainpop Easiteach Interactive whiteboard Kutasoftware www.shodor.org/interactive Edhelper.com MathSnacks.com Quia.com Assessments Quizzes Formal assessments Participation Homework Rubric specific activities or projects Teacher created assessments Open-ended assessments Math journal Teacher observation during cooperative learning groups
Unit Title: Geometry Standard ROCKAWAY TOWNSHIP PUBLIC SCHOOLS MATHEMATICS UNIT GUIDE GRADE 8 MATH Time Frame: Third and Fourth Marking Period 21 st Century Theme 8.G Geometry 9.1.A Critical Thinking and Problem Solving 9.1.B Creativity and Innovation Enduring Understandings: Reflections, rotations, and translations are rigid transformations and maintain congruence. Dilations may change the size of the object being transformed, but not the shape. There is a special relationship between the side lengths of a right triangle that states that the sums of the squares of the legs equal the square of the hypotenuse. There is a relationship between the volumes of cylinders, cones, and spheres. Essential Questions: How can a coordinate grid be used to model and describe the results of various transformations? How can transformations and symmetry be used to investigate and describe geometric situations? What occurs when two parallel lines are cut by a transversal? How can you use any two sides of a right triangle to find the third side? What is the relationship between the legs and the hypotenuse in a right triangle? What is the connection between the volume of cylinders and cones? Cumulative Progress Indicator Number(s): 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. 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 8.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. 8.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.
8.G.6 Explain a proof of the Pythagorean Theorem and its converse. 8.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. 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. 8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Unit Learning Targets: Suggested Activities: Vocabulary The student will be able to. Identify relationships of angles formed by two parallel lines cut by a transversal Examine angle relationships formed by parallel lines and a transversal Graph translations on the coordinate plane Graph reflections on the coordinate plane Identify rotational symmetry Use the scale factor to graph dilations on the coordinate plane Draw compositions of translations, reflections and rotations Find relationship among the sides of a right triangle Use Pythagorean Theorem Solve problems using the Pythagorean Theorem Find the distance between two points on the coordinate plane Find volume of cylinders, cones and spheres Including Differentiated Strategies (DI) Use graph paper to estimate the distance between two points on the coordinate plane Use the Pythagorean Theorem to calculate the length of a diagonal Write a short story using the Pythagorean Theorem to solve a real-world problem Use boxes to find the measurement of diagonals of the top, side and front faces Use an animated picture and graph paper to transform (translation, reflection, rotation, dilation) the picture in the coordinate plane Use a picture to compare and contrast angle measures and corresponding lengths Given two parallel lines, students will create their own transversals and determine the measures of all interior and exterior angles Have students bring in a cylindrical object to determine its volume Volume Area Circle Cone Cylinder Sphere Composite solids Cubic units Transformations Rotations Center of rotation Reflections Translations Symmetry Line of symmetry Similar triangles Congruent Pre-image Image Interior angles Exterior angles Alternate interior angles Alternate exterior angles Corresponding angles Parallel lines
Transversal Vertical angles Supplementary angles Complementary angles Adjacent angles Pythagorean theorem Converse Right triangle Legs Hypotenuse Square Square root Area Distance formula Resources and Technology Integration Current textbook and textbook resources Graph paper Illuminations.nctm.org Mathwire.com www.brainpop Easiteach Interactive whiteboard Kutasoftware www.shodor.org/interactive Edhelper.com Eduplace.com/kids/mw Geometer s Sketchpad MathSnacks.com Quia.com nlvm.usu.edu Mirra Assessments Quizzes Formal assessments Participation Homework Rubric specific activities or projects Teacher created assessments Open-ended assessments Math journal Teacher observation during cooperative learning groups
ROCKAWAY TOWNSHIP PUBLIC SCHOOLS MATHEMATICS UNIT GUIDE GRADE 8 MATH Unit Title: Time Frame: Fourth Marking Period Investigate Patterns of Association in Bivariate Data Standard 8.SP Statistics and Probability 21 st Century Theme 9.1.A Critical Thinking and Problem Solving 9.1.B Creativity and Innovation Enduring Understandings: Identifying outliers and other data characteristics allows for meaningful data interpretation and analysis. Linear trends can be identified as positive or negative, while some trends have no correlation. The line of best fit represents the data set as a whole. Essential Questions: What kind of relationships can be found when comparing two sets of data? How can mathematics help you make good decisions? How can predictions be made based on data and probability? How do people use data to influence others? How can a linear equation be used to model bivariate data? Cumulative Progress Indicator Number(s): 8.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. 8.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. 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. 8.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.
Unit Learning Targets: The student will be able to. Study Measures of Central Tendency as Well as Fair and Unfair Distribution Explore How an Outlier Affects the Three Measures of Central Tendency Gain an Intuitive Understanding of How to Construct Scatter Plots and how to write an equation of the Line of Best Fit Construct and interpret two-way tables using categorical data Suggested Activities: Including Differentiated Strategies (DI) Using the height of each student in class find the measures of central tendency Make and identify whether a scatter plot has a positive, negative, or relatively no relationship using real-world data Create a scatter plot, graph and write an equation of best fit line. Use that equation to make predictions Compare and contrast data with other classes Vocabulary Scatter-plot Outlier Quantitative variable Slope y-intercept Positive association Negative association Relative frequency Bivariate data Venn diagram Two-way tables Resources and Technology Integration Current textbook and textbook resources Graph paper Illuminations.nctm.org Mathwire.com www.brainpop Easiteach Interactive whiteboard Kutasoftware www.shodor.org/interactive Edhelper.com Eduplace.com/kids/mw Geometer s Sketchpad MathSnacks.com Quia.com nlvm.usu.edu Assessments Quizzes Formal assessments Participation Homework Rubric specific activities or projects Teacher created assessments Open-ended assessments Math journal Teacher observation during cooperative learning groups