The Ins and Outs of Functions by Jason Kozel

Transformation 2013 PBL 5E Planning Form Guide Design Challenge Title: The “Ins and Outs” of Functions Teacher(s):Cassandra Ricks School: Education Service Center, Region XIII Subject: Algebra 1 Abstract: In this design challenge, students will learn the foundations for functions. They will also write and solve word problems involving functions.

Last Updated: 9/25/08

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: (A.1) Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to: (A) describe independent and dependent quantities in functional relationships; (B) gather and record data and use data sets to determine functional relationships between quantities; (C) describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations; (D) represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities; and (E) interpret and make decisions, predictions, and critical judgments from functional relationships 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 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 work for “Mathematics Today” publishing company and have been asked to assist in writing real-world problems for their new project-based learning textbook. The unit that you and your team have been assigned to is “Foundations for Functions”. Your first task will be to review the concepts behind functions to deepen your understanding of the concepts. You will then write a series of five (5) word problems to be incorporated into the unit that your team has been assigned. By the way, the book is being sent to the publisher next week, so you don’t have much time. Let’s get started!

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Map the PBL Already Learned

Performance Indicators

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 5. Write and solve word problems

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5E Lesson Plan PBL Title: The “Ins and Outs” of Functions TEKS/TAKS objectives: A1, 8.14, 8.15, 8.16 Engage Activity Ask the students to complete the following statements: What goes up must ______________ (come down). For ever action, there is _____________ (an equal but opposite reaction). For every input, there is ___________ (an output). We will be focusing on functions, which are nothing more than inputs and outputs, over the next few class periods through a design challenge. You work for “Mathematics Today” publishing company and have been asked to assist in writing real-world problems for their new project-based learning textbook. The unit that you and your team have been assigned to is “Foundations for Functions”. Your first task will be to review the concepts behind functions to deepen your understanding of the concepts. You will then write a series of five (5) word problems to be incorporated into the unit that your team has been assigned. By the way, the book is being sent to the publisher next week, so you don’t have much time. Let’s get started! Introduce the students to the web sites that they will be using for the review portion of the design challenge and define the products of the Engage Activities (see below). Engage Activity Products and Artifacts Students will work in teams by creating tables of values using the large chart paper and markers to document input and output from the Shodor web site. They will also develop a rule for each pattern. Students will share their information with the rest of the class. Students will complete the quiz at the end of each brain pop video. Engage Activity Materials/Equipment Computer Lab, Internet Access, Large Chart Paper, Markers Engage Activity Resources http://www.shodor.org/interactivate/activities/FunctionMachine/?version=1.6.0_02&bro wser=MSIE&vendor=Sun_Microsystems_Inc. http://www.brainpop.com/math/dataanalysis/problemsolvingusingtables/ http://www.brainpop.com/math/dataanalysis/wordproblems/

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Explore Activity Have the students work in pairs or teams to complete the explore activity (see attached guide). Explore Activity Products and Artifacts Students will use large chart paper and markers to record the results of their investigations. They will also identify the “rule” that is created by each table. Explore Activity Materials/Equipment Attached Explore Guide, Large Grid Paper, Markers Explore Activity Resources Explore Guide Explain Activity Have the students present their findings from the Explore Activity to the rest of the class. Upon the conclusion of all group presentations, have the students discuss observations, ideas, questions, and hypotheses with the rest of the class. Act as the facilitator, clear up any misunderstandings, and broaden the students vocabulary base…input (independent variable), output (dependent variable), function, rule, patterns, tables, equations, domain, range. During the discussion, create a word wall for the students to be able to refer back to during the project. Have the students reflect in their journal regarding the concepts and vocabulary that have been discussed during the Explain phase. Explain Activity Products and Artifacts Word Wall, Journal Explain Activity Materials/Equipment Journal, pencil, word wall, construction paper, markers Explain Activity Resources None

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Elaborate Activity

Share an example of an exemplar word problem involving functions (see attached exemplar word problem). Work through the word problem with them, and model how you would create this particular word problem. Provide the students with a list of “real-world” examples of how patterns are used in business, design, education, etc (see attached list for a starting point). Ask the students if they can think of other areas in which patterns are used to guide work. Make a list of their responses and post them in the classroom to be used as ideas for “starting points”. Ask the students to work together to create their five (5) pattern problems to be submitted to the publishing company. Students will type up their problems to submit to teacher and then students will create large posters of their 5 problems to post in the classroom. Elaborate Activity Products and Artifacts List of real-world examples, 5 problems to be submitted to publishing company The teacher will need to work through all of the problems to determine whether or not the problems work and solutions to the problems. Elaborate Activity Materials/Equipment Exemplar word problem, real-world application list, large grid paper, markers, computers, word processing program, scratch paper, pencils Elaborate Activity Resources None Evaluate Activity Have the students create posters of their five (5) respective word problems, and ask them to post them around the classroom. Re-group the students in pairs and assign them two word problems per group (assuming that the original group size was 4 and the class size is 16…adjust to meet your students’ needs). The students must be assigned to problems that were not created by their original groups. Have the students work through their assigned problems (documenting on a flow chart) and present their findings to the class. Have them talk about what they discovered, whether or not the problems worked, what types of patterns emerged, and whether or not they got the right answer. If the problems did not work, why didn’t they? What could have been done differently so that the problem would work?

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Evaluate Activity Products and Artifacts Presentation, assigned problems, flow charts for problem solving, write-up regarding the word problems they were assigned, and report back on assigned problems Evaluate Activity Materials/Equipment Exemplar word problems posted around the classroom, scratch paper, pencils Evaluate Activity Resources None

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Explore Activity 1. Supply the missing entries in the following tables. Then write out a rule for each table that tells what to do with the input to get the output. Express each rule in sentence form; for example, “The output is one more than three times the input.” Be as clear as you can. Also, identify the domain and range for each function. Record the table, your results, and your rule on a large piece of grid paper and be prepared to share your results with the class. Input (x) Output (y) 2 4 3 6 11 22 27 ? ? 18 ? 50

Input (x)

Output (y) 3 11 15

Input (x) Output (y) 2 7 4 13 7 22 10 31 12 ? ? 76

Input (x) Output (y) House 4 Cup 2 Writer 5 Elephant 7 Spin ? Mathematics ? ? 3 ? 8 ? 0

Input (x) Output (y) Division I Ever E Opportunity O Toast A Safe E People O Mathematics ? ? A Internet ?

2. Create two (2) input/output tables based on rules of your own. Plan your tables below, and write your final tables on two (2) separate sheets of large grid paper. Be sure to write your rule in pencil on the reverse side of your grid paper.

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3. Assuming the pattern continues in the same way, build the next one. Describe verbally the 10th one. How many cubes would you need to build the 10th one? The 20th one? Can you generalize for any building number? How do you know? Record your responses below, and then record them on large grid paper to share with the rest of the class.

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Exemplar Word Problem Example Cassandra’s Chocolate Have you ever really been in the mood to eat chocolate? There are some pretty amazing things that can get in the way of that pursuit. Cassandra was walking home from work one cool afternoon. She had just bought a big bag of chocolate candies and was going home to share them with her daughter…Alexandra loves chocolate, too. On the side of the road, she saw two people collecting food for nursing home residents. Well, Cassandra decided that she had quite a few pieces of chocolate in her bag, and she and Alexandra really didn’t need that much chocolate. “Here,” said Cassandra, “you can have half of my chocolate for the nursing home residents.” The people were very happy to get all of that chocolate. Cassandra thought for a moment and then said, “Aw, take one for each of yourselves, too.” So there went two more. As Cassandra continued walking, her son, Caleb, and his friend, Michael, came riding up on their bikes. They saw the bag of chocolate that she had and asked if they could have some, too. They pleaded, “We are so-o-o hungry. We really need a snack. Pleaaaase!” As you might imagine, Cassandra was a very understanding mother who just couldn’t resist her baby boy’s face. She handed her bag to the boys and they took half of the chocolate and then, just as they were about to hand the bag back, they took two more. Cassandra thought that she had enough chocolate left to make Alexandra happy, so she continued walking home. As you may have already guessed, Cassandra didn’t get very far before she had another encounter. Just before she reached home, her friend Stephanie approached. After exchanging greetings, Stephanie explained that she was on her way to get some chocolate for her family, but was in a bit of a rush because her son, Jaden, had soccer practice in 30 minutes. Generosity took over Cassandra and she found herself saying, “Why don’t you save yourself a trip and take some of my chocolates? As you can see, I’ve got several.” So Stephanie took half of what Cassandra had in the bag then two more. Cassandra finally finished her walk home without further interruption. When she opened the once bulging bag of chocolates, she discovered that there were only two left. She had one chocolate and Alexandra had one chocolate and then there were none! After supper, Alexandra asked her mother how many chocolates were in the bag to begin with. Cassandra told her the story of her walk and then said that if Alexandra could figure it out herself, Cassandra would take her to the park the next day. Alexandra took awhile, but she figured it out and got her park outing. What was Alexandra’s answer?

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Once you have found your answer, work on extending the problem and exploring different values when Cassandra gets home, like 0, 1, 3, 4, or 5. Assume that everything else in the problem remains the same except for the ending number of chocolates. Use the blank space below to work through the process and create an input/output table.

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Functions in the Real World Patterns are found all around us and people use patterns everyday to guide their work. Here are some examples of when patterns are used. Retail Management: Managers look at trends from the previous year at the same time period to decide how much staff is needed to support the number of customers who will be shopping during the current work week. Architecture: Architects look at form versus function when designing buildings, the use of materials and their failure rates, planning where plumbing will go, patterns in landscaping, patterns in shapes, etc. Educators: Educators look at patterns in TAKS data and use it for staff allocation, focusing on areas of strengths and weaknesses, class size, scheduling, etc. Personal: Parents look at cell phone usage patterns to determine the best, most economical rate plans to meet their teenagersâ&#x20AC;&#x2122; needs. Teenagers look at gas prices, miles per gallon, use of air conditioning, speeding, etc. to budget for price increases as seen in patterns. Scientists: Scientists look at patterns in nature to determine coastline erosion, volcanic cycles, the number of hurricanes that could occur this year, increases in deer populations, etc.

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Plan the Assessment Engage Artifact(s)/Product(s): Tables, BrainPOP Quizzes

Explore Artifact(s)/Product(s): Completed Explore Activity on large grid paper

Explain Artifact(s)/Product(s): Vocabulary word wall, Journal entry regarding the key concepts behind functions

Elaborate Artifact(s)/Product(s): List of real-world examples to use as problem starters, five (5) real-world word problems using functions

Evaluate Artifact(s)/Product(s): Presentation, assigned problem flow charts with solutions, write-up regarding the word problems they were assigned, report back on the assigned word problems

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Rubrics Scoring Rubric Word Problems 4 points _____ Completely explains steps to solve the problem. _____ Shows all work for solving the problem _____ Uses a problem solving method _____ Includes clear, simple, well- labeled diagrams _____ Includes all the needed formula(s) _____ Includes correct solution WITH labels _____ Uses complete sentences with capitals and punctuation _____ Word problems are flawless, understandable, and applicable to real-world scenarios

3 Points _____ Completely explains the steps to solve the problem _____ Shows most of the work for solving the problem _____ Uses four of the 5 steps of our problem solving process _____ Includes a diagram of the plan which may contain some inaccuracies (inc labels, dimensions, etc) _____ Includes most needed formula(s) _____Solution is nearly correct WITH labels _____ Uses complete sentences with some mechanical errors _____ Word problems have some mechanical errors, but are understandable, and applicable to real-world scenarios

2 points _____ Attempts to explain the steps for solving the problem _____ Shows some work _____ Uses 3 of the 5 step problem solving process _____ Includes a diagram which does may not correspond to the problem _____ Includes some formula(s), however, some may be wrong _____ Solution is nearly correct WITHOUT labels _____ Uses some sentence fragments with some mechanical errors _____ Word problems have errors, are difficult to understand, but an attempt at applicable realworld scenarios is made

1 point _____ Does not attempt to explain the steps of the solving the problem _____ Shows no work _____ Does not use the 5 step problem solving process _____ No diagram included ÂŠ 2008 Transformation 2013

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_____ No formula(s) are uses _____ Solution has many errors _____ Uses many sentence fragments and mechanical errors _____ Word problems are full of errors, not understandable, and are not applicable to real-world scenarios Adapted from: http://traverschool.badger.k12.wi.us/jennifer%20schaible/wordproblemrubric.html

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http://www.nden.k12.wi.us/tlcf/prob3.htm MATH PROBLEM-SOLVING RUBRIC Given a story problem, the student will use appropriate strategy to find the solution and use graphs and/or tables to illustrate and explain. MINIMAL(1)

BASIC(2)

PROFICIENT(3)

ADVANCED(4)

Understands The Problem

Student can't read Student can problem- not sure read problemwhat to do. can pull out numbers, not sure what to do.

Student can read problem, pulls out numbers and operation may or may not be correct.

Student reads problem, pulls out numbers and uses correct operation.

Computation

Pulls out numbers which may or may not be correct.

Pulls out numbers, operation and answer may or may not be correct.

Pulls out numbers operation correct, answer may or may not be correct.

Pulls out numbers, operation and answer are correct.

Use of Tables/Graphs

Unsure of numbers involved, can't demonstrate how to use in table/graph.

Knows numbers involved, unsure of table /graph to use.

Knows numbers Knows numbers involved and put into involved and table/graph correctly. table/graph to use, may or may not be completed correctly.

Explanation

Cannot explain why he/she did what they did.

Unclear, not concise, or incomplete.

Clear with some appropriate math language, not concise.

Clear, concise, and uses appropriate math language.

Satisfies All Requirements

Does not satisfy requirements of the problem.

Satisfies all requirements of problem, answer may or may not be correct.

Satisfies all requirements of the problem, answer partially correct.

Satisfies all requirements of the problem with correct answer.

SCORE TOTAL _________ Scoring Process Advanced: 19 -20 Proficient: 15 -18 No score of 2 in any area.

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Basic: 10-14, no more than one area with a score of 1. Minimal: score<10

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Story Board 

Week 1 Activities

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Day 1 Engage: Introduction of DC, Shodor website activities, BrainPOP website activities (45 minutes) Explore: 3 problem activity (45 minutes)

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Day 2 Explain: Present findings from Explore Activity, Interactive Discussion (30 minutes) Elaborate: Real-world examples, Exemplar Word Problem (60 minutes)

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Day 3 Elaborate: Students write 5 word problems to be turned in by the end of class (90 minutes) Teacher works all word problems and provides feedback to student groups next class

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Day 4 Elaborate: Students use teacher feedback to improve word problems (45 minutes) Evaluate: Students create posters of their word problems to post throughout classroom (45 minutes)

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Day 5 Evaluate: Students solve assigned word problems, make presentations incorporating flow charts and evaluation of the word problems they were assigned (90 minutes)

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