Title: Tying Knots: An Introductory Activity for Writing Equations in Slope-Intercept Form Prior Knowledge Needed: Grade: 8 Author(s): Hope Phillips BIG Idea: Linear Equations how to analyze data from a scatter plot (see M7D1f) how to apply linear equations in one variable (see M7A2) given a problem, how to define a variable, write an equation, solve the equation, and interpret the solution (see M7A2a) given a problem, how to set up an x/y chart how to look for patterns in data how to find the average of a set of data rounding to the nearest ¼ GPS Standards: M8A4. Students will graph and analyze graphs of linear equations. a. Interpret slope as rate of change b. Determine the meaning of slope and y-intercept in a given situation c. Graph equations in the form y= mx + b f. Determine the equation of a line given a graph, numerical information that defines the line, or a context involving a linear relationship. M8P1. Students will solve problems (using appropriate technology). Objectives: 1. Students will recognize a pattern given a table of data. 2. Students will interpret the slope of the line in the context of a problem using the appropriate units. 3. Students will interpret the y-intercept of the line in the context of a problem using the appropriate units. 4. Students will develop a linear equation in slope-intercept form to describe a set of data. 5. Students will predict how the slope and y-intercept would change given different, but related scenarios. M8P3. Students will communicate mathematically. M8P4. Students will make connections among mathematical ideas to other disciplines. M8P5. Students will represent mathematics in multiple ways.
Materials: Graphing calculators or online graphing calculators (see websites below in Resources section) String or rope measuring at least 1.5 to 2 feet. Be sure whatever material is selected is not slick because the knots will slip and not be as uniform as they need to be for this activity to be a success. meter/yard sticks Knot Tying Investigation student handouts guided and open versions Task: Investigate the relationship between the number of knots in a rope and the length of the rope. Gather data, look for patterns, and present your data using multiple representations, including tables, graphs, and equations. Description and Teacher Directions: This activity is designed to introduce the writing of equations in slopeintercept form and interpret the meanings of slope and y-intercept in a real-world problem solving situation. This activity can be worked by pairs of students to facilitate discussion and discovery and reduce the number of materials required. Teacher Commentary: The comments below are from a teacher who piloted this lesson. The alphabetic letters correspond to the questions on the Knot Tying Investigation student handouts. Given a rope of pre-cut length (determined by the teacher), students will tie eight knots that are approximately the same size. They do not have to be equally spaced. However, they do not need to overlap each other. After each knot is tied, students will measure the length of the rope and record it in their chart. Remind students to measure the length of the rope before tying the first knot. This pre-cut length does not have to be the same for all student groups. Class discussion will be richer if the lengths are not the same. The remaining lengths of rope will be different as each group will tie knots differently.
change in the length of the rope Many students used two points on the line All length measures should be rounded to the nearest tenth of a centimeter or fourth of an inch. We stick to metrics! On the graph of the data, N is the number of knots, and L is the length of the rope. The length of the rope depends on the number of knots. As the number of knots increases, the length of the rope decreases. The graph can be described as having a negative relationship, or negative slope. It is important, however, that this not be referred to as an inverse variation/proportion because 1. There is a value for L when N is zero (see explanation of the y- intercept below). 2. The graph of the relationships touches the y-axis because it has a y-intercept. 3. The value of L N is not a constant. This means the relationship between N and L is not proportional. Teachers should lead students through questioning to help them understand the meanings of the slope and y-intercept in the context of the problem. The y-intercept is the value of y when x is zero. When there are no knots tied, the rope is at its original length. C. We used sample data and the graphing calculators to view and discuss the graph as a class before students made a scatter plot on graph paper. Even in going over the window settings on the calculator to set the scale and x- and y- min and max, some students still had a hard time setting up their axes, but I think it helped. E., F., G. During instruction, I really emphasized the standards M8A4a & b assessed in this section. Some students may struggle because they may not have had practice with similar problems. Teachers may have to ask leading questions. The slope is the rate at which the length of the rope changes. Theoretically, each knot is the same size, which means that the rate of change is constant. In this realworld problem, however, the size of the knot will vary slightly. Even given the slight variations in size, the slope can be interpreted as the amount of rope it takes to make each knot. The slope can also be described as the distance between each set of data points on the graph. It is written as follows:
1 knot The units of the slope are cm or in. knot Students can find the value of the slope using several methods. 1. Select two ordered pairs from their chart and use the slope formula y 2 y 1 x 2 x 1 2. Find the differences of each successive y value in their chart. There may be a slight difference between each value because of human error involved in tying each knot. Students can average their differences to arrive at one value. 3. Using a graphing calculator, enter the data and have the calculator find the slope. of fit they generated which may or may not have been part of the data they recorded. I was pleased that the students recalled the method of averaging the, almost, constant differences without my having to re-tell them. I don t share this method with students until they ve demonstrated mastery of slope concepts. I suggest integrating section G4 with section H to better lead into the verbal description. Lead students through a series of questions that would allow them to develop the equation to represent the length of the rope given any number of knots. Students should begin with describing the equations in words and then proceed to a symbolic representation. For example, rope length = original length of the rope - amount of rope consumed per knot (y) = b - m(x) When students are novices in writing equations, it is best to use variables that represent the meaning of the problem situation. Instead of y for the rope length, the use of L would be more intuitive. Instead of x for the number of knots, N would be more appropriate. Allow students to use letters that are meaningful to them. Later, discuss how all versions of their equations are just examples of y = mx + b. For my less advanced classes, I modeled the change in length per knot, and matched it with a numeric expression showing the difference, then substituted variables for the numbers instead of vice versa. While slope-intercept form traditionally places the slope first and then the y-intercept, in this problem situation, this is not the best
representation for conceptual understanding. Because the rope starts out without any knots in it, it is more intuitive for students to begin their equation with the rope s original length. This length decreases, noted by subtraction, at a constant rate as each additional knot is tied. Placing the slope at the end of the equation conceptually (and visually) reinforces the change taking place as knots as tied. For example, the equation above can be written with (sample) values. 3.5cm l = 80 - k where l is knot the length of the rope, k is the number of knots 3.5cm is the slope and 80 cm is knot the original length of the rope Notice that the equation suffices even when no knots have been tied. By substituting in 0 for x we find, 3.5cm l = 80 - ( ) knot 0 l = 80 Where 80 is the length of the rope when no knots have been tied. Using a graphing calculator or online graphing application, students can graph their data points and determine whether their equation is a good representation of the graph. Students can see how many of the data points are on or very near the line. If the data points do not fall on or very near the line, students will know their equations were not good representations of the graphs and must try again. This worked so well with my students. That s why I mentioned starting with sample data, initially, to view the graph as a class. F. We ran into trouble here because all our ropes were about the same thickness. Ideally, a variety of rope thicknesses should be used. I can really see the value of comparing the graphs of ropes of varying thickness. The wording of question F may confuse some students. If the rope used were thicker than the current rope, then the graph of the line would be steeper than the current line because the rate of change of the rope s length would increase. A thicker rope would
consume more space per knot. The slope of the line with this thicker knot would be larger than the slope of the current graph. The space between each data point would be greater than the current graph. The opposite is true if the material were thinner than the current rope. The graph of the line would be less steep than the current line because the rate of change of the rope s length would decrease. A thinner rope would consume less space per knot. The slope of the line with this thinner knot would be smaller than the slope of the current graph. The space between each data point would be less than the current graph. It is important to remember that there is no actual x-intercept, but you could theorize or predict at what number of knots the rope length would be zero. When analyzing the graph, the data points are discrete, not continuous. One can only tie a knot that is an integer value. There is no such thing as one-half or one-quarter of a knot. If you are just graphing the data, you should not draw a line. A line is the graph of the equation. In the real word, one reaches a point where one cannot tie another knot without the knots overlapping. In the real world, the length of the rope will never be zero. Theoretically, there is no limit to what the value of x can be. For some value of x, the value of y would be zero and the line would cross the x-axis. By giving student groups different lengths of rope, the y-intercept will be different. If the length of one rope is longer than another, the graph would shift up on the y axis compared to the graph of the current length. If the length of one rope is shorter than another, the graph would shift down on the y axis compared to the graph of the current length. If all students ropes are about the same length, there won t be much variance when the students compare graphs. Also, their scales were different. I think maybe a good visual of the y-intercept for the whole class would be to stand a meter stick upright vertically at the board as the scale and place various groups ropes alongside to mark and compare y- intercepts. Modifications/Extensions:
Modifications: Two versions of the student handout exist. One is scaffolded, helping students navigate the activity through a series of directions and questions. The other is less structured and designed for students to discover the mathematical concepts largely on their own. Resources: If you do not have access to graphing calculators, select one of the following online graphing applications: www.geogebra.org http://my.hrw.com/math06_07/nsmedia/tools/graph_calculator/graphcalc.html http://nces.ed.gov/nceskids/createagraph/default.aspx?id=7054cadef046448ab4 c275a89c8940db http://www.shodor.org/interactivate/activities/graphit/