8.5 Training Day Part II
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- Ariel Bridges
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1 Training Day Part II A Solidify Understanding Task Fernando and Mariah continued training in preparation for the half marathon. For the remaining weeks of training, they each separately kept track of the distance they ran during the week. Since they ran together at the same rate on Saturdays, they took turns keeping track of the distance they ran and the time it took. So they would both keep track of their own information, the other person would use the data to determine their own total distance for the week Week 2: Mariah had completed 15 more laps than Fernando before they trained on Saturday. a. Complete the table for Mariah. Time (in minutes on Saturday) Fernando: Distance (in laps) Mariah: Distance (in laps) b. Write the equation for Mariah as a transformation of Fernando. Equation for Mariah: m(t) = f(t) 2. Week 3: On Saturday morning before they started running, Fernando saw Mariah s table and stated, My equation this week will be f(t) = m(t) a. What does Fernando s statement mean? b. Based on Fernando s translated function, complete the table. Time (in minutes on Saturday) Fernando: Distance (in laps) Mariah: Distance (in laps)
2 27 c. Write the equation for both runners in slope-intercept form: d. Write the equation for Mariah, transformed from Fernando. e. What relationship do you notice between your answers to parts c and d? 3. Week 4: The marathon is only a couple of weeks away! a. Use Mariah s graph to sketch f(t). f(t) = m(t) 10 Distance Time (in minutes on Saturday) b. Write the equations for both runners in slope-intercept form. c. What do you notice about the two graphs? Would this always be true if one person ran k laps more or less each week?
3 28 4. Week 5: This is the last week of training together. Next Saturday is the big day. When they arrived to train, they noticed they had both run 60 laps during the week. a. Write the equation for Mariah on Saturday given that they run at the same rate as the week before. b. Write Fernando s equation as a transformation of Mariah s equation. 5. What conjectures can you make about the general statement: g(x) = f(x)+k when it comes to linear functions?
4 8.5 Training Day Part II Teacher Notes A Solidify Understanding Task Purpose: Students will solidify their understanding of vertical transformations of linear functions in this task. Goals of this task include: Writing function transformations using function notation. Recognizing that the general form y = f(x)+ k represents a vertical translation, with the output values changing while the input values stay the same. Understanding that a vertical shift of a linear function results in a line parallel to the original. Core Standards Focus: F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. (Note: Focus on vertical translations of graphs of linear and exponential functions.) F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F.BF.1 Write a function that describes a relationship between two quantities. Related Standards: F.IF.1, F.IF.2, A.CED.3 Standards for Mathematical Practice of Focus in the Task: SMP 2 Reason abstractly and quantitatively SMP 3 Construct viable arguments and critique the reasoning of others SMP 7 Look and make use of structure
5 The Teaching Cycle: Launch (Whole Class): Read the initial story and ask students to clarify what this means. If not stated, clarify that the two runners are going the same rate each time they run together on Saturday morning. Students should be able to get started on this task without additional support, since it is similar in nature to the work they did on Training Day. Explore (Small Group): Watch for students who confuse input/output values as well as for students who struggle with making sense of using function notation in the first two problems (weeks 2 and 3). Listen for students who make the connection that the shift is about adjusting the output values and are visually showing the connection to the table and the equations (the distance between the number of laps of the two runners is the same in the table as it is in the shift or +k value). Week 4 has students visually see the shift of k units with a graphical representation. This is another way students see that each output value is exactly k units away from the other function. It can be noted that the lines are parallel, however, make sure the discussion talks about the distance of the output values (since with future functions, they will not be parallel lines but that they do maintain a distance of K units. Discuss (Whole Class): The goal of this whole group discussion is to highlight the different ways to see vertical translations of linear functions. Have different students go over each week of training, showing how the vertical shift of one function relates to the other. For each week, have students show connections between the context, the mathematical representation, and the transformation function. Aligned Ready, Set, Go: Connections 8.5
6 29 CONNECTING ALGEBRA & GEOMETRY READY, SET, GO! Name Period Date READY Topic: Describing spread. 1. Describe the spread in the histogram below Describe the spread in the line plot below. 3. Describe the spread in the box and whisker plot.
7 30 CONNECTING ALGEBRA & GEOMETRY SET Topic: Writing functions in translation form. You are given information about!! and!!. Rewrite!! in translation form:!! =!! +! 4.!! = 7! + 13!! = 7! 5 5.!! = 22! 12!! = 22! !! = 15! + 305!! = 15! x f(x) g(x) x f(x) g(x) x f(x) g(x) GO Topic: Vertical and horizontal translations. f (x) 10. Use the graph of!! =!" to do the following: a. Sketch the graph of!! =!"! on the same grid. b. Sketch the graph of!! =!!! c. Describe how!!,!!,!"#!! are different and how they are the same. d. Explain in what way the parentheses affect the graph. Why do you think this is so?
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