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1 M O R R I S S C H O O L D I S T R I CT M O R R I S T O W N, N J GRADE 8 MATHEMATICS CURRICULUM MAP 1

2 Grade 8, focus on: (1) 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; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem. Grade 8 Overview The Number System Know that there are numbers that are not rational, and approximate them by rational numbers. Expressions and Equations Work with radicals and integer exponents. Understand the connections between proportional relationships, lines, and linear equations. Analyze and solve linear equations and pairs of simultaneous linear equations. Functions Define, evaluate, and compare functions. Use functions to model relationships between quantities. Geometry Understand congruence and similarity using physical models, transparencies, or geometry software. Understand and apply the Pythagorean Theorem. Solve real-world and mathematical problems involving volume of cylinders, cones and spheres. Statistics and Probability Investigate patterns of association in bivariate data. 2

3 Curriculum Map Content/Objective Solving Equations Give examples of linear equations in one variable, and transform the equation into simpler forms (8.EE.7a) Formulate and use different strategies to solve linear equations with rational number coefficients, including use of distributive property and combining like terms. (8.EE.7b) Graph proportional relationships, interpreting the unit rate as slope of the graph. Compare two different proportional relationships represented in different ways. (8.EE.5) How can you test a rule? How can you check the reasonableness of your solution? What makes an algebraic algorithm both effective and efficient? How can you use a formula for one measurement to write a formula for a different measurement? How can you convert form one measurement system to another : Rules of arithmetic and algebra can be used together with equivalence to transform equations so solutions can be found to solve problems. Solving equations using addition and subtraction, using inverses, properties of equalities, combining like terms, distributive property, using formulas to find the unknown, converting forms of measurement Sum of the Areas of a Triangle: Gather data on triangle measures in table format. Look for patterns and develop a rule. Explore the sum of the angle measurements of a triangle to develop an understanding of writing and solving simple equations. Writing a rule Applying your rule: Solve for missing angles of a triangle. Solving for the Angles of a Triangle (unknowns) Problem solving puzzles involving circle graphs and six triangles grouped to form a rectangle. Perimeter and Area - write equations for figures that have perimeter equal to area, solve for missing sides Using perimeter and area formulas solve for unknown Using volume formulas solve for unknown Convert units of measure related to perimeter and area. Describe the pattern. Olympic scoring translating scoring systems, find the radius of ferris wheels given the circumference Pre-Assessments, DoNows, and correct, year to date cumulative assessments, unit tests, standardized test practice Understand the connections between proportional relationships, lines, and linear equations. Know how to strategically choose and efficiently 3

4 implement procedures to solve linear equations. Graphing Linear Equations and Linear Systems Write, solve, and graph onestep and two-step linear equations. Use the equation of a linear model to solve problems in the context of bivariate data, interpreting the slope and intercept (8.SP.3) Construct and analyze tables, graphs, and equations to describe linear functions and simple relations. Graph proportional relationships and identify the rate as slope of linear functions. (8.EE.5) Use similar triangles to explain why the slope m is the same between two distinct points on a non-vertical line in the coordinate plane (8.EE.6) Use tables, graphs, and models to represent, analyze, and solve real world problems related to systems of linear equations. Understand that solutions to a system of linear equations in two variables correspond to points of intersection of their graphs (8.EE.8a) How can change be best represented mathematically? How can the collection, organization, interpretation, and display of data be used to answer questions? How can the representation of data influence decisions? How can you recognize a linear equation? How can the slope of a line be used to describe the line? How can you describe the graph of y = mx +b? How can you describe the graph of ax. By = c? How can you solve a system of linear equations? What does the solution mean? : Understand the connections between proportional relationships, lines, and linear equations. Graphing linear equations, horizontal lines, slopes, intercepts, solving systems by graphing Graph a linear equation, select points on the line and substitute point into equation Real-life application graph the equation for a tropical storm, determine when the storm becomes a hurricane. Taking math deeper determine how many pictures can be taken on a digital camera vs. number of videos that can be stored Find the slope of a line using various points on a line Drawing lines given slopes Taking math deeper determine slope and changes in slope of a slide Find slopes and y-intercepts explore the connection between the equation of a line and its graph Inductive reasoning graph equations, then describe, list slope, and point of intersection with y-axis. Describe patterns, make connections. Taxi problem, discuss flat fees vs. fares Taking math deeper creating a website and determining money earned based on clicks Use a table to plot points sold certain \$ amount of tickets for a concert, but lost track of how many of each type was sold (adult vs. child) Rewriting an equation cheese sold, forgot how many pounds of each kind sold Writing a system of linear equations bed and breakfast revenue vs costs / break even point Use graphing calculator to graph systems of equations Pre-Assessments, DoNows, and correct, year to date cumulative assessments, unit tests, standardized test practice 4

5 Solve systems of two linear equations in two variables algebraically and estimate solutions by graphing (including real-world problems)(8.ee.8b, 8c) Understand slope is a constant rate of change. Recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b). Understand linear equations describe the association between two quantities in bivariate data. Writing Linear Equations and Linear Systems Use similar triangles to explain why the slope m is the same between two distinct points on a non-vertical line in the coordinate plane (8.EE.6) Construct a function to model a linear relationship between two quantities. Determine and interpret the rate of change (8.F.4) Solve systems of two linear equations in two variables algebraically and estimate solutions by graphing (including real-world problems)(8.ee.8b, 8c) What information is needed to write an equation of a line? How can you use a linear equation in two variables to model and solve a real-life problem? : Graphs and equations are alternative (and often equivalent) ways for depicting and analyzing patterns of change. Write equations based on slope and point, two points, write systems of equations Writing equations of a line from graphs, what do these lines have in common? Describing a parallelogram write equations for each side of the parallogram, compare slopes Interpreting slope and y-intercept make inferences from a time vs difference graph Writing linear equations based on real life graphs (balance vs time, population vs time) Writing and using linear equations for hot air balloons and roller coaster rides (height vs time) Write a story based on graphs Draw graphs based on real life problems Writing a system compare enrollment of two schools over time Pre-Assessments, DoNows, and correct, year to date cumulative assessments, unit tests, standardized test practice Functions Construct and analyze tables, How can you find the domain and range of a function? Activities Evaluate ticket sales of child and adult tickets. Determine the Domain and Range of a Function Pre-Assessment, DoNows, 5

6 graphs, and models to represent, analyze, and solve problems related to linear equations including analysis of domain, range, and difference between discrete and continuous data. Translate among representations of linear functions in words, tables, graphs, and equations. Compare graphs of linear and non-linear functions. 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.1) Compare properties of two functions each represented in a different way (algebraically, graphically, numerically, verbal descriptions (8.F.2) Interpret the graph of 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.3) Construct a function to model a linear relationship between two quantities. Determine and interpret the rate of change (8.F.4) Graph proportional relationships and identify the How do you decide whether the domain of a function is discrete or continuous? How can you use a linear function to describe a linear pattern? How do you know when a pattern is linear or nonlinear? Enduring Understanding: Functional relationships can be expressed in real contexts, graphs, algebraic equations, tables, and words; each representation of a given function is simply a different way of expressing the same idea. Finding Domains and Ranges using input /output tables and graphs Finding Domain and Range from a Graph Finding the Range of a Function given a table Real-Life Application- What percent of the moon was visible on January 26, 2011? Interpret table of days and percent visible. Manatee eats 12% of its body weight each day, Find the function for manatee s weight Discrete and continuous domains (write functions to describe hotel costs vs number of rooms and weight of luggage vs pounds per piece) Graph Discrete Data and Continuous Data Taking Math Deeper Write and graph a function for the arrangements of picture frames. Finding linear patterns plot points from a table, write an equation, and describe the pattern Find a linear function using a graph Find a linear function using a table Real-life Application Write a linear function relating calories burned to hours kayaking. Taking math deeper Find the correlation between years of education and salary by writing a function Find patterns for similar figures using a sequence of similar rectangles (area and perimeter) Compare linear and nonlinear functions Identify functions from tables Identify functions from graphs Real-life application Compare simple and compound interest Compare proportional relationships Flick a paper football and measure the length and height the football travels. Write domain and range of both the ascent and descent travelled. Write a linear function to approximate the ascent and one to approximate the descent. and correct. 6

7 rate as slope of linear functions. (8.EE.5) Describe qualitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function that has been described verbally. (8.F.5) Angles and Similarity Classify and determine the measures of angles, including angles created when parallel lines are cut by transversals. Demonstrate the sum of angles of a triangle is 180 degrees and apply this fact to find unknown measures of angles, and the sum of the angles in polygons. Determine height and distance using similar triangles. Use informal arguments to establish facts about the angle sum and exterior angles 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.5) How can you classify angles? How can you classify triangles? How can you find a formula for the sum of the angle measures of any polygon? Which properties of triangles make them special among all other types of polygons? How can you use properties of parallel lines to solve real-life problems? : Understand congruence and similarity using physical models, transparencies, or geometry software. Explore the properties of complementary and supplementary angles by completing a table of angles and their complimentary or supplementary angle, then graphing the function. Taking Math Deeper- Two angles are vertical angles. What are their measures if they are also complementary angles? Supplementary angles? Exploring the Angles of a Triangle Draw an obtuse triangle, cut it out and tear off the 3 corners of the triangle. Draw a straight line and fit the 3 torn corners on the straight edge. Try with other types of triangles. What do you notice? From an abstract painting of geometric shapes, trace four triangles and classify them. Then design your own painting. How many different types can you use in your painting? Taking Math Deeper- Stack cards with different angles. How did you do this? Is the value of the 2 base angles fixed? Find the Sum of the Angle Measures of different types of polygons by dividing the shapes into triangles. Identifying Convex and Concave Polygons Taking Math Deeper- Create a table listing the sides of a polygon and the measure of one angle. Pre-Assessment, DoNows, and correct. 7

8 Is this a linear function? Cut out a triangle. Try to make a larger triangle that has the same angle measures as the first triangle. Measure the lengths and compare the rations. Taking Math Deeper- Using a Shuffleboard diagram with side lengths given, find the lengths of the smaller interior triangles. A Property of Parallel Lines draw 2 parallel lines, the draw a transversal line. Evaluate the angles to determine which have similar measures. Square Roots and Pythagorean Theorem Use square root, and cube root symbols to represent solutions to equations. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. (8.EE.2) Explain a proof of the Pythagorean Theorem and its converse (8.G.6) Apply the Pythagorean Theorem to determine unknown side lengths in right triangles applying to realworld problems (8.G.7) Know that numbers that are not rational are irrational. Understand informally that every number has a decimal expansion, rational number repeat eventually (8.NS.1) Use rational approximations How can you find the side length of a square when you are given the area of a square? How are the lengths of the sides of a right triangle related? How can you find decimal approximations of square roots that are irrational? How can you use square roots to describe the golden ratio? How can you use the Pythagorean Theorem to solve real life problems? : The Pythagorean Theorem can be derived / explained by decomposing a square in two different ways. Finding Square Roots using square models and area Real-Life Application- Find the radius of a Crop Circles given the area. Discovering the Pythagorean Theorem using grid paper, triangles, and quadrilaterals Approximating Square Roots try to approximate the square root of radical 3 following in Archimedes footsteps using calculators. How did Archimedes do this without a calculator? Approximating Square Roots Geometrically using grid paper, a straight edge, and compass. Constructing a Golden Ratio using grid paper, compass, and Pythagorean theorem Work with a partner to gather ratios of a the human body like Leonardo da Vinci and approximate the Golden Ratio There is a fire in a building of a certain size, and the recommended angle for the ladder is 75 degrees. Given specified height of the ladder, and how far from the base the ladder should be placed, how high will the ladder reach? Finding a Volume Taking Math Deeper- Ice Blocks Pre-Assessments, DoNows, and correct, year to date cumulative assessments, unit tests, standardized test practice 8

9 of irrational numbers to compare the size of irrational numbers, locate the on a number line and estimate the value (8.NS.2) Apply the Pythagorean Theorem to find the distance between two points in a coordinate system (8.G.8) Data Analysis and Displays Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive/ negative association, linear association, and nonlinear association. (8.SP.1) Know that straight lines are widely used to model relationships between two quantitative variables. Informally fit a straight line. (8.SP.2) Use the equation of a linear model to solve problems in the context of biariate measurement data, interpreting slope and intercept. (8.SP.3) Understand patterns of association can also be seen in bivariate categorical data by displaying frequencies and How can you use measures of central tendency to distribute an amount evenly among a group of people? How can you use a box and whisker plot to describe a population? How can you use data to predict an event? How can you display data in a way that helps you make decisions? : Variables are symbols that take the place of numbers or ranges of numbers; they have different meanings depending on how they are being used. Activities Exploring Mean, Median and Mode using coins. Stack 45 coins into 9 stacks. Record the stack number and number of coins in the stack. Find the mean, median, and mode. Move coins from one stack to the other. Will this change the mean? Median? Mode? Fair and Unfair Distributions distribute 45 coins to 9 people. How many different ways can you distribute them to create a fair distribution? How is each distribution related to the mean? Look at outliers in data. How do they affect the mean, median, and mode? Look at a dataset of hourly wages. Increase each hourly wage by 40 cents. How does this increase affect the mean, median, and mode? Create a Box-and-Whisker Plot based on the number of cousins each student in class has. Construct a plot to evaluate the data on a strip of grid paper. Taking math deeper Evaluate box and whisker plots comparing battery life of two brands of cell phones. Determine which battery has the longer battery life and why. Graph data points on the measures of a baby alligator s growth over time. Try to construct a linear equation to predict the alligator s growth in two years. Find the height and arm span of three people. Pre-Assessments, DoNows, and correct, year to date cumulative assessments, unit tests, standardized test practice 9

10 relative frequencies in a twoway table (8.SP.4) Make a scatter plot and draw a line of best fit. Is there a relationship between height and arm span? Use the internet to find information on an endangered species. Use graphical displays of the data you gather. Why did you choose the types of data displays? Exponents Know and apply the properties of integer exponents to generate equivalent numerical expressions (8.EE.1) Use numbers expressed in the form of a single digit times and 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.3) 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. Interpret scientific notation that has been generated by technology (8.EE.4) How can we model situations using exponents? How can you use exponents to write numbers? How can you multiply two powers that have the same base? How can you divide two powers that have the same base? How can you define zero and negative exponents? How can you read numbers that are written in scientific notation? How can you write a number in scientific notation? : Real world situations involving exponential relationships can be solved using multiple representations. Construct a table showing the power and value of a series of exponents (-3) to the first, second, third, power... How can you find the value of (-3) to the n power. Finding Products of Powers Using a Calculator Multiplying Powers with the Same Base Raising a Product to a Power Finding Quotients of Powers Compare volumes of various cubes. Compare larger to smaller cubes volumes. What patterns are seen? A drop of water leaks from a faucet every second. How many liters of water leak from the faucet in 1 hour? Use a calculator experiment with multiplying very large numbers until you get a number that is not in standard form. What does the e mean on the calculator? Can you explain it mathematically? Try the same thing with very small numbers. Use a table of distances and masses from the sun. Match each planet to the distance. Then write in scientific notation. Multiplying Numbers in Scientific Notation Adding Numbers Written in Scientific Notation Subtracting Numbers Written in Scientific Notation Dividing Numbers Written in Scientific Notation 10

11 Volume of cylinders, cones, and spheres Solve real world and mathematical problems involving volume of cylinders, cones, and spheres (including knowing the formulas) (8.G.9) What is the relationship of the volume of a sphere, cone, and cylinder? What is similar? What is different? : Measurements can be used to describe, compare, and make sense of phenomena. Construct cylinders What shapes are needed to construct a cylinder? How many blocks are needed to fill the cylinder? How could find the volume mathematically? Construct a cone with the same height as the cylinder. How many blocks are needed to fill this shape? Construct a sphere of similar height to the cone and the cylinder. How many blocks are needed to fill the sphere? Rotations, reflections, and Transformations Understand congruence and similarity using physical models, transparencies or geometry software. Verify experimentally the properties of rotations, reflections, and translations (8.G.1) How are similarity, congruence, and symmetry related? What situations can be analyzed using transformations and symmetries? How can transformations be described mathematically? Activities Translating a Figure on the coordinate plane a specified number of units to the left and down. Record the new coordinates. Translate the figure again. What do you notice about the coordinates each time? Reflecting a few figures on the coordinate plane, recording the coordinates before and after. What do you notice? Rotate a few figures on the coordinate plane, recording the coordinates before and after. What do you notice? Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations (8.G.2) Describe the effect of dilations, translations, : Shape and area can be conserved during mathematical transformations. 11

12 rotations, and reflections on two-dimensional figures using coordinates (8.G.3) Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations (8.G.4) 12

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