This is the first of 11 units in the 4MAT 4 Geometry Curriculum Series from About Learning.
These skills are essential for communicating geometrically: Identify points, lines, planes Use segments and congruence Use midpoint and distance formulas Measure and classify angles Describe angle pair relationships Classify polygons Find perimeter, circumference, and area Using inductive reasoning Analyze conditional statements Apply deductive reasoning Use postulates and diagrams Reason using properties of Algebra Prove statements about segments and angles Prove angle-pair relationships
Unit 1 - Reasoning
Research confirms that the brain is not likely to remember new information and even more unlikely to recall that information if the human brain cannot associate this information to personal experiences in a meaningful and relevant way. As an educator, it would be egregious not to recognize the implications of this research and act on it. The following introduction gives explanations of how to create meaning for students through an established context.
The gateway to any content area is vocabulary. Setting this up in a meaningful and relevant way for the students is critical and will minimize the number of times students question why they are learning this new information. It is our intention to set up the importance vocabulary plays in being able to reason and logically support your hypothesis. This is essential to not only Geometric Proofs but ALL reasoning in the real world. We would challenge anyone to find a task that does not have reasoning behind it.
This first unit begins with an activity called "Friplet". Providing students a made up word, students will not be able to efficiently or effectively reason because no contextual reference exists. By entering into this experience, students begin to see the challenge of vocabulary that is foreign to them. When the teacher finally provides students with a definition of this made up word, students now have a contextual reference. We believe this is an effective connect activity because it would be extremely rare to have any student who is not familiar with what a cell phone is and the possible uses of a cell phone. By beginning a unit with something that teenagers are highly familiar with and can easily connect to personally, we have engaged all students, created a contextual reference, and put all students on a level playing field.
This activity provides a seamless and natural progression from Friplets to Geometry vocabulary. Teachers will need to be reminding students how easy it was to reason (prove) when you understand the vocabulary. We would then recommend telling students that this course is designed so that students build knowledge in Geometry while learning how to argue and reason using supporting details and facts. To do this we will start the year by building our understanding of vocabulary. As you introduce vocabulary to students, keep reminding them of the Friplet activity and how the vocabulary that is going to be introduced will enhance our ability to effectively argue (reason).
Example: Yesterday we discussed the vocabulary words complementary and supplementary. It is much easier to reason (argue) when we understand vocabulary. It is difficult to master skills when you don't know what the words mean. We found this out with the Friplet activity. Today we are going to define a few more vocabulary words such as polygon, convex, concave, and regular. Understanding them will help us to use them in our future reasoning.
This dialogue should happen as you bring in new vocabulary throughout this unit. In doing so you keep students connected to why vocabulary is essential to their learning.
The Concept is Reasoning. The Content is the essentials of Geometry and proof. There is a relationship between reasoning and proof and the relationship is vocabulary and logic!
Learning Outcomes
Students will: Explore terminology that is not familiar to them. Gain insight on the importance of understanding and using common vocabulary. Understand that to communicate effectively, vocabulary must be defined and commonly understood. Understand the vocabulary necessary to reason effectively in Geometry. Practice the concepts taught. Apply their knowledge of the vocabulary in this unit by completing an ABC book. Collaborate to fine-tune projects. Share their projects.
Teacher Notes
This unit is authored by Michael Arlien and Colleen Hodenfield, authors of the 4MAT 4 Geometry Curriculum Series and 4MAT 4 Algebra Curriculum Series. This is the first unit of 11 conceptual Geometry units.
"l'll help with driving my sister to practice and school." "I'll get groceries for you when you need something quick." "I'll get myself to my job so you don't have to take me." "I'll..." After Mikey spent 30 minutes explaining the reasons for why he should get a car of his own, his parents realized he had some legitimate points and went ahead and got him one. Knowing the facts, laying them out in a orderly fashion with a good conclusion are the essentials of good reasoning and proof.
Why is it important to understand common Geometric vocabulary and logic when reasoning?
Time Frame
Total time : 217 minutes
Section 1: Design Framework
Concept :
Reasoning
Content :
Reasoning and Proof
Bridge :
Vocabulary and Logic
Essential Question :
Why is it important to understand common Geometric vocabulary and logic when reasoning?
Outcomes :
Students will understand the vocabulary associated with reasoning and proof in Geometry. Students will be able to reason and show proof for Geometric problems. Students will be able to reason through problems in Geometry and transfer that process to real life.
Section 2: Instructional Delivery
I need a friplet!
I need a friplet discussion
Revisit friplets; Ma and Pa Kettle video
Kinesthetic representation; Frayer's Model, Fold and mold; Exit slips
1. Connect: Connecting to the Concept Experientially
4MAT 4 Geometry Student Activity Book page 3
Use this student page for this "Connect" activity.
Objective: Students explore terminology that is not familiar to them.
Activity: Students prepare an argument for why their parents should buy a “Friplet” for them.
Assessment: Student participation
Getting ready
This is the “Connect” octant of the unit, Step One. This activity connects learners directly to the concept in a personal way.
Activity directions
Have students turn to Student Activity Book p. 3. Ask students to prepare an argument stating why their parents should buy a “Friplet” for them.
Be prepared for students to want to know the definition of a “Friplet”. When students ask and they will, tell the students you will reveal the definition in good time. Add that if they don’t know what a Friplet is, they need to “bluff” their way through their argument. This activity is really powerful if you are very vague at this point.
Notes
1. Not every student will have something written down at this point.
2. It is not important that every student finish this, only that they “entered into” the experience.
3. Don’t allow more than a couple of minutes at this point.
2. Attend: Attending to the Connection
4MAT 4 Geometry Student Activity Book page 3
Use this student page for this "Connect" activity.
Objective: Students gain insight on the importance of understanding and using common vocabulary.
Activity: Have some students report what they wrote down. What might make this easier for them to do this assignment?
Assessment: Student participation
Getting ready
This is the “Attend” octant of the unit, Step Two. This activity examines what just took place and begins to clarify the reason
behind the learning.
Activity directions
1. After students have struggled for a few minutes, have some students report what they wrote down.
2. Ask students what might make this easier for them to do this assignment. Students use the bottom of Activity Book p. 3 to reflect on this question.
3. Image: Creating a Mental Picture
4MAT 4 Geometry Student Activity Book pages 4-5
Use these student pages for this "Image" activity.
Objective: Students understand that to communicate effectively, vocabulary must be defined and commonly understood.
Activity: Friplet is defined. Students revisit their arguments with this new knowledge. Show video.
Assessment: Student participation
Getting ready
This is the “Image” octant of the unit, Step Three. This step allows students to create a personal representation of the concept by drawing on their past experiences.
1. Define “Friplet” for student. Tell them that a “Friplet” is the newest, high-tech cell phone on the market. Have the students repeat the activity of writing down an argument of why their parents should buy them a “Friplet”.
2. At the conclusion, have them complete the statement at the bottom of the Activity Book p. 4, “To argue effectively, you must…” Have them list at least 3 responses.
3. Show the Ma and Pa Kettle video to show the importance of knowing your facts before you reason with someone!
4. Inform: Receiving Facts & Knowledge
4MAT 4 Geometry Student Activity Book pages 6-13
Use these student pages for this "Inform" activity.
Objective: Students understand the vocabulary necessary to reason effectively in Geometry.
Activity: Information delivery, exit slips, kinesthetic representations, Frayer’s Model, fold and mold activities
Assessment: Students complete notes, share thoughts and reactions through oral discussion, and journal their understandings.
Getting ready
This is the “Inform” octant of the unit, Step Four. This is when students learn the information, knowledge base, the content, using a variety of delivery systems.
Activity directions
1. Kinesthetic representation of vocabulary with hands/arms (eyes closed)
Suggested vocabulary to check: acute, right, obtuse, straight, congruent, supplementary, and complimentary angles
concave and convex polygons i.e. “Students, I want you to close your eyes and use your hands and/or arms to show me what an acute angle looks like.”
Continue with the rest of the vocabulary. By having them close their eyes you will have less peeking for help. When they do peek, you know that they don’t understand the vocabulary yet.
2. Kinesthetic representation of midpoint and angle bisector on paper: Students will need 2 pieces of paper, a straightedge and a protractor for these activities. Have them follow the directions in their book. [Activity Book p. 6-7]
3. Notes pages
Have students use Activity Book pages 8-9 for note taking during this unit's information delivery.
4. Frayer Model
The Frayer Model is for vocabulary development. This model helps develop content by contrasting what it is from what it isn’t which leads to a better understanding of the content. The model is included in this unit only, but could be done in each unit. Vocabulary study pages can be found at the back of the Activity Book to be used with each unit.
Steps to the Frayer Model:
Explain the Frayer model to the class. Go through the first key word with them to demonstrate the various parts of the table. Explain the difference between a definition and a characteristic. See examples provided in the Activity Book. [Activity Book p. 8] Tell students which of the key words they will be working on. Write this list on the board and review it with the class before students read the section. Another option would be to have the students decide which terms to put in the model. Divide the class into student pairs. Assign each pair one of the key concepts and have them read the section carefully to define this concept. Have these groups complete the row in the model for this concept. You may also assign these individually instead.
Ask the student pairs to share their conclusions with the entire class. Use these presentations to review the entire list of key concepts.
Alternate Steps to the Frayer Model:
Explain the Frayer model to the class. Go through the first key word with them to demonstrate the various parts of the table. Explain the difference between a definition and a characteristic. See examples provided in the Activity Book. [Activity Book p. 10]
Give the students the list of all key words you would like them to understand for the entire unit. Another idea would be to brainstorm a list. This could be done at the beginning of the unit as a preview or at the end of the unit as a review.
Students work individually or as a group as described above.
Notes
There are two forms that could be used for this activity. A chart form and a box and oval form. The chart form has been included for use in the student Activity Book. If using the box and oval form, the vocabulary word would go inside the oval in the middle.
5. Fold and Mold Notes
Use Student Activity Book page 13 for this activity.
6. Exit Slip with follow up (Law of Syllogism)
You will need to copy and cut out the exit slips for each student. Lecture on the Law of Syllogism and then before the students leave class, hand out the exit slip and tell them to fill it in and hand it in. Once all the slips have been handed in you will need to recreate the question as a multiple choice question with their answers as the choices. Have the problem on the board when they walk into class the next day. After the bell rings, have the students decide which is the correct answer. The discussion here will be a valuable tool to assess where the students are on this concept, both as a group and individually. You can do this for any concept you are teaching.
Blank exit slip has been provided for you to use on other content if you so desire. [See Teacher Book addendum]
5. Practice: Developing Skills
4MAT 4 Geometry Student Activity Book pages 15-37
Use these student pages for this "Practice" activity. Choose any or all of the activities suggested.
Objective: Students practice the concepts taught.
Activity: Students complete assignments, explore applications, complete activities to practice the concepts they are learning.
Assessment: Check for mastery of concepts using worksheets, book problems, and a quiz or test.
Getting ready
This is the “Practice” octant of the unit, Step Five. These activities provide opportunities for the students to practice in multiple ways.
Activity directions
Examples of activities that may be used here include but are not limited to:
1. Book assignments and worksheets.
2. Geometric property workplace applications
This activty guides the learner to an understanding about logic vocabulary. [Activity Book p. 15]
Geometric Properties Answer Key
1. E
2. A
3. D
4. B
5. C
3. TRY-angles Practice
(Angles and Triangles)
Materials: red, blue, and yellow
colored pencils/markers
[Activity Book p.16-17 ]
TRY-angles Practice Answer Key
Name 1 right angle: ___________
Possible answers: ∠FDE, ∠ADG, ∠ADE
Name 2 acute angles: _________, _________
∠BDC, ∠BCD, ∠EAF, ∠BAE, ∠GDB, ∠FED, ∠EFD
Name 1 obtuse angle: ___________
∠EFA, ∠BCA
Name 2 straight angles: _________, _________
∠CFD, ∠ABG, ∠AFD
Name a pair of angles that are adjacent but NOT supplementary or complimentary: ___________, ___________
∠AEF, ∠FED or
∠ABC, ∠DBC or
∠CBD, ∠DBG
4. It Figures! Activity (Angles)
Students draw a figure that demonstrates the angle relationships discussed, based on the following criteria:
At least two supplementary angles
At least four complementary angles
At least two congruent angles
At least one obtuse angle
At least one acute angle
At least one right angle
At least one linear pair
At least one set of vertical angles
When they have completed the assignment, have them turn it in. Make a copy of it. Students will then label the copy with all the different angles included (as many times as was required in the directions).
Have the students color the original and mount the original on one side of construction paper and the copy/answer key on the other side. Hand it in. The next day, hand out the figures as students walk into the classroom, making sure no one gets their own. Have the students try to find all the angles in their classmates figure. [Activity Book p. 18]
5. Rotation Station (Area and Perimeter)
Perimeter and Area practice worksheet. [Activity Book p.19-21]
This activity can be set up different ways.
1. Use the blown up problems as a rotation station. [Teacher Book p. ] Place the problems around the room and have the students rotate after a set time. Students bring their answer sheet to each station. [Activity Book p. 21] If a student does not finish, they would have the problems in their Activity Book p. to complete for
homework.
2. Students could do this at their desk individually or as a small group using the worksheets [Activity Book p.19-21]
Rotation Station Answer Key
perimeter
area
1
20.0m
14.9m2
2
26.1m
31.0m2
3
19.7m
10.7m2
4
23.3m
25.9m2
5
17.8m
15.6m2
6
24.0m
31.5m2
7
24.2m
31.5m2
8
15.0m
9.4m2
Rotation Station Teacher Book Pages
Notes
This worksheet requires students to use the Pythagorean Theorem. Review may be necessary.
6. I Have Who Has... Game (Conditionals)
The “I Have...Who Has” strategy works well with all grades and can be done as a whole group activity or with small groups. This activity is designed to be completed in small groups.
Small groups:
Using the cut-apart cards in the back of their Activity Book, have the members of the group cut out the cards. [Activity Book p. ] On each card is an answer and a new question (the answer and question on the card do not match) Have the group pass them out to each other so each has an equal amount if possible.
The student who receives the “Start Here” card will lay down the card in the middle of the group and say, “Who has ___?”
The student who has the answer that matches the question will say, “I have _____”. They will then lay their card down and say, “Who Has ___?”
This will continue until all the cards are laid down.
Whole group:
The teacher could give each student a card then display the START card. The student who has that answer would call out “I have... who has...” The class would continue until all cards are read. This could serve as a quick review at the end of the day or the beginning of class the following day. The drawback of doing this as a whole group is that once the student has read their card, they may stop paying attention. Try to create an atmosphere where that is not an option.
7. Tic-Tac-Toe games (Angles)
Students must complete three in a row in this activity (horizontally, vertically, or diagonally. This activity will give teachers another alternate activity for students to practice their skills. [Activity Book p. 22]
Tic-Tac-Toe Answer Key
Never
An obtuse angle is larger than 90°, and it is not possible to have a compliment of an angle that is greater than 90°.
95°
Never
A straight angle is 180°, and it is not possible to have a complement of an angle that is 180°.
28°
78°, 12°
Sometimes
A straight angle does not have a supplement.
58°, 122°
Always
An acute angle is less than 90°, since the sum of supplementary angles is 180° and one of the angle sis less than 90°, the other angle must be larger than 90°, which makes it obtuse.
132°, 48°
8. Law of Syllogism Activity
Students will cut out strips and arrange them into the proper order and fill in the worksheet. They will then make up their own statements using the Law of Syllogism. [Activity Book p. 23, 25, 27]
9. Quiz-Quiz Trade (Angles)
Quiz-Quiz Trade is a strategy that involves all students and gets them moving. This activity can effectively be used as a starter activity. It can also be used as a way to review previous benchmarks or check for mastery at the end of a lesson.
Before starting, cut the Questions and Answer (Q&A) cards out along the solid lines [Teacher Book p. ]; fold on the dashed line.
Option #1: Photocopy the problem set you wish to work with so that each student gets a Q&A card.
Option #2: Mix all the problem sets up and use as a review near the end of the unit.
Hand out the questions to the students, answers will be on the back. Allow students time to review their problem and answer.
1. Stand up, Hand up, Pair up
When a student is ready, he/she should stand up and put their hand up and pair up with another student with their hand in the air.
2. Partner A asks their question and listens to Partner B’s answer; Partner B then asks his question and listens to Partner A’s answer. The teacher can tell the student with the shortest hair, closest birthday, etc. that they will be partner A and will go first.
3. Partners trade cards, raise their hand again, and walk around looking for another student with his hand raised.
4. Continue with steps 1-3 until the teacher has everyone return to their desks.
To make this activity more structured, try inside/outside circles. Half the students form a circle facing out and the other half of the class forms a circle facing in. They should be facing a partner. After students trade cards, the outside circle can move to the left or right. Now, they have a new partner and quiz-quiz-trade again.
( See Attachments page of this unit for full size cards to print out.)
Quiz-Quiz Trade Teacher Book Pages
10. Error Analysis, Fill in the blanks and Cut Out Proofs
This activity gives the students extra practice in writing proofs. You can differentiate the cut out proofs in this activity by cutting along different lines. If you have the students cut along the solid lines only, you will have a lower level activity since the statement and reason will not be separated. If you have the students cut along all lines, dotted and solid, and then shuffle the cards, you will have a more difficult task since the statement will be apart from the reason. The key will be to make sure the students shuffle their cards right away before they realize they have some of the answers!! [Activity Book p. 29-37 ][Teacher Book p. 14-18 answer keys]
Fill in the Blank Proofs A-B
Cut Out Proofs A-D
Proof Error Analysis A-C
Teacher Book Answer Keys p. 14-18
6. Extend: Extending Learning to the Outside World
4MAT 4 Geometry Student Activity Book pages 38-45
Use these student page for this "Extend" activity.
Objective: Students apply their knowledge of the vocabulary in this unit by completing an ABC book.
Activity: Students create an ABC book using vocabulary from the first unit.
Assessment: Rubric for project
Getting ready
This is the “Extend” octant of the unit, Step Six. This activity provides opportunities for learners to design their own open-ended exploration of the concept.
Activity directions
Students create an ABC book using vocabulary from the first unit. The student must have the letter, the vocabulary word, a definition and an illustration for each letter of the alphabet
These were student examples found in the Authors’ Geometry Textbook in the first two chapters.
Notes
Teachers may have to be flexible and allow students to be creative to complete their lists. In some instances, the authors would allow students to use a letter not at the beginning of the word. {See example below.}
P: Perpendicular, parallel, point, postulate, protractor, proof, pattern
Q: Question, Quadrilateral, Quadrant
R: Reasoning, ray, right, ruler, reflexive
S: Segment, straight, symmetric, supplementary
T: Theorem, triangle, transitive
U: Undefined terms
V: vertical, verify
W: width, walls, within (interior), write (writing a proof)
X: x-axis
Y: y-axis
Z: zero
7. Refine: Refining the Extension
4MAT 4 Geometry Student Activity Book pages 45-47
Use this student page for this "refine" activity.
Objective: Students collaborate to “fine-tune” projects.
Activity: Students critique and refine projects.
Assessment: Student participation
Getting ready
This is the “Refine” octant of the unit, Step Seven. This activity helps turn mistakes into learning opportunities.
Activity directions
Edit/Revise Project
When students are finished writing the words and definitions but not the illustrations, they should trade books with someone else who is finished. Students will then make correction suggestions using Activity Book p. 45-46. The students will then finish their books, adding the illustrations. Non artistic students may be allowed to use technology to help them write and illustrate their book. These would have to be taped or glued into the book.
Essential Question
Have students use page 47 in their Activity Book to answer this unit’s Essential Question: Why is it important to understand common Geometric vobabulary and logic when reasoning?
8. Perform: Creative Manifestation of Material Learned
Unit 1 Reasoning Rubric
See the rubric for this unit on the Attachments page.
Objective: Students share their project. Activity: Projects are shared in the classroom. Assessment: Teacher assesses projects using a pre-agreed upon rubric.
Getting ready This is the “Perform” octant of the unit, Step Eight. This activity gives the learner the support they need to teach and share with others.
Activity directions Have students exchange their Activity Books as many times as time allows to share their ABC Projects. The teacher should also assess the completed projects using the rubric. (see Attachments page of this unit)
Use this student page for the Project Performance Assessment. It can also be found on the Attachments page of this unit.
Students use the Reasoning Unit Rubric found on the Attachments page of this unit to peer edit their projects. Then the teacher uses the Rubric to assess the student's work for this unit.
