Powerful Stuff by Jason Kozel

Transformation 2013 Design Challenge Planning Form Guide Design Challenge Title: “Powerful” Stuff Teacher(s): Bonnie McClung School: Transformation 2013 T-STEM Center Subject: Algebra 1 Abstract: Exponential notation and working with polynomial functions is a very difficult concept to make relevant and interesting to the Algebra I student. This PBL unit will supply practice and relevancy so the student can continue with quadratic factoring and application problems.

MEETING THE NEEDS OF STEM EDUCATION THROUGH DESIGN CHALLENGES

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Begin with the End in Mind The theme or “big ideas” for this design challenge: Students need to learn that polynomials can represent actual “things” in real life. This is one of the hardest topics to teach with relevancy so this design challenge is intended to show the student that there are valid reasons for studying polynomials. TEKS/SEs that students will learn in the design challenge: (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: (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. (A.4) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to: (A) find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations; (B) use the commutative, associative, and distributive properties to simplify algebraic expressions (A.11) Quadratic and other nonlinear functions. The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. The student is expected to:

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(A) use patterns to generate the laws of exponents and apply them in problemsolving situations; (C) analyze data and represent situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods. Key performance indicators students will develop in this design challenge: Use patterns to determine rules of exponents, and apply rules of exponents to solve problem situations; Simplify polynomial expressions by applying the properties of algebra and using addition, subtraction, and multiplication; Apply polynomial operations to solve problem situations; Factor polynomial expressions concretely and by patterns; Verify multiplication; Apply factoring of polynomials to problem situations 21st century skills that students will practice in this design challenge: www.21stcenturyskills.org Articulating thoughts and ideas clearly; Demonstrating ability to work with others in a group to complete a project; Assuming shared responsibility in group work STEM career connections and real world applications of content learned in this design challenge:

Students will see the connection between exponents and the medical field, rocketry, and package design. They will use technology to practice skills (computer and graphing calculator) and deepen their understanding of the real world applications of polynomials.

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The Design Challenge You and your team have been hired by Mona Nomial’s Packaging Company to construct an open box with a square base that will serve to transport plants from the owner’s husband’s nursery, Pauly Nomial’s Trees & More, to customer’s homes. Because the Nomials are environmentally conscious, they would like the boxes to double as planters, so you will be constructing the open box out of 24 inch square metal. The design needs to yield a maximum volume. Knowing that you will be working with exponents, you must refresh yourselves on the mathematics involved with exponential expressions and functions before you can begin tackling the design. Mona and Pauly are sticklers for accurate mathematical calculations, so you must run your calculations and a scale drawing by them before they will give you materials for creating a prototype of your design. Upon completion, your team must formally present your finished product to the Nomials.

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Map the Design Challenge Taught before the project

Taught during the project

2. Understand the vocabulary of polynomials (monomial, binomial, trinomial, polynomial)

3. Understand how to combine like-terms when adding/subtracting polynomials

Already Learned

Performance Indicators

1. Apply the laws of exponents

4. Understand how to establish a viewing window in the graphing calculator by interpreting the problem situation

5. Solve problems involving polynomials

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5E Lesson Plan Design Challenge Title: “Powerful” Stuff TEKS/TAKS objectives: A.1B,C,D,E; A.4A,B; A.11A,C Engage Activity Students should be introduced to the laws of exponents prior to beginning this PBL. Begin by reviewing the laws of exponents with the students, and share with them that even though it seems like exponents is a subject that has no relevancy to the real-world, the contrary is true. Introduce them to the following scenario and guide them through the activity. Influenza (commonly know as the flu) isn’t always thought of as a serious or lifethreatening illness; however, the dangers and complications it can have on older adults, children, and people with health problems is quickly changing that perception. Each year in the United States, an estimated 25-50 million cases of flu are reported, leading to 150,000 hospitalizations and 30,000-40,000 deaths. If we were to look at the world statistics, we would see an estimated 1 billion cases of flu resulting in 3-5 million cases of severe illness and 300,000-500,000 deaths. The spread of flu is exponential. When infected people cough or sneeze, they spray drops of the virus into the air. When others breathe in these drops or come in contact with something that the drops land on, they are likely to become sick, too. Just to show you how quickly the virus spreads, we are going to simulate the spread of the virus. Follow these directions to conduct the simulation:  Draw the following table on the board: Time (Days) 0 1 2 3 4 5 6 7 8 9 10 11 Number of Sick People  The time (“days”) will be considered “fast” days. The teacher announces “time” to fast-forward to the next time interval.  At time = 0, choose 1 person in the class to be sick, and give this person a large piece of clay. Everyone else in the class is healthy, but has a 100% chance of getting sick if a sick person gives them the virus.  At time = 1, the sick person will split the clay into two halves. He or she will keep one half and give the other half to someone in the class. This person is sick, so at time = 1, number sick = 2. Keep track of the number sick on the board.  Emphasize that students must accept the clay. They do not have the choice to refuse.  At time = 2, both sick people will split their clay in half and give one half away to infect someone new, so at time = 2, number sick = 4.  Continue this process until everyone in the class is sick. If you want to keep the © 2008 Transformation 2013

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 

simulation going one time interval longer, tell all sick students that their right hands represent a sick person and their left hands represent a healthy person. Their right hands can infect their left hands. Debrief with the students regarding the exercise. Ask the class to note any patterns in the growing number of sick people in the table. Ask them how the pattern will continue. Have the students copy the table into their journal and reflect on the pattern. Also, ask the students to represent the pattern they found using multiple representations (pictures, symbols, variables, sketch). Have them predict how many people will be sick when t=20 and ask them to explain how they determined their prediction. Engage Activity Products and Artifacts

Participation in the virus simulation, Reflection in journal Engage Activity Materials/Equipment 1 lump of clay, notes from Engage Activity, journal, pencil, chalkboard/dry erase board, chalk/dry erase markers Engage Activity Resources http://www.flufacts.com/impact/statistics.aspx Building Math: Amazon Mission by Dr. Peter Wong and Dr. Barbara M. Brizuela Explore Activity Prior to this activity, have the students write their names on craft sticks and draw sticks to determine groups. Please note that you are really in control of the how groups are assigned…the students really don’t know whose stick has been drawn, so if you see that the groups are unbalanced, feel free to mix them up for the benefit of the class. Introduce the students to the design challenge: “You and your team have been hired by Mona Nomial’s Packaging Company to construct an open box with a square base that will serve to transport plants from the owner’s husband’s nursery, Pauly Nomial’s Trees & More, to customer’s homes. Because the Nomials are environmentally conscious, they would like the boxes to double as planters, so you will be constructing the open box out of 24 inch square metal. The design needs to yield a maximum volume. Knowing that you will be working with exponents, you must refresh yourselves on the mathematics involved with exponential expressions and functions before you can begin

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tackling the design. Mona and Pauly are sticklers for accurate mathematical calculations, so you must run your calculations and a scale drawing by them before they will give you materials for creating a prototype of your design. Upon completion, your team must formally present your finished product to the Nomials.” Have the students use the “Explorations” worksheet (Feel free to add additional problems to support this exercise) below to expose them to additional real-world examples, solidify the concepts involving exponents, and introduce them to the use of the graphing calculator to solve exponential problems. Upon completion of the activity, have the students reflect in their journals regarding the following:  What values determine the WINDOW for your each of the problems?  What are the values of X called?  What are the values of Y called?  Solutions to the problems  Correct window settings for each of the problems  Patterns seen within each of the problems  Solution to the bacteria problem after 4 hours The word “polynomial” has been used. In your own words, write the meaning of each of the following:  monomial  binomial  trinomial  polynomial Explore Activity Products and Artifacts Explorations worksheet, journal reflections Explore Activity Materials/Equipment Explorations worksheet, journal, pencil, graphing calculator Explore Activity Resources None Explain Activity Upon completion of the “Explorations” activity, have each group post their results around

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the classroom using large chart paper. Provide the students with an opportunity to do a “Gallery Walk” around the classroom to note the similarities and differences in each teams’ work in their journals. Debrief the “Gallery Walk” and clear up any misunderstandings from the “Explorations” activity. Now, we will expand our work with exponents to include adding and subtracting polynomials. Give them several examples of polynomials. Ask the students to brainstorm and come up with things that they have in common (eye color, hair color, age, sex, etc.) and place these on large chart paper, projector, or board as they come up with commonalities. Tell them that polynomials also share things that are alike and give them several examples (6x3, -5x3, 40x3...). Ask them to describe what the polynomial has in common. At this time, reinforce the vocabulary by having the students complete a flow chart that represents the “members” of the polynomial family and give examples of each (see attached scheme). Explain that the only way to combine terms is if their exponents are exactly alike. Make a wall hanging of the flow chart. Debrief with the students regarding the activity and have them reflect in their journals regarding the following:  List things you had in common with the class.  Did ALL students share each trait?  Give five examples of polynomials that are alike. If students have already made Algebra Tiles from a previous unit, they can use them. If not, have the students make tiles or take them to the computer lab to complete the activity from the following site: http://www.regentsprep.org/regents.cfm http://www.regentsprep.org/Regents/math/polyadd/Tadd_til.htm http://www.regentsprep.org/Regents/math/polyadd/sprac_a.htm These sites have activities that involve Algebra Tiles and operations with polynomials. If, for some reason, you do not have access to a computer lab, you can use the site for demonstration or giving the students instruction on using the tiles. Tiles offer a great way to visualize otherwise difficult and abstract concepts such as operations with polynomials. They are excellent for all learners. Upon successful completion of the activity, give the students a handout of problems that involve adding and subtracting polynomials. Give them time to work within their groups to solve the problems. Sample handout is attached. When students have completed the handout, have them reflect in their journals regarding the following:  List at least three things you learned about adding and subtracting polynomials from using Algebra Tiles.  What problems did you encounter as you solved the handout problems?  Solutions to handout problems  Flow chart of vocabulary

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Explain Activity Products and Artifacts Journal entries from Gallery Walk, classroom activity, and handout; Algebra Tiles activity; Adding & Subtracting Polynomials handout Explain Activity Materials/Equipment Large chart paper, markers, journal, pencil, computers with internet access, card stock, scissors, projector (optional) Explain Activity Resources http://www.regentsprep.org/regents.cfm http://www.regentsprep.org/Regents/math/polyadd/Tadd_til.htm http://www.regentsprep.org/Regents/math/polyadd/sprac_a.htm

Elaborate Activity

Debrief with the students regarding the concepts that have been studied up to this point and present them with a review problem before allowing them to solve the design challenge. “Athletes are trying out for the Summer Olympics right now, and diving is one of the most popular Olympic sports to watch. A diver is standing on the platform which is about 33 feet above the water. He/she performs a dive modeled by the polynomial -16t2 + 33 where t is in seconds. How far above the water is the diver after 1.2 seconds? How long will it take the diver to hit the water? Draw a graph showing the path of the diver and include the window you used.” Check the progress of the students and continue when you are assured they have learned to use the calculator effectively. Re-introduce the design challenge: “You and your team have been hired by Mona Nomial’s Packaging Company to construct an open box with a square base that will serve to transport plants from the owner’s husband’s nursery, Pauly Nomial’s Trees & More, to customer’s homes. Because the Nomials are environmentally conscious, they would like the boxes to double as planters, so you will be constructing the open box out of 24 inch square metal. The design needs to yield a maximum volume. Knowing that you will be working with exponents, you must refresh yourselves on the

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mathematics involved with exponential expressions and functions before you can begin tackling the design. Mona and Pauly are sticklers for accurate mathematical calculations, so you must run your calculations and a scale drawing by them before they will give you materials for creating a prototype of your design. Upon completion, your team must formally present your finished product to the Nomials.” You have already spent quite a bit of time refreshing yourselves on the mathematics involved with exponents, so it’s time to tackle the challenge. You don’t want to work with metal and waste it so your team will use card stock to experiment with. Your model does not have to be 24” x 24”, but you are expected to have a scale model, so make sure that you calculate the sides and height of the box appropriately. You will cut identical squares from each corner of the square card stock. The volume of the box in cubic inches can be represented by the polynomial: 4x3 – 96x2 + 576x where x is the side length of the missing squares. As you construct your box, answer the following questions in your journal:  What is the volume if the length of the square you cut out is 1 in.?  What is the volume if the length of the square you cut out is 3 in.?  What is the volume of the box if the length of the square you cut out is 12 in?  Considering your answer when you cut out a square of length 12 in., does this make sense?  Draw a graph of the polynomial that represents the volume and collaborate with your group to determine a good viewing window. Include this viewing window in the write-up of your design.  Use a table on your graphing calculators to approximate the size of the square that needs to be removed in order to have the maximum volume.  Use the TRACE function on your calculator and trace to the highest point on your graph. Is it close to the same number you approximated from the Table?  Now use TRACE on your calculator and trace until you are close to the highest point. Press CALC: 4 on your calculator and make sure your cursor is to the left of the highest point. Press ENTER for the left bound. Continue tracing until you are past the highest point and press ENTER again for the right bound. Press ENTER a third time and this will give you the length “x” that needs to be cut out in order to have a maximum volume.  Document the scale factor that you will use when creating your prototype out of an 8.5” x 11” piece of cardstock.  Check your group’s calculations with your teacher before moving on.  Use the cardstock to create a prototype of your design. Feel free to decorate the outside of the box for bonus points.  On the inside of your cardstock, write the dimensions of your length, width, and height, and show the calculation for volume on the base of the box. Also, show the scale factor.  What is the volume of the prototype? © 2008 Transformation 2013

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Teacher will act as facilitator during the Elaborate Activity making sure the students stay on task and all are participating. Some may get frustrated trying to get a good viewing window and will need help. They will not be expected to find the exact maximum volume but should be able to approximate from the table of Y values. Elaborate Activity Products and Artifacts Journal entry, prototype Elaborate Activity Materials/Equipment Journal, pencil, graphing calculator, card stock, ruler, scissors, tape, markers Elaborate Activity Resources None Evaluate Activity Groups will detail the work that they did during the Elaborate Activity on large chart paper. They will present their finished product to the class, making sure that they discuss the answers posed during the Journal Reflection Activity. Use the Rubric below to grade the studentsâ&#x20AC;&#x2122; performance. Evaluate Activity Products and Artifacts Presentation Evaluate Activity Materials/Equipment Large chart paper, markers, prototype from Elaborate Activity Evaluate Activity Resources None

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Explorations 1. A heavy object has been tossed from the top of a building 200 feet tall and you are standing directly below. The path the object follows is given by –16t2 + 200 where t is in seconds. How far above you is the object after 2 seconds? How many seconds do you have to avoid being hit by the object? Using your graphing calculator  Enter the polynomial in Y1  Set the WINDOW as follows XMIN 0 XMAX 4 XScale 1 YMIN 0 YMAX 220 YScale 1  Graph the polynomial  Using CALC enter an X value of 2 and record your answer  Trace until you get close to Y=0. Using CALC: 2, ENTER a value above the x-axis and then TRACE until you are below the x-axis. Enter that number and press ENTER again. You will get a guess and when you press ENTER a second time that value will be your x-intercept.  This value of x tells you how long you have before the object will hit you. 2. A toy rocket is launched from a platform that is 6 feet above the ground at a rate of 90 feet per second. The polynomial -16t2 + 90t +6 gives the rocket’s height in feet after t seconds. Make a table showing the rocket’s height after 1 second, 2 seconds, 3 seconds, and 4 seconds. At which of these times will the rocket be the highest? What is an appropriate adjustment for the calculator’s window settings? When does the rocket hit the ground after being launched?

3. A biologist notices that a certain bacterium he found on a slide in the park doubles every 10 minutes. If there was one bacterium on the slide to begin with, how many would there be one hour from now? Set up a table to solve the problem and see if you can develop a pattern that will find out how many bacteria are present 4 hours from now.

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Flowchart: polynomials

monomial

binomial

trinomial

Sample box for design challenge with squares of length “x” cut out:

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Explain Handout Problems

(12m3 + 4 n2 – 3n) + (3m3 -5n2 + 4n – 8)

(5p4 – 3p3 + 7p2 – p + 10) – (2p4 + 5p3 – 9p2 – 2p – 1)

(6y3 + 4y2) – (7y2 + 10y)

(- 9t3 + 8t2 – 7t + 15) + (-4t3 – 17t2 – 3)

(15p5 – 2p4 + p2 – 12) + ( -13p5 – 3p3 + 8p2 – p – 13)

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Plan the Assessment Engage Artifact(s)/Product(s): Participation in the virus simulation, Reflection in journal

Explore Artifact(s)/Product(s): Explorations worksheet, journal reflections

Explain Artifact(s)/Product(s): Journal entries from Gallery Walk, classroom activity, and handout; Algebra Tiles activity; Adding & Subtracting Polynomials handout

Elaborate Artifact(s)/Product(s): Journal entry, prototype

Evaluate Artifact(s)/Product(s): Presentation

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Rubrics 20 points each Student demonstrates understanding of like terms. Student can perform algebra of adding/subtracting polynomials with no mistakes. Student relates understanding of the domain to X-Min and X-Max calculator values and understanding of the range to Y-Min and Y-Max calculator values and demonstrates mastery of adjusting the calculator’s window settings.

Student was on task throughout the project and did a great job redirecting team mates when they were off task. Student was extremely comfortable presenting in front of the class, maintained eye contact with the audience, spoke clearly, and used the correct terminology during his/her portion of the presentation.

17 points each Student demonstrates understanding of like terms with some guidance. Student can perform algebra of adding/subtracting polynomials with few mistakes. Student relates understanding of the domain to X-Min and X-Max calculator values and understanding of the range to Y-Min and Y-Max calculator values but needs limited guidance adjusting the calculator’s window settings.

14 points each Student indicates no understanding of like terms.

Student was on task throughout most of the project.

Student can perform algebra of adding/subtracting polynomials but makes mistakes with signed numbers. Student indicates little understanding of the domain and its relationship to X-Min and X-Max calculator values and little understanding of the range and its relationship to Y-Min and Y-Max calculator values and needs a great deal of guidance adjusting the calculator’s window settings. Student was somewhat on task, but had to be redirected several times.

Student was somewhat comfortable presenting in front of the class, made eye contact frequently with the audience, spoke clearly, and used the correct terminology during his/her portion of the presentation.

Student was extremely uncomfortable presenting in front of the class, made no eye contact with the audience, did not speak clearly, and did not use the correct terminology during his/her portion of the presentation.

Provide the students with additional bonus points for artwork that is drawn on the outside of their boxes.

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Story Board  Week 1 Activities

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Day 1 Engage (40 minutes) Explore (50 minutes)

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Day 2 Explain (90 minutes)

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Day 3 Elaborate (90 minutes)

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Day 4 Evaluate (90 minutes)

Day 5

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