MATH 106 Course Packet, Study notes of Geometry

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MATH 106 Course Packet

Version Posted July 1, 2003

Contents

• 1 Problem-solving introduction and basic terms
  • 1.1 Photo Layouts
  • 1.2 Reflection on Teacher Preparation
  • 1.3 A Short Problem Solving Self-Help Checklist
  • 1.4 Problem Report Tips
  • 1.5 Basic Terms
• 2 Area
  • 2.1 Pizzas
  • 2.2 Paper Shapes
  • 2.3 Geoboard Areas
  • 2.4 Measuring a Sector
  • 2.5 Area Formulae
  • 2.6 Picture Proofs
  • 2.7 The Apothem
• 3 Length, area and volume
  • 3.1 Need for Standard Units
  • 3.2 Units of Measurement
  • 3.3 Surface Area
  • 3.4 Length, Area and Volume
  • 3.5 Volume: Eureka!
  • 3.6 Volume: Prisms and Cones
  • 3.7 The Length of a Square
  • 3.8 Perimeter
  • 3.9 The Hungry Cow
• 4 Scaling
  • 4.1 Scaling worksheet
  • 4.2 Changing units
  • 4.3 Reflection on Units
• 5 Angle
  • 5.1 Units of Angle
  • 5.2 Angle Measurement
  • 5.3 Angles of a Polygon
  • 5.4 Trisection
  • 5.5 Olentangy River
• 6 Deductive reasoning
  • 6.1 Pythagorean Theorem
  • 6.2 Rigidity
  • 6.3 Similarities
  • 6.4 Applying Postulates and Theorems
  • 6.5 Proofs
  • 6.6 First Constructions
  • 6.7 Circle Construction
  • 6.8 False Proofs from True
  • 6.9 Postulates and Theorems
  • 6.10 Postulates and Theorems FAQ
• 7 Symmetry and rigid motions
  • 7.1 Symmetry of Planar Figures
  • 7.2 Rigid Motions
  • 7.3 What symmetries are possible?
  • 7.4 Composing Rigid Motions
  • 7.5 Three Flips
  • 7.6 Three Flips, continued
  • 7.7 Tesselations of the Plane
  • 7.8 Symmetries of Solids
• 8 Supplements
  • 8.1 Excerpts from the NCTM Standards,
  • 8.2 Excerpts from the MAA recommendations
  • 8.3 Supplemental readings

Math 106 (5 Credits)

Fundamental Math Concepts for Teachers, II: geometry

Winter 2003

Prerequisite: Continuation of Math 105 Instructor: Robin PEMANTLE Office: 636 Math Tower, 292-1849, pemantle@math.ohio-state.edu Office hours: M 2:30, Tu 11:30 by appointment, and any other time by appointment

Co-Instructor: Vic FERDINAND Office: 336 Math Building (connected to Math Tower), 292-2074, ferdinand@math.ohio-state.edu Office Hours: MWF 2:30 and by appointment

Text: The only text is the course packet available at the University Bookstore. It is packaged with other equipment that you will need (compass, protractor, graph paper, and so forth); please buy the packet ASAP.

Grading: Your grade in this course will be based on:

Attendance and participation 15%. Being late or leaving early counts as half an absence. Each of the 48 days (10 weeks minus 1 midterm and 1 holiday) day counts as 1/3 of 1%. That means anyone who misses at most 3 days can get 100% of the grade for attendance, assuming their participation is satisfactory. Because we allow you 3 absences without penalty, we will not have make-ups. Written work (45%) including problem writeups, group writeups, reflec- tions and quizzes. For your problem writeups, you will be on four aspects:

1 Problem-solving introduction and basic terms

1.1 Photo Layouts

Billie has just bought a digital camera and is planning to put together her photos into a scrapbook with square pages. She will be able to put photos of different sizes together, but needs all the photos to be the same shape or else she will lose picture quality when uploading the images. Her photo software does allow her to rotate the photos 90◦^ if necessary, and to expand or shrink by any factor. Her goal is to have the photos on a page fit exactly, so they completely cover the page and don’t overlap. She also refuses to put more than four on any page.

Find all the aspect ratios she could choose for her photos so that some number of photos of that shape, but no more than four, will exactly fit on a square page. Once you have found these, which among them is closest to the aspect ratio of a standard 312 × 5 photograph? Shown below is an example of how, if she were willing to fit five photos on a page, she could use the aspect ratio 2.

2 inches

2 inches

4 inches

1.3 A Short Problem Solving Self-Help Checklist

1. Understand the problem. It is foolish to answer a question you do not under- stand. It goes without saying that you need to understand all the terminology. In addition, you need to know what is given. Make a sketch, if appropriate, to see that all the given data makes sense. Also, be sure you understand what you are supposed to find or determine. A good way to check this is to ask yourself whether you could verify the answer if someone gave it to you. For a true/false or a prove/disprove question, ask what would constitute a counterexample. You should be especially aware of whether you are trying to show that something al- ways holds, or whether you are you are being asked to find a specific case where something holds. Another question to ask yourself in this phase is whether the data really determine the unknown. Is there really a solution? More than one solution?
2. Devising a plan. If a road to the solution occurs to you naturally, you don’t need this checklist. Assuming you find yourself somewhat stuck, here are a few general procedures you can follow.

(a) Try a few examples. Trial and error is the single best problem-solving method. If you are supposed to find the relation between coordinates in a before and after picture, try listing a few pairs of points and looking for a pattern. If you are supposed to find all semi-regular polyhedra, start listing them. If there are variables in the problem, replace them with numbers and see how the problem goes then. (b) Work from both ends. When you cannot deduce any more from what is given, work backwards from the goal, for example: if you are supposed to find the area of something, perhaps you see that part of it is a square with known side, so you just need the area of the rest; perhaps after you do this a few times, you will see that what remains is a familiar shape which you had not recognized was embedded in the problem before. (c) Can you solve a special case? Perhaps if you assume that a triangle is isosceles, or that a certain angle is a right angle, you can solve it. Perhaps if you find an analogous two-dimensional problem and solve that, you will gain insight into the three-dimensional problem you are actually trying to solve.

(d) Use physical intuition – after all the main reason for learning geometry is to develop and harness physical intuition. See if you can imagine the line segments in a geometry problem as made of steel or wood, the vertices as flexible joints, the two-dimensional components as sheets of stiff material with weights proportional to their areas,

3. Carrying out the Plan. If you are using variables, make sure you understand what they mean. If you get stuck, ask yourself whether you need any more variables. Once you have an idea you think is right, and are trying to prove it, examine why you believe it. Probe at it: why can’t this line segment be longer than that other one? How would that contradict the data given? Try to recall facts and theorems you already know, so you don’t have to prove every little bit from scratch. In order to have a better sense of what assertions require proof, try to think of situations where a similar-sounding assertion might be wrong. The reason we wait until the worksheet “False from True” to have you rigorously prove what you discover about the Pythagorean Theorem is that we want you first to see similar arguments that have flaws, then to prove those flaws do not arise in your argument.
4. Checking It Over. You should always ask yourself whether the answer is plausible. In addition, you should ask whether you used all the data. If you did not, perhaps you made a mistake, or perhaps the problem did not require all the data; in this case, you should try to understand why some of the data was irrelevant, and mention this in your report. If you used some algebra you are not sure of, test it out on a calculator with some numerical examples. If you used variables, make sure you believe any equations you wrote relating them. If you are able to solve a problem more general than the one stated, please by all means include this in your writeup.

• Here’s what I’m going to tell you about
• Content
• This is what we can come away with

and not

• Here’s what I’m going to tell you about
• Content
• Here’s what I just told you
• Use paragraphs, one for each idea. It’s not a rule, but usually you should have more than one paragraph per page.
• A good report is exhaustive but not verbose, and could be understood by a friend who is not in this class.

1.5 Basic Terms

The list of geometric vocabulary on the next few pages is mean to serve as a common base for class discussions. Some of the terms you may already know; these you may ignore. Others you may have some idea of but lack a precise definition; for these, you will want to look at the definitions. Some you will never have heard of; after reading the definitions you may want to discuss these in class.

I will tell you in class which terms you need to learn, a day or two in advance of when we will be using them. So there is no need to go over the whole list now. Rather, this list is meant to serve as a reference for you. Many of the definitions you need may also be found in Supplements A and C, and in the other materials at the end of this packet. Chapter 12 of Burger/Musser/Peterson, which many of you are familiar with, contains some definitions as well.

In addition to the basic terms, there are a few standard notations I would like to point out. If A and B are points, then we use AB for the line segment between them. Sometimes this is abbreviated as AB. We use

AB for the ray from A through B and ←→ AB for the line through A and B. We use 4 ABC for the triangle with vertices A, B and C, and 6 ABC for the angle made by the rays

BA and

BC. We use the symbol ∼= for congruence. Although some people maintain a distinction between congruence

and equality, in this class you will be permitted to say that two line segments with the same lengths are equal (for instance, AB = CD) or that two angles with the same measure are equal (for instance, 6 ABC = 6 DBC).

VOCABULARY

• acute angle: an angle that is less than 90◦. (Euclid first defined it as an angle that lay inside a right angle.)
• adjacent angle: two angles are adjacent if the two rays that bound the second angle include one of the rays that bound the first angle and one ray that does not bound nor lay inside the first angle.

• circumscribed circle: given a polygon, if there is a circle passing through all the vertices of the polygon and containing the polygon in its interior, this circle is called the circumscribed circle.
• collinear points: a set of points all lying on some line. Thus any two points are collinear, but only certain sets of three points are.
• complementary angles: Euclid defined these as adjacent angles that togther form a right angle. Once we have defined degree measure, we may define com- plementary angles as any angles summing to 90◦.
• composition of rigid motions: to do one rigid motion, then another, is to com- pose them, and the resulting rigid motion is called the composition.
• cone: a finite cone is defined like a pyramid, except that any plane figure may be used instead of a polygon. A cone is called a circular cone if the plane figure is a circle; the common usage of the word cone usually assumes it is a circular cone. An infinite cone is the set of points in space that lie on any ray passing through a given circle and emanating from a given pont not in the plane of the circle.
• congruent: two objects are congruent if their points can be placed in one-to-one correspondence so that the distance between two points of one object is always the same as the distance between the corresponding two points of the other. One way to test this is to see if one object can be picked up and placed on the other so the two match perfectly. For two-dimensional objects, this is a good test, but for three-dimensional objects, one ust be willing to pass an object through parts of another to do this test, and one must allow mirror reflection (which in two dimensions is just turning the object over). Once we have defined rigid motions, we may say that two objects are congruent if there is a rigid motion transforming one into the other.
• convex: a plane figure is convex if any line segment whose endpoints are con- tained in the figure is wholly contained in the figure. The same definition may be used to define convexity of an object in space.
• coplanar: a set of objects is said to be coplanar if there is one plane containing them all.
• cylinder: the technical definition is more general than the one in common usage. A cylinder is any three dimensional object consisting of two congruent bases and

the lines connecting corresponding points of the two bases, provided the two bases are subject to the conditions of the two bases in the definition of a prism, though not necessarily polygons. The cylinder is a right cylinder if one base lies directly above the other. The common usage of the term “cylinder” is to mean a right circular cylinder.

• decagon: a polygon with ten sides.
• dihedral angle:
• dilation: a dilation by a factor of x with center P is a transformation mapping each point Q to a point on the ray

P Q that is precisely x times as far from P as was the original point Q. So for example, a dilation by a factor of two doubles all lengths.

• distance: the distance between any two points may be defined as the length of a straight line between them. Conceptually, it may be preferable to think of distance as defined even when a straight line cannot be drawn between the points, but we will not give a definition of this kind since it requires coordinate geometry.
• dodecagon: a polygon with twelve sides.
• dodecahedron: a regular dodecahedron is a regular polyhedron with twelve pentagonal faces.
• edge: see the definition of polygon.
• equiangular: a polygon is equiangular if all interior angles are equal.
• equilateral: a polygon is equilateral if all edges are equal.
• face: see the definition of polyhedron.
• flip: synonym for reflection in two dimensions.
• flip-glide: a rigid motion which is a flip composed with a translation along the axis of the flip.
• hexagon: a polygon with six sides.
• hypercube: a four-dimensional figure analogous to a cube.

• octahedron: a regular octahedron is a regular polyhedron with faces that are equilateral triangles.
• order of a rotational symmetry: if P is a center of rotational symmetry of some object, then there is a minimal angle of rotation possible before the object is mapped to itself. The order of rotational symmetry about P is the number of times the symmetry of minimal angle must be composed before each point is moved back to where it was initially. An analogous definition holds for the order of a rotational symmetry about an axis in three-dimensional space.
• orientation-preserving: a rigid motion of the plane is orientation-preserving if it can be done by composing translations and rotations. Informally, one must be able to carry it out without “picking up the figure and flipping it over”. A rigid motion of space is orientation-preserving if it does not mirror-reverse objects. Informally, again, it can be carried out without the object being “picked up into four-dimensional space and flipped over”.
• orientation-reversing: oppostie of orientation-preserving.
• parallel lines: coplanar lines that do not intersect.
• parallelogram: a quadrilateral whose opposite (nonintersecting) sides are par- allel.
• pentagon: a polygon with five sides.
• perpendicular bisector: the perpendicular bisector of a line segment is the line passing through the midpoint of the segment making a right angle with the segment.
• perpendicular lines: lines making a right angle where they cross.
• plane figure: anything that can be drawn in a bounded (finite) region in a plane.
• point: this is taken by Euclid to be one of the undefined terms. Informally we mean a dot representing a single point in a plane or in space. Euclid called it “that which has no length width or breadth”.
• polygon: a polygon can be defined in terms of its edges, or in terms of the region it covers. Definition 1 (in terms of edges): a polygon must have at least 3 edges (sides); if it has n edges, it is called an n-gon. The edges must form a closed

chain, meaning that there are n points, call them X 1 ,... , Xn, and the edges are the line segments X 1 X 2 , X 2 , X 3 ,... , Xn− 1 Xn, XnX 1. A polygon is a plane figure, so all the points X 1 ,... , Xn, which are called the vertices of the polygon, must lie in one plane. Furthermore, no two of the vertices can be equal, and no two of the edges may cross – the only point they may share is their common endpoint. Definition 2 (in terms of the region covered): a polygon is a region in a plane, whose boundary consists of some number n, at least 3, of line segments, meeting the criteria in Definition 1.

• polyhedron: this term is very difficult to define precisely. We will settle for an imprecise definition. A polyhedron is a finite figure in space, such as a cube or prism, whose boundary is made of polygons, called its faces. The rules for the way faces are allowed to intersect are not commonly agreed upon, but faces must meet each other at their edges and vertices only, and in the end, the polyhedron must have an inside and an outside. The plural is polyhedra.
• postulate: same as axiom.
• prism: To construct a prism, begin with two parallel, unequal planes. We will think of these planes as defining the horizontal direction, so we may talk of the “upper” and “lower” plane. First, the bases of the prism must be two congruent polygons, one in each plane. The upper polygon may lie directly above the lower polygon (in which case the prism is said to be a right prism), or may be moved from this position to another in the upper plane as long as it is not rotated. The remaining sides are gotten as follows. For each edge XY of the lower base, there corresponds an edge X′Y ′^ of the upper base. The rules by which this is constructed ensure that all four points X, X′, Y and Y ′^ lie in one plane, so they form a quadrilateral XY X′Y ′^ which is then taken to be a face of the prism. A triangular prism is a prism whose bases are triangles, and this kind of phrasing can be used for other base shapes as well.
• pyramid: Given a polygon and a point not in the plane of the polygon, the pyramid they form is the set of all points on line segments one of whose endpoints is the given point and the other of whose endpoints is a point in the polygon. The given point is said to be the apex of the pyramid.
• quadrilateral: a polygon with four sides.
• ray: a half line, including the point at which it starts; the ray is said to emanate from this point.