Transformation 2013 PBL 5E Planning Form Guide PBL Title: Simply Logical, My Dear Watson Teacher(s): Chris Fancher School: Manor New Tech High School Subject: Geometry Abstract: In this unit, students will explore ideas in logic and will create logic puzzles.
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Begin with the End in Mind The theme or “big ideas� for this PBL: Students will design logic puzzles while looking at the basic structure of logic. TEKS/SEs that students will learn in the PBL: (G.1) Geometric structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to: (A) develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems; (G.3) Geometric structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to: (A) determine the validity of a conditional statement, its converse, inverse, and contrapositive; (C) use logical reasoning to prove statements are true and find counter examples to disprove statements that are false; (D) use inductive reasoning to formulate a conjecture; and (E) use deductive reasoning to prove a statement. Key performance indicators students will develop in this PBL: Develop vocabulary (biconditional statement, conclusion, conditional statement, conjecture, contrapositive, converse, counterexample, deductive reasoning, definition, hypothesis, inductive reasoning, inverse, negation, proof, theorem, truth value, Venn diagram), formulate conjectures, create rules for using deductive reasoning and inductive reasoning, determine validity, write and solve word problems, create rules for making hypotheses and drawing conclusions. 21st century skills that students will practice in this PBL: www.21stcenturyskills.org Critical thinking, problem solving, communication, collaboration
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STEM career connections and real world applications of content learned in this PBL:
Careers: Law, Mathematics, Science, Politics Connections: Many students love to argue about anything and everything. Now, we are arming them with the proper procedures for arguing a point!
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The Problem We, at Puzzles and Games Publishing, need your expertise in the design of our new Logic-Puzzles Magazine. Because we want to ensure our readers are receiving puzzles from the best high school mathematicians, there are some requirements you will have to meet prior to signing your extended contract for a total of 30 puzzles, over a twelve month period. The total financial award is $100 for each completed puzzle. This means that your team could get a contract worth $3000 (or more). Here is what we need from you: 1. A document containing – a. a conditional statement with its hypothesis and conclusion identified. b. the converse, inverse, and contrapositive of the original conditional statement. c. Three examples in advertising, politics, or journalism where conditional statements and their converses and inverses are used to mislead people (sometimes intentionally, sometimes unintentionally) or where a conditional statement is just misinterpreted. Explain your example fully in 3-4 paragraphs. 2. A second document containing two logic puzzles with the following constraints: a. Story-like and based around a topic that you think would be interesting to middle school students. b. Include a key and some instructions and hints for learning to solve these types of puzzles. c. Have equal numbers of people and facts. For example, 3 people with 3 things, each a different color. d. The first must have 3 people as the main characters and the second must add at least one thing‌an additional character or an additional characteristic.
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Map the PBL Taught before the project
Taught during the project
1. Vocabulary: biconditional statement, conclusion, conditional
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statement, conjecture, contrapositive, converse, counterexample, deductive reasoning, definition, hypothesis, inductive reasoning, inverse, negation, proof, theorem, truth value, Venn diagram 2. Create rules for using inductive and deductive reasoning
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3. Create rules for making hypotheses and drawing conclusions
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Already Learned
Performance Indicators
about the hypotheses 4. Write and solve word problems
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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. Cup Stack Team-Building Activity Objectives: Participants work together in teams to accomplish a timed task. Participants practice effective communication skills. Participants reflect on one’s participation in a teamwork setting. Group Size: 3 to 4 participants (ideal is 4 participants) Materials: You will need a watch or clock with a second hand or a timer/stopwatch to time 1 minute 15 seconds. Each team will need 15 foam cups and a rubber band with 4 strings attached like rays of sun.
string
Setup: Cut string into 2-foot lengths. Tie four strings to the rubber band evenly spaced around the circle. It should look like a sun with four rays coming out. Rubber band Divide the cups into stacks of 15. Procedures: Explain to the class that they will participate in a team-building activity that focuses on accomplishing a task and communication. Distribute a set of materials to each team. Explain that the task is to build a pyramid using the cups with a 1 minute 15 second time limit. The pyramid will begin with 5 cups in a row at the base, 4 cups on the next row, 3 cups in the middle row, then 2 and finally 1 cup at the top. Group members may not touch the cups with their hands or any part of their body, even if the cups fall. Each person may only hold the end of one string attached to the rubber band (unless group size is 3 and then one participant may hold 2 strings). Group members must work together to stretch and relax the rubber band to grab each cup and place the cup in the right place. When groups are ready, give them 30 seconds to practice and plan. At the end of 30 seconds, have them disassemble their practice pyramid. When they are ready, start timing 1 minute 15 seconds. When time is up, stop the activity and check each team’s progress. Debrief the activity with these questions: o Was anyone frustrated at all during the activity? If so, how was it handled?
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Why is teamwork so important for this activity? Did any team come up with a strategy for working together as a team? If so, what was the strategy? o Are you ever in a situation where you must use teamwork? Is it always easy for you? Why or why not? o What are some skills needed to be good at teamwork? o How did you contribute to your team? Did you give suggestions? Lead or follow? Encourage or cheer? o How would you do the activity differently if you were asked to do it again? ď ˛ Reset and repeat the activity. Give teams 30 seconds to strategize before starting the time. After the task, debrief with these questions: o Did your teamwork improve this time? How and why did it improve? o Why is good communication important to accomplishing this task? o How would you use this in your classroom, on your campus, or with other teams? o o
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5E Lesson Plan PBL Title: Simply Logical, My Dear Watson TEKS/TAKS objectives: G.1A; G.3A,C,D,E Engage Activity Pose the following questions to your students and have them reflect on them in their journals: Have you ever worked any type of puzzle? What type was it? Were you successful at completing the puzzle? What strategies did you use to solve the puzzle? Follow up the journal reflection with a discussion…have the students share out regarding their reflections. Introduce your students to the challenge: We, at Puzzles and Games Publishing, need your expertise in the design of our new Logic-Puzzles Magazine. Because we want to ensure our readers are receiving puzzles from the best high school mathematicians, there are some requirements you will have to meet prior to signing your extended contract for a total of 30 puzzles, over a twelve month period. The total financial award is $100 for each completed puzzle. This means that your team could get a contract worth $3000 (or more). Here is what we need from you: 3. A document containing – a. a conditional statement with its hypothesis and conclusion identified. b. the converse, inverse, and contrapositive of the original conditional statement. c. Three examples in advertising, politics, or journalism where conditional statements and their converses and inverses are used to mislead people (sometimes intentionally, sometimes unintentionally) or where a conditional statement is just misinterpreted. Explain your example fully in 3-4 paragraphs. 4. A second document containing two logic puzzles with the following constraints: a. Story-like and based around a topic that you think would be interesting to middle school students. b. Include a key and some instructions and hints for learning to solve these types of puzzles. c. Have equal numbers of people and facts. For example, 3 people with 3 things, each a different color. d. The first must have 3 people as the main characters and the second must add at least one thing…an additional character or an additional characteristic. Have your students complete the Cup Stack team-building exercise (see above) and debrief the activity with them. Tell them that it is going to be extremely important that © 2008 Transformation 2013
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all group members work in conjunction with one another to tackle this project. Everyone has strengths that they can contribute to the group that will only serve to make the group stronger. Logic puzzles are not easy! They really make you think, so teamwork is vital to the group’s success. Engage Activity Products and Artifacts Journal reflection, team-building activity Engage Activity Materials/Equipment Journals, pencils, materials for team-building activity Engage Activity Resources Cup Stack Team-Building activity: Building Math: Everest Trek © 2006, Boston Museum of Science Explore Activity Have students work in teams to complete the Explore Activity and the brain teaser. Explore Activity Products and Artifacts Explore Activity, Brain Teaser Explore Activity Materials/Equipment Explore Activity, Brain Teaser, pencil, sticky notes, index cards Explore Activity Resources None 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. During the discussion, create a word wall or have students take notes regarding the vocabulary in their journals. The students can then refer back to these during the project. Have the
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students reflect in their journal regarding the concepts and vocabulary that have been discussed during the Explain phase. Consider awarding prizes to the groups who arrive at the correct conclusion for the Brain Teaser…free homework pass, candy, a small container of play dough, etc. Explain Activity Products and Artifacts Journal entry, word wall Explain Activity Materials/Equipment Journal, pencils, card stock, markers, prizes (optional) Explain Activity Resources None Elaborate Activity Reintroduce students to the problem: We, at Puzzles and Games Publishing, need your expertise in the design of our new Logic-Puzzles Magazine. Because we want to ensure our readers are receiving puzzles from the best high school mathematicians, there are some requirements you will have to meet prior to signing your extended contract for a total of 30 puzzles, over a twelve month period. The total financial award is $100 for each completed puzzle. This means that your team could get a contract worth $3000 (or more). Here is what we need from you: 1. A document containing – b. a conditional statement with its hypothesis and conclusion identified. c. the converse, inverse, and contrapositive of the original conditional statement. d. Three examples in advertising, politics, or journalism where conditional statements and their converses and inverses are used to mislead people (sometimes intentionally, sometimes unintentionally) or where a conditional statement is just misinterpreted. Explain your example fully in 3-4 paragraphs. 2. A second document containing two logic puzzles with the following constraints: a. Story-like and based around a topic that you think would be interesting to middle school students. b. Include a key and some instructions and hints for learning to solve these types of puzzles. c. Have equal numbers of people and facts. For example, 3 people with 3 things, each a different color.
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d. The first must have 3 people as the main characters and the second must add at least one thing…an additional character or an additional characteristic. Provide students with an opportunity to research brain teasers/logic puzzles on the Internet and assign them additional logic problems to solve. Continue to support the academic vocabulary throughout group discussions, expand on the concepts being discussed, and solidify the mathematics being discovered. Students should complete their project by the end of this lesson cycle. Check all logic puzzles to ensure that there are not any errors…that the students’ problems and answers are correct. Have students write each logic puzzle onto a piece of chart paper and post them around the classroom. Elaborate Activity Products and Artifacts Completed project Elaborate Activity Materials/Equipment Computers with Internet access, additional brain teasers/logic problems as needed, pencils, large chart paper, markers Elaborate Activity Resources http://brainden.com/ http://www.braingle.com/Logic.html http://www.braingle.com/Logic-Grid.html http://www.marcopoloeducation.org/SearchShell.aspx?filename=/searchresults.aspx&site_area=teacher&subjec t=math&partner=all&resource_type=all&q=logic&grade=9,12&WebSiteArea=educator Evaluate Activity Set up stations around the classroom with one problem posted at each station (pick the best problem from each group or allow each group to pick what they feel is their best problem). Place logic grids or note cards and sticky notes at each station. Send the groups around the room to solve each problem (except their own). Have the groups report back their solutions to each problem. Allow the creators to demonstrate the logic necessary to arrive at the correct conclusions. Reward the members of the group that arrived at the most correct conclusions a prize…blue ribbons for each group member, stickers, etc.
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Evaluate Activity Products and Artifacts Completed logic puzzles, demonstration of the solutions Evaluate Activity Materials/Equipment Pencils, note cards, sticky notes, logic grids (optional) Evaluate Activity Resources http://www.mathwire.com/templates/logicgrid1.pdf http://www.mathwire.com/templates/logicgrid2.pdf Sample Logic Grid: http://www.braingle.com/brainteasers/logicgrid.php?id=27770
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Explore Activity 1. Identify the hypothesis and conclusion of this conditional statement: If two lines intersect at right angles, then the two lines are perpendicular. 2. Write this statement as a conditional in if-then form: All triangles have three sides. 3. Use the Venn diagram to illustrate this conditional: Cars are motor vehicles. 4. Another name for an if-then statement is a ____. This type of statement has two parts. The part following if is the ____ and the part following then is the ____. 5. A conditional can have a ____ of true or false. 6. What is the conclusion of the following conditional? A number is divisible by 3 if the sum of the digits of the number is divisible by 3. 7. What is the converse of the following conditional? If a point is in the first quadrant, then its coordinates are positive. 8. What are the converse and the truth value of the converse of the following conditional? If an angle is a right angle, then its measure is 90. 9. For the following true conditional statement, write the converse. If the converse is also true, combine the statements as a biconditional. If x = 3, then x2 = 9. 10. Determine whether the conditional and its converse are both true. If both are true, combine them as a biconditional. If either is false, give a counterexample. If two lines are parallel, they do not intersect. If two lines do not intersect, they are parallel. 11. Determine whether the conditional and its converse are both true. If both are true, combine them as a biconditional. If either is false, give a counterexample. If an angle is a right angle, its measure is 900. If an angle’s measure is 900, the angle is a right angle. 12. Write the two conditional statements that make up the following biconditional. I drink juice if and only if it is breakfast time. 13. When a conditional and its converse are true, you can combine them as a true ____. 14. Decide whether the following definition of perpendicular is reversible. If it is, state the definition as a true biconditional. Two lines that intersect at right angles are perpendicular. 15. Is the following statement a good definition? If not, find a counterexample. A square is a figure with two pairs of parallel sides and four right angles.
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Venn Diagram
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Brain Teaser Break students up into groups of 3 or 4 and provide them with 5 index cards and a pad of sticky notes to use while solving the problem. There are 5 houses in 5 different colors that make up the north side of Hoops Lane. In each house lives a man who is a fan of a different basketball team. The 5 owners drink a certain type of beverage, eat a certain type of food, and keep a certain pet. Using the clues below, can you determine who owns the fish? The Lakers fan lives in a red house. The Knicks fan keeps dogs as pets. The Mavericks fan drinks tea. The green house is on the immediate left of the white house. The green house owner drinks coffee. The person who eats popcorn rears birds. The owner of the yellow house eats hot dogs. The man living in the house right in the middle of the block drinks milk. The Spurs fan lives in the first house. The man who eats pickles lives next door to the one who keeps cats. The man who keeps horses lives next door to the man who eats hot dogs. The owner who eats pizza drinks soda. The Bulls fan eats burgers. The Spurs fan lives next door to the blue house. The man who eats pickles has a neighbor who drinks water.
Position
1 Hoops Lane
2 Hoops Lane
3 Hoops Lane
4 Hoops Lane
5 Hoops Lane
Team Color Food Drink Pet
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Brain Teaser Solution
Position Team Color Food Drink Pet
1 Hoops Lane Spurs Yellow Hot Dogs Water Cats
2 Hoops Lane Mavericks Blue Pickles Tea Horses
3 Hoops Lane Lakers Red Popcorn Milk Birds
4 Hoops Lane Bulls Green Burgers Coffee Fish
5 Hoops Lane Knicks White Pizza Soda Dogs
The Bulls fan owns the fish!
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Plan the Assessment Engage Artifact(s)/Product(s): Journal reflection, team-building activity
Explore Artifact(s)/Product(s): Explore Activity, Brain Teaser
Explain Artifact(s)/Product(s): Journal entry, word wall
Elaborate Artifact(s)/Product(s): Completed project
Evaluate Artifact(s)/Product(s): Completed logic puzzles, demonstration of the solutions
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Rubric STUDENT:_______________________________ EVALUATOR:___________________________
COURSE: GEOMETRY PROJECT: Simply Logical, My Dear Watson EVENT: DUE: Presentation Day
CRITERIA
UNSATISFACTORY PROFICIENT (Minimal Criteria)
(Below Performance Standards)
Title/Outline:
Has spelling errors
No spelling errors
Students discuss the format of their presentation
Missing title and/or date of presentation
Includes title and date of presentations
Written Communication/Critical Thinking
Missing names of student presenters
ADVANCED (Demonstrates Exceptional Performance) In addition to meeting all proficient criteria, student: Illustrations/graphics suggest contents of talk
Includes names of students giving presentation
Outline is missing
Includes an outline of the presentation in list form
Other
Other Group grade
Background: Students provide information about the design challenge. Written Communication/Critical Thinking NOTE: If a model is required then use the last criteria mentioned in each column.
Group grade
Method: Students state their process for gathering data
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Background information is missing or has unreasonable estimates for key information including: Summary of the final puzzles Time required to complete the puzzles. There are more than 2 spelling errors on presented material. Font size and/or color are difficult to read
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Missing or incomplete information.
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Each group gives basic details of their puzzles including: Summarize the content of the puzzles. Time required to complete the product. 2 or less spelling errors on presented material. Font size and/or color are easy to read 16
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Complete information as stated in the design challenge.
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In addition to meeting all proficient criteria, student: Puzzles are of high quality and include drawings/diagrams giving the audience a better picture of the end product.
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In addition to meeting all proficient criteria, student: Includes appropriate TEKS in presentation and is able to explain verbally why these TEKS go with
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TEKS: (G.3.A) (G.3.C) (G.3.D) (G.3.E)
Group grade
this design challenge
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Conclusions/Summary
Does not state a clear decision
Students summarize main points of their decision to complete their drawings in the manner chosen.
Does not refer to puzzles in justifying the decision to complete the puzzles.
States a decision about the result of the project.
Refers to final puzzles in justifying the decision
Does not identify possible flaws in the puzzles.
Group grade
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In addition to meeting all proficient criteria, student: Justifies decision using additional evidence Includes appropriate TEKS in presentation and is able to explain verbally why these TEKS go with this design challenge
Identifies a possible flaw in the puzzles and shows ways the flaw could be corrected 8
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Identifies multiple flaws in their evidence and discusses ways to correct these flaws
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Story Board Week 1 Activities
Week 2 Activities
Day 1 Engage (20 minutes) Explore (30 minutes) Day 6 Evaluate (50 minutes)
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Day 2 Explore (50 minutes)
Day 7 Evaluate (50 minutes)
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Day 3 Explain (30 minutes) Elaborate (20 minutes) Day 8 Evaluate (50 minutes)
Day 4 Elaborate (50 minutes)
Day 9
Day 5 Elaborate (50 minutes)
Day 10
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