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.