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 Helping Students Understand Slope The Principles and Standards for School Mathematics, published by the National Council for Teachers of Mathematics, states that students in grades 6-12 should be able to "understand patterns, relations, and functions." The ability to represent linear functions in various forms is a critical skill, but helping students understand slope can be a challenge. Fortunately, there are several steps you can take to help students conceptualize slope. Prepare them with introductory activities. Conduct appropriate developmental activities. Carefully introduce y-intercept and negative slope concepts. Generate predictions using linear functions. Informal Introductory Activities When introducing slope, use linear functions that are direct variations, that is, linear functions passing through the origin. Incorporate the following activities into your classes. Have students measure various staircases and compare rise to run. Discuss the steepness of the staircases. Ask students to examine the grade of a road. For example, if a highway has a 6% grade for the next 5 miles, how far does the road drop vertically over the 5 miles traveled horizontally? If students are interested, measure ski slopes or skateboard runs. Prepare a handout showing graphs of various linear functions. Discuss the steepness and direction of the lines. Take your class to a large room, such as the gym, to create a human slope. Use students as points on the graphs of various linear functions, moving them to suggest lines of different steepness and direction. Developmental Activities Have students describe functions with words, tables, and graphs. Help students become adept in converting between these representations. Determine real-world situations that interest your students and develop activities to match these interests. Consider the following activities. Select an average constant speed for a vehicle. Have students make a table showing hours and total distance traveled. After graphing the function, they should predict how long different lengths of trips will take. This demonstrates positive slope, with slope being the speed. Choose an item and its cost. Have students make a table with the number of items as the independent variable and total cost as the dependent variable. They should graph the function and describe the meaning of the slope. Make a table and graph the relationship between two units of measure, such as centimeters per inch. Students should realize that the slope is the conversion factor. Introducing y-Intercept and Negative Slope After students have demonstrated an understanding of direct variations, introduce linear functions not passing through the origin. Ask what the slope and the y-intercept represent in each situation. Give students a handout in which you have provided various linear functions by hand or using technology. Have them write the equations, checking their equations on a graphing calculator. Have students identify the slope and y-intercept for the following applications: converting temperatures from Celsius to Fahrenheit and vice versa; relationship between increasing altitude and decreasing temperature for aircraft; relationship between the depth of a diver and increasing water pressure. Additionally, have students make tables and graphs for the following: A situation with an initial charge and an additional constant charge per unit of time, such as a telephone service or a car rental. Students should understand that the y-intercept is the initial charge and the slope is the charge per unit. A situation in which they are a particular distance from a destination and are traveling at a constant rate toward the destination. Have students determine the time when they will reach the destination (represented by the x-intercept). Another application demonstrating negative slope would be a full swimming pool draining at a constant rate. Have students interpret the y-intercept, slope, and x-intercept. Prediction Using Linear Functions After students learn to represent linear functions with tables and graphs and understand slope, y-intercept, and x-intercept as attributes of these functions, have them generate their own data and make predictions using their graphs. This type of data may not always be strictly linear-students will need to approximate a linear graph for the data or use a graphing calculator or graphing software to generate a line of best fit. Consider the following activities. Have all students in the class measure their height and arm span in either inches or centimeters, record the data in a table, and graph the data. Use the graph to predict arm span given height or vice versa. Have students measure the diameters and circumferences of a number of circular objects, record the measurements in a table, and graph the data. Have students make conjectures about the graph. Have students make a conjecture about two data sets that they think may have a linear relationship, such as year and population of the world. They should then use resource books or the Internet to find the data. Students can make graphs and discuss the relationship, if any, shown for the two data sets. Students can also predict using their graphs, if a relationship is found. Team with a science teacher and gather scientific data using a Calculator-Based Laboratory™. Have students make graphs and use the graphs to make predictions. Further Study As students become more adept with slope, be sure to discuss 0 slope and no slope. When assessing students' mastery of slope, always include graphs and equations having positive and negative slopes in addition to cases of 0 and no slope. This article was contributed by Teri Willard, a mathematics teacher and writer from Belgrade, Montana.

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