Fleet footed fauna In this task students manipulate coefficients of linear, quadratic and exponential functions to match given conditions. The task makes use of a narrative to describe the functions and uses the desmos graphing calculator website. Mathematical Focus Manipulating coefficients for linear functions, quadratics and exponentials MA5.3-9NA A student sketches and interprets a variety of non-linear relationships Students describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (ACMNA267) Incidental connected content in context: 5.2 Rates and ratio MA5.2-5NA A student recognises direct and indirect proportion, and solves problems involving direct proportion Groundwork This activity would be planned to occur after students have attained familiarity with non-linear functions. Students interpret graphs and manipulate coefficients to attain required conditions. The activity has been designed using the desmos website (www.desmos.com) as the tool to generate graphs. Other digital technologies could be used. Preparation: Students need internet access to use desmos unless using the app. Students readily learn how to enter and edit functions in desmos. It does not matter if they haven’t used the site before. Print and cut out the cards from the template Matching animals to distance v time graphs. Timing 60-90 minutes Reference This task was adapted from:Williams, G., Menzel, B. & Sheridan, B. (2009) Suddenly, We Had Engaged Middle Years Students! How Did That Happen? Mathematical Association of Victoria The Task 1 The teacher may choose to provide a narrative to engage the students with the problem to solve. A possible narrative to introduce the investigation is below. Narrative Out in the far distant jungles of Kurvytopia, three animals are competing in the 1 kilometre race. Animal A races at a constant speed. Animal B gets faster the further they runs. Animal C gets slower the further they runs. Your group needs to choose three appropriate animals to represent animals A, B and C. What different animals did the students choose? What have the students noticed? Where do all three animals start from? Can they tell which animal wins? Did they notice the scales are missing from the graphs making it impossible to know which animal wins? Students match the cards from the template Matching animals to distance v time graphs. Questions One of the functions is linear. Discuss in your group which animal has a linear function? (animal A) One of the functions is a quadratic. Discuss in your group which animal has a quadratic function? (Possibly B or C) The third animal’s function is an exponential. The Task 2 At this point the students are to use desmos to create a linear function (red), a quadratic function (blue) and an exponential function (green). The graphs produced do not have to match the functions described by the fast fauna at this stage. The amount and nature of teacher intervention at this point will vary depending on prior understanding of students. The ability of students to create examples of linear and non-linear functions will provide information of their understanding. Narrative It is now time to announce the winner. It was in fact, a three way tie. They all finished the 1km length race in one minute. Animal C not only slows down but starts running backwards at the 1km mark. We now need to adjust our three functions to represent this new information. The Task 3 The students now have to modify their functions so that they all pass through the starting point (0, 0) and the finish point (1, 1). Typically there will be many students keen to immediately return to the computers and take a guess and check approach. At this stage of the task it is important to slow students down to think through possible strategies. Have students discuss in small groups the things they know already that will help them modify their functions. You can also generate a list of questions that they have about the task. Prompts If the students are finding it difficult, ask them how the formula y=mx+b relates to the task. Ask them what role the coefficients have in the formula. Clearly students will make faster progress with the linear function. If necessary suggest the students start with the linear function. Once the linear function is found, telling students that animal C, who slows down, is a quadratic function may be required. Ask whether the parabola for this quadratic would be concave up or concave down. Reflection It is assumed that not all groups of students will modify all three curves successfully. The task, by nature, allows the students to pursue the investigation after the lesson. The reflection discussion should draw together the different findings that the class had in trying to modify their functions under the given conditions. Adaptations If students need a greater challenge, change the distance to 2km in one minute.