Stretching It. Suggested Grade Range: 6-8. Approximate Time: 1 hour. State of California Standards:

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1 Students will measure the stretch of a rubber band using varying weights and find a linear relationship, in the y = mx + b form, between the rubber band s length and the weight used to stretch it. This activity is meant to allow students to apply their knowledge of linear relationships in a practical setting. Suggested Grade Range: 6-8 Approximate Time: 1 hour State of California Standards: California Standards Grade 6: Algebra and Functions; Statistics, Data Analysis, and Probability AF 1.0 Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results. AF 2.0 Students analyze and use tables, graphs, and rules to solve problems involving rates and proportions. SDAP 1.0 Students compute and analyze statistical measurements for data sets. California Standards Grade 7: Algebra and Functions; Statistics, Data Analysis, and Probability AF 1.0 Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs. AF 3.0 Students graph and interpret linear and some nonlinear functions. SDAP 1.0 Students collect, organize, and represent data sets that have one or more variables and identify relationships among variables within a data set by hand and through the use of an electronic spreadsheet software program. California Standards Grade 8: Algebra I AF 6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality. AF 7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. California Science Standards Grade 8: Physical Science 2. Unbalanced forces cause changes in velocity. As a basis for understanding this concept: a. Students know a force has both direction and magnitude. b. Students know when an object is subject to two or more forces at once, the result is the cumulative effect of all the forces. c. Students know when the forces on an object are balanced, the motion of the object does not change. 8-1

2 d. Students know how to identify separately the two or more forces that are acting on a single static object, including gravity, elastic forces due to tension or compression in matter, and friction. Relevant National Standards (Mathematics): Mathematics Common Core State Standard: 6RP 3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Mathematics Common Core State Standard: 7EE 4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Mathematics Common Core State Standard: 8EE 5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Mathematics Common Core State Standard: 8F 3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. 4. Construct a function to model a linear relationship between two quantities. Next Generation Science Standards: 5-PS2-1. Support an argument that the gravitational force exerted by Earth on objects is directed down. [Clarification Statement: Down is a local description of the direction that points toward the center of the spherical Earth.] [Assessment Boundary: Assessment does not include mathematical representation of gravitational force.] MS-PS2-2. Plan an investigation to provide evidence that the change in an object s motion depends on the sum of the forces on the object and the mass of the object. [Clarification Statement: Emphasis is on balanced (Newton s First Law) and unbalanced forces in a system, qualitative comparisons of forces, mass and changes in motion (Newton s Second Law), frame of reference, and specification of units.] [Assessment Boundary: Assessment is limited to forces and changes in motion in one-dimension in an inertial reference frame and to change in one variable at a time. Assessment does not include the use of trigonometry.] HS-PS2-1. Analyze data to support the claim that Newton s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration. [Clarification Statement: Examples of data could include tables or graphs of position or velocity as a function of time for objects subject to a net unbalanced force, such as a falling object, an object rolling down a ramp, or a moving 8-2

3 object being pulled by a constant force.] [Assessment Boundary: Assessment is limited to one-dimensional motion and to macroscopic objects moving at non-relativistic speeds.] Lesson Content Objectives: Construct rubber band stress apparatus to collect and analyze data. Measure and record the length of a rubber band by stretching it with various weights. Construct a scatter plot to represent the data and make a conjecture about the relationship between the variables. Determine an equation that models the observed data in the y = mx + b form. Materials Needed: One meter stick per group of three to four students One plastic bag stapled to one end of a heavy duty rubber band per group One C-clamp per group One nickel and ten quarters per group (washers or any other uniformly massed objects can be used in place of quarters) Adapted from: Laub, M. (2011). Linear relationship activities. CMC ComMuniCator, 36(2),

4 Summary of Lesson Sequence Introduce the lesson by describing the set up of the experiment and asking discussion questions about the relationship between the variables involved. Guide students through their experiment as they make measurements and organize and analyze their data. Check for students understanding by asking the provided key questions. Allow students to write a summary of the relationship between the variables independently. Close the lesson with a discussion about why there might have been variances in the groups data and by allowing students to summarize the relationship between the length of the rubber band and the weight attached to it. Assumed Prior Knowledge Prior to this lesson students should know how to create a scatterplot from data and should be able to determine the linear relationship between variables in the form of y = mx + b from a graph. Classroom Set Up To set up the rubber band stress apparatus, clamp the rubber band to a rigid table so that the plastic bag that is stapled to it suspends below the edge of the table. Insert one quarter into the plastic bag to eliminate any slack in the rubber band before beginning. 8-4

5 If C-clamps are not available, a large index card or piece of cardboard may be used. Staple a rubber band to the card. Use an open paperclip and hook one end on the rubber band and place the coins on the other end of the hook or in a plastic bag attached to the hook. Students can mark the displacement of the rubber band directly on the card. This activity works well for groups of three to four students. Lesson Description Introduction Inform students that they will be conducting an experiment to determine the relationship between two variables. Describe the set up for the experiment and provide the necessary materials to each group of three or four students and allow them to set up the experiment. Before permitting students to begin, ask the following questions: What do you think will happen if you add a quarter (or other objects with uniform mass such as washers, nickels, pennies, etc.) to the plastic bag? Two quarters? Three? [The rubber band will stretch and be longer each time a quarter or other chosen object is added.] What method will you use to measure the rubber band? [If students measure the entire rubber band they will find that b (in y = mx + b) equals the length of the unstretched rubber band. If students measure the amount the rubber band stretches they will find b=0 (in y = mx + b).] 8-5

6 What are the two variables we will be measuring and keeping track of? [The number of objects added and the length of the rubber band.] How should you keep track of your data? [With a table; the x value is the number of objects and the y value is the length of the rubber band.] What would be a good way to analyze the relationship between the variables once you have collected your data? [Construct a scatterplot.] What kind of relationship do you think there is between the variables? [Answers may vary. Depending on their prior knowledge, students should recognize the relationship as linear or describe the relationship as a direct variation.] Guide Students Through Their Practice Ensure each group has set up their apparatus correctly and they have measured the length of the rubber band without any of the uniform objects in the bag (one nickel or other weight should be in the bag at the start to ensure there is no slack in the rubber band before beginning). Model for each group how they should be measuring the length of the rubber band: from the top of the table to the end of the rubber band that is attached to the plastic bag. Remind students that it is important to make the measurement using a consistent method. Allow students to add one quarter and measure the length of the rubber band, then add a second quarter and measure again. Students should continue until they have added all ten quarters to the bag and measured the length of the rubber band. Every student should create a table of values for the data with the number of quarters as the x value and the length of the rubber band as the y value. Students should then construct a scatterplot of the data to make a conjecture about the relationship between the variables. Encourage students to try to determine a linear relationship in the form of y = mx + b that models their observed data. Ask groups to predict what the length of the rubber band would be if 13 quarters were in the plastic bag and report their prediction by writing it on the board. Check for Understanding While students are working, check for understanding by moving around to each group and noting their progress. Ask the following key questions: What do you notice about the relationship between the number of quarters and the length of the rubber band? Was there a time in your experiment when the x value was zero? What about the y value? 8-6

7 With a linear relationship, describe what may be found when the x value is zero. If there is a data point on your scatter plot that does not seem to fit with the rest of your data in the linear model, what might you do? If you look at your data and your friends data you will likely see that they are different. What are some reasons you would get different data? [Different rubber bands will stretch differently some are easy to stretch, others hard, some of the teams may have measured the total length while others only measured the change in length, some teams may have used quarters while others used pennies or washers.] Closure Prompt a whole class discussion of the predictions each group has made for the length of the rubber band with a weight of 13 quarters and why they might vary. Ask a volunteer group to verify in front of the class the length of the rubber band after adding 13 quarters to the bag. Discuss with the class how their graph and this information can be used to predict length for a very large amount of quarters (e.g., 100). Suggestions for Differentiation and Extension Different rubber bands will have different spring constants. This will result in some difference between results. It could be an interesting discussion or extension activity to try this with thin and thick rubber bands. It will take more quarters (or washers) to stretch a thick rubber band the same distance as a thin rubber band. This experiment demonstrates Hook s Law, which states that the restoring force of a spring is directly proportional to a small displacement. In equation form, we write: F = -kx where x is the size of the displacement. The proportionality constant, k, is specific for each spring. Students may explore the different proportionality constants of different rubber bands as an extension to this activity. Graphing calculators may be used to find the line of best fit for the scatterplot if they are available. Technology extension activity: The PHET Simulation, Masses & Springs, allows students to explore different masses. Each spring has its own spring constant. Students can collect and graph data and find the mass of unknown objects using the data they collect with known masses. The simulation is free and downloadable and sample lesson plans for use with the simulation are included on the website: 8-7