AIMS Common Core Math Standards Alignment

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1 AIMS Common Core Math Standards Alignment Eighth Grade The Number System (8.NS) Cluster Standard Activity To copy the files from this disc to your computer, create a new folder and drag all of the files and folders over together. Changing the organization of the files will cause the links in this alignment to break. 8.NS.A Know that there are numbers that are not rational, and approximate them by rational numbers. 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. 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 ). 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. Developing the Concept of Square Root Models of Square Numbers Expressions and Equations (8.EE) Cluster Standard Activity 8.EE.A Work with radicals and integer exponents. 1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, = 3 3 = 1/33 = 1/ 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. 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. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger. 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. Eighth Grade Alignment AIMS Education Foundation

2 Expressions and Equations, continued (8.EE) Cluster Standard Activity 8.EE.B Understand the connections between proportional relationships, lines, and linear equations. 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 distancetime equation to determine which of two moving objects has greater speed. 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. Functions in Circles Tunnel Vision Nuts and Bolts Shrink Art Shadow Lines Time Trials 8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations. 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. 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. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. c. 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. Equality in the Balance What s in a Case? Pogo Stick Paths Calling Long Distance Sales Calls Eighth Grade Alignment AIMS Education Foundation

3 Functions (8.F) Cluster Standard Activity 8.F.A Define, evaluate, and compare functions. 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. 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. 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. For example, the function A = s 2 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. Let s Party! Books Upon Books How Cups Function Functioning With Knots Tile Pattern Zap It! 8.F.B Use functions to model relationships between quantities. 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. 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. Bungee Dolls Cliff Diving Ready, Set, Flip Rocket Shot Temperature Conversion The Thick of Things Eighth Grade Alignment AIMS Education Foundation

4 Geometry (8.G) Cluster Standard Activity 8.G.A Understand congruence and similarity using physical models, transparencies, or geometry software. 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. 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. 3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 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. 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. 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. Transformers Capture the Flag Growing Designs 8.G.B Understand and apply the Pythagorean Theorem. 6. Explain a proof of the Pythagorean Theorem and its converse. 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. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Essential Math: The Pythagorean Relationship book 8.G.C Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. 9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Essential Math: The Effects of Changing Lengths book Essential Math: Measurement of Prisms, Pyramids, Cylinders, and Cones book Eighth Grade Alignment AIMS Education Foundation

5 Statistics and Probability (8.SP) Cluster Standard Activity 8.SP.A Investigate patterns of association in bivariate data. 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. 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. 3. 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. 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? Close to Ideal? Hats, Heads, and Heights Head Size to Hat Size This Is So Typical Eighth Grade Alignment AIMS Education Foundation

6 30 Topic Linear functions Key Question How can you determine how long to make a bungee cord so that the first time you drop the fashion doll her hair touches the ground but her head does not hit? Learning Goals Students will: determine the length of bungee cord needed to give a fashion doll a safe drop from various heights, graph the data to establish a linear pattern, develop an equation to model the simulation, use an equation to predict the appropriate length of bungee cord required, and test their predictions. Guiding Documents Project 2061 Benchmarks Mathematics is helpful in almost every kind of human endeavor from laying bricks to prescribing medicine or drawing a face. In particular, mathematics has contributed to progress in science and technology for thousands of years and still continues to do so. In the absence of retarding forces such as friction, an object will keep its direction of motion and its speed. Whenever an object is seen to speed up, slow down, or change direction, it can be assumed that an unbalanced force is acting on it. NCTM Standards 2000* Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope Relate and compare different forms of representation for a relationship Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships Math Measurement linear Algebra graphing equations writing equations solving equations linear functions FALL 2006 Science Physical science forces elasticity gravity Integrated Processes Observing Comparing and contrasting Generalizing Predicting Materials For each group: 1 yard elastic cord fashion doll 150 cm tape measure 2" x 4" lumber scrap, four inches long 2 small nails For the class: hammer 3 yards elastic cord 30 m tape measure Background Information The extreme sport of bungee jumping is gaining popularity, and students may have already experienced it. The type and length of the bungee is of extreme importance to the safety of the jumper. If the cord is too strong for the jumper, it will act as a rope and injure or kill the jumper as the bungee is stretched to its full length. If it is too weak and it stretches to its maximum load, it will snap. A bungee cord that is too long will result in the jumper hitting the ground before the bungee is stretched so its force stops the fall. Using a doll, students simulate a jump, observe the forces involved, and gather data. Two forces are interacting in a bungee jump. Gravity attracts the jumper to Earth while the elasticity of the bungee pulls the jumper skyward, breaking the fall. As soon as a jumper steps off the platform, she is in free fall. Gravity accelerates the speed of the fall until the bungee is extended fully to its relaxed length. From this point on, the rubber in the bungee resists the force of gravity. This resistance, or counterforce, grows greater as the rubber continues to stretch. When the elastic force becomes great enough, it balances the force of gravity and the jumper stops falling and is suspended above Earth. The elastic force stopping the fall is greater than the gravity of the static body, and AIMS Education Foundation

7 the elastic force drags the jumper skyward until the elastic and gravitational forces are balanced, leaving the jumper suspended above the ground again. An elastic cord is well suited for the simulated fashion doll bungee jump. In a real bungee jump, the bungee cord is matched to the human s weight. The momentum of the falling body should not cause the cord to reach its maximum length where the jumper would be jolted by a sudden stop. If the fashion doll were a human, the elastic cord would be too weak. Even on a short jump it reaches its maximum length and in real life would at worst rip or dislocate a leg and at best cause discomfort. However, since in the simulation every jump reaches maximum stretch length, the stretch length of the cord is always a constant factor of the cord s relaxed length. For the bungee jump simulation, the only thing that affects the length of the jump is the length of cord. By changing the length of the cord, the platform s height needs to change to get the doll s hair, but not her head, to touch the ground. As students graph data comparing the appropriate platform heights for different cord lengths, a line forms. The rate of change, or slope, is the factor of the maximum stretched length to the relaxed length of the cord. The y-intercept of the graph provides the starting condition, or height of doll including the extended hair. This provides an excellent opportunity to recognize how each of the constants in the linear equation represents an aspect of the situation. When a higher jump from a platform such as a ladder or balcony is considered, students just need to apply the equation. By substituting the height of the platform and working backwards, students can determine the required length of a safe, but exciting bungee cord. Management 1. Gather the necessary number of fashion dolls before beginning this simulation. Students of middle school age are very willing to bring in their dolls voluntarily to be involved in this exciting investigation. The dolls should have long hair. If not, hair can be simulated by tying or gluing strings or ribbon to the head of the dolls. 2. Purchase enough standard elastic cord. It is inexpensive and available at fabric and discount stores. Make sure to purchase standard cord, as elastic bands are too stiff for success in this investigation. 3. Before the investigation, attach a yard of elastic cord to each doll by tightly winding and tying it around the dolls ankles. A strip of tape can be wound on top of the cord to secure the knots. 4. Prepare a platform for each doll from a scrap of 2" x 4" board about four inches long. The top of the platform is any narrow side, and the broad sides become the front and back that is placed against a wall. Drive a small nail into both the front and back of the top of the platform. The students will secure the cord to the platform by first wrapping it several times around the front nail and then the back nail. Students should hold the free end taut as the doll drops. 5. Groups of five work well for this investigation. Assign each student a task: platform and cord holder, doll dropper, measurer, hit sighter, and recorder. 6. If you anticipate having the class predict a final high jump, find an appropriate location a ladder, stairwell, second story window, or balcony. Prepare a doll with a cord that is longer than required. The length of the cord should be no less than half the distance from the platform to the ground. 7. The student page has assigned cord lengths to provide direction that gets more consistent data for most students. To make the investigation more open-ended, omit the assigned lengths and let students choose random lengths. Procedure 1. Discuss with students their experiences and understanding of bungee jumping and explain that they will be using dolls and elastic cord to simulate the experience and gather data. Hold a class discussion about the Key Question and how they might go about gathering the necessary information. 2. Distribute the materials and explain the procedure. 3. For each of the assigned cord lengths, have the students measure and record the appropriate platform height. 4. When a team has determined the platform height for each of the assigned cord lengths, have the recorder share the data with the team. Instruct the students to use their data to make a graph and recognize the linear nature of the data. 5. Discuss with the students how they can use their graphs to predict the appropriate length of the cord for the high jump and how to use the graph to make an approximation of an equation that models the simulation. 6. Provide students with time to use their data and graph to develop a more precise equation that models the simulation. AIMS Education Foundation FALL

8 7. Inform the students of the height of the high platform and have them use their equations to predict the length of bungee cord required to have the doll s hair touch the ground but not her head. 8. Measure out the length of cord the students predict, put student sighters on the ground, and have the doll jump to confirm the accuracy of the students predictions. Connecting Learning 1. What pattern do you see in the data points on your graph? [The points form a line.] 2. How can you use the graph to predict the height required for an assigned length of cord? [Go vertically from the assigned length until you hit the data line. From that point, move horizontally to find the predicted height of the platform. The teams might want to check a length that was not assigned that falls within the range of the graph to determine if their data are accurate.] 3. What does the straight line tell you about how the cord stretches? [It is consistent in the amount it stretches.] 4. Using your graph, how could you determine how much the cord stretches for each unit of length added to the cord? [Estimates will vary. Divide change in platform height by change in cord length. The elasticity factor of the bungee cord is seen as the slope on the graph.] 5. According to the graph, how high above the floor should the platform be even if the bungee cord has no length? What does this height relate to in the simulation? [Answers will vary, but it should be about the height of the doll including extended hair.] 6. Write an equation that models the length of fall for any length of bungee cord. [For some students it will be appropriate for them determine the slope and y-intercept from the data points. For others, the approximate equation from the graph will be sufficient. Equations will vary depending on the data.] 7. How long a cord is needed for the high platform with a height of (height to be inserted)? 8. How accurately did you equation predict the length of the cord? What could have caused errors? Extension Have students compare the jumps of different height and weight dolls and when different types of elastic are used. Have the students gather and graph data, generalize the patterns and make predictions, and confirm the accuracy of the predictions. * Reprinted with permission from Principles and Standards for School Mathematics, 2000 by the National Council of Teachers of Mathematics. All rights reserved. 32 FALL 2006 AIMS Education Foundation

9 Key Question How can you determine how long to make a bungee cord so that the first time you drop the fashion doll her hair touches the ground but her head does not hit? Learning Goals Students will: determine the length of bungee cord needed to give a fashion doll a safe drop from various heights, graph the data to establish a linear pattern, develop an equation to model the simulation, use an equation to predict the appropriate length of bungee cord required, and test their predictions. AIMS Education Foundation FALL

10 Before starting the investigation, assign each team member one of the following tasks: Measurer: Measure out the assigned length of bungee cord and secure it to the platform by first winding the loose end of the cord around the front nail of the platform several times and then around the rear nail several times. Measure the height of the platform on the final drop as determined by the Sighter. Holder: Place the platform against the wall a bit higher than the group is confident it needs to be so the doll will not hit the ground. While keeping the platform against the wall, hold the free end of the cord taut. Move the platform as instructed by the Sighter. Dropper: Place the doll s feet on the top of the platform for each trial and drop the doll. Sighter: Put your eyes at ground level and watch to see if the hair of the doll touches the ground. If the hair does not touch the ground, advise the Holder to lower the platform a small distance. When the hair just touches the floor, have the Measurer record the height of the platform above the floor. Recorder: Record the height of the platform to the floor on the table. From the starting height, drop, sight, and lower the platform until the doll s hair just touches the ground. When the doll s hair touches the ground, measure and record the height of the top of platform above the ground. Repeat the procedure for each of the assigned bungee cord lengths. Share the data with the team, graph the data, and determine a way to predict the safe height for different cord lengths. Bungee Length (cm) Jump Height (cm) 34 FALL 2006 AIMS Education Foundation

11 Graph your team s data to see if it helps you predict how long a 165 bungee cord is needed Jump Height (cm) Bungee Length (cm) AIMS Education Foundation FALL

12 Connecting Learning CONNECTING CONNECTING LEARNING LEARNING 1. What pattern do you see in the data points on your graph? 2. How can you use the graph to predict the height required for an assigned length of cord? 3. What does the straight line tell you about how the cord stretches? 4. Using your graph, how could you determine how much the cord stretches for each unit of length added to the cord? 36 FALL 2006 AIMS Education Foundation

13 Connecting Learning CONNECTING CONNECTING LEARNING LEARNING 5. According to the graph, how high above the floor should the platform be even if the bungee cord has no length? What does this height relate to in the simulation? 6. Write an equation that models the length of fall for any length of bungee cord. 7. How long a cord is needed for the high platform with a height of? 8. How accurately did you equation predict the length of the cord? What could have caused errors? AIMS Education Foundation FALL