Spring 02


First Week (Problem Solving)

You learned how to work on a problem for which both methods and precise goals were not given. You began to learn to discuss and write mathematics in a comprehensive manner. Specific elements included addressing when you were not able to prove some aspects of your answers (on both The Cube and Photo Layouts), using algebra on a geometry problem (Photo Layouts), and understanding some new terminology (the term "aspect ratio" was new to most of you, as were "upper bound" and "lower bound").

A second emphasis was on meeting some of the new recommendations and standards. Your reflections were meant to give you a better understanding of what was being asked of future teachers and why. Your specific responses, which I don't consider to be right or wrong, raised some issues which I addressed in my posted response to your reflections.

We discussed some vocabulary words as well, after which I decided you will need some definitions, which will be provided soon.


Second Week (Area)

Most of you already knew all the "facts" you needed about area. While the Area Formulae worksheet outlined some possible further formulaic knowledge, the content of the unit on area was mainly on how to better understand, explain, develop and apply knowledge. Thus, very few of the formulae you found were deemed sufficiently important to remember, including the one formulae you re-discovered and derived in the Apothem worksheet.

In the Pizza problem you learned when to apply your knowledge of area to a physical problem.

In Geoboard Areas and Measuring a Sector you learned where our knowledge about area comes from, and how you might build that up when you teach it.

In Picture Proofs, you learned about how we use basic facts about area (area formulae for simple shapes such as rectangles) together with our conceptual understanding of area (conservation under rearrangement) to build more advanced knowledge.

In The Apothem you put the methods you learned in Picture Proofs into practice and derived a fairly advanced formula for the area of a regular polygon. You then applied it to computing the area of a circle.

Next week, we start the unit on Length, Area and Volume, which covers aspects of quantitative measurement in all three dimensions.


Third Week (Length, Area and Volume)

A major theme for this week was dimensionality: what things are 1, 2 and 3-dimensional, and how do we measure size in each of the three corresponding ways. We concentrated on formulating and applying good defintions: definitions that are more informative than the definition in the dictionary or in a textbook.

In Length, Area and Volume, you learned how to generalize the notions of length and distance to curved settings. For area this is harder and I only hinted at how it might be done; nevertheless, this is an important tool for understanding length and area measures as something more that "the length is what my ruler tells me".

In The Length of a Square, you found that an object can have a measure in more than one dimension: the string had length, but it also could be viewed as having area.

Many children believe area and perimeter to be almost the same thing: although the measures are different, they believe one increases as the other does, and that they tell you the same thing. The Perimeter worksheet, was designed to illustrate one path to curing this misconception: seeing one of the two quantities vary while the other remains fixed. We also practiced using a graph to bolster an argument.

In Volume: Eureka, we took our first crack at the notion of scaling: how does one quantity change when I double another? This has a lot to do with the meaning of dimension, and of the notion of proper multiplication of physical units in an equation, which we have touched on before. We will come back to this again in the Scaling worksheets and those immediately following.

In Volume: Prisms and Cones, you learned to apply the knowledge you found in Part 4 of Volume: Eureka, by identifying various physical objects as prisms.

Your take-home assignment on Surface Area was meant to give you a chance to complete a write-up completely outside of class and to check that you know the definition of the surface area of an object.


Fifth Week through Monday (Review of units; beginning angle)

Because of the midterm we spent time reviewing units. We did also get to begin working on angle. The review of units centered around how measurements change when either the object changes by a scale factor, or the units used to measure the object change. It turned out, on the midterm exam, that fewer than half of you absorbed this sufficiently well to apply it to a problem with a 1:50,000 scale map.

In the Mental Math problem about the Tower of Pisa, we reviewed how volume and area changed with a change of scale factor. When the question was asked in terms of weight rather than volume, most people were unsure of what dimension the quantity was. You did better on the second question, realizing that the quantity of paint was measuring surface area, a two-dimensional quantity.

The Tower of Pisa also demonstrated that angle is dimensionless - it is unaffected by changes of scale. More importantly, the converse is true: if all angels are preserved then figures are proportional. We discussed the generality in which this was true: all angels of triangles must be preserved, not just angles of, say, quadrilaterals in the figure.

In Measuring angles, we discussed the fine points of protractor use, and the conceptual basis for it. Analogously to the videotape of linear measure, there are hidden points: alignment of the device; extension of lines when necessary; knowledge of special angles such as right angles and straight angles; the fact that adjacent angles add; ability to choose between the two sets of numbers on the protractor.

We also agreed on a definition of dihedral angle: the angle between two polygons in different planes that share an edge is the angle between the planes containing the polygons. This is called the dihedral angle and is defined as the angle between two rays, one in each plane, both emanating from the same point on the line of intersetion, and each perpendicular to the line of intersetion.


Sixth Week (Congruent and similar triangles)

This week was spent on the identification and uses of congruence. We also spent a little time on coordinate geometry.

The Olentangy River problem developed the idea of indirect measurement. This is one of the main uses of congruence. When you want to measure a distance but there are obstacles to direct measurement, one must figure out the distance via an alternative measurement that can more easily be carried out. One way is to create a new length congruent to the old and measure that. To do this, one needs a method for constructing the congruent length, a means of measuring that, and a justification for the fact that it is congruent. Another way is to create a new shape that is similar to the old one. One needs then to measure and justify, but also to measure the proportionality constant. One way to do this is to measure some pair of corresponding parts of the two figures. The worksheet caused you to think of many ways to do this. In addition, we discussed how the introduction of trigonometry was nothing other than a formalization of facts you already knew concerning proportions arising in similar triangles.

Rigidity was all about how to identify congruence: when are two triangles or two more general figures congruent? You have read about various congruence theorems, such as SSS and ASA. The Rigidity worksheet explored the physical reasons behind these theorems, which is important since we will use them but not spend class time proving them (proofs may be found in geometry textbooks). Harnessing your physical intuition is a very important component to learning, in all of mathematics but particularly in geometry.

Trisection investigated what kinds of facts you CANNOT conclude about congruence and similarity.

The Challenger II worksheet served as a checkpoint: do you understand the standard conventions for the use of rectangular coordinates, and can you find and express a rudimentary formula for a change of scale and orientation? It also allowed you to explore the relation between the geometry and algebra of dilations and rigid motions.


Seventh Week (Deductive reasoning)

We began the unit on deductive reasoning. The early worksheets were designed to make you feel the need for some reliable way of knowing that an argument is sound.

The worksheet on rigidity explored intuitive notions of whether a figure has degrees of freedom, versus whether the shape is completely determined by the information you have. You figured this out for many scenarios involving triangles. This is intended to show you that the list of theorems (SAS, SSS, and so on) about congruent triangles is not arbitrary, and that you could reconstruct it if you had to. The last problem taps your physical intuition, showing one reason that congruence is an important physical concept.

The Pythagorean Theorem worksheet showed you something most people don't remember ever seeing: a proof of the Pythagorean Theorem. The False Proofs worksheet showed you a very similar but bogus proof. Figuring out what is wrong with the bogus proof is an important step in guarding against it. Now we want you to be worried enough about whether the Pythagorean proof is bogus so that you go back to it and try to make it airtight.

Applying Postulates and Theorems was a quick check on whether you understood how to apply something like the SAS theorem. You all did. :)

The Proofs worksheet was the first where you had to use the Euclidean axioms and theorems. The key to success here is to figure out why you believe something is true: usually the formal proof stems from an intuitive reason. You also had to add points or lines to the given figure, and prove some intermediate results. This challenges (and improves) your problem-solving skills. Please make sure you know how to write your proof properly once you have it! You get a chance to practice on this worksheet, then use what you learned again on First Constructions.

You began First Consructions, but we'll delay the recap of that until next week.


Eighth Week (Constructions)

We spent most of this week on the dual tasks of inventing constructions to accomplish certain geometric tasks, and proving that these and various other constructions do what they claim. The hardest part, for most of you, was specifiying how new points and lines were added to your figure. If you overspecify, you end up assuming something that requires proof. For example, if you say to "draw a line through points A, B and C", then you need to have already shown that A, B and C are collinear. The creative aspect is also difficult -- figuring out how you can accomplish the task -- but most of you in the end did quite well at that.

In First Constructions, you determined how to accomplish some of the basic moves that we often refer to as obvious: copy, bisect, construct a perpendicular or parallel, and so forth. I haven't graded these yet, but I hope that you got from this an idea of how rigidity can replace measurement as a device for creating diagrams. It is interesting to note that for the ancient Greeks, bisecting an angle by measuring, dividing by two, and drawing with a protractor was considered far more unreliable than a careful straightedge and compass construction. For one thing, they didn't have good protractors (no mass production), and for another, a protractor involves rounding to the nearest degree. Another thing you should come away with is the idea that every construction, if it is to be useful, goes along with a verification. This is true of every aspect of math and science.

The other major project for the week was a verification that a parallelogram may indeed be rearranged to form a rectangle with the same base and altitude. There were three different constructions, leading to three different proofs. By the end, every group got all or nearly all the steps, including some lemmas not apparent at first. We hope the experience of determining all these steps and logically ordering them so as to make an airtight proof will be a valuable experience reinforcing your geometric intuition.


Ninth Week (Rigid motions and symmetries)

This week we studied rigid motions. These are very important for several reasons: (1) you need this concept to define symmetry; (2) the vocabulary associated with them is very helpful in describing design and mechanical motion; (3) viewing planar and spatial motion as a function helps to mathematize pre-existing intuitive notions of space. A symmetry of an object is a rigid motion that "leaves the object the same", or in other words, that returns the object to the exact same space it previously occupied. Symmetry is a common mathematical topic at all grade levels.

The worksheet Symmetry of planar figures was intended mainly to convey by example what the three types of symmetry are, and which symmetry properties are compatible with which others.

Rigid Motions explored more deeply each type of symmetry. For each one, you had to understand it algebraically (as a function mapping points to algebraically related points), constructively (through straightedge and compass construction) and as it relates to the dissection of a familiar figure. Also, you needed to learn how to reverse the process of consructing a rigid motion: you had to name a given rigid motion and you had to identify a rigid motion that produced a given result.

In What Symmetries are Possible you found out how different types of symmetry are related by finding when the existence of two symmetries causes a third symmetry to arise automatically. This is a lead-in to the composition of rigid motions (the science of computing the result of doing one rigid motion then another), a topic we will not adequately cover this quarter.

In Symmetries of Solids, you continued hunting for symmetries of objects, only now in three dimensions. It is a lot trickier to visualize and to count in a methodical way in three dimensions, so in addition to learning about three-dimensional symmetry, this strengthens your spatial reasoning skills.


Winter 03


First Week (Problem Solving)

You learned how to work on a problem for which both methods and precise goals were not given. You began to learn to discuss and write mathematics in a comprehensive manner. Specific elements included addressing when you were not able to prove some aspects of your answers (Photo Layouts), using algebra on a geometry problem (Photo Layouts), and understanding some new terminology (the term "aspect ratio" was new to most of you). and "lower bound").

A second emphasis was on meeting some of the new recommendations and standards. Your reflections were meant to give you a better understanding of what was being asked of future teachers and why. Your specific responses, which I don't consider to be right or wrong, raised some issues which I addressed in my posted response to your reflections.


Second Week (Area)

Most of you already knew all the "facts" you needed about area. While the Area Formulae worksheet outlined some possible further formulaic knowledge, the content of the unit on area was mainly on how to better understand, explain, develop and apply knowledge. Thus, very few of the formulae you found were deemed sufficiently important to remember, including the formula from the Apothem worksheet and the one from the worksheet on measuring a sector. worksheet.

In the Pizza problem you learned when to apply your knowledge of area to a physical problem. The idea of how area scales made its first appearance. You all did pretty well on this problem, immediately recognizing the importance of using area for the proportionality even if you first tried linear proportionality.

In Geoboard Areas and Measuring a Sector you learned where our knowledge about area comes from, and how you might build that up when you teach it.

In Picture Proofs, you learned about how we use basic facts about area (area formulae for simple shapes such as rectangles) together with our conceptual understanding of area (conservation under rearrangement) to build more advanced knowledge.

Next week, we start the unit on Length, Area and Volume, which covers aspects of quantitative measurement in all three dimensions.


Third Week (Length, Area and Volume)

A major theme for this week was dimensionality: what things are 1, 2 and 3-dimensional, and how do we measure size in each of the three corresponding ways. We concentrated on formulating and applying good defintions: definitions that are more informative than the definition in the dictionary or in a textbook.

In The Apothem you put the methods you learned in Picture Proofs into practice and derived a fairly advanced formula for the area of a regular polygon. You then applied it to computing the area of a circle.

In Length, Area and Volume, you learned how to generalize the notions of length and distance to curved settings. For area this is harder and we only hinted at how it might be done; nevertheless, this is an important tool for understanding length and area measures as something more that "the length is what my ruler tells me".

Many children believe area and perimeter to be almost the same thing: although the measures are different, they believe one increases as the other does, and that they tell you the same thing. The Perimeter worksheet, was designed to illustrate one path to curing this misconception: seeing one of the two quantities vary while the other remains fixed. We also practiced using a graph to bolster an argument. Some of you may have seen how to prove your answersw to parts 2 and 3 of this problem in a way that is pretty much airtight. We hope that everyone had a chance to appreciate this idea, even if they did not come up with it on their own.

The worksheet on Surface Area dealt with the same theme only in two and three dimensions rather than one and two dimensions: three-dimensional objects have surface area as well as volume, these two measures are independent, and compact shapes such as cubes and balls have the most volume per surface area.

In Volume: Eureka, we took our first crack at the notion of scaling: how does one quantity change when I double another? This has a lot to do with the meaning of dimension, and of the notion of proper multiplication of physical units in an equation, which we have touched on before. We will come back to this again in the Scaling worksheets and those immediately following.

Fourth Week (Scaling)

After this week you are supposed to understand how measures in various dimensions change when you change the scale of an object. This is one of the fundamental pieces of knowledge in the course. The other major theme for the week was the understanding of how one might measure areas and volumes of irregular shapes, and how one might develop formulae for some classes of not so irregular shapes.

In Volume: Prisms and Cones, you learned to apply the knowledge you found in Part 4 of Volume: Eureka, by identifying various physical objects as prisms. An important class discussion concerned the reason that the ``area times height'' formula works for some objects, and exactly which objects it works for. We would like you to understand and remember the argument about solid objects being like reams of paper, whose volume does not change when the papers are shifted to form a leaning tower.

In The Length of a Square, you found that an object can have a measure in more than one dimension: the string had length, but it also could be viewed as having area. Another important message in this worksheet is that keeping track of the units of your answer gives a clue as to what you are computing and whether your computation is valid.

The Scaling worksheet occupied the remainder of the week. The first two problems emphasized putting quantitative knowledge about scaling into words and finding airtight proofs using algebra. The third problem was similar, and was intended as well to elicit a pictorial proof to compare to the algebraic proofs from problems 1 and 2. Neither the algebraic nor the "9 shapes fit together" argument works on Problem 4, so this part forces you to come up with a new argument. Please remember these arguments -- they are very important to your understanding of two-dimensional scaling and your ability to explain it to others. The two arguments that are easiest to use are (1) the "nine times as many little squares" argument and the "each little square increases in size by 9" argument (use the first if you chose for the grid size to remain the same when the object grew, and the second if your grid expanded along with the object).

The Hungry Cow assignment was there mainly to give you a chance to work on a problem entirely outside of class. It reinforced the notion of computing areas by adding together areas of pieces which can be put together to make up a whole. A significant part of the task was to understand the set-up and what was being asked, on which you did generally quite well.