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Axiomatic Development Module

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Contents

1 Axiomatic development 31.1 Viewing tubes - Day 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Viewing tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2 Formal Writing Assignment - Viewing Tubes . . . . . . . . . . . . . . 71.1.3 Rubric: Formal Writing Assignment - Viewing Tubes . . . . . . . . . 81.1.4 Homework - Introduction Survey . . . . . . . . . . . . . . . . . . . . 9

1.2 What is geometry - Day 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.1 What is geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.2 Homework - Constructions . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Pre-test: LessonSketch - Day 3 . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Lines on a sphere - Day 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.1 Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.2 Homework - Train tracks . . . . . . . . . . . . . . . . . . . . . . . . 191.4.3 Homework - Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5 Lines on a sphere - Day 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.1 Homework - Triangle Exploration . . . . . . . . . . . . . . . . . . . . 251.5.2 3100f15 Triangle: LessonSketch Experience . . . . . . . . . . . . . . . 26

1.6 Triangles on the sphere & Hilbert’s axioms - Day 6 . . . . . . . . . . . . . . 271.6.1 Incidence geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.6.2 Homework - Interpretations, models, and proofs . . . . . . . . . . . . 31

1.7 Incidence geometry - Day 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.7.1 Hilbert’s Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.7.2 The SMSG Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.7.3 Homework - Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.8 Parallel postulate - Day 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.8.1 Formal Writing Assignment - Parallel Lines; LessonSketch Experience 41

1.9 Complete - Day 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.9.1 Homework - Parallel postulate . . . . . . . . . . . . . . . . . . . . . . 46

1.10 Midterm Exam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.10.1 Part 1: Constructing and critiquing arguments . . . . . . . . . . . . . 471.10.2 Experience 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.10.3 Part 2: Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . 491.10.4 Experience 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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1 Axiomatic development

Our goals:

1. Understand the importance of definitions, axioms, and undefined terms.

2. Understand how axioms build geometry.

3. Become aware and appreciate geometries other than Euclidean.

4. Develop ideas of intrinsic and extrinsic geometry, ideas of straightness.

5. Understand the difference between axiomatic system and a model.

6. Understand historical importance of parallel postulate.

7. Be able to articulate differences between neutral, Euclidean, spherical, and hyperbolicgeometries.

8. Develop skills for writing proofs - understand the role of axioms and rules of logic.

9. Develop ability to make sense of other’s reasoning

In addition to writing a textbook, students are given a variety of assignments. Here is acode:

• Writing Assignment: to be graded and feedback provided. These assignments will belooked at and commented on before a final version is submitted for evaluation.

• Homework Assignment: verified that it is completed, as a preparation for class orplanning information for the instructor.

• Ask your question Important part of doing mathematics is to learn to ask one’s ownquestions. You might not be able to answer them, but you can always learn morethrough seeking the answer to your questions. Which questions has this work madeyou ask? Record all the questions that you’d like to investigate on the class website.

• Preparation for next day: Print the handout Viewing tubes. Prepare a google form forIntroduction Survey’.

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1.1 Viewing tubes - Day 1

Geometry means Earth Measurement and it seems the best way to start the class witha measurement activity. This activity is based loosely on Field of Vision p117 of NCTMNavigating Through Geometry, 9-12, as an example of best practices lesson. This can bepointed out to the students once the activity is completed. Perhaps number the tubes insuch a way that it is clear that there are 3 tube lengths for each diameter width tube.

We provided the paper only, and did not provide the actual tubes. It seemed impor-tant for them to figure out what they need and how many measurements to make, whatwas important to measure and how. In addition, many students repeated experimentsat home, once they figured out what else they needed, or how to be more accurate andprecise.

The activity will roughly unfold as follows:

1. Materials: three different sized papers (thicker), tape, tape measure, rulers

2. Watch youtube video: videoBig question is “Is this an example of machismo or a reasoned concerned? How wouldyou attempt to answer this question?”

3. Let the students discuss in the large group discussion what exactly they think thisquestion is asking. Or, what are the mathematical questions worth investigating?Collect some thoughts, then agree on the question we want to answer, and ask themto come up with a plan for answering the question in small groups.

4. Our hope is that they will proceed in a manner such as:

• First, try to identify all the variables that you think might effect how much of avertical wall is visible through a viewing tube. These may be properties of whatyou’re looking at (e.g., how far away is it), or properties of the particular tubethat you’re using.

It may be necessary to pull the students back together after they start think-ing about the experiment and discuss which variables they are interested in,and how they’d go about collecting data. Have a class wide discussion inwhich the students share their understanding of the problem. The instruc-tor’s role is that of a facilitator of discussion, and not arbitrator of truth. Themain reason for this is to encourage students to listen to each other’s ideasand reasoning, to compare them to their own and to convince and be con-vinced by each other. Further, they will be writing up their solutions and itis important to have them write their own ideas down as opposed to “correct”solution given by the instructor.

• Next, pick a tube and gather data with your particular tube. Since you are usingone particular tube, some of your variables will be fixed. Your goal is to see whatpatterns you can observe. With your group you should try to develop a methodof making measurements that is as careful and consistent as possible. Make surethat you take enough measurements with your tube so that you will be able toidentify patterns. If you only have a couple of measurements, then you won’t beable to look for patterns.

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• After you have looked for patterns with an initial tube, try other sizes of viewingtubes and try to see how this changes the patterns that you find. See whatconjectures you can make. Our ultimate goal is to be able to predict the field ofvision for any viewing tube. That is, if someone shows you a tube and tells youit is a certain distance from the wall, can you find a method of predicting howmuch of the wall will be visible?

5. Discuss syllabus. It is important to start the year with the actual mathematics - setsthe tone and expectations. This might be a good time to discuss the technicalities.There is no reason to expect the students to be completed with the assignment, so youcan break when you think appropriate headway has been achieved.

6. Preparation for next day:

• Students should fill out a form in which they provide an email address withwhich they can access google docs and google sites. Listed below as Homework- Introduction Survey. They should download Geogebra and MikTeX on theircomputers and bring them in the next class.

• Instructor should print out handouts What is Geometry, Euclid. Several otherhandouts could be either printed or provided electronically: Hilbert’s, SMSG,UCSMP axioms.

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1.1.1 Viewing tubes

In your group you should use the tubes provided to gather data about the size of the fieldof vision for tubes. After you have gathered the data, you should find a meaningful way topresent the data in the form of a table and a graph. Be sure to collect enough data to showrelationships among the variables. Once you have done this, you should investigate the datafor a pattern as well as find a geometrical justification for the pattern. At that point youshould be able to answer the following questions:

1. What did you do to mitigate error in your data collection? What could you have done?

2. How does your model take units into account? Is it necessary to convert the units ofthe tube into the same units as the distance?

3. If you had a tube with diameter 2 inches and length 14 inches, how far would youneed to stand from the wall to view a 3 × 4 foot painting on the wall? Is the problemmaking an assumption on where the painting is hanging?

4. For a given tube of diameter d and length l, at what distance from the wall wouldyour formula for the size of field of vision break down because the ground came intoview if a person’s eye level was h? Create a piecewise function that accounts for thisphenomenon.

5. How does the problem change if the tube is not held level, but inclined at an angle θ?

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1.1.2 Formal Writing Assignment - Viewing Tubes

This assignment is a formal description of the work you started with your classmates on thefirst day of class. Use the handout from class to guide your writing. Explain what you did,what you found, the conjectures that you made, and whatever evidence you have that theyare correct. You should try to write your explanation so that it would convince one of yourpeers that your conclusions are correct.

In order to be considered complete, a your write up must contain:

• a reasonable description of your method of collecting data;

• a chart of your data that includes a sufficient number of data points;

• a graph of your data;

• a correct formula for computing the viewing diameter;

• a complete proof of why your formula must always work that proceeds from reasonableassumptions that are explicit; and

• an analysis of how closely your measured values and predicted values agree (for exampleby computing percentage error for each measurement). Explain why your methods ofcomparison is sensible.

You will receive back comments and feedback on your work. You will then have an oppor-tunity to revise your assignment before it is assigned the final score.

Portfolio entry: Both the original and the final version of this assignment should beincluded in your portfolio. In addition, include a reflection of what you learned through therevision process: reflect on the feedback you received and how that influenced the changesyou made. You should start building the portfolio as soon as you have completed thisassignment.

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1.1.3 Rubric: Formal Writing Assignment - Viewing Tubes

Quality ofExperiment

0 - 1 points

Experiment isnot well designedand an insufficientamount of data iscollected

2 - 3 points

Experiment islacking either indesign or collectionof data.

4 points

Experiment iswell designed. Asufficient numberof data points arecollected.

MathematicalGeneralization

0 - 1 points

No mathemati-cal generalizationis provided oris incorrect. Nojustification isprovided.

2 - 4 points

The mathemat-ical generalizationhas minor errors orthe justification isweak.

5 - 6 points

A mathemati-cal generalizationof the situationis presented andjustified with amathematicalargument.

Discussionand

Interpretation

0 - 1 points

The experiment isnot described. Theprovided questionsare not addressed.

2 - 4 points

The experiment isvaguely describedor the discussiondoes not includeresponses to theprovided questions.

5 - 6 points

The experiment isclearly describedand includes re-sponses to theprovided questions.

OverallPresentation

0 - 1 points

The paper islacking in three orfour areas: Displayof data, typed, elec-tronically createdfigures, grammar

2 - 3 points

The paper islacking in one ortwo areas: Displayof data, typed, elec-tronically createdfigures, grammar

4 points

Data is displayedin graphical form.The entire writingassignment is typedand all figures arecreated electron-ically. There areno grammaticalerrors.

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Transformational Geometry Module

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Contents

2 Transformational geometry 462.1 Using transformations - Day 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.1.1 What’s the shortest way? . . . . . . . . . . . . . . . . . . . . . . . . 492.1.2 Homework - Classifying functions . . . . . . . . . . . . . . . . . . . . 51

2.2 Transformations, distance preserving and angle preserving - Day 2 . . . . . . 532.2.1 Homework - Definitions! . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3 Distance preserving transformations - Day 3 . . . . . . . . . . . . . . . . . . 552.3.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3.2 Homework - Reflections . . . . . . . . . . . . . . . . . . . . . . . . . 592.3.3 Formal Writing Assignment - Composing Isometries . . . . . . . . . . 61

2.4 Transformations and congruence - Day 4 . . . . . . . . . . . . . . . . . . . . 632.4.1 Homework - Knot Symmetry . . . . . . . . . . . . . . . . . . . . . . . 65

2.5 Transformations and congruence - Day 5 . . . . . . . . . . . . . . . . . . . . 662.5.1 Homework - How to get from here to there? . . . . . . . . . . . . . . 68

2.6 Compositions - Day 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.6.1 Homework - When does the order not matter? . . . . . . . . . . . . . 72

2.7 Fixed points - Day 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.7.1 Homework - What students understand? - Lesson Sketch . . . . . . . 74

2.8 Applications: Congruence theorems - Day 8 . . . . . . . . . . . . . . . . . . 752.9 Optional explorations - Day 9 . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.9.1 Coordinate geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.9.2 Graph transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 782.9.3 Slopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.9.4 Formal Writing Assignment - Transformations . . . . . . . . . . . . . 80

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2 Transformational geometry

Our goals: The students will

1. know the definitions of rigid transformations and will be able to use coordinates todescribe them.

2. be able to describe axioms in which we postulate that reflections, rotations and trans-lations are distance preserving and prove theorems using this set of axioms.

3. be able to describe the compositions of isometries

4. know that reflections generate the group of isometries of the plane. They will furtherbe able to explain that any isometry is a composition of at most three reflections.

5. know definition of congruent triangles using isometries

6. be able to describe symmetries of finite plane figures

7. be able to prove SAS, SSS, ASA.

Preparation for next class

• Students

• Instructor: Print the handout What’s the shortest way?

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2.1 Using transformations - Day 1

In this lesson will attempt to solve several problems which are much more easily solvedif one uses transformations. We are going to use these as a motivation for investigating(rigid) transformations. It is unlikely that we will get past these three problems, one ofwhose goals is to also try to pose questions

1. Lead with: We have been talking about straight in several different ways, one of whichis the shortest path between two points. In the following problems we will attemptto solve some problems whose goal is to find a shortest path, although the constraintsposed exclude a straight line between them.

2. Sam was on her way to camp from a long hike and notices that a fire is raging in thegeneral direction of her tent. She figures she better run to the river and get some waterto put out a fire. Which path should she take if she wants to save her tent?

This problem has two issues: shortest distance and shortest time. The questionis phrased ambiguously so we could get at both of those things at the same time.Students generally want to pick out the mid point along the river. This does notgive the shortest distance which can be easily obtained by reflecting the tent overthe river, then simply finding the shortest path between Sam’s position and thereflected tent. We need to prove that this position gives the shortest distance.

3. These problems allow an opportunity to discuss CCSS.MATH.PRACTICE.MP1 Makesense of problems and persevere in solving them as well as modeling standard onceagain. We recommend having a discussion about students’ attempts and struggleswith the problem before addressing the solutions and solution strategies.

4. Two cities, Parallelofield and Circleville, are separated by a river. They’d like to builda road between the two for the most efficient travel and you’re the engineer. Or who-ever builds bridges. At this particular moment, the context is totally unimportant :)So, umm, where would you build the bridge?

We assume that the banks of the river are parallel - see figure below. From cityA, draw a line AA that is equal and parallel to the width of the river. Such anact can successfully virtually move city A to location A′. The shortest distancebetween the 2 cities is now A′B (the green line). This line somehow will cut theriver bank at Y . From Y , build the perpendicular bridge Y X. The highway is nowAX+XY +Y B. The students generally want to “average” the river. Their solutionis oh so close to the actual solution, but it’s not the shortest path!

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• Students Homework - Classifying functions. The students are likely to strugglewith this handout. They are likely to try to graph a function R → R. You maywant to include a short segment in which you discuss one of the problems and howthey might represent and get ideas about the effect of these functions - studentsshould understand that they’re not to produce a graph of the given function. Itmight be valuable to let them work in groups on this at first since the tools theychoose to use might spark a discussion about their value.

• Instructor

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2.1.1 What’s the shortest way?

1. Sam was on her way to camp from a long hike and notices that a fire is raging in thegeneral direction of her tent. She figures she better run to the river and get some waterto put out a fire. Which path should she take if she wants to save her tent?

2. Two cities, Parallelofield and Circleville, are separated by a river. They’d like to builda road between the two for the most efficient travel and you’re the engineer. Or who-ever builds bridges. At this particular moment, the context is totally unimportant :)So, umm, where would you build the bridge?

Extension What if there are several rivers between the two cities?

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CCSS.MATH.PRACTICE.MP1 - Make sense of problems and persevere in solvingthem.

Mathematically proficient students start by explaining to themselves the meaning of aproblem and looking for entry points to its solution. They analyze givens, constraints,relationships, and goals. They make conjectures about the form and meaning of the so-lution and plan a solution pathway rather than simply jumping into a solution attempt.They consider analogous problems, and try special cases and simpler forms of the orig-inal problem in order to gain insight into its solution. They monitor and evaluate theirprogress and change course if necessary. Older students might, depending on the con-text of the problem, transform algebraic expressions or change the viewing window ontheir graphing calculator to get the information they need. Mathematically proficientstudents can explain correspondences between equations, verbal descriptions, tables,and graphs or draw diagrams of important features and relationships, graph data, andsearch for regularity or trends. Younger students might rely on using concrete ob-jects or pictures to help conceptualize and solve a problem. Mathematically proficientstudents check their answers to problems using a different method, and they contin-ually ask themselves, ”Does this make sense?” They can understand the approachesof others to solving complex problems and identify correspondences between differentapproaches.

MODELING:

The basic modeling cycle is summarized in the diagram. It involves (1) identifyingvariables in the situation and selecting those that represent essential features, (2) for-mulating a model by creating and selecting geometric, graphical, tabular, algebraic, orstatistical representations that describe relationships between the variables, (3) ana-lyzing and performing operations on these relationships to draw conclusions, (4) inter-preting the results of the mathematics in terms of the original situation, (5) validatingthe conclusions by comparing them with the situation, and then either improving themodel or, if it is acceptable, (6) reporting on the conclusions and the reasoning behindthem.

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2.1.2 Homework - Classifying functions

We can use coordinates to describe functions on the plane. For each of the given func-tions describe their effect on points of the plane, coordinate axes, and characteristics theyleave invariant. To pre-emptively strike: Yes, I know. I should have two sets of parentheses.But why bother?1Try to make predictions about what will happen before you ”graph” these.2

Function Prediction Diagram

f(x, y) = (x3, y3)

g(x, y) = (2x, 3y)

h(x, y) = ( 3√x, ey)

i(x, y) = (cos x, sinx)

j(x, y) = (−x, x+ 3)

l(x, y) = (x3 − x, y)

m(x, y) = (3y, x+ 2)

1Actually, no one ever does... I wonder why.2Why is graph in quotations?

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Function Prediction Diagram

n(x, y) = (x+ 2, y − 3)

p(x, y) =(0.6x− 0.8y, 0.8x+ 0.6y)

q(x, y) =(−0.6x+ 0.8y, 0.8x+ 0.6y)

If you were to group these functions into two disjoint groups, what would your classifi-cation be:

• Group 1:

• Group 2:

What is your rationale for this classification?

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Similarity Module

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Contents

4 Similarity 684.1 Transformations that aren’t as nice - Day 1 . . . . . . . . . . . . . . . . . . . 69

4.1.1 Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 714.1.2 Homework - Pythagorean Theorem . . . . . . . . . . . . . . . . . . . 72

4.2 Area vs calculating area - Day 2 . . . . . . . . . . . . . . . . . . . . . . . . . 734.2.1 Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3 Applications - Day 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.3.1 Area of a spherical triangle . . . . . . . . . . . . . . . . . . . . . . . . 774.3.2 Homework - Make it bigger . . . . . . . . . . . . . . . . . . . . . . . 78

4.4 Similarity - Day 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4.1 Find the center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5 Similarity - Day 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.5.1 We are similar, can you tell? . . . . . . . . . . . . . . . . . . . . . . . 86

4.6 Similarity - Day 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.7 Similarity - Day 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.8 Lines - Day 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Final project - to be included in portfolio 94

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4 Similarity

Our goals: The students will

1. develop understanding of similarity transformation.

2. will know how to perform dilations, as well as locate the center of a given dilation andthe scale factor.

3. know area axioms.

4. be able to find areas of irregular shapes.

5. be able to explain formulas for areas of polygons and circles.

6. be able to explain how dilations impact lengths and areas.

7. will be able to prove Pythagorean theorem in several different ways

8. will be understand that all circles and parabolas are similar.

Preparation for next class

• Students

• Instructor: Print the handout Pythagorean theorem

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4.1 Transformations that aren’t as nice - Day 1

1. We worked with distance and angle preserving transformations. Part of doing mathe-matics is asking “What if...?” So, what if we wanted to consider transformation thatpreserve only one of those attributes?

(a) Can you find transformations that preserve distances, but not angles?

(b) Can you find transformations that preserve angles, but not distances?

2. Students can do an exploration in dynamic geometry software, however this is mostly athinking/imagining experiment. Students could argue that SSS criterion for congruenceessentially gives them the proof that distance preserving must be angle preserving.Another venue they might be inclined to explore is using the Pythagorean theorem,and or trigonometry.

3. There are no transformation that belong to the first class, and the only transforma-tions that belong the second category are similarity transformation. The students areunlikely to think about similarity transformations in general. They are more likely tothink about the dilations. If we’ve done a good job of thinking about compositionsthey might go to the compositions of dilations and isometries.

4. This conversation will lead into the discussion of the scaling of distance, and we finallynote that we don’t actually know how to measure the said distance.

5. How do we get the expression for distance? Students might recall the distance formula,and we push them to articulate the source of the formula, Pythagorean theorem. Whatdoes it say?

Students often forget to give precise assumptions - right triangle and what the vari-ables represent.

Include the usual figure of the statement and point out that this is not the actual proofof the theorem.

6. We will discuss numerous proofs of Pythagorean Theorem; Proofs. Students are totake 5 minutes to work on each on their own, then compare their ideas with partnersand write a formal proof. See handout Pythagorean Theorem

7. It would be very helpful to complete at least one proof presentation so that the studentscan produce reasonable proofs in their homework. Especially useful would be one of

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the most obvious proofs. Students are generally very good about arguing algebraicallythat the areas must be equal, however they neglect to show that the shapes are whatthey assume they are (either we start with right triangles and have to show that theresulting shape is a square, or we start with a square and need to show that theresulting triangles are right).

8. Prove the converse of PT: The ancient Egyptians used to form right angles by takinga rope with 12 equally spaced knots. They would form a triangle with sides of lengthof 3, 4 and 5 knots. Why?

9. Preparation for next class

• Students: Homework - Pythagorean Theorem

• Instructor: Note that this is the last module in the course. The students will beworking on their final projects and portfolios and will need some time to completethose. The daily homeworks will become sparse towards the end of the module.Make sure to assign the portfolio and projects with sufficient time for completion.Independent research will be necessary. Print Areas.

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4.1.1 Pythagorean Theorem

1. Adapted from the Chou pei suanching author unknown, circa 200BCE:

2. Bhaskara (12th century)

3. Adapted from Euclid 4. James A. Garfiled (1876), 20th Pres-ident of the US

5. Similarity(a) (b)

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4.1.2 Homework - Pythagorean Theorem

1. Write a formal proof of Pythagorean theorem you worked on in class.

2. Consider the following story:

3. Construct√n for any whole n.

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