When Paths Meet by Jason Kozel

Transformation 2013 Design Challenge Planning Form Guide Design Challenge Title: When Paths Meet Teacher: Bonnie McClung School: Transformation 2013 T-STEM Center Subject: Algebra 1 Abstract: In this design challenge, students will learn to solve systems of equations graphically, analytically, and with tables. They will also produce a set of their own systems of equations modeling a real-life situation.

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 will develop a system of linear equations and linear inequalities and understand how this concept can be applied to the real world. TEKS/SEs that students will learn in the design challenge: (A.6) Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to: (A) develop the concept of slope as rate of change and determine slopes from graphs, tables, and algebraic representations; (D) graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept. (A.8) Linear functions. The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: (A) analyze situations and formulate systems of linear equations in two unknowns to solve problems; (B) solve systems of linear equations using concrete models, graphs, tables, and algebraic methods; and (C) interpret and determine the reasonableness of solutions to systems of linear equations. Key performance indicators students will develop in this design challenge: Develop the concept of slope as a rate of change; determine slope from graphs, tables, and algebraic representations; graph and write linear equations using two points, a point and a slope, or a slope and y-intercept; analyze situations involving two unknowns; formulate systems of linear equations in two unknowns; solve systems of linear equations using different representations; determine the reasonableness of solutions

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

This challenge is built so it incorporates relevance to designs used in the real world in applications that involve tracking and locating objects or people.

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The Design Challenge You and your brother are going to spend a few days with your grandfather, affectionately called “Mister”, who was a civil engineer. You have been allowed to also bring two friends. “Mister” welcomed you and announced that he has sprung for some very special activities the following day. However, there is a catch. He wants you and your friends to discover the surprises he has in store and gives you some clues to find the surprises. Mister owns a 2-acre farm which is approximately 400 ft. by 400 ft. His house is built 120 ft from the lower boundary of the property and he has a fence located 200 ft from the boundary. He keeps a few horses on that part of the property. He tells the boys that he began at the fence and walked at about a 15o angle from the fence toward the magic “spot”. At the same time, Grandmom began at the house and walked at about a 10o angle toward the mark. When they met, he hid the surprises he has in store for you. Begin the search and find the surprise Mister has for you and your friends.

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Map the Design Challenge Already Learned

Performance Indicators

Using a protractor to measure an angle

Taught before the project

Taught during the project

2. Using a ruler to take measurements and approximate the slope of a line

3. Draw a graph of the situation indicating correct x- and y-intercepts

4. Shade the region (feasibility region) in which the surprise may possibly be

5. Find the equation of the two lines represented by the problem

6. Use a graphing calculator to represent the equations

8. Use a table to determine the approximate value of x such that the two functions are equal

9. Solve a system of equations algebraically

10. Interpret the results in context of the problem

7. State the domain and range of the function and use it to find a suitable window to represent the problem

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Team-Building Activity It is important that teachers provide team-building activities for students to help build the 21st Century Skills that are necessary for success in the workforce. Team-building helps establish and develop a greater sense of cooperation and trust among team members, helps students adapt to new group requirements so that they can get along well in a new group, serves to bring out the strengths of the individuals, helps identify roles when working together, and leads to effective collaboration and communication among team members so that they function as an efficient, productive group. Our students are often not taught how to work in groups, yet we assume that they automatically know how. Use team-building activities with your students so that you can see the benefits which include improvement in planning skills, problem solving skills, decision making skills, time management skills, personal confidence, and motivation and morale. Tower of Power Team Building Activity

Purpose: The purpose of this activity is to get the participants to work together as teams to accomplish a timed task. They will need to focus on brainstorming skills, communication skills, and engineering design skills as well. The participants will then reflect on their participation in a teamwork setting. Group Size: 3 to 4 participants (ideal is 4) Materials:  100 3x5 index cards per group  Small stuffed animal to serve as the artifact  Meter stick  Stopwatch or watch with a second hand Procedures: The workshop facilitator will set the scene: “You work for the Boston Museum of Science and have been asked to design a tower that will display an ancient artifact that must be 30 inches tall when sitting on a table. Each group will be given 100 3x5 index cards to use to design a prototype for the actual tower. The tower must be able to support the weight of the “ancient artifact” (hold up the small, stuffed animal) for a minimum of 10 seconds. You will be given 15 minutes to complete the design challenge, and, upon completion, a member of your group must test the design for the rest of the participants. Are there any questions?” ***The participants may tear the cards, but they are not allowed to use scissors, tape, etc. They must use only the index cards to complete the challenge.***

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5E Lesson Plan Design Challenge Title: Input Design Challenge title here TEKS/TAKS objectives: (From Step 1, Step 2) Engage Activity Teacher will identify the concept to be explored through asking a leading question such as “Two planes are headed toward the same location on different paths—what would happen if something is not done?” Draw a picture of two airplanes on different paths or let students use a vector simulation activity located at www.mped.org. Enter keyword(s): vector investigation. Warm up activity: (see cards below): Give each student a linear equation to solve (make two to four depending upon the size of the class). Have students work the equations and then play “I have—who has” to match up answers to the equations. Explain that this forms the group they will work with on the focus activity. Introduce the students to the design challenge: “You and your brother are going to spend a few days with your grandfather, affectionately called “Mister”, who was a civil engineer. You have been allowed to also bring two friends. “Mister” welcomed you and announced that he has sprung for some very special activities the following day. However, there is a catch. He wants you and your friends to discover the surprises he has in store and gives you some clues to find the surprises. Mister owns a 2-acre farm which is approximately 400 ft. by 400 ft. His house is built 120 ft from the lower boundary of the property and he has a fence located 200 ft from the boundary. He keeps a few horses on that part of the property. He tells the boys that he began at the fence and walked at about a 15o angle from the fence toward the magic “spot”. At the same time, Grandmom began at the house and walked at about a 10o angle toward the mark. When they met, he hid the surprises he has in store for you. Begin the search and find the surprise Mister has for you and your friends.” Before we get started with the design challenge, let’s work a practice problem to deepen our understanding of word problems and systems of linear equations.

Engage Activity Products and Artifacts Students will answer the following questions in their math journals:  Describe two major approaches to solving systems of equations that we have discussed in class  Write a problem demonstrating each of the approaches  What relationship did you observe about the velocity of the planes in the activity

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to the slope of a line? Students will solve the problems provided in the warm-up. Engage Activity Materials/Equipment Computer, projector, screen, cards for the warm-up, chart paper/chalk board/dry marker board Engage Activity Resources www.mped.org Explore Activity Give the following problem to the groups and provide them with graph paper, ruler, and a graphing calculator. An antique car show was in town for five days and the small auditorium would hold a capacity of 700 people before the fire marshals closed the doors. It has been estimated that about 50 people were in line each day at the opening time of 10:00 a.m. The show closes after 8 hours and, even though traffic tends to slow down in the afternoon, it is estimated that people enter the show at an average rate of 80 per hour. If people did not leave, the small auditorium could not hold all of the people. So the director did some calculations and figured the model showing people leaving to be -20t + 250. These cars are very valuable and the director of the event wants maximum security for the peak crowds. Your task is to determine at what time the number of people entering and the number of people leaving is exactly the same. In executing your task you will provide the following information.  A linear equation that represents people entering the event. Enter this as Y1 in your graphing calculator  A linear equation that represents people leaving the event. Enter this as Y2 in your graphing calculator  Make a table of values showing the number of people entering at each hour interval and the number of people leaving at each out interval  Draw a graph of the situation and find the intersection point algebraically  Interpret the meaning of the point in the context of the problem (teacher should be a facilitator during the process and pose the question “How much time passes before some of the people begin to leave?”)  Interpret the meaning of the slope in the model of people leaving the event (this is a good teaching moment—students with the DEEP understanding will conclude that nobody leaves the first two hours and this helps distinguish a 4 from a 5 on the rubric)  Interpret the meaning of the y-intercept in the model of people leaving the event.

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The directors of the car show only travel to mid-size cities so they don’t have to adjust their plans each time the show closes and then moves to another city. What considerations would have to be made if they did not have this restriction? Write a complete paragraph explaining your design for this direction. Presentation of Problem (Question: Did the student go above and beyond in the presentation and clarity of presenting his problem and solution—again, the difference between a 4 and a 5 on the scoring rubric) Lead the students into a discussion involving curves other than straight lines and give some examples, particularly quadratic and exponential curves, and show how they also intersect.

Explore Activity Products and Artifacts Equations representing the problem situation, Graph of the problem, Algebraic solution, Interpretation of solution as a journal entry

Explore Activity Materials/Equipment Graph paper, ruler, graphing calculator, journal, pencil, map pencils (optional) Explore Activity Resources None Explain Activity Have the students present their findings from the explore activity to the rest of the class. Focus on the vocabulary and content that are evolving from the presentations. Develop a word wall with key vocabulary so that the students can refer back to it for the remainder of the unit. Clear up any misunderstandings and maybe work through the problem after all groups have presented their findings to bring closure to the problem (do this only if all groups were unsuccessful reaching a common conclusion...if only one of two groups are unsuccessful, host a “workshop” for those students and provide them with additional support). Have students get in their groups that were formed the previous activity. Give each student a copy of the “Estimating the Slope of a Line” handout and follow the directions to estimate the slope of a line through measurement. Give each student several sheets of blank graph paper in case of mistakes. They will also need colored pens, large chart paper, rulers, and protractors. Pass out prepared worksheets that have a line drawn (example below) and make an overhead transparency for you to demonstrate how to

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estimate slope using a ruler and emphasize that slope is a rate of change. The worksheet should have both negative and positive slopes. Proceed with directions as to estimating slope using a ruler. Choose a beginning point on the demonstration line and measure a horizontal length of your choice. Place a second point on the ending spot. Then measure from that spot vertically to the given line and record that length. Students should be recording all of the information in their math journal which will be a part of the evaluation. Questions that could be asked are: 1. What is your known formula for slope? 2. Tell me other ways of describing slope (rise/run, change in y over change in x, etc. ) 3. Which measurement that we took is the rise? Which is the run? 4. What would our approximate slope be? 5. Is it a positive or negative slope? Why? (because it rises/falls) Clear up any misconceptions you think they might have at this point. Explain Activity Products and Artifacts Presentation of explore activity, Estimating the Slope of a Line handout, journal entry Explain Activity Materials/Equipment Estimating Slope handout, pencil, ruler, graph paper, colored pens, large chart paper, transparency, protractors Explain Activity Resources Teacher may want to refer to http://www.mi.sanu.ac.yu/vismath/gompa/index.html for a refresher on using a ruler to estimate slope

Elaborate Activity

Present the students with the design challenge: Pass out copies of the challenge and graph paper. Ask the students to read the challenge together and make notes of key information. Once they have done this, walk through the steps to find the surprises. 1. On your graph paper, make a grid that represents 400 by 400â&#x20AC;&#x201D;you might need to count by 10â&#x20AC;&#x2122;s to do this. 2. From (0,0) , draw a horizontal and a vertical line to the points (0,400) and (400,0). Connect the lines to make a square representing the plot of land. 3. Mark the house at the point (0, 120). (Debrief to see if students understand why) 4. Draw a fence horizontally halfway on the property. What coordinates will be used? 5. From the point (0, 200), use your protractor and measure a 15o angle away from the fence. Draw a line across your grid with this angle. 6. From the point (0, 120), use your protractor and measure a 10o angle toward the fence. ÂŠ 2008 Transformation 2013

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Draw a line across your grid with this angle. 7. Take your rulers and estimate the slopes of your lines just as we did in the previous activity. 8. Using your slope and y-intercept, write equations representing your two lines. 9. Lightly shade the region in which the surprise could be located according to the information in the problem. 10. Put the two equations into your graphing calculator and , using the dimensions of the farm, graph the system of equations. 11. Using the calculator’s trace feature, trace to approximate the intersection point. Then use CALC—intersection to find the actual point (Your solution may not be the same as all groups since the slopes may differ somewhat). 12. Explain how a table of values could help them find the approximate location of the surprise. 13. Solve the system of equations algebraically. 14. Discuss approaches used to solve the problem. Elaborate Activity Products and Artifacts Solutions to the design challenge, Journal entry regarding the approach used to solve the problem Elaborate Activity Materials/Equipment Graphing calculators, large chart paper, graph paper, colored pens, rulers, protractors, journals Elaborate Activity Resources None Evaluate Activity

After debriefing the students, have each group list 5-10 instances in which solutions of this nature would be found in real-life. Take time to discuss these briefly and record them on the board for all students to see. Lead a discussion in which the students talk about the similarities and differences in the types of problems. Then, using these instances as guides, have each group write two real-life problems that have to be solved using a system of equations. They may use chart paper, power point, or handouts for this part of the problem. Once they have written the problems, ask them to solve both of their problems, too. Check their problems and solutions for accuracy. Have the students post their problems in the room and assign each group to a new problem set. Ask the groups to evaluate the problems they were assigned using a rubric And have them solve each of the assigned problems. Ask the students to document the

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problem solving process in their journals. Have each group display their solutions to the assigned problems. The students must show their work and detail the steps they took to solve the assigned problems. Ask each group to talk about the problems they solved, what they did to solve the problems (key information, method used, etc.), and how they determined the final solution. Probe the students regarding the reasonableness of their solutions. Ask the students to turn in their evaluation of the problems that they worked to you. Evaluate Activity Products and Artifacts Two real-world word problems involving the use of systems of equations with solutions, Solutions and evaluations of the assigned problems, Journal entry of problem solving process, Presentation of their solutions, Peer evaluations Evaluate Activity Materials/Equipment Graphing calculators, large chart paper, graph paper, colored pens, rulers, protractors, computers, projector, PowerPoint, copy machine, journals Evaluate Activity Resources None

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Solve the Equations (Engage Warm-Up Activity) Make cards with the following problems (these will form four groups of three each).

6x + 3 – 8x = 13

9 = 6 – (x + 2)

2.4 = 3 ( m+4)

-3x – 8 + 4x = 17

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Estimating the Slope of a Line Place point A at the beginning of the line and point B at the end of the line. Place your ruler at point A and draw a horizontal line across the page. Measure one inch on the horizontal line and mark it as point C. Then measure another inch and mark it point E. From point C measure vertically to the line AB. Call the point of intersection D. From point E measure vertically to the line AB. Call the point of intersection F. Use your ruler to measure the length of CD and the length of EF (both in inches) Make a fraction from the measurement CD/1 and EF/1â&#x20AC;&#x201D;these two fractions should be the same and be a good approximation of your slope. Is your slope positive or negative? Explain why.

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Plan the Assessment Engage Artifact(s)/Product(s): Journal reflection, Solutions to warm-up activity

Explore Artifact(s)/Product(s): Equations representing the problem situation, Graph of the problem, Algebraic solution, Interpretation of solution as a journal entry

Explain Artifact(s)/Product(s): Presentation of explore activity, Estimating Slope handout, journal entry

Elaborate Artifact(s)/Product(s): Solutions to the design challenge, Journal entry regarding the approach used to solve the problem

Evaluate Artifact(s)/Product(s): Two real-world word problems involving the use of systems of equations with solutions, Solutions and evaluations of the assigned problems, Journal entry of problem solving process, Presentation of their solutions, Peer evaluations

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Rubric Systems of Equations Level

Understanding

Strategies/reasoning

Communication

+There is no solution + The solution does not address the mathematical components presented in the task

+ No strategy to help solve the problem + No evidence of mathematical reasoning + Errors in the mathematical procedures that cannot be resolved

+ Explanation does not relate to task assigned + No use or inappropriate use of graphs, tables, other representation + No use, or mostly inappropriate use, of mathematical notation

+The solution is not complete indicating that parts of the problem are not understood +The solution addresses some, but not all of the components are presented in the task

+Uses a strategy that is partially +Incomplete explanation, it useful, leading some way toward may not be clearly presented a solution, but not a full solution +Some use of appropriate +Some evidence of mathematical mathematical representation reasoning + Some use of mathematical +Could not completely carry out terminology and notation mathematical procedures appropriate of the problem +Some parts may be correct, but a correct answer is not achieved

+The solution shows that the has a broad understanding of the problem and the major concepts necessary for its solution +The solution addresses all components presented in the task

+Uses a strategy that leads to a solution of the problem +Uses effective mathematical procedures +All parts are correct and a correct answer is achieved

+There is a clear explanation +There is appropriate use of accurate mathematical representation +There is effective use of mathematical terminology and notation

+The solution shows a deep understanding of the problem including the ability to identify appropriate mathematical concepts and the information necessary for its solution +The solution completely addresses all mathematical components presented in the task +The solution puts to use the underlying mathematical concepts upon which the task is designed

+Uses a very efficient and sophisticated strategy leading to a solution +Employs refined and complex reasoning +Applies procedures accurately to correctly solve the problem and verify the results +Verifies solution and/or evaluates the reasonableness of the solution +Makes mathematically relevant observations and/or connections

+There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made +Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem +There is precise and appropriate use of mathematical terminology and notation.

2-3 = 70% to 80% 3-4 = 80% to 90% 4-5 = 90% to 100%

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Story Board 

Week 1 Activities

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Week 2 Activities

Day 1 TeamBuilding Activity (20 minutes) Engage (45 minutes) Explore (25 minutes) Day 6 Elaborate (90 minutes)

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Day 2 Explore (65 minutes) Explain (25 minutes)

Day 7 Evaluate (90 minutes)

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Day 3 Explain (65 minutes)

Day 8

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Day 4 Elaborate (90 minutes)

Day 9

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Day 5 Elaborate (90 minutes)

Day 10

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