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GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 UNIT 1 – NUMBER SENSE AND OPERATIONS REVIEW Critical Area Time Frame Connections to prior learning Connections to future learning Additional Notes Practice Standards Essential Questions -Number Sense & Operations 10 Days -All units previously covered -This unit will be taught in 2012 – 2013 and will be good review in future years -Additional tasks and test items can be found at www.map.mathshell.org -SMP1 – Problems -SMP6 – Precision What in the world are rational numbers and how and when do we use them? Content Standards 6.NS.4a 7.NS.1b Apply number theory concepts, including prime factorization and relatively prime number, to the solution of problems. Impact 2004 Impact 2009 Other Greatest Common Factor Lesson Impact Course 2 2004 3.1.1 Vocab Assessment ESSENTIAL: Integer, Absolute Value, Rational Numbers, Expressions, Equations Understand p + q as the number located a distance |q| from p, in the positive to negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of o (additive inverses). Interpret sums of rational numbers by describing real-world contexts. Page 1 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 7.NS.1 7.EE.3 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. NLVM - Circle 3 - Circle 21 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations as strategies to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies Illustrative Mathematics Tasks - Discounted Books - Shrinking Illustrative Mathematics Tasks - Comparing Freezing Points, - Distances on the Number Line 2, - Operations on the Number Line Page 2 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 UNIT 2 – RADICALS AND EXPONENTS Critical Area Time Frame Connections to prior learning Connections to future learning Additional Notes Practice Standards Essential Questions - 8NS1 and 2 are supporting clusters while the rest are major clusters. 20 days 6th grade: whole numbers exponents, rational numbers, absolute value 7th grade: apply and extend understanding of rational numbers Additional tasks and test items can be found at www.map.mathshell.org SMP1 – Problems SMP8 – Repeated Reasoning 8NS1 and 8NS2: How do we recognize and know value of rational numbers? 8EE1 and 8EE2: What effects do exponents and roots (radicals) have on numbers? 8EE3 and 8EE4: How can I make very large and very small numbers more manageable? Content Standards 8.NS.1 Impact 2004 Understand informally that every number has a decimal expansion; the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat. Know that other numbers are called irrational. Course 2: 7.1 & 7.2 Impact 2009 Other Illustrative Math Task -Converting Decimal Representations of Rational Numbers to Fraction Representations -Identifying Rational Numbers Vocab Assessment ESSENTIAL: Radicals, Irrational Numbers IMPORTANT: Exponent, Base, Power Page 3 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 8.NS.2 8.EE.2 8.EE.1 8.EE.3 Illustrative Math Task -Estimating Square Roots 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., p2). 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 on to get better approximations. 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 is irrational. Many Tasks on this link: Comparing rational and irrational numbers, Irrational numbers on the number line. Course 3 4.3 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/(3)3 = 1/27. Course 3 4.1 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or Course 3 4.1 Illustrative Math Task - The Ant and Page 4 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 8.EE.4 very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger. Elephant 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. Powers Of Ten – Outer Space to Atom Course 3 4.1 Illustrative Math Task -Giant Burgers Page 5 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 UNIT 3 – CONGRUENCE AND SIMILARITY USING TRANSFORMATIONS Critical Area Time Frame Connections to prior learning Connections to future learning Additional Notes Practice Standards Essential Questions Understand congruence and similarity using physical models, transparencies, or geometry software. 10 Days Grade 7: Students develop understanding of the special relationships of angles and their measures (complimentary, supplementary, adjacent, vertical) -This is a major cluster. -7G2 is a gap that needs to be taught in 2012-13 Course 1 Impact -Additional tasks and test items can be found at http://map.mathshell.org -8G5 is not being tested in 2012-13 -SMP3 – Arguments; SMP4 – Modeling; SMP5 - Tools How can we use transformations to show similarity or congruence of two-dimensional figures? Content Standards 8.G.1 8.G.1a 8.G.1b 8.G.1c Verify experimentally the properties of rotations, reflections and translation Impact 2004 Impact 2009 Other Course 3 6.1, 6.2, 6.3 NLVM: Triangles – Build similar triangles by combining sides and angles. a. Lines are taken to lines and line segments to line segments of the same length NLVM: Geoboard – Coordinate – Rectangular geoboard with x and y coordinates b. Angles are taken to angles of the same measure c. Parallel lines are taken to Parallel lines Course 3 6.1, 6.2, 6.3 NLVM: CompositionDilation Reflection Rotation Translation Vocab Assessment Resources ESSENTIAL: Transformation, Rotation, Dilation, Translation, Similar, Congruent, Transversal IMPORTANT: Interior Angles, Exterior Angles, Alternate Interior Angles, Alternate Exterior Angles, Vertical Angles, Corresponding Angles, Complementary Angles and Supplementary Angles. Page 6 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 8.G.2 8.G.3 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 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 angleangle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the three angles appear to form a line, and give an argument in terms of transversals why this is so. Course 3 6.1, 6.2, 6.3 Course 3 6.1, 6.2, 6.3 6.3 Course 1 TE 1.2 Extension Problem Course 3 2.2 Illustrative Math Task: -Find the missing Angle -Find the Angle Page 7 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 UNIT 4 – LINEAR FUNCTIONS Critical Area Time Frame Connections to prior learning Connections to future learning Additional Notes Practice Standards Essential Questions -Grasp the concept of a function and use to describe quantitative relationships - This is a MAJOR cluster 25 Days Grade 5: Graphing and plotting points on a co-ordinate plane Grade 6: Represent and analyze quantitative relationships between dependent and independent variables Grade 7: Analyze and solve problems involving ratios and proportions including input and output tables and graphs Grade 8: Solving systems of equations Additional test and task items can be found at http://map.mathshell.org S.MP.2 – Reasoning; S.MP.4 – Modeling; S.MP.7 – Structure 8F1, 2 and 4: What is a function and how can it be modeled? 8EE5 and 6: What mathematical ideas are represented by slope? SP 1, 2, 3, 4: Why use a scatter plot? Content Standards 8.F.1 Impact 2004 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. (Footnote: Function notation is not required in Grade 8.) Impact 2009 Other NC - Develop an understanding of function. NC – Matching Linear Functions n/a Course 3 10.1 Vocab Assessment Resources ESSENTIAL: Function, Linear, Slope, Y-Intercept IMPORTANT: Proportional Relationships, Origin Page 8 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 8.F.2 8.F.4 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. 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. Illustrative Math Task F2 Battery Charging Course 3 10.1, 10.2 EXPOSURE: Scatter Plot, Cluster, Outlier, Association, Correlation, Line Of Best Fit Illustrative Math Task F4: Baseball Cards (good intro task) Course 3 1.1, 1.2, 1.3 Chicken and Steak Variations 1(good instructional tool) A Picture Tells a Linear Story Illustrative Math Task F4: Video Streaming 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. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Illustrative Math Tasks -Coffee by the Pound (good intro), -Comparing Speeds in Graphs and Equations, Course 3 1.1 -Peaches and Plums, -Sore Throats Var. 2, -Who has the Best Job? Page 9 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 8.EE.6 8.SP.1 8.SP.2 8.SP.3 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. 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. 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. Finding Slope of a Line from Graphs, Tables and Ordered Pairs Course 3 1.2 Similar triangles Glued to the Tube or Hooked on a Book Course 3 2.1 Investigation 4 Impact of a Superstar Exploring Linear Data Scatter plots, cluster, outlier, association/correlatio n (positive, negative, none, linear, nonlinear), Line of Best Fit Video 26 Course 3 2.1 Inv. 4 Illustrative Math Task -Birds' Eggs Line of best fit Course 3 2.1 Inv. 4 Page 10 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 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. 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? Course 3 2.1 Inv. 4 Page 11 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 UNIT 5 – FUNCTIONAL RELATIONSHIPS Critical Area Time Frame Connections to prior learning Connections to future learning Additional Notes Practice Standards Essential Questions - Formulating and reasoning about expressions and equations 7 days - * vertical alignment question - 7EE4c. Extend analysis of patterns to include analyzing, extending, and determining an expression for simple arithmetic and geometric sequences (e.g. compounding, increasing area), using tables, graphs, words, and expressions. -This is a gap that needs to be addressed in 2012-13. Impact Course 2 2004 Inv. 5.1. -Additional tests and tasks can be found at http://www.map.mathshell.org SMP4 – Modeling; SMP7 – Structure - 8F3: How is a non-linear function different from a linear function? - 8F5: What is the story behind the graph? Content Standards 8.F.3 8.F.5 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 = s2 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. 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. Impact 2004 Impact 2009 Other Illustrative Math Task -Introduction to Linear Functions Vocab Assessment Resources ESSENTIAL: Non-Linear Course 3 10.1 Course 3 10.1 Illustrative Math Tasks -Bike Race -Riding By The Library -Tides -Velocity vs. Distance Page 12 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 UNIT 6 – SOLVING EQUATIONS WITH ONE VARIABLE Critical Area Time Frame Connections to prior learning Connections to future learning Additional Notes Practice Standards Essential Questions - Strategically choose and efficiently implement procedures to solve linear equations in one variable. 20 days Grade 6: Reason about and solve one-variable equations and inequalities. Grade 7: Solve real life and mathematical problems using numerical and algebraic expressions and equations. - Acts as a stepping stone to solving simultaneous linear equations - This is the culmination of linear equations. Fluency is expected. - Additional tasks and test items can be found at http://www.map.mathshell.org SMP1 – Problems; SMP2 – Reasoning; SMP6 - Precision 8EE7: How does solving one variable linear equation apply to real world problem? Content Standards 8.EE.7 Solve linear equations in one variable. 8.EE.7a 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. 8.EE.7b Impact 2004 Course 2, Chapter 6 Impact 2009 Other Course 3 7.1 Course 2 Chapter 6 Assessment Resources ESSENTIAL: Coefficient, Solution, Like Terms Illustrative Math Task Solving Equations Course 2 Chapter 6 Vocab IMPORTANT: Distributive Property Course 3 7.1 Illustrative Math Task -Coupon vs. Discount Course 3 7.1 -The Sign of Solutions Page 13 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 UNIT 7 – PYTHAGOREAN THEOREM Critical Area Time Frame Connections to prior learning Connections to future learning Additional Notes Practice Standards Essential Questions -Understanding and applying the Pythagorean Theorem -10 Days -6th and 7th grade: Solve real life and mathematical problems involving area -This unit supports grade level work with irrational numbers. - Additional tasks and test items can be found at http://www.map.mathshell.org -SMP3 – Arguments; SMP5 - Tools -How can our understanding of the Pythagorean Theorem affect our understanding of the world around us? Content Standards 8.G.6 Impact 2004 Explain a proof of the Pythagorean Theorem and its converse. Course 2 7.3 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. Impact 2009 Other Proofs of the Pythagorean Theorem Vocab Assessment Resources ESSENTIAL: Hypotenuse, Leg IMPORTANT: Right Triangle Right Triangle Solver – Practice finding unknown sides Course 2 7.3 NC – Applying the Pythagorean Theorem NC – Real World Pythagoras Pythagorean Theorem in a Math Context (IT) 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Course 2 7.3 Page 14 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 UNIT 8 – SIMULTANEOUS LINEAR EQUATIONS Critical Area Time Frame Connections to prior learning Connections to future learning Additional Notes Practice Standards Essential Questions - Use systems of linear equations to represent, analyze and solve a variety of problems. - 25 days - Grade 8: Solving linear equations. Additional tests and tasks can be found at http://www.map.mathshell.org -SMP1 – Problems; SMP4 - Modeling; SMP6 - Precision - 8EE8: How can two linear equations simultaneously solve one real world problem? Content Standards 8.EE.8 Analyze and solve pairs of simultaneous linear equations. 8.EE.8a 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. 8.EE.8b Impact 2004 Impact 2009 Other Supply and Demand Tasks Vocab Assessment ESSENTIAL: Simultaneous, System, Point of Intersection IMPORTANT: Parallel Course 3 7.3 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. Page 15 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 8.EE.8c 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. Page 16 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 UNIT 9 – Volume Unit and Post MCAS Material Critical Area Time Frame Connections to prior learning Connections to future learning Additional Notes Practice Standards Essential Questions - Students complete volume work by solving problems involving cones, cylinders and spheres - 9 days - Grade 6: Area of a circle - Grade 7: Surface area of a sphere, volume of a cube, and a right prism -This is a culminating standard and fluency is expected. -Additional tasks and test items can be found at www.map.mathshell.org -6.SP.4 and 6.SP.4a to be done after MCAS in 2012-13 and 2013-14 -6.G.4 needs to be reviewed in 2012-13 and 2013-14 -7.SP.1 and 7.SP.8b need to be covered for 2012-13 after MCAS SMP4 - Modeling SMP5 - Tools SMP6 - Precision How can the volumes of cones, spheres and cylinders help us solve real world problems? Content Standards 8.G.9 Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real world and mathematical problems. Impact 2004 Impact 2009 Chapter 2.3 & The Soft drink Promotion Course 2, Chapter 5.1 Other Finding Surface Area and Volume Blue Cube, 27 Little Cubes (Stella Stunner) Vocab Assessment Resources ESSENTIAL: Cone, Sphere IMPORTANT: Volume, Cylinder Volume of Spheres and Cones (Rich Problem) 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Use supplementary materials Page 17 of 18 August 16, 2012 GRADE 8 – CURRICULUM MAP – CCSS – 2012-13 6.SP.4a Read and interpret circle graphs Use supplementary materials 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface areas of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Use supplementary materials 7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produces representative samples and support valid inferences. Use supplementary materials 7.SP.8b Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”) identify the outcomes in the sample space which compose the event. Use supplementary materials Page 18 of 18 August 16, 2012