plot those points on a graph. 5.Evaluate the equation y = 225 x 2 at the points x={ 15, 12, 9, 0, 9, 12, 15}. Then

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1 Construct a Concept Lesson Goal: Students are able to construct a concept for a working definition of What is a function? in terms of input and output by evaluating and comparing graphs of example functions and non example functions. Unit Objective: A. Students will be able to discriminate between relations that are functions and relations that are not by defining a function as a rule that assigns to each input exactly one output. (Construct a Concept) Lesson Plan: 1. Sorting and Categorizing: (15 min) Students will work in 5 groups of 4 to 6. Each students receives a tasksheet Each group is assigned two items from the tasksheet, one example and one non example of a function. They are instructed to graphically represent them. For example: Examples: 1.Archimedes is draining his 30 gallon bathtub. Every 3 minutes, 5 gallons of water are drained. Make a table of values of the amount of water w at time t and then plot those values on a graph. 2.Salt Lake City s pollution level this morning at 6 a.m. was 30 parts per million, and it grows by 25 parts per million every hour. Make a table of values of the pollution level y at time t and then plot those values on a graph. 3.Graph the set of ordered pairs: (1,2) (2, 2) (3,5) (4,1) (5,7) (6,4) (7,3) 4.Evaluate the equation y = 2x 2 1 at the points x ={ 4, 3, 2, 1, 0, 1, 2, 3, 4}. Then plot those points on a graph. 5.Evaluate the equation y = 225 x 2 at the points x={ 15, 12, 9, 0, 9, 12, 15}. Then plot those points on a graph. Nonexamples: 1. A ball is thrown at a wall at a constant speed of 2 meters per second (m/s), hits the wall 4 meters away, and then bounces back at a constant speed. The graph of the ball s position d at time t is shown below. Fill in the table for the values of time t at distance d. 2. Graph the following representation of the number of people to the color of their shirt.

2 3. Graph the set of ordered pairs: (1,2) (1,3) (3,3) (4,2) (4,1) (5,7) (6,4) (7,5) 4. Evaluate the equation y 2 = x at the points x = {1, 4, 9, 16}. Then plot those points. (Remember to include both positive and negative roots of x) 5. Evaluate the expression y = ± 25 x 2 at the points x={ 5, 4, 3, 0, 3, 4, 5}. Then plot those points on a graph. (You will have twice as many points as in problem 1) Students are instructed to find, discuss, and make conjectures about the differences and similarities of their given items. Instructor writes on the board the following questions for students to consider when the are done with their tasksheet. o What are some of the similarities between the example and the non example? o What is different between the example and the non example? After about 10 minutes, have one person from each group come to the board and draw their graphs in assigned spot. Board is divided into 2 parts one labeled example and the other non example. Then, a different person from the group will briefly explain their graphs and the differences they found to the class. Students are instructed to jot down all of the graphs and proposed differences in their notebooks. 2. Reflecting and Explaining (15 min) Students then engage in a large group discussion of how to categorize all of the examples and non examples, and explain their rationale for doing so. Use the Keep it going strategy, with the caveat that the next person to speak must be from a different group than the previous person to speak (i.e. the next person chosen cannot be from either of the two preceding groups) The teacher asks questions to help students through the thought process (note that mistakes made on graphs or groups that did not finish their graphs will be used for learning in the discussion: What is similar between the values of all the Examples?

3 What is similar between the values of all the Non Examples? What makes an example, an example or a non example a non example? What if we try connecting the dots in one line from left to right, like Taylor s group did? Introduce concept of input and output: For a function there is only one output for every unique input Generalizing and Articulating (10 min) Students construct a concept of how their examples or non examples might relate to one another, i.e. what distinguishes a function from something that is not a function. Students create a name for the concept (does not have to be function ) Ask the students: What is a function? Can we define it in terms of input and output? Write the agreed upon definition on the board. Students write in their journals what the class decided on as criterion for examples and nonexamples. 4. Verifying and Refining (10 min) Students are given additional examples and nonexamples to test their concept, which will be taken home as homework. Explain that while we called our examples, the name we use in the math world and that is used in the textbook is function. If their definition does not hold up, they can revisit the previous steps the next day in class to further refine the definition. A brief review will be given at beginning of class the next day as an into to next lesson. *Note the official definition of a function: A function is a set relationship that assigns each input to exactly one output. Miniexperiment: (5 min) Prompt: Is the following relation a function? Why or why not? What is the relation between the input and the output? Emma borrows $15 from John, promising to pay him back that amount, plus 5 cents per dollar every week until she pays him back. Observer s Rubric: A maximum of 3 points is possible depending on the following criteria: +1 For answering yes, it is a function +1 For explaining why or why not this is a function +1 For explaining that there is one output for every input

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5 Tasksheet: Group 1 Create a graph and table of values for the following example and non example: 1. Archimedes is draining his 30 gallon bathtub. Every 3 minutes, 5 gallons of water are drained. Make a table of values of the amount of water w at time t and then plot those values on a graph. 2. A ball is thrown at a wall at a constant speed of 2 meters per second (m/s), hits the wall 4 meters away, and then bounces back at a constant speed. The graph of the ball s position d at time t is shown below. Fill in the table for the values of time t at distance d.

6 Tasksheet: Group 2 Create a graph and table of values for the following example and non example: 1. Salt Lake City s pollution level this morning at 6 a.m. was 30 parts per million, and it grows by 25 parts per million every hour. Make a table of values of the pollution level y at time t and then plot those values on a graph. 2.

7 Tasksheet: Group 3 Create a graph and table of values for the following example and non example: 1. Graph the set of ordered pairs: (1,2) (2, 2) (3,5) (4,1) (5,7) (6,4) (7,3) 2. Graph the set of ordered pairs: (1,2) (1,3) (3,3) (4,2) (4,1) (5,7) (6,4) (7,5)

8 Tasksheet: Group 4 Create a graph and table of values for the following example and non example: 1. Evaluate the expression y = 2x 2 1 at the points x ={ 4, 3, 2, 1, 0, 1, 2, 3, 4}. Then plot those points on a graph. 2. Evaluate the expression at the points x = {1, 4, 9, 16}. Then plot those points. (Remember to y 2 = x include both positive and negative roots of x)

9 Tasksheet: Group 5 Create a graph and table of values for the following example and non example: 1. Evaluate the expression y = 225 x 2 at the points x={ 15, 12, 9, 0, 9, 12, 15}. Then plot those points on a graph. 2. Evaluate the expression y = ± 25 x 2 at the points x={ 5, 4, 3, 0, 3, 4, 5}. Then plot those points on a graph. (You will have twice as many points as in problem 1)

10 Part Two Homework 1. Juanita is a pilot and just received her flight schedule for the next two weeks. The first week she is scheduled to take off out of Texas at 8:00 am landing in Ecuador at 4:30 pm. Tuesday she leaves Ecuador at 10:00 am and arrives in the United Kingdom at 7:00 pm. Wednesday is her day off. Thursday she departs the United Kingdom at 12:00 pm has a connecting flight in Egypt then lands in Russia 9 hours later. Friday she leaves Russia at 2:00 pm and arrives in Vietnam just in time for dinner. Saturday she plans on flying to Australia where she can lay on the beach until her next weeks schedule begins. Graph Juanitas flight schedule for week 1. Week 1: 2. Does Juanita s flight schedule represent a function? Provide an explanation of why or why not.

11 3. Juanita is up bright and early for her flight out of Australia which leaves Monday morning at 4:00 am. She arrives in Russia at noon. She is scheduled to depart on Tuesday at 9:00 am and land in India at 2:00 pm. She is scheduled to be in France at 11:00 pm that same day. From India she leaves on Wednesday morning at 11:00 am and arrives in France at 5:00 pm. Thursday her flight is schedules to leave at 6:00 am and arrives in Bolivia at 8:00 pm. She also is scheduled to leave France at 4:00 pm and arrive in Maine, USA at 8:00 pm. Friday morning she departs Maine and flies home to Canada where she looks forward to a weekend relaxing at home. Graph Juanitas flight schedule for week 2. Week 2: 4. Does Juanita s flight schedule represent a function? Provide an explanation of why or why not.