The Exercise Bike Problem Jennifer Bostic Schmalzbach MATH 565 Adapted from a lesson idea from Dr. Bardzell

Standards SC : 8 th grade: 8.1.A.1.c Determine whether relationships are linear or nonlinear when represented in words, in a table, symbolically, or in a graph AL- Use a graph to determine if a relationship is linear or nonlinear SC : Algebra 1: Goal 1 The student will demonstrate the ability to investigate, interpret, and communicate solutions to mathematical and real-world problems using patterns, functions and algebra. Expectation 1 The student will analyze a wide variety of patterns and functional relationships using the language of mathematics and appropriate technology. Indicator 1 The student will recognize, describe and/or extend patterns and functional relationships that are expressed numerically, algebraically, and/or geometrically. AL: The student will represent patterns and/or functional relationships in a table, as a graph, and/or by mathematical expression. Core Standards: 8.F.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.

Student Outline Objective: Students will explore the relationship between work and “steepness” of the landscape by sketching a graph of the elevation required to achieve a sample exercise bike workout Warm-up: Sketch a graph that demonstrates a positive or negative correlation between two variables. Discussion: exercise bikes, effort, and reading the screen of an exercise machine Activity: You will be asked to sketch the landscape that would be responsible for the workout that you are presented Discussion: the relationship between effort and elevation, the relationship between effort and “steepness”

Lesson Plan Details Warm-up: Have students sketch a graph that demonstrates a positive or negative correlation between two variables. Since not all students have experience with exercise bikes, students and the teacher will engage in a discussion about exercise bikes, effort, and reading the screen Students will be asked to sketch the landscape that would be responsible for the workout that they are presented – Teacher notes: in most cases, students will create parabola that looks like a hill; if this does not happen the next step may not be as vital The teacher and students will “ride” their bikes, this interactive experience is designed to make students really think about the incline and effort that is required for each stage (1 minute block) of the exercise profile; students will describe the landscape in terms of “steepness” The teacher will allow students to re-sketch their graphs The teacher will float around the room having small group discussions with students about their graphs and how they came up with their elevation growth/decline

Questions: – Does your graph represent the effort required for the stationary bike workout? – Should your landscape have any downward sloping sections? Why? Why not? – What does each increase in the stationary bike workout mean for the elevation of your landscape? The whole class will have a discussion about the elevation and the rate at which it increases or decreases Students will create a table that demonstrates the relationship between time, effort level, and elevation of the real life hill Students will use this information to infer information about the “rate of increase” between each stage (minute) 6 th and 7 th grade students can complete this activity without having a clear understanding of slope because the language can be centered in the real world 8 th grade and Algebra 1 students can use their understanding of slope to explore the linear and quadratic relationship between the variables (see lesson wrap up)

Summary Question Create a sketch of the landscape elevation that would be required for this exercise profile.

Differentiation There are times throughout the lesson for the teacher to have a 1 on 1 conversation and lesson with students that struggle Students that lack the vocabulary and background knowledge for a discussion on slope can discuss the activity in terms of “steepness” Alternate Activity: Some students may find it difficult to go from the abstract (bike screen) to the real world (elevation). Alternatively, some students could be given the assignment in reverse. With a given elevation sketch, they would have to chart the effort required over time to ride the terrain

Reflection At the end of this lesson, I felt my students were _____________________. Positive parts of the lesson: Negative parts of the lesson: Improvements for next time: