Course: Grade 8 Mathematics Year: 2013 2014 Teachers: MaryAnn Valentino, Cheryl Flanagan Unit 1: Equations Approximate Time Frame: # of Weeks 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. How can inverse operations are used to solve equations? What does it mean for equations in one variable to have one solution, no solutions or infinite solutions? How can I use equations to represent and solve real world problems? Expressions can be manipulated (decomposed and recomposed) in different ways to generate equivalent expressions to simplify the problem. The equal sign demonstrates equivalence. Ex: 2x + x = 3x (equivalent expressions) 2x + x and 3x + 4 are not equivalent expressions Inverse operations are used to solve equations and inequalities. Solutions to an equation are the values of the variables that make the equation/ true. SOLVE linear equations including equations with fractional number coefficients EXPAND expressions and equations by using the distributive property and combining like terms SOLVE multi step equations including distributive property, combing like terms and with variables on both sides of the equations UNDERSTAND solutions Inverse operations are used to solve equations. Equations in one variable can have one solution, no solutions or infinite solutions. Combine Like Terms Distributive Property Inverse Operations Infinitely Many Solutions/Identity
Unit 2: The Number System Approximate Time Frame: # of Weeks 8.NS.1. Know that numbers Why do I need to In the real number KNOW rational and irrational numbers There are numbers that Real numbers that are not rational are called understand the types of system, numbers can be are not rational called Rational numbers irrational. Understand informally that every number numbers found in the real defined by their decimal UNDERSTAND decimal expansion irrational. Irrational numbers has a decimal expansion; for number system? representations. Repeating decimals rational numbers show that SHOW decimal expansion repeats Irrational numbers are a Terminating the decimal expansion repeats How do I determine the subset of the Real eventually, and convert a decimals decimal expansion which best numerical CONVERT repeating decimal Number System. Decimal expansion repeats eventually into a representation for a given expansion to a rational number Square root rational number. situation? Every number has a Perfect square USE rational approximations of decimal representation: Cube root irrational numbers Irrational decimals Perfect cube are non repeating COMPARE sizes of rational numbers and non terminating 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., 2 ). For example, by truncating the decimal expansion of 2, show that 2 is between 1and 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 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 irraonal. LOCATE rational numbers approximately on a number line ESTIMATE value of expressions USE square root and cube root symbols EVALUATE square roots of perfect squares and cube roots of perfect cubes Rational number decimals eventually terminate or repeat. Irrational numbers can be approximated for comparing and ordering them. A perfect square is a number in which the square root is an integer. A perfect cube is a number in which the cube root is an integer. The 2 is irrational. Equivalent forms of an expression allows for efficient problem solving. Estimation as a means for predicting & assessing the reasonableness of a solution.
Unit 3: Exponential Expressions Approximate Time Frame: # of Weeks 8.EE.1. Know and apply the Why do I need to The properties of USE integers with a power of 10 to Properties of integer Exponent properties of integer exponents to generate understand the types of number systems and ESTIMATE large or small exponents. Scientific equivalent numerical numbers found in the their relationships quantities and express magnitude notation expressions. real number system? remain consistent of numbers using powers of 10 Perfect squares and Square root when applied to perfect cube numbers Perfect square How do I determine the integer exponents USE square root and cube root up to 100. Cube root best numerical symbols Perfect cube representation for a given situation? 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. USE scientific notation to REPRESENT very large and very small numbers and choose units of appropriate size EVALUATE square roots of perfect squares and cube roots of perfect cubes KNOW/APPLY properties of integer exponents and generate equivalent numerical expressions CALCULATE/CONVERT numbers expressed in scientific notation/decimal form The base ten number system can be applied to represent very large and very small numbers using powers of 10. Equivalent forms of an expression allows for efficient problem solving. Strategies for computing with numbers expressed in scientific notation. Estimation as a means for predicting & assessing the reasonableness of a solution.
Unit 4: Pythagorean Theorem Approximate Time Frame: # of Weeks 8.G.6 Explain a proof of the How can our The Pythagorean APPLY the Pythagorean Theorem In a right triangle, the Base Pythagorean Theorem and its converse. understanding of the Theorem is essential legs are the side Cartesian Plane Pythagorean Theorem for solving real world DETERMINE unknown side lengths that form the Converse affect our problems because it lengths in right triangles right angle and the 8.G.7 Apply the Pythagorean Coordinate System Theorem to determine understanding of the can be applied to hypotenuse is the Distance Formula unknown side lengths in right world around us? many situations where FIND distance between two points diagonal length that triangles in real world and Exponents mathematical problems in two a missing length of a in a coordinate system connects the legs Hypotenuse and three dimensions. Why is it necessary to right triangle needs to Irrational Number prove formulas true? be calculated. EXPLAIN a proof of the Pythagorean Theorem: Legs 8.G.8 Apply the Pythagorean Pythagorean Theorem and its a 2 + b 2 = c 2 where a Perfect Square Theorem to find the distance The Pythagorean converse and b are the legs of between two points in a Power coordinate system. Theorem relates the the triangle and c is Pythagorean areas of squares on USE square root and cube root the hypotenuse Theorem the sides of the right symbols; REPRESENT solutions to triangle. equations using radical symbols Proof of the Pythagorean triples Pythagorean Theorem (e.g. 3, 4, 5 & 5, 12, 8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irraonal. = The Pythagorean Theorem is a relationship among the sides of a right triangle. There is more than one way to prove the Pythagorean Theorem and its converse. EVALUATE square roots of small perfect squares; cube roots of small perfect cubes Converse of the Pythagorean Theorem The Pythagorean Theorem only applies to right triangles Pythagorean triples, such as (3, 4, 5) and (5, 12, 13) 2 is irrational 13) Radicals Rational Number Right Triangle Square Root
Unit 5: Congruence and Similarity Approximate Time Frame: # of Weeks 8.G.2. Understand that a twodimensional figure is congruent How are spatial Spatial relationships VERIFY experimentally the Transformations Similar to another if the second can be relationships used help to make sense of properties of include: reflections, Congruent obtained from the first by a to represent real the physical space o Rotations translations, rotations, Reflection sequence of rotations, situations? around us. o Reflections and dilations reflections and translations; Translation given two congruent figures, o Translations Rotation describe a sequence that Why does shape o Dilations Reflections, rotations, Dilation exhibits the congruence between them. and size matter? and translations are Scale Factor OR DESCRIBE a sequence of rigid transformations Corresponding Angles Why do geometric rotations, reflections, and that maintain Complementary relationships and translations that exhibits congruence Angles measurements congruence between two figures matter? and dilations that exhibits similarity between two figures. 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.4. Understand that a twodimensional 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 twodimensional figures, describe a sequence that exhibits the similarity between them. 8.G.3. Describe the effect of dilations, translations, rotations, and reflections on two dimensional figures using coordinates. 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. DESCRIBE the effect of dilations, translations, rotations and reflections on two dimensional figures USE coordinates to describe the effect of dilations, translations, rotations and reflections on twodimensional figures PROVE informally o angle relationships in parallel lines cut by a transversal o sum of angles in a triangle = 180 FIND missing angle measures of a triangle FIND missing angle measures when two parallel lines are cut by a transversal DETERMINE if two triangles are similar using the Angle Angle criterion of similar triangles Congruence and similarity in terms of transformations When parallel lines are cut by a transversal, it creates congruency of angles Two triangles are similar if at least two pairs corresponding angles are congruent (AA postulate for similarity)
Unit 6: Functions Approximate Time Frame: # of Weeks 8.F.1 Understand that a How can mathematics Students will INTERPRET the form y= mx + b as Properties of functions Linear function is a rule that assigns to each input exactly one be used to measure, understand that the defining a linear function One input for Ordered Pairs: output. The graph of a function model and calculate various methods to one output is the set of ordered pairs Slope: change? display linear GIVE example of linear and Linear vs. consisting of an input and the corresponding output. relationships provide nonlinear equations non linear Y intercept How can linear opportunities to relationships influence explain a setting in COMPARE functions your real world multiple o Algebraically decision making? representations. o Graphically 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. 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. Why is it valuable to understand a situation in multiple representations? Students will understand that there are multiple ways to model and compare linear relationships that arise in everyday life. o o Numerically in tables Verbal descriptions CONSTRUCT functions o Algebraically o Graphically o Numerically in tables o Verbal descriptions o Model linear relationships DETERMINE and INTERPRET the rate of change DETERMINE initial value of a function INTERPRET initial value of a linear function o in terms of the situation it models o in terms of its graph or a table of values READ a description of a relationship from a table or a graph DESCRIBE a functional relationship between two quantities Functional Representations (linear and nonlinear): Algebraic (y = mx and y = mx+b) Graphic Table Verbal Rate of change/slope Initial value/yintercept Slope of a line is a constant rate of change The y intercept (initial value) is the point at which a line intersects the vertical axis (yaxis) A graph of a linear function is a straight line
SKETCH a graph (linear or nonlinear) from a verbal description Unit 7: Proportional Relationships and Linear Equations Approximate Time Frame: # of Weeks 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 for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. 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 systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. How is mathematics used to measure, model and calculate change? How are the characteristics of a linear relationship relevant in a real world situation How is determining the solution for a system of linear equations valuable in describing relationships in real world situations? What are the benefits of having different types of strategies to solve systems of linear equations related to real world situations? The connection between linearity and proportionality (as a special case of linearity) is based on an understanding of slope as the constant rate of change and the y intercept. Systems of linear equations can be used to analyze and solve real world problems. Systems of linear equations can be analyzed and solved graphically and algebraically. Solving a system of two linear equations requires a point that satisfies both equations simultaneously. This point may or may not exist in certain situations. A system of linear equations is an algebraic way to model a situation and find an exact solution to the constraints of the situation. GRAPH proportional relationships INTERPRET unit rate as slope COMPARE different representations of proportional relationships EXPLAIN why slope is the same between any two points on a nonvertical line DERIVE linear equations (y = mx and y = mx + b SOLVE linear equations including equations with fractional number coefficients EXPAND expressions and equations by o Using the distributive property o Combining like terms GRAPH sets of ordered pairs ANALYZE linear equations UNDERSTAND solutions SOLVE systems of equations ESTIMATE solutions Proportional relationships can be represented symbolically (equation), graphically (coordinate plane), in a table, in diagrams, and verbal descriptions. In a proportional linear relationship, the point (0, 0) is the y intercept and (1, r) is the slope, where r is the unit rate. Slope of a line is a constant rate of change. The y intercept is the point at which a line intersects the vertical axis (y axis). One form of an equation for a line is y=mx + b, where m is the slope and b is the y intercept. A special case of linear equations (proportional relationships) are in the form of y/x = m and y = mx. The formula Distance = Rate Time Linear Non Linear Ordered Pairs Proportional Relationships Similar Triangles Slope Unit Rate: Y intercept System of Equations Point of Intersection Consistent System Inconsistent System Dependent System Independent System Solution to a system of linear equations Graphing method Substitution method Elimination method Viable solutions Nonviable solutions Constraints Intersection Linear Programming
A system of linear inequalities is an algebraic way to model a situation and find viable solutions to the constraints of the situation. System of Equations is represented by two or more linear equations. Solutions to a system of equations are the values of the variables that make both equations true (one point of intersection). Systems of equations that have infinitely many solutions are equivalent forms of the same equation, and represent the same line in a plane (all points intersect because they are the same line). Systems of equations that have no solution are parallel lines having equivalent slopes (no point of intersection).
Unit 8: Patterns and Data Approximate Time Frame: # of Weeks 8.SP.1. Construct and interpret Why is data collected The same Scatter Plots Bivariate data scatter plots for bivariate measurement data to and analyzed? characteristics used to Construct and Interpret scatter Patterns plots for bivariate Scatter plot investigate patterns of describe linear o Clustering association between two measurement data to Outlier quantities. Describe patterns How do people use data relationships allow us o Outliers Investigate patterns of Quantitative such as clustering, outliers, to influence others? to describe, classify, positive or negative association, association between two o Positive or and analyze the linear association, and quantities. Negative Variable nonlinear association. How can predictions be association of bivariate o Linear Qualitative Data made based on data? measurement data. Describe patterns such as o Nonlinear Slope 8.SP.4. Understand that patterns of association can also clustering, outliers, positive Frequencies Y intercept be seen in bivariate categorical or negative association, Two way table Positive Association data by displaying frequencies linear association, and and relative frequencies in a o variables How do people use two way table. Construct and nonlinear association. Negative Association Equation of Linear interpret a two way table data to influence Relative Frequency summarizing data on two Know that straight lines are Model categorical variables collected others? Histogram widely used to model from the same subjects. Use relative frequencies calculated relationships between two o Slope and Two Way Table for rows or columns to How can predictions quantitative variables Intercept Cluster describe possible association The line of best fit between the two variables. be made based on Distribution Use the equation of a linear represents the data 8.SP.3. Use the equation of a data? model to solve problems in Five number set as a whole, linear model to solve problems the context of bivariate summary fitted through the in the context of bivariate measurement data, Line of best fit measurement data, interpreting majority of points. the slope and intercept interpreting the slope and The characteristics of a Mean absolute intercept. linear relationship deviation Understand that patterns of Standard deviation association can also be seen Strategies for modeling Symmetric in bivariate categorical data a line of best fit given a Univariate by displaying frequencies and set of data. relative frequencies in a twoway table. 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. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to Describe possible Linear trends can be identified as positive or negative, while some trends have no correlation. Identifying outliers and other data characteristics allows for meaningful data
association between the two variables. interpretation and analysis. Unit 9: Volume Approximate Time Frame: # of Weeks Content Vocabulary Skills 8.G.9 Know the formulas for How can the volume of KNOW the formulas for volume of Formulas for volume the volumes of cones, cylinders and spheres and 3 D objects be used to cones, cylinders, and spheres of cones, cylinders, use them to solve real world solve real world and spheres. and mathematical problems problems? The volume of any three dimensional shape is dependent on the area of its base, height of the shape and the number of parallel bases (layers) that the shape has. The relationship between the volumes of cylinders, cones, and spheres. The volume of a cone is 1/3 of the volume of a cylinder (with same radius and height). The volume of a sphere is 4/3 the volume of a cylinder (with same radius and height). USE the formulas for volume of cones, cylinders, and spheres to solve real world problems SOLVE the formulas for volume of cones, cylinders, and spheres within the context of a problem Volume is measured in cubic units. The volume of a cylinder is the area of the base multiplied by the height. The volume of a cone is 1/3 of the area of the base multiplied by the height. The volume of a sphere is 4/3 times π multiplied by the radius cubed. Volume Pi (π) Cylinder Cone Sphere Hemisphere Diameter Radius Height Cubic Units Squared (raised to 2 nd power) Cubed (raised to 3 rd power)