Functions are Fun by Jason Kozel

Transformation 2013 PBL 5E Planning Form Guide PBL Title: Functions are fun! Teacher(s): Bonnie McClung School: Transformation 2013 T-STEM Center Subject: Algebra 2 Abstract: In this project, students will learn the foundations for functions. They will also write and solve word problems involving functions.

MEETING THE NEEDS OF STEM EDUCATION THROUGH PROBLEM BASED LEARNING ÂŠ 2008 Transformation 2013

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Begin with the End in Mind The theme or “big ideas” for this PBL: Students will develop a basic understanding of functions and patterns drawn from functions, reflecting on how numbers behave, rather than just using them in computations. TEKS/SEs that students will learn in the PBL: (2A.1) Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations. The student is expected to: (A) identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations; and (B) collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments. (2A.4) Algebra and geometry. The student connects algebraic and geometric representations of functions. The student is expected to: (A) identify and sketch graphs of parent functions, including linear (f(x) = x), quadratic (f(x) = x2), exponential (f(x) = ax), and logarithmic (f(x) = logax) functions, absolute value of x (f(x) = |x|), square root of x (f(x) = √x), and reciprocal of x (f(x) = 1/x). Key performance indicators students will develop in this PBL: Develop vocabulary (functions, independent and dependent variables, equations, input, output, domain, range), create tables, find patterns, create rules for functions, write and solve word problems

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21st century skills that students will practice in this PBL: www.21stcenturyskills.org Critical thinking, problem solving, communication, collaboration STEM career connections and real world applications of content learned in this PBL:

Careers: Retail management, science, architecture, education, personal finance, investments Connections: Patterns are found all around us and people use patterns everyday to guide their work.

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The Problem You have been called in for an interview at a mathematics textbook publishing company and will have to demonstrate your writing skills and computational skills in order to be considered for the job. The tasks will include evaluating some existing functions they consider important and you will be asked to construct five of your own function problems.

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Map the PBL Taught before the project

Taught during the project

1. Vocabulary: function, dependent, independent, equations, input, output, table, patterns, domain, range

2. How to create tables

3. How to find patterns and create rules for functions

4. Concrete understanding of functions and their application in real-world scenarios

Knowledge and Skills Needed Elaborate on the knowledge and skills (TEKS student expectations) required to accomplish each step of the task.

5. Write and solve word problems

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Already Learned

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5E Lesson Plan PBL Title: Functions are fun! TEKS/TAKS objectives: 2A1.A,B; 2A4.A Engage Activity Using the following website, show the students the introduction to a function machine. http://207.207.4.198/pub/flash/9/9.html Using dry erase boards, ask the students to create a mapping diagram by writing the name of their grandmother (or another relative) in an oval box (you may need to illustrate this). Then have them draw another oval box to the right and list all of the names this person is called depending upon who is addressing him/her. Have them draw arrows from the name in the left oval to all of the names they have in the right oval. Some students will have more names in the right oval than others. You have just created a set of ordered pairs in mapping form called a relation. A relation is just a set of ordered pairs with no particular rule governing them. In this unit, we will be focusing on a special relation called a function. During our journey, you will be using some web sites to reinforce the concepts. http://en.wikibooks.org/wiki/Algebra/Functions#Function_as_Relation http://www.shodor.org/interactivate/activities/PossibleOrNot/ http://www.shodor.org/interactivate/activities/PossibleOrNot/ You will be introduced to several types of functions in this PBL and these will be referred to as parent functions. Introduce the students to the project: You have been called in for an interview at a mathematics textbook publishing company and will have to demonstrate your writing skills and computational skills in order to be considered for the job. The tasks will include evaluating some existing functions they consider important and you will be asked to construct five of your own function problems. Introduce the students to the web sites they will be using as an introduction to the concept of functions (see resource section below). Engage Activity Products and Artifacts Students will write in their math journals addressing the following questions: • What other relationships might you think of that you encounter every day? (associating numbers with the alphabet in texting for example) • Write five examples of relations using mapping notation

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Engage Activity Materials/Equipment Dry erase boards, computer, LCD projector, Internet Engage Activity Resources http://en.wikibooks.org/wiki/Algebra/Functions#Function_as_Relation http://www.shodor.org/interactivate/activities/PossibleOrNot/ http://www.shodor.org/interactivate/activities/PossibleOrNot/ http://www.purplemath.com/modules/fcns.htm http://www.shodor.org/interactivate/activities/FunctionMachine/ http://nlvm.usu.edu/en/NAV/frames_asid_191_g_3_t_1.html Explore Activity The most important relation we study about in mathematics is a function. Instead of writing our relationship as y=, we use the functional notation f(x) to denote that f is a function of x. Have students work in pairs and complete the Relations and Functions practice sheet (see below). Ask each pair of students to define a function in their own words and share that definition with the class. Put all definitions on chart paper before providing them with a simple definition (A function is a relation in which the first coordinate (domain) is never repeated.). The domain is the independent variable while the range is dependent (ie. the value in the output depends upon what is placed in the input). To reinforce this concept, have students make a t-chart with names of five friends in the x (domain) column. Then have them put the phone number of each friend in the y (range) problem. The output depends upon which friend they call. Two major parent functions studied in Algebra II are the linear function (y = x) and the quadratic function (y=x2). If the domain and range is given, the student should be able to describe the function. The following websites have useful activities for the students to use to solidify concepts: http://nlvm.usu.edu/en/NAV/frames_asid_191_g_3_t_1.html http://www.shodor.org/interactivate/activities/FunctionMachine/ http://score.kings.k12.ca.us/lessons/functions/machine.html After a class discussion about student observations and a summary of what they learned through the explorations, follow with a reinforcement activity (see sample problems below). • Use popsicle sticks (from a craft store). • On one set of sticks write a domain of [-5,5] in any color you wish. • On another set of the same amount of sticks, write the range from a set of functions using a different color.

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Give students a copy of the set of problems which are labeled 1-10 and have them work in pairs. • Give each pair a stick with the domain and have them choose a stick with a range. • They have three minutes to come up with the “rule” describing the function. • At the end of the three minutes, they get another stick and this continues until all pairs have had a chance to find the rule for all problems. The winning team may be awarded candy or just recognized. Explore Activity Products and Artifacts Math journals should include: • definition of a function • examples of functions • vocabulary • t-chart example Completion of functions and relations worksheet Completion of “Find the Rule” game Explore Activity Materials/Equipment Provided worksheet, chart paper, math journals, popsicle sticks, markers, pencils Explore Activity Resources http://en.wikibooks.org/wiki/Algebra/Functions#Function_as_Relation http://nlvm.usu.edu/en/NAV/frames_asid_191_g_3_t_1.html http://www.shodor.org/interactivate/activities/FunctionMachine/ http://score.kings.k12.ca.us/lessons/functions/machine.html Explain Activity Take time to debrief students and see if they have any questions. Encourage them to share tricks they used to find the rule in the game. Explain that functions do not always have to be expressed in terms of x and y as they have seen in the exploration. Functional notation is the most common way of expressing a function and makes it very easy to evaluate a function. Illustrate this with a few examples. Output Value Input Value Ordered Pair f(x) = 2x – 3 f(1) = 2(1) – 3 (1, -1) f(-1) = 2(-1) – 3 (-1, -5)

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Explain to the students that they are going to participate in an activity that visually describes the attributes of a function using the vertical line test. Either have students work in a computer lab or print the pictures following the directions on the website. http://www.shodor.org/interactivate/activities/PossibleOrNot/ Bring to their attention the labels of the axes as distance versus time. Have students choose three of the graphs, draw the graph and write a story that could describe what is happening in the graph. Debrief and have students share what they have learned about functions so far. Explain how functions are used in real-life on a daily basis to help them get started on their challenge problem. Some examples are: 1. Monte, an experienced car salesperson, makes a base salary of $2800 per month plus a 5% commission on each car he sells. Write a function representing his gross income in terms of the number of cars he sells. 2. If a rock falls from a height of 20 meters on Earth, the height H (in meters) after x seconds is approximately H(x) = 20 â&#x20AC;&#x201C; 4.9x2. What is the height of the rock after 1.1 seconds? 3. Using a table of values such as one showing how the distance from the top of a building to the horizon depends on the height of the building, graph the relationship and identify which parent function best describes the graph. Height (m) 5 Distance (km) 8.0

10 11.3

20 15.9

40 22.5

100 35.6

Use your graph to estimate the distance to the horizon from a height of 80 m.

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Explain Activity Products and Artifacts Math Journal Entries: • Graphs with stories • Is it possible to rewrite some relations to make them functions? Explain your reasoning. Answers to class questions Explain Activity Materials/Equipment Graph paper, graphing calculators, math journal, computer access Explain Activity Resources http://www.shodor.org/interactivate/activities/PossibleOrNot/

Elaborate Activity

Tell the students that many companies require the applicant to complete some sort of evaluation to make sure they are competent to be trained for the job. Also, college entrance exams require the student to recall knowledge of all subject content. Have them work in pairs and present them with their challenge problem. You have been called in for an interview at a mathematics textbook publishing company and will have to demonstrate your writing skills and computational skills in order to be considered for the job. The tasks will include evaluating some existing functions they consider important and you will be asked to construct five of your own function problems. The problems must represent a variety of situations and include both linear and quadratic functions. Good luck to you. This is an entry level position with room to move up very quickly! Give the students a copy of the group work rubric that will be a part of their evaluation. Elaborate Activity Products and Artifacts Five problems to be presented to the interviewer Elaborate Activity Materials/Equipment Markers, pencil, graph paper, graphing calculators, computer word processing program, scratch paper, pencils

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Elaborate Activity Resources None Evaluate Activity Teacher will review all problems for accuracy, complexity, and understanding to ensure that the problems are on level, relevant to the content, and workable. Use the rubric to evaluate each groupsâ&#x20AC;&#x2122; work. Students will post their final problems up around the classroom (one problem per chart page). They will then rotate through and complete all problems with their groups to turn in for a test grade. Evaluate Activity Products and Artifacts Completed problems Evaluate Activity Materials/Equipment Rubric, chart paper, markers, notebook paper, pencils, graphing calculators Evaluate Activity Resources None

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Plan the Assessment Engage Artifact(s)/Product(s): Students will write in their math journals addressing the following questions: • What other relationships might you think of that you encounter every day? (associating numbers with the alphabet in texting for example) • Write five examples of relations using mapping notation Explore Artifact(s)/Product(s): Math journals should include: • definition of a function • examples of functions • vocabulary • t-chart example Completion of functions and relations worksheet Completion of “Find the Rule” game Explain Artifact(s)/Product(s): Math Journal Entries: • Graphs with stories • Is it possible to rewrite some relations to make them functions? Explain your reasoning. Answers to class questions Elaborate Artifact(s)/Product(s): Five problems to be presented to the interviewer

Evaluate Artifact(s)/Product(s): Completed problems

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Rubric CATEGORY Grammar & spelling (conventions)

Writer makes 3-4 errors in grammar and/or spelling

Writer makes more than 4 errors in grammar and/or spelling.

Writer makes 1-2 errors in capitalization and punctuation.

Writer makes 3-4 errors in capitalization and punctuation.

Writer makes more than 4 errors in capitalization and punctuation.

Writer makes no Writer makes 1-2 errors in grammar or errors in grammar spelling. and/or spelling.

Writer makes no Capitalization and Punctuation errors in

capitalization and punctuation.

Neatness

Journal entry is written neatly and clearly. It is not crumpled. Handwritting is exceptional and was done with care.

Journal entry is written neatly and clearly. It is not crumpled. Handwritting is good.

Journal entry is somewhat neat. It is partly crumpled. Handwritting is average.

Journal entry is not neat. It is very crumpled. Handwritting is poor and looks like it was done in a hurry.

Directions

Journal entry contains all directions in the correct order for making dye. Directions are written in sentences in a paragraph.

Journal entry contains all directions, with 1-2 out of order. Directions are written in a paragraph or listed.

Journal entry contains all directions with 3 out of order. Directions are written in a paragraph or listed.

Journal entry is missing directions and/or has 4 or more out of order. Directions are written in a paragraph or listed.

Creativity

Journal entry was organized and creatively written.

Journal entry was mostly organized and creatively written.

Journal entry was somewhat unorganized and lacked creativity.

Journal entry was very unorganized and not creatively written.

Sentences & Paragraphs

All sentences and paragraphs are complete.

All sentences are complete and paragraphs are generally good.

A few sentences are incomplete and paragraphs are decent.

Many sentences are incomplete and paragraphing needs a lot of work.

Equations relative to functions

All functions include equations Solutions are correct.

All functions include equations. Minor errors in solutions.

Some functions include equations. Minor errors in solutions.

Equations given no solutions.

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Story Board Week 1 Activities

Week 2 Activities

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Day 1 Engage (90 min.)

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Day 2 Explore (90 min.)

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Day 6 Evaluate (90 min.)

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Day 7 Evaluate (90 min.)

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Day 3 Explain (90 min.) Day 8

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Day 4 Elaborate (90 min.) Day 9

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Day 5 Elaborate (90 min.) Day 10

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Relations and Functions We call the input value the domain of a function and the output value is the range. In each of the following, identify the domain and range. 1.

2. x -1 0 1 2 3. x f(x)

-3 5

F(x) 4 3 2 1

-2 5

-1 5

0 5

1 5

2 5

3 5

f(x) = { ( 3,2), ( 5, 6), ( 7,9), (7, 10) (9,9)

D(t) = 3t + 4 Build a table of values for this function and graph it on graph paper.

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All functions are relations. Not all relations are functions. Function as Box Functions are another way of describing certain things mathematically. They are often described as a machine in a box open on two ends; you put something in one end, something happens to it in the middle, and something pops out the other end. The function is the machine inside, and it's defined by what it does to whatever you give it. Let's say the machine has a blade that slices whatever you put into it in two and sends one half out the other end. If you put in a banana, you'd get back half a banana. If you put in an apple, you'd get back half an apple.

Since this is algebra, the things that go in and come out of functions will be numbers. Let's define the function to take what you give it and cut it in half, that is, divide it by two. If you put in 2, you'd get back 1. If you put in 57, you'd get back 28.5. Functions are typically named with a single letter. We'll call this one h for half. (There's nothing special about the letter we choose--we could just as well called this function f. The letter doesn't have to stand for anything.) Now we need the notation. To put 2 into the function, we write h(2) (read h of 2). We know that h(2) = 1, and h(57) = 28.5 but to completely explain this function, we can write this:

What this means is that for any value of x, h(x) equals x divided by two. This form is called a function definition. http://en.wikibooks.org/wiki/Algebra/Functions#Function_as_Relation

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