Section 10-7: Volumes of Pyramids and Cones (Day 138)

Today is an official state holiday here in California, but it's one of the least known holidays. Today is Cesar Chavez Day, to celebrate the 20th century farm worker and labor leader.

Some schools -- and most notably universities -- here in California observe Chavez Day. As I mentioned earlier, there's a trend to sever the link between spring break and Easter, in order to avoid having the school holidays be tied to a holiday that can vary by over a month like Easter. In the University of California and California State University systems, spring break is tied not to Easter, but to Chavez Day.

Notice that for schools whose the first day of school is after Labor Day, that end-of-summer holiday is the most important holiday on the calendar. The first day of school is either early or late depending on whether Labor Day is early or late, which means that the last day of every quarter and trimester, including the last day of school, is early or late depending on the previous Labor Day date. In 2014, Labor Day fell on its earliest possible date, September 1st, while this year, Labor Day will fall on its latest possible date, September 7th. This means that dates for the end of each semester will fall later during the 2015-6 school year than in the 2014-5 school year. Schools on an Early Start calendar may choose a different holiday to be the most important holiday on the calendar. For example, some Early Start schools are set up so that the last day of school is before Memorial Day, so that holiday is the most important. Others set up the year so that a fixed number of weeks occur before winter break, thereby making Christmas the most important holiday. The LAUSD Early Start calendar appears to have a fixed number of weeks before winter break -- but winter break itself is defined to begin three weeks after Thanksgiving. I won't know for sure until the holiday reaches its earliest possible date again on November 22nd, 2019, but it appears that Thanksgiving is the most important holiday on the LAUSD academic calendar.

Well, the most important holiday for the UC and Cal State college is Chavez Day. For example, at my alma mater, UCLA, the school year is divided into three quarters -- fall, winter, and spring. Spring break contains the observed Chavez Day and separates winter quarter from spring quarter. Since each quarter has a fixed length (ten weeks plus a finals week), and the length of the breaks is also fixed, all the dates in the UCLA academic calendar can be determined starting from Chavez Day.

There is a slight difference between the Cal State and UC calculations of the Chavez Day date. In the Cal State system, Chavez Day is on the labor leader's actual birthday, March 31st, and so spring break is the week containing the last day of March. Since most Cal States operate on a semester system, the spring break week is actually halfway during the spring semester, so that midterms can be given the week before the holiday. But at UC's -- well, at least at UCLA -- Chavez Day is defined to be the last Friday in March -- so that this year, March 27th was Chavez Day Observed. This means that at the Cal States, this week is spring break, while at the UC's, spring break was last week (just as it was on the blog) and spring quarter has already begun. Notice this means that there are often classes held at UC on the labor leader's actual birthday on the 31st, while there is never school this day at Cal State. I point out that in either case, it's more convenient, especially at UC, to tie spring break to Chavez Day rather than to Easter, in order to avoid quarters of differing lengths in years when the Christian holiday is exceptionally early or late.

LAUSD also observes a Chavez Day holiday. It is observed on the actual date, March 31st, in some years, while in others it's moved to the nearest Monday or Friday. In years in which March 31st is close to Easter -- recall that the district's spring break coincides with Holy Week -- Chavez Day is pushed to Easter Monday. So this year, LAUSD observes Chavez Day on Monday, April 6th -- six days after the labor leader's actual birthday and ten days after the UCLA holiday.

Two years ago, Easter actually fell on Chavez Day. There was a Google Doodle that day celebrating Cesar Chavez -- which angered some Christians expecting to see a Google Doodle for Easter. I point out that Chavez was a devout Catholic, and so one could argue that Chavez himself would have rather seen Google celebrate Easter than himself.

Section 10-7 of the U of Chicago text is on the volumes of pyramids and cones. And of course, the question on everyone's mind during this section is, where does the factor of 1/3 come from?

The U of Chicago text provides two ways to determine the factor of 1/3, and these appear in Exploration Questions 22 and 23. Notice that without the 1/3 factor, the volume formulas for pyramid and cone reduce to those of prism and cylinder, respectively -- so what we're actually saying is that the volume of a conic surface is one-third that of the corresponding cylindric surface. So Question 22 directs the students to create a cone and its corresponding cylinder and see how many conefuls of sand fill the cylinder. The hope, of course, is that the students obtain 3 as an answer. This is the technique used in Section 10.6 of the MacDougal Littell Grade 7 text that I purchased last weekend as well.

But of course, here in High School Geometry, we expect a more rigorous derivation. In Question 23, students actually create three triangular pyramids of the same base area and height and join them to form the corresponding prism, thereby showing that each pyramid has 1/3 the prism's volume. But this only proves the volume formula for a specific case. We then use Cavalieri's Principle to show that therefore, any pyramid or cone must have volume one-third the base area times height -- just as we used Cavalieri a few weeks ago to show that the volume of any prism, not just a box, must be the base area times height.

I decided not to include either of the activities from Questions 22 or 23. After all, there was just an activity yesterday and I wish to avoid posting activities on back-to-back days unless there is a specific reason to, such as during CAHSEE week.

But instead, I do wish to honor Cesar Chavez in one of the questions. Notice that Chavez dropped out of school early in order to work in the fields, so he most likely never made it to Geometry class. Still, Chavez was of Mexican descent, and there just so happens to be a question from this section of the text in which Mexico is mentioned. (A recent movie features a group of students working the fields in California at about the same age that Chavez once did, albeit some four decades later. But these students notably didn't drop out -- instead, they attended McFarland High School.)

In Question 11, we learn that the largest monument ever built is the Quetzalcoatl at Cholula de Rivadabia, a pyramid about 60 miles southeast of Mexico City. So in particular, the Quetzalcoatl is larger than any of the more famous Egyptian pyramids mentioned in yesterday's lesson. (But apparently, the Quetzalcoatl is well-known enough to avoid being flagged as a spelling error here on the Blogger editor.) Naturally, the students are asked to determine the volume of this large pyramid.

Question 15 invokes one of the simplest solids of revolution -- a BC Calculus staple. Of course, we only need to use the cone formula to find the volume and not anything from Calculus.

Finally, in the lesson I incorporated some of Section 10-6, "Remembering Formulas." I like the idea of having a section devoted solely to remembering these formulas -- since these are notoriously difficult for students to remember. The problem is that the way my lessons are set up, there is precious little time to devote an entire day's lesson just to remembering formulas. But I squeeze it in here by showing the same hierarchy of three-dimensional figures that appear in Section 10-6.

Section 10-2: Surface Areas of Pyramids and Cones (Day 137)

Spring break is over -- for me, at least. The geometry student I tutor actually attends a religious school that takes two weeks off for Easter, so he's still off another week. Yeah, he's lucky!

Last weekend, the county library held its biannual book sale. Usually, book sales are held the first Saturday in April and October, but the spring sale moves up to the last Saturday in March if necessary to avoid Easter weekend.

As I mentioned after the last book sale in October, I often purchase used math textbooks at these book sales, especially if they are texts for classes that I may consider teaching some day. It's been a while since I've last found some geometry texts, but I did see some pre-algebra texts which may be intended for seventh graders. Let me discuss these texts, especially the geometry units.

One of the texts was published by Merrill, the other by McDougal Littell. I ended up purchasing the latter, which is dated 2001. I actually recognize this text from when I spent one month in an advanced seventh grade math classroom back in 2012. Geometry is covered in Chapters 8 through 10. Chapter 8 covers points, lines, polygons, transformations, and similarity. The transformation section covers reflections and translations (but not dilations in the similarity section), but of course, this is an old pre-Common Core text, so transformations aren't used to define congruence. Chapter 9 is officially called "Real Numbers and Solving Inequalities," but the real numbers portion of the chapter segues from square roots to the Pythagorean Theorem and to the Distance Formula.

That takes us to Chapter 10. As it turns out, much of Chapter 10 of this seventh-grade text matches up with the same numbered chapter of the U of Chicago geometry text. Here are the sections:

Section 10.1: Circumference and Area of a Circle

Section 10.2: Three-Dimensional Figures

Section 10.3: Surface Areas of Prisms and Cylinders

Section 10.4: Volume of a Prism

Section 10.5: Volume of a Cylinder

Section 10.6: Volumes of Pyramids and Cones

Section 10.7: Volume of a Sphere

Section 10.8: Similar Solids

It's often interesting to see how much surface area and volume appears in pre-algebra texts. Wee see that this text gives all of the volume formulas, while only the cylindric solids have their surface areas included in the text.

But let's keep in mind that this text was specifically written for the old California state standards that we had before the Common Core -- and these standards are still relevant only in that they appear in the CAHSEE exit exam that students must still take to graduate. Surface area and volume appear on the CAHSEE only because they are technically seventh grade standards. In practice, only boxes and their unions have their surface areas and volumes appear on the test. As we can see, this corresponds to Sections 10.3 and 10.4 on the test. The second half of Chapter 10 doesn't appear on the CAHSEE.

The final chapter, Chapter 12, of this text is on polynomials. This chapter actually goes a bit beyond the seventh grade standards -- most notably, Section 12.5 is "Multiplying Polynomials" and actually teaches the FOIL method of multiplying two binomials. I was only in the classroom that taught using this text for a month, but I was told that the honors class would cover Chapter 12 around the start of the second semester, with the rest of the chapters taught in numerical order. (Non-honors classes would not cover Chapter 12 at all.) The next section, Section 12.6, may also seem a bit advanced for a pre-algebra class -- "Graphing y = ax^2 and y = ax^3" -- but not only does it appear in the seventh grade standards, but also on the CAHSEE. Algebra I teachers may cover the graph of y = ax^2 slightly after the date of the CAHSEE exam, but it nonetheless appears on the test because it's not an Algebra I standard, but a seventh grade standard! Even when we look at the CAHSEE Released Test Questions, we see that y = ax^2 and y = ax^3 appear not in the Algebra I section, but actually the Algebra and Functions (i.e., seventh grade standards) section.

If we compare this to the Common Core Standards, we see that much of Chapter 10 of the McDougal Littell text corresponds to an eighth grade standard in Common Core:

CCSS.MATH.CONTENT.8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

This is to be expected. The Common Core Standards are based on Algebra I in ninth grade, while the California Standards were based on Algebra I in eighth grade. So many eighth grade Common Core Standards must have been seventh grade standards in California.

Before we leave the McDougal Littell text, let me note that Section 4.3 is on "Solving Equations Involving Negative Coefficients." I point this out only because, if you recall, last month I subbed for a sixth grade teacher who unwittingly assigned a worksheet with a few negative coefficients, and the sixth graders were confused -- as they should have been, since this was a seventh grade standard.

I didn't purchase the Merrill Pre-Algebra text, so I don't recall how old the text is. But I glanced at it and noticed that all of the equations that appear in Chapter 10 of the U of Chicago Geometry text also appear in this text, with the exception of the equations involving a sphere. That is, the surface area formulas of all cylindric and conic solids appear in this text. This is unusual since, as we've seen, neither the CAHSEE nor the Common Core Standard expect students to learn the more complex surface area formulas before high school Geometry. Since today's lesson is Section 10-2 of the U of Chicago text, which is on surface areas of pyramids and cones, I want to discuss what I remember about the Merrill lesson on these surface areas.

Both Merrill and the U of Chicago give the lateral area of the pyramid as the sum of the areas of its triangle lateral faces. But only the U of Chicago gives the formula for a regular pyramid, which it defines in Section 9-3 as a pyramid whose base is a regular polygon and the segment connecting the vertex to the center of this polygon is perpendicular to the plane of the base. The formula for the lateral area of a regular pyramid is LA = 1/2 * l * p.

But now we must consider the surface area of a cone. The Merrill text does something interesting here, as it considers the area of the net of the cone. We cut out the circular base and a slit in the lateral region, and then flatten this lateral region. What remains is a sector of a circle. Then the Merrill text simply gives the area of this sector as pi * r * s (where s, rather than l, is the slant height) without any further explanation.

The U of Chicago text, meanwhile, gives a limiting argument for the surface area of the cone, as its circular base is the limit of regular polygons as the number of sides approaches infinity. But there is Exploration Question 25, where the Merrill demonstration is done in reverse -- we begin with a sector of a disk and fold it into a cone.

But neither tells us why the area of the sector (and thus the lateral area of the cone) is pi * r * l. Let me give a demonstration of why the area of the sector is pi * r * l.

We begin with the area of a circle, pi * R^2. The reason why I used a capital R is to emphasize that the radius of the circle that appears in Question 25 is not the radius r of the base -- indeed, it's easy to see that the radius of the circle becomes the slant height l. So the area of the circle is pi * l^2 -- that is, before we cut out the sector. We want to fit the area after we cut it.

Let's recall another formula for the area of a circle given by Dr. Hung-Hsi Wu: A = 1/2 * C * R -- and once again, R = l, so we have A = 1/2 * C * l. But neither one of these gives us the circumference or area of a sector. If we let theta be the central angle of a sector, we obtain:

x = theta / 360 * C

L.A. = theta / 360 * A

        = theta / 360 * 1/2 * C * l

For lack of a better variable, I just let x be the arclength of our sector. But here I let L.A. be the area of the sector, since these equals the lateral area of the cone we seek. The big problem, of course, is that we don't know what angle theta is for the cone to have a particular shape. But we notice that we can simply substitute the first equation into the second:

L.A. = 1/2 * theta / 360 * C * l

        = 1/2 * x * l

And what exactly is the arclength x of our sector? Notice that once we fold the sector into a cone, the arclength of the sector becomes the circumference of the circular base of the cone! And this we know exactly what it is -- since the radius of the base is r, its circumference must be 2 * pi * r:

L.A. = 1/2 * (2 * pi * r) * l

        = pi * r * l

as desired. QED

I incorporate this demonstration into my lesson. As this is the first lesson after a vacation, I wanted to give the students something fun to do -- so let's have them cut out sectors and make cones!

Section 9-5: Reflections in Space

A commenter responded to last post, when I was discussing how the Common Core Standard for polynomials made a hidden reference to the polynomial ring:

What possible value is there in teaching ring theory to high school students, particularly, if by doing so, topics vital to the preparation for calculus are abandoned?

Well, I was discussing the wording of the Common Core Standard, where the strange wording "the polynomials form a system analogous to the integers" appears. I found a commenter on another blog who had something to say about this particular standard, from four years ago:

http://tcmfa.blogspot.com/2011/03/april-6.html

With an eye towards my own high school experience, it's interesting to see a standard like A-APR.1: "Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials." Are students really being asked to see the underlying ring structure, or is it enough for them simply to demonstrate that addition, etc. of two polynomials will produce another? I'm sure I could have done the latter, but explicit, verbal knowledge of the "system" way of seeing things definitely came later.

I decided to check the practice PARCC exams for Algebra I and II to see whether any questions about ring theory actually appear on the PARCC. And, as I expected, there are no questions that ask about this "system analogous to the integers" on the PARCC (and I doubt that there are any such questions on the SBAC either) -- which is all that really matters. So standard A-APR.1 might as well have omitted all but the last five words of the standard:

CCSS.MATH.CONTENT.HSA.APR.A.1
[A]dd, subtract, and multiply polynomials.

The polynomial questions that I saw on that practice PARCC were on the difficult side, and so I definitely feel that some of these questions are flawed. But fortunately, our fears that the Common Core would expect students to master ring theory are unfounded, and we don't have to worry about ring theory taking time away from topics that prepare students

I only brought up ring theory because I wanted to tie the Common Core Standards to the Google Doodle celebrating the birthday of mathematician Emmy Noether, and since it was during a spring break post, I wanted to discuss topics that aren't directly related to what should be taught in an actual high school classroom.

And that takes us into our next spring break topic. I mentioned in the last post that Emmy Noether studied symmetry -- including translational and rotational symmetry -- as applied to physics. We already discussed translations and rotations for Common Core Geometry, but those transformations applied to the two-dimensional plane. The physical world has three dimensions, and so it's the 3D transformations that actually apply to physics.

Section 9-5 of the U of Chicago text is on reflections in space. I find this to be an interesting topic, but since it doesn't appear on the PARCC, it would be a waste of time to cover it in class. There will be no worksheet for this lesson, since it's not intended to be taught in class. This is why I waited until spring break to blog about this topic.

The text begins by defining what it means for a plane in 3D, rather than a line in 2D, to be a perpendicular bisector:

"In general, a plane M is the perpendicular bisector of a segment AB if and only if M is perpendicular to AB and M contains the midpoint of AB."

Now we can definition 3D reflections almost exactly the same way we define 2D reflections:

"For a point A which is not on a plane M, the reflection image of A over M is the point B if and only if M is the perpendicular bisector of AB. For a point A on a point M, the reflection image of A over M is A itself."

If you think about it, when you (a 3D figure!) look at yourself in a mirror, the mirror itself isn't a line, but rather a plane. Mirrors in 2D are lines, while mirrors in 3D are planes. So now we can define what it means for two 3D figures to be congruent:

"Two figures F and G in space are congruent figures if and only if G is the image of F under a reflection or composite of reflections."

Most texts don't actually define what it means for two 3D figures to be congruent. We know that the traditional textbook definition, that congruent figures have corresponding sides and angles congruent, only applies to polygons. It doesn't even apply to circles, much less 3D figures. But we can simply use the Common Core definition -- two figures are congruent if and only if there exists an isometry (i.e., a composite of reflections) mapping one to the other -- and it instantly applies to all figures, polygons, circles, and 3D figures.

In 2D there are only four isometries -- reflections, translations, rotations, and glide reflections. An interesting question is, how many isometries are there in 3D?

Well, for starters, translations and rotations exist in 3D. We can define both of these exactly the same way that we do in 2D -- a translation is the composite of two reflections in parallel planes, while a rotation is the composite of two reflections in intersecting planes.

Notice that every 2D rotation has a center -- the point of intersection of the reflecting lines. The same thing happens in 3D, except that the intersection of two planes isn't a point, but a line. Therefore, a 3D rotation has an entire line as its center -- every point on this line is a fixed point of the rotation. But usually, instead of calling the line the center of the rotation, we call it the axis of the rotation. One 3D object that famously rotates is the earth, and this rotation has an axis -- the line that passes through the North and South Poles. Confusingly, the mirror of a 2D reflection is often called an axis -- but in some ways, these two definitions are related. One can perform a 2D reflection by taking a 2D figure and rotating it 180 degrees about the axis in 3D.

Glide reflections also exist in 3D -- although these are often called glide planes in 3D. A glide reflection is the composite of a reflection and a nontrivial translation parallel to the mirror. Notice that there are infinitely many directions to choose from for our translation in 3D, whereas if this were a 2D glide reflection there are only two possible directions for a translation that's parallel to the mirror.

But are there any other isometries in 3D? Well, we notice that a glide reflection is the composite of two other known isometries, a reflection and a translation. So the next natural possibility to consider is, what if we find the composite of the other two combinations? What is the composite of a reflection and a rotation, or a translation and a rotation?

In 2D, the composite of a translation and a nontrivial rotation is another rotation. This is possible to prove, as follows: let T be a translation and R be a nontrivial rotation, and G be the composite of T following R. Both translations and rotations preserve orientation, and so their composite G must preserve orientation as well. In 2D, three mirrors suffice -- that is, every isometry is the composite of at most three reflections. Since G preserves orientation, it must be the composite of an even number of rotations. Therefore G is the composite of two reflections (or the identity transformation -- the transformation that maps every point to itself), and so G is either a translation or a rotation.

Let's try an indirect proof -- assume that G is a translation. That is:

T o R = G

Since T is a translation, it has a translation vector t, and as we're assuming that G is a translation, it must also have a translation vector g. Now let U be the inverse translation of T -- that is, the translation whose vector is -t, the additive inverse of t. We now compose U on both sides:

U o T o R = U o G

Now since U and T are inverses, U o T must be I, the identity transformation:

I o R = U o G

Since I is the identity transformation, I o R must be R:

R = U o G

Notice that G and U are both translations, whose vectors are g and -t, respectively. Then their composite must be another translation V whose vector is g - t. So now we have:

R = V

that is, a rotation equals a translation. Now no rotation can equal a translation (as the former has a fixed point, while the latter has no fixed point) unless both are the identity -- which contradicts the assumption that R is a nontrivial rotation (i.e., not the identity). Therefore, the composite of a translation and a nontrivial rotation isn't a translation, so it must be a rotation. QED

But this proof is invalid in 3D. This is because the proof uses a step that only works in 2D -- namely that three mirrors suffice. We must show how many mirrors suffice in 3D.

Let's recall why mirrors suffice in 2D. Let G be any 2D isometry, and let A, B, and C be three noncollinear points whose images under G are A', B', and C'. The first mirror maps A to A', the second mirror fixes A' and maps B to B'. It could be that the image of C under both mirrors is already C', otherwise a third mirror maps it to C'. Notice that the proof of the existence of these mirrors is nontrivial and depends on theorems such as the Perpendicular Bisector Theorem, since reflections are defined using perpendicular bisectors.

As it turns out, four mirrors suffice in 3D. To prove this, we let G be any 3D isometry, and let A, B, C, and D be four noncoplanar points. The first mirror maps A to A', the second mirror fixes A' and maps B to B', the third mirror fixes A' and B' and maps C to C', and the fourth, if necessary, fixes A', B', and C' and maps D to D'.

And so this opens the door for there to be a new transformation in 3D, one that is the composite of a translation and a rotation as well as the composite of four reflections. We can imagine twisting an object like a screw. A screwdriver rotates the screw about its axis, but then it's being translated into the wall in the same direction as that axis. And because of this, this new transformation is often called a screw motion.

We still have one last combination, the composite of a reflection and a rotation. It is subtle why the composite of a reflection and a rotation in 2D is usually a glide reflection -- why should the composite of a reflection and a rotation equal the composite of a (different) reflection and a translation? And it's even subtler why the composite of a reflection and a rotation may be a new transformation in 3D.

But the simplest example of this roto-reflection is the inversion map. In 3D coordinates, we map the point (x, y, z) to its opposite point (-x, -y, -z). This map is the composite of three reflections -- the mirrors are the three coordinate planes (xy, xz, and yz). As the composite of an odd number of reflections, it must reverse orientation. Yet it can't be a reflection, since it has only a single fixed point (0, 0, 0) and not an entire plane. Similarly, it can't be a glide plane because glide planes don't have any fixed points at all.

Roto-reflections are formed when the axis of the rotation intersects the reflecting plane in a single point -- and of course, this single point is the only fixed point of the roto-reflection.

So now we have six isometries in 3D -- reflections, translations, rotations, glide planes, screw motions, and roto-reflections. Are there any others? As it turns out, these six are all of them -- and the proof depends on the fact that four mirrors suffice in 3D.

Returning to the U of Chicago text, we have the definition of a reflection-symmetric figure:

"A space figure is F is a reflection-symmetric figure if and only if there is a plane M such that the reflection of F in M is F."

Similarly, a figure can be rotation-symmetric, as well as roto-reflection-symmetric. In the text, figures such as the right cylinder have reflection, rotation, and roto-reflection symmetry.

But a figure can't have translation symmetry unless it's infinite, as translations lack fixed points. So likewise, figures that have glide reflection or screw symmetry must also be infinite, as these transformations are based on translations. The translational symmetry mentioned by Noether refers to infinite space.

Of course, there exist dilations in 3D space as well. There is very little discussion of similarity in 3D, except to compare the surface areas and volumes of 3D figures.

Thus concludes my spring break post. I return to posting for our Common Core Geometry course on Monday, March 30th, which will be Day 137 on our calendar. The topic will be the surface areas of pyramids and cones, from Section 10-2 of the U of Chicago text.

What If We Had a German-like Education System?

This week is spring break, for both the school district where I sub and the school where the geometry student I tutor attends. As I mentioned last week, my plan was to have a topic this week comparing our system to the German education system -- and with both my school and my student's school closed, I can devote the entire post to that topic. My plan wasn't to post this topic on Monday of spring break week. Usually, I rest for a few days and post the vacation topic a few days into the week.

But it's rare when Google features a mathematician in its Doodle. But today, Monday, March 23rd, Google celebrates the 133rd birthday of Emmy Noether, who was not only a mathematician, but a German mathematician. And so I couldn't resist making my planning about the German education system on the day that one of the most famous products of that education system was born. I wrote this post at the last minute, and so even though I'm posting this late in the evening (Pacific Time) of March 23rd, by the time most of you read this it will be March 24th and no longer Noether's birthday.

But who, exactly, was Emmy Noether? As usual, whenever I want a biography about a mathematician, I link to the University of St. Andrews website:

http://www-history.mcs.st-and.ac.uk/Biographies/Noether_Emmy.html

This page describes her as an abstract algebraist:

"At Göttingen, after 1919, Noether moved away from invariant theory to work on ideal theory, producing an abstract theory which helped develop ring theory into a major mathematical topic. Idealtheorie in Ringbereichen (1921) was of fundamental importance in the development of modern algebra."

Here "algebra" refers not to high school Algebra I or II, but Abstract Algebra, a college class that's far beyond Calculus. Nonetheless, this "ring theory" actually appears, believe it or not, in the Common Core Standards!

CCSS.MATH.CONTENT.HSA.APR.A.1

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Now, as it turns out, this system that's analogous to the integers -- a system where one can freely add, subtract, and multiply -- is formally called a ring! The word "ring" is the hidden word behind this Common Core Standard. The standard could easily have been written:

CCSS.MATH.CONTENT.HSA.APR.A.1
Understand that polynomials form a ring, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Similarly, we notice the following standard:

CCSS.MATH.CONTENT.HSA.APR.D.7
(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

The formal term for such a system that's analogous to the rational numbers is a field:

CCSS.MATH.CONTENT.HSA.APR.D.7
(+) Understand that rational expressions form a field, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Notice that there are four different fields mentioned in the Common Core Standards -- the field Q of rational numbers, the field R of real numbers, the field C of complex numbers, and the field of rational expressions mentioned in HSA.APR.D.7, often written as qf(R[x]). There are six rings mentioned in the standards -- all fields are rings, so the four mentioned fields are rings. The two additional rings mentioned in Common Core are the ring Z of integers and the polynomial ring mentioned in HSA.APR.A.1, often written as R[x]. But the standards avoid the words "ring" and "field" in order to avoid confusing the math teachers who never took Abstract Algebra. So instead of "ring" we have the verbose phrase "system analogous to the integers."

Not only did Emmy Noether study ring theory, but there is a category of rings that is named after Noether herself. A Noetherian ring is a ring that is simple -- in a precise sense that can only truly be defined in an Abstract Algebra course. Suffice it to say that all of the rings that appear in the Common Core standards are Noetherian -- as it turns out, all fields are Noetherian, and the two extra rings mentioned in the standards, Z and R[x], are also Noetherian. One must study advanced college-level math before ever encountering a ring that isn't Noetherian.

But we can appreciate Noether's impact on mathematics without knowing anything about Abstract Algebra, but instead sticking to Physics and even Geometry! Here is another link to an article about Emmy Noether:

http://www.vox.com/2015/3/23/8274777/emmy-noether

This article tells us that Noether studied "invariants in algebra," and provides us with the following example of what "invariants" exactly are:

Her early work focused on invariants in algebra, looking at which aspects of mathematical functions stay unchanged if you apply certain transformations to them.(To give a very basic example of an invariant, the ratio of a circle's circumference to its diameter is always the same — it's always pi — no matter how big or small you make the circle.)

And of course, we discussed and celebrated the invariance of that ratio last week. Here's another example of an invariant: the article states that an invariant is something that stays unchanged when when certain transformations are applied to them. So let's consider a certain type of transformation that appears throughout Common Core Geometry -- the isometry. What are the invariants of a figure when we apply an isometry? That's easy -- angle measure, betweenness, collinearity, and distance -- and there's even a theorem that states that these four are invariants, the A-B-C-D Theorem found in Section 6-5 of the U of Chicago text. Of course, if we replaced "isometry" with "similarity transformation," then there are only three invariants, as distance is no longer invariant. This is implied by the Size Change (Dilation) Theorem of Section 12-5.

But the importance of Noether's work is revealed when we apply these invariants to physics. As it turns out, these invariants in geometry correspond to Conservation Laws in physics. (Recall that Stephen Hawking was also looking for invariants in searching for his Theory of Everything.)

These correspondences are known as Noether's Theorems. Let's return to the article for a description of Noether's Theorems:

Here's an example: Let's say we conduct a scientific experiment today. If we then conduct the exact same experiment tomorrow, we'd expect the laws of physics to behave in exactly the same way. This is "time symmetry." Noether showed that if a system has time symmetry, then energy can't be created or destroyed in that system — we get the law of conservation of energy.

NOETHER HAD LINKED TOGETHER CONCEPTS AS DIFFERENT AS ENERGY AND TIME

Likewise, if we do an experiment, and then do the exact same experiment again 20 miles to the east, that shouldn't make any difference — the laws of physics should work the exact same way in both places. This is known as "translation symmetry." Noether showed that translation symmetry leads to the law of conservation of momentum.

Finally, if we put our experiment on a table and rotate the table 90 degrees, that shouldn't affect the laws of physics, either. This is known as "rotational symmetry." But if rotational symmetry holds in a system, then angular momentum is always conserved. (That is, if you have a spinning bicycle wheel, it should spin in the same direction forever unless friction slows it down.)

Aha! Now the phrases "translation symmetry" and "rotational symmetry" should sound very familiar to anyone who reads this blog, because translations and rotations appear all over the Common Core Geometry standards. And so we see that the Common Core's focus on transformational geometry -- derided by its critics as "fluff" -- is in fact the key to understanding the laws of physics. I can only imagine someone trying to tell Noether that all of her work is based on "fluff"!

So far, this post that I wanted to devote to the German education system discusses precious little about German education system per se, but about a particular German mathematician. It's now time to move on to a discussion of the education system.

Here's finally what I wanted to discuss. There are problems that our current education system under Common Core -- and considering that there are so many walkouts at schools due to the Common Core tests, that fact alone is evidence that the system is flawed. What I want to know is, what would happen if the U.S. were to adopt certain facets of the German education system?

There are some parts of German education that would be universally unpopular if they were to be adopted here. For example, many German high schools go up to the 13th grade. I can't see anyone, no matter the political persuasion, who would consider 13th grade a good idea to have here. Indeed, many German schools are dropping the 13th grade, so it would make no sense to bring the extra year here.

I wish to compromise and accept parts of the German education system that liberals or Democrats would support as well as parts that conservatives or Republicans would like. Yes, it's difficult to avoid making this into another political topic -- which is why I buried it here in a spring break post rather than add politics to the posts that are supposed to focus on geometry.

But before I give any proposals based on the German education system, let's recall that the German model is not my preferred model for education. I've stated my own preferred ideas -- for example, replacing grade-levels with "paths," and using the full potential of computer-based tests like the PARCC and SBAC to meet student needs. These were mentioned in previous posts. Instead, I mention the German model because I was intrigued by it when I subbed in a German class, and I wanted to consider whether adopting parts of the German model would solve some of the issues that parents, students, and teachers have with Common Core. 

In Germany, many universities are tuition-free, and there was a backlash when the universities tried to charge tuition. It is probably impossible to make the colleges free in the U.S., but the closest we can get to Germany would be to adopt the president's proposal that community college tuition be free.

But of course, this isn't the most controversial part of the German education system. That, of course, is the rigid tracking system -- the division of all students into three educational paths. Here is a link to an article that discusses the three-track German system:

http://www.pearsonfoundation.org/oecd/downloads/Germany_strong_performers_US.pdf

As the 19th century came to a close and the 20th was opening, Germany had uniform elementary schooling with compulsory elementary education for all children aged 6 through 10, providing four years of basic education. This demonstrated Germany’s commitment to a state-run system of basic education. Following completion of elementary school, students were streamed into one of three types of school: 

• Volksschule – The students thought to be of low ability (the majority), were streamed into the Volksschule (People’s School, later call the Main School or Hauptschule) where they would get a few more years of education, and receive a qualification entitling them to apply for training leading to working-class jobs in Germany. 

• Realschule – Students thought to be of higher ability were streamed into the Realschule, where they would prepare for a qualification entitling them to apply for more training that would lead to more prestigious jobs such as clerks, technicians and lower-level civil servants. 

• Gymnasium – Those thought to be of the highest ability were streamed into the Gymnasium, where they would be given a broad preparation in the humanities and prepared to take examinations for the Abitur, which was the sole gateway to the professions, teaching and the upper levels of the civil service.

Again, the word "Gymnasium" clearly doesn't mean the same in English that it does in German. We notice that, as is common in European countries, the entire Gymnaisum is geared towards a single goal -- passage of a big test known as the Abitur. And we see how important the Abitur was, as university entrance or a professional career were impossible without it. Certainly there were never huge walkouts at the Abitur the same way there are walkouts at PARCC or SBAC now.

According to the article, the three educational racks began around the end of the 19th or beginning of the 20th century. That is, they were formed around the time that Emmy Noether was a student. Let's go back to the St. Andrews website and learn more about Noether's secondary education:

"After elementary school, Emmy Noether attended the Städtische Höhere Töchter Schule on Friedrichstrasse in Erlangen from 1889 until 1897. She had been born in the family home at Hauptstrasse 23 and lived there until, in the middle of her time at high school, in 1892, the family moved to a larger apartment at Nürnberger Strasse 32. At the high school she studied German, English, French, arithmetic and was given piano lessons. She loved dancing and looked forward to parties with children of her father's university colleagues. At this stage her aim was to become a language teacher and after further study of English and French she took the examinations of the State of Bavaria and, in 1900, became a certificated teacher of English and French in Bavarian girls schools. She was awarded the grade of "very good" in the examinations, the weakest part being her classroom teaching."

Notice that Noether was born in 1882, so she was clearly a prodigy -- she was admitted to secondary school at age 7 and left at age 15.

The Pearson Foundation article tells us that students are tracked around age 10 -- that is, as they enter 5th grade. But -- since once again, the education system is run by the states -- individual states may choose different ages to enter secondary school. Some states don't begin tracking until the students enter 7th, rather than 5th, grade.

If we were to adopt this division in the U.S., it would make the most sense to track as the students make the transition from elementary to middle school -- but this is state-dependent, as many states have middle schools start at different grades. Here in California, middle school generally starts with the 6th grade, although I know of at least three local school districts where middle school doesn't begin until the 7th grade. Beginning middle school with 5th grade is rare in California, though I do know of at least one charter school network that includes 5th grader in middle school. So most likely, a German tracking system applied to California would track entering 6th graders. In states where 5th graders in middle school is more common, tracking may occur with students entering 5th grade, just as in most German states.

Of course, the early age at which tracking occurs is only part of the controversy. The real problem occurs when the tracks line up with other demographics, such as race/ethnicity or social class -- or even to a lesser extent, gender. We can observe the inequalities that were present in the German system by looking back at Emmy Noether's education.

Undoubtedly, Noether faced gender discrimination. But notice that according to the St. Andrews link, Noether attended a "Tochter Schule." As "Tochter" means daughter, this implies that Noether attended an all-girls school. The discrimination that she would face due to her gender would occur after she left secondary school and headed for the university:

"Instead she decided to take the difficult route for a woman of that time and study mathematics at university. Women were allowed to study at German universities unofficially and each professor had to give permission for his course. Noether obtained permission to sit in on courses at the University of Erlangen during 1900 to 1902. She was one of only two female students sitting in on courses at Erlangen and, in addition to mathematics courses, she continued her interest in languages being taught by the professor of Roman Studies and by an historian. At the same time she was preparing to take the examinations which allowed a student to enter any university. Having taken and passed this matriculation examination in Nürnberg on 14 July 1903, she went to the University of Göttingen. During 1903-04 she attended lectures by Karl Schwarzschild, Otto Blumenthal, David Hilbert, Felix Klein and Hermann Minkowski. Again she was not allowed to be a properly matriculated student but was only allowed to sit in on lectures. After one semester at Göttingen she returned to Erlangen."

Eventually, Noether would meet the prominent mathematician David Hilbert. I mentioned back around the first day of school, and again in November, that Hilbert was the one who wrote a rigorous formulation of Euclid's original axioms of geometry. As it turned out, Hilbert allowed Noether to lecture with him, and the link above shows an advertisement for a Noether lecture:

"Mathematical Physics Seminar: Professor Hilbert, with the assistance of Dr E Noether, Mondays from 4-6, no tuition."

Now Hilbert was sharply criticized for allowing a woman to lecture with him. Hilbert's response to his critics was, "We are a university, not a bath house."

So we see the uphill battle that Emmy faced as a woman. But it is noticeable that nowadays, it is the male students who are struggling to get into Gymnasium, not the female students. This phenomenon occurs here in the U.S., where there are more female college students than males. Sometimes I refer to this as the "Bart and Lisa Problem," referring to two children on The Simpsons. Notice that it's not the son Bart, but the daughter Lisa who is more academically minded, while Bart couldn't care less about getting an education at all. (I point out the character Gaston in Disney's Beauty and the Beast, which takes place in the 19th century. Gaston notices that Belle is always reading books and tells her that it's improper for women to read. Compare this with the attitude of Bart and many boys in the 21st century, who believe that it's improper for men to read!)

Sometimes I wonder how we got from the 19th century, when women like Noether had trouble getting an education, to the 21st century, when men have trouble getting an education. I suspect that it has something to do with traditional gender roles -- males, of course, have always preferred to be the more active of the two genders. In the 19th century, when women were expected to work in the home, getting an education was an active thing to do, and so men preferred that they themselves be the only ones entitled to an education. But in the 21st century, women are no longer assumed to be housewives and technology has increased to the point that there are so many other things to do in the world besides get an education. So reading and education were no longer considered active things to do, and so males like Bart reject them!

On the other hand, Emmy Noether escaped the class divisions associated with German tracking, for her own father Max Noether was a mathematician in his own right. From Max's biography, also from the St. Andrews site:

"Max attended school in Mannheim but his studies at the gymnasium were interrupted in 1858. He suffered an attack of polio when he was 14 years old and it left him with a handicap for the rest of his life. For two years he was unable to walk and was unable to attend the Gymnasium. However, his parents arranged for him to receive lessons at home and so he was able to complete the Gymnasium curriculum without returning to school. At this stage Noether was interested in astronomy, so before beginning his university studies he spent a short period at Mannheim Observatory."

Notice that Max's family were iron wholesalers, they so most likely had a vocational education. But see that Max, fortunately, was able to enter Gymnasium -- his main problem was his infection with polio and his inability to walk (similar to FDR).

The last problem associated with tracking is, of course, race and ethnicity. We return to the Noether family and notice that they were Jewish. This wasn't a problem for Emmy until, of course, the rise of the Third Reich in the 1930's:

"Further recognition of her outstanding mathematical contributions came with invitations to address the International Congress of Mathematicians at Bologna in September 1928 and again at Zurich in September 1932. Her address to the 1932 Congress was entitled Hyperkomplexe Systeme in ihren Beziehungen zur kommutativen Algebra und zur Zahlentheorie. In 1932 she also received, jointly with Emil Artin, the Alfred Ackermann-Teubner Memorial Prize for the Advancement of Mathematical Knowledge. In April 1933 her mathematical achievements counted for nothing when the Nazis caused her dismissal from the University of Göttingen because she was Jewish."

Of course, the Third Reich no longer exists in Germany. But we can see that many immigrant groups are being placed on the lowest track:

"In this way the old tripartite system was quietly transformed. In the past, most of the graduates of the Hauptschule went on to apprenticeships and had a good shot at a decent job and a good career. When the transition was complete, the Hauptschule had become, in some schools and in some parts of the country, a dumping ground for students who would find it hard to get a qualification of any kind – these included immigrants and native Germans from lower class families alike."

The article tells us that many immigrants and other low-class students are locked onto a dead-end track that doesn't lead to success. This is why many people oppose such rigid tracking systems, in both Germany and the U.S.

Is it possible to have tracking without such significant racial and class-based inequalities? I say that it's possible, but politically difficult. It's hard for me to avoid politics when discussing the possibilities here.

Someone on the liberal or Democratic side of the political spectrum might say that if only certain tracks lead to success, then there should be some sort of Affirmative Action to help those groups that are less represented on the tracks that lead to success. I'm not sure whether hard quotas or extra points would increase access to success for the underrepresented groups. It might depend on the degree to which these groups are underrepresented, as well as the disparity between the levels of success between the highest and lowest tracks. Notice that the Common Core already contains a longitudinal data system, P20, which may be used to determine which tracks lead to success and which groups are represented on these tracks.

But someone on the conservative or Republican side of the spectrum generally opposes Affirmative Action and so would be against this plan. Instead, one might point to the preceding paragraph:

"As more students went to Gymnasium, and passed the Abitur, more decided to then enter the dual system and get a qualification, as a form of insurance against unemployment, before proceeding to university. More students who would formerly have gone to the Hauptschule put in the extra effort needed to get into the Realschule, which improved their chance of getting an apprenticeship from a good employer. But employers offering the best apprenticeships, who had earlier taken students only from Realschule, began to see a steady stream of Gymnasium students, some of whom already had an Abitur, knocking on their doors. Others who had formerly recruited only from the Hauptschule, found that they could get better candidates from the Realschule. Increasingly, the Hauptschule became a giant storage locker for the students who had no future, a road to nowhere for those students."

This is often known as "credential inflation." It's said to happen in the U.S. as well -- so people talk about college-educated taxi drivers and bellhops.

So a conservative might say that there's nothing wrong with certain groups being underrepresented on various tracks as long as all tracks lead to success. Instead, the goal should be to stop and reverse credential inflation so that all tracks lead to success. My main problem with this idea is that many articles discuss the existence of credential inflation, rather than a solution to credential inflation.

Thus a solution to track without inequality is politically difficult -- but then again, it would be politically difficult to implement any part of the German education system. In conclusion, emulating the German system may be a good idea in theory, but not in practice, and so this is why doing so is not my preference for how to improve the American education system.

END

Activity: Pythagorean Theorem Practice (Day 136)

Today is the last day of school before spring break. Thankfully, our Long March through a tough part of the school year is finally over.

Now many of you are saying -- what? Easter isn't until April 5th, and most schools take either the week before or the week after the holiday off. And even a school taking the week preceding Easter off still has a full week left before spring break. So how can today be the last day before the vacation?

As it turns out, the school calendar that I'm following happens to be one where spring break isn't tied to the Easter date at all. Many schools around the country wish to avoid the high variability associated with the Easter date and the havoc it wreaks on the school calendar. For example, consider a school that takes the week after Easter off. A few years ago, Easter was so early that it was still March when the students returned from spring break. Students were greeted with end-of-quarter exams (at schools with Labor Day starts) and state testing right when they came back. But then in 2011, Easter was so late that it was already May when the students returned from spring break. Students were greeted with AP exams right when they came back -- indeed, the AP Chemistry exam was the very first morning after the students had taken a week off from thinking about academics.

If one wants to avoid the volatility of the Easter date and select a fixed week in the school calendar during which to schedule spring break, a natural choice would be at the end of the third quarter. In fact, this makes very much sense at schools on the Early Start Calendar, since just as the winter break divides the year into semesters, the spring break can divide the second semester into quarters.

But one may recall that the fourth quarter of my calendar began earlier this week. If the school's spring break was intended to divide the quarters, spring break should have been scheduled this week, not next week. My guess is that the CAHSEE exam prevented this week from being spring break, since it would be awkward to make sophomores come to school to test during a vacation week. So next week is the closest that spring break can get to the actual midpoint of the semester without interfering with the state test.

So what does this say about my so-called "Long March"? I spent the day after President's Day lamenting the tough stretch of the school year, as there would be no holidays until spring break. Well, our Long March is five weeks long. This is still longer than the maximum of three weeks that occurred between holidays from Veteran's Day to President's Day, but it is shorter than the six- or seven-week Long Marches that occur at schools that observe spring break at Easter.

What this does mean is that the longest stretch of my school year without a holiday is actually from Easter to Memorial Day. Many schools that don't tie spring break to Easter still have a long weekend for the holiday, including mine. This year, there is a seven-week stretch from the long Easter weekend to the long Memorial Day weekend. But "Long April/May" doesn't have the same ring to it as "Long March" does.

With spring break -- and the spring equinox -- upon us, now let's consider what effect this has on our math lessons. This was, in fact, part of the reason that I originally wanted to do Chapter 15 of the U of Chicago text this week. I knew that there would be only one week left before spring break, and the last important section of the chapter was Section 15-3 -- nothing after this lesson usually appears on standardized tests, including the practice PARCC exam that we've been discussing all week. I could have covered the lessons up to Section 15-3 by Tuesday, and then have a short quiz covering the last part of Chapter 8 (on pi) and the first part of Chapter 15 (on circles and tangent lines) by Thursday -- this would have been very similar to Chapter 9 of Dr. Franklin Mason's text, as Dr. M also combines pi and tangent lines in a circle chapter. Then we could have introduced surface area and volume after spring break without being interrupted by the vacation period.

But I changed my plans after taking a peek at that practice PARCC exam -- especially after I had compared the PBA and EOY versions of the test. Combining pi and tangent lines into a single circle chapter, as Dr. M does, is no longer feasible because pi, along with surface area and volume formulas involving pi, appear on the PBA, while tangent lines don't appear until the EOY. So Chapter 15 needs to be the final unit and be taught between the PBA and EOY, while pi, surface area, and volume should be taught before the PBA.

Notice that I want to time my lessons to both CAHSEE, for the sake of Californian readers, and the PARCC, for the sake of readers from PARCC states. But the dates aren't quite compatible, as for CAHSEE, all I'd have to do is introduce surface area and volume of boxes (and their unions) by the second week of March, before the CAHSEE. But PARCC requires that I teach surface area and volume before the PBA, which could be given as early as March 1st.

Instead, my plan for next year is to compromise -- I teach the surface area and volume of boxes at the end of February, and finish pi, surface areas, and volumes by the second week in March. Then the last week before spring break could be a full activity week. The Dan Meyer lesson that I posted earlier could become an opening activity for surface area and volume in late February, while this week I could have posted another lesson -- Meyer also has a volume activity involving meatballs.

But that's next year. This year, I'm left with a need to give a lesson for today. Just as I wrote before Thanksgiving break, I want to avoid teaching a brand new lesson right before a week-long break. So instead, just as I did before Thanksgiving, today's lesson is a preview activity of what's to come after the vacation. Coming up will be the surface areas and volumes of pyramids, cones, and spheres.

Today I ended up subbing in a special education course -- but it was in fact a Geometry class. The class is now in Chapter 8 of the AGS text that I mentioned during the second week in January -- and this is on the Pythagorean Theorem. Considering that Chapter 12, the last chapter of the text, is not covered on standardized tests, Chapter 8 is more or less on pace to reach Chapter 11 by the end of the year, although I'm not sure about reaching it before the SBAC for juniors.

This is significant because Chapter 11 is on volume and surface area -- that is, it corresponds to Chapter 10 of the U of Chicago. The volume lesson will be given too late to help sophomores on the CAHSEE, so let's see whether it will be taught in time to help juniors on the SBAC.

Indeed, as we look at the surrounding chapters of the AGS text, we notice that Chapter 10, on circles, includes both pi and tangents -- both the tangent of a circle and the trig tangent! Indeed, the two types of tangent appear in back-to-back sections -- Section 10-7 for tangent of a circle, and Section 10-8 for the trig ratios. There is a brief mention of a unit circle to justify having the trig functions in the same chapters as circles. Spheres also appear in Chapter 10 -- leading to the awkward situation that the surface area and volume of a sphere are known before the surface area and volume of a cube!

Chapter 9 of the AGS is on perimeter and area -- it therefore corresponds to the sections of Chapter 8 of the U of Chicago that appeared on the last test posted to the blog. Chapter 7 is on similarity -- it therefore corresponds to the same numbered chapter in Glencoe and Chapter 12 of the U of Chicago.

But Chapter 8 -- a chapter solely devoted to the Pythagorean Theorem -- doesn't appear in any of the other texts that we've discussed here on the blog. The AGS does the exact opposite of what we did here on the blog. On this blog we rushed through the Pythagorean Theorem, whereas in AGS we slow it down and spend an entire chapter on Pythagoras.

Here are the sections of Chapter 8 of the AGS text:

Section 8-1: Pythagorean Triples
Section 8-2: Pythagorean Triples and a Proof
Section 8-3: Pythagorean Demonstration
Section 8-4: Pythagorean Theorem and Similar Triangles
Section 8-5: Special Triangles
Section 8-6: Pythagorean Proof and Trapezoids
Section 8-7: Distance Formula: Pythagorean Theorem
Section 8-8: Converse of the Pythagorean Theorem
Section 8-9: Algebra Connection: Denominators and Zero

So notice that we begin with Pythagorean Triples -- numbers that satisfy a^2 + b^2 = c^2 -- before saying anything about right triangles. Sections 8-3 and 8-4 both contain area-based proofs of the Pythagorean Theorem -- a bit awkward, since area doesn't appear until Chapter 9 of this text. I believe that the proof mentioned in Section 8-6 is President Garfield's proof. Section 8-9, as well as the last section of every chapter of the AGS text, is an "Algebra Connection" or algebra review.

In some ways, maybe spending extra time on the Pythagorean Theorem is worthwhile. Most texts include it as part of other chapters, such as the area or similarity chapters -- and often separate the Distance Formula by including it in a different chapter. Perhaps my blog represents a compromise -- we introduce it quickly, but we remind students of the theorem whenever they need to remember it.

And one of those times is now. I post these worksheets based on AGS's Sections 8-1 and 8-2 in order to remind students of the Pythagorean Theorem, just ahead of the lessons on pyramids and cones. After all, the radius, vertical height, and slant height of a cone form a right triangle, and so one might need the Pythagorean Theorem to calculate their lengths before finding the surface area or volume of the cone. As I said, the "Chapter 8" at the top of this posted worksheet refers to the AGS text, although one might assume that it refers to the U of Chicago text since the Pythagorean Theorem appears in Section 8-7 of that text and we just covered the rest of Chapter 8 last week.

In most classes, I ended up only discussing Pythagorean Triples from the Section 8-1 worksheet -- which I don't post to this blog -- and not any triangles at all. I played a modified version of the usual game that I give because some of the special ed classes had only four or five students.

Only in third period did I actually reach the triangle worksheet that I posted here. Many of the students were confused when they were asked to solve for a side of the triangle. Part of this is because the first two problems involve integers, and then everything jumps into decimals. Students were allowed to use a calculator, but many of them had only a simple online calculator that lacked a square root function.

Perhaps if I were the one preparing this lesson, I would have included more simple integer questions before jumping into decimals with irrational square roots so quickly, but this was a lesson that was already prepared by the regular teacher. Then again, we can't shield the students from decimals and irrational square roots forever. A few of the problems that appear in Sections 10-2 and 10-7 of the U of Chicago text on pyramids and cones require decimal square roots, and of course we might expect such questions on PARCC or SBAC as well.

And so this concludes my last post before spring break. Once again, I plan on making one or two posts next week, during spring break itself. One of the posts may be a continuation of what I posted in February about the German education system and what would happen if one tried to apply their system to the U.S. And the other may be about Section 9-5 of the text -- a lesson that we'll skip as it's not important to PARCC or SBAC, but it's interesting to me. The Common Core standards devote much time to reflections and other transformations of the plane, and now Section 9-5 discusses what happens with reflections in 3D space. After the break, we will continue with pyramids and cones.

Section 10-5: Volumes of Prisms and Cylinders (Day 135)

Last night I tutored my geometry student. Section 7-4 of the Glencoe text is on the Side-Splitting Theorem -- well, Glencoe calls it the "Triangle Proportionality Theorem," but we know it from the U of Chicago text as the Side-Splitting Theorem.

So of course I showed my student my worksheet on the Side-Splitting Theorem, which we first encountered in Section 12-10 of the U of Chicago. Since we had a little extra time, and as I mentioned before, Glencoe's 7-5 has no counterpart in the U of Chicago (but I may consider including an extra lesson on the Angle Bisector Theorem next year), I moved straight on to Glencoe's 7-6, which is on similarity transformations, including dilations.

Once again, the closest counterpart to Glencoe's 7-6 is U of Chicago's 12-1. That is, dilations using coordinates are emphasized. My worksheet emphasizes dilations without coordinates and is based on U of Chicago's 12-2. For many students, it's far easier to use coordinates to explain what dilations are, and the dilation problems on the practice PARCC exam also use dilations. So, you may ask, why did I focus on Section 12-2 over 12-1?

It's to avoid circularity in the proofs. In Section 12-1, we prove the theorem that the transformation mapping (x, y) onto (kx, ky) maps lines to parallel lines and segments to other segments whose lengths are multiplied by k. But the proof that a line is parallel to its image involves comparing their slopes -- and later on we prove the slope formula using similarity, and of course we prove the properties of slope using dilations. The only way to avoid this circularity is to introduce dilation without using coordinates, and that's exactly what we do in Section 12-2.

Perhaps it might be better to use coordinates for introducing dilations as per Section 12-1, but avoid proving anything about these transformations until we reach similarity and slope. This may help students understand dilations better.

Since we've been looking at the PARCC questions all week, we notice that there are two questions involving dilations on the practice PARCC exam. Question 5, the easier of the two questions, uses a dilation centered at (0, 0) and asks whether a triangle and its image are similar, as well as whether the scale factor of the dilation is less than, equal to, or greater than 1.

But Question 16 asks the students to prove a theorem, which the U of Chicago text calls the "Size Change Distance Theorem" in Section 12-3 (as "dilation" doesn't appear in the U of Chicago). It is actually uncertain what the PARCC is expecting students to write here. The U of Chicago proves the theorem in 12-3 by going back to the previous theorem from 12-1. But that theorem was proved using the Distance Formula, which is in turn proved using the Pythagorean Theorem, which is in turn proved in Common Core using similarity, which is in turn proved using dilations. So we find ourselves stuck in circularity once again!

Dr. Hung-Hsi Wu breaks free from this circularity by giving a proof based on induction on the scale factor k -- but this seems a bit too advanced even for a PARCC question. It's possible that PARCC expects the students to use SAS Similarity to prove the theorem -- which would be the traditionalist way of proving it. Dr. Wu's writing implies that this is a problem with similarity that congruence avoids, and it's unfortunate that this is the proof that appears on the PARCC practice test.

OK, that's enough about dilations. As I mentioned yesterday, we are moving on to Section 10-5 of the U of Chicago text, which is on volumes of prisms and cylinders. As we proceed with volume, let's look at what some of our other sources say about the teaching of volume. Dr. David Joyce writes about Chapter 6 of the Prentice-Hall text -- the counterpart of U of Chicago's Chapter 10:

Chapter 6 is on surface areas and volumes of solids. It is strange that surface areas and volumes are treated while the basics of solid geometry are ignored.

There are 11 theorems, the only ones that can be proved without advanced mathematics are the ones on the surface area of a right prism (box) and a regular pyramid. Most of the theorems are given with little or no justification. The area of a cylinder is justified by unrolling it; the area of a cone is unjustified; Cavalieri's principle is stated as a theorem but not proved (it can't be proved without advanced mathematics, better to make it a postulate); the volumes of prisms and cylinders are found using Cavalieri's principle; and the volumes of pyramids and cones are stated without justification. The only argument for the surface area of a sphere involves wrapping yarn around a ball, and that's unlikely to get within 10% of the formula. Finally, a limiting argument is given for the volume of a sphere, which is the best that can be done at this level.

In summary, there is little mathematics in chapter 6. Most of the results require more than what's possible in a first course in geometry. Surface areas and volumes should only be treated after the basics of solid geometry are covered. Alternatively, surface areas and volumes may be left as an application of calculus.

There are a few things going on here. First of all, Joyce writes that surface areas and volumes should be treated after the "basics of solid geometry," but he doesn't explain what these "basics" are. It's possible that the U of Chicago's Chapter 9 is actually a great introduction to the basics of solid geometry -- after all, Chapter 9 begins by extending the Point-Line-Plane postulate to 3D geometry, then all the terms like prism and pyramid are defined, and so on. Only afterward, in Chapter 10, does the U of Chicago find surface areas and volumes.

So I bet that Joyce would appreciate Chapter 9 of the U of Chicago text. Yet most texts, like the Prentice-Hall text, don't include anything like the U of Chicago's Chapter 9, and I myself basically skipped over most of it and went straight to Chapter 10. Why did I do this?

It's because most sets of standards -- including the Common Core Standards, and even most state standards pre-Common Core -- expect students to learn surface area and volume and essentially nothing else about solid geometry. Every single problem on the PARCC and CAHSEE exams that mentions 3D figures involves either their surface area or volume. And so my duty is to focus on what the students are expected to learn and will be tested on, and that's surface area and volume.

Joyce writes that something called "Cavalieri's principle" is stated as a theorem but not proved, and it would be better if it were a postulate instead. Indeed, the U of Chicago text does exactly that:

Volume Postulate:
e. (Cavalieri's Principle) Let I and II be two solids included between parallel planes. If every plane P parallel to the given plane intersects I and II in sections with the same area, then
Volume(I) = Volume (II).

According to the text, Francesco Bonaventura Cavalieri was the 17th-century Italian mathematician who first realized the importance of this theorem. Cavalieri's Principle is mainly used in proofs -- as Joyce points out above, the volumes of prisms and cylinders is derived from the volume of a box using Cavalieri's Principle, and the U of Chicago also derives the volumes of oblique prisms and cylinders using Cavalieri.

The text likens Cavalieri's Principle to a stack of papers. If we have a stack of papers that fit in a box, then we can use the formula lwh to find its volume. But if we shift the stack of papers so that it forms an oblique prism, the volume doesn't change. This is Cavalieri's Principle.

Notice that we don't need Cavalieri's Principle if one is simply going to be handed the volume formulas without proving that they work. But of course, doing so doesn't satisfy Joyce. Indeed, the U of Chicago goes further than Prentice-Hall in using Cavalieri to derive volume formulas -- as Joyce points out, we can find the volume of some pyramids without advanced mathematics (Calculus). But then the U of Chicago uses Cavalieri to extend this to all pyramids as well as cones. Finally -- and this is the grand achievement -- we can even use Cavalieri's Principle to derive the volume of a sphere, using the volumes of a cylinder and a cone! As we'll soon see, Joyce is wrong when he says that a limiting argument is the best that we can do to find the volume of a sphere. Cavalieri's Principle will provide us with an elegant derivation of the sphere volume formula.

Here are the Common Core Standards where Cavalieri's Principle is specifically mentioned

CCSS.MATH.CONTENT.HSG.GMD.A.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

CCSS.MATH.CONTENT.HSG.GMD.A.2
(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

CCSS.MATH.CONTENT.HSG.GMD.A.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*


But not everyone is excited about the inclusion of Cavalieri's Principle in the standards. The following link is to the author Dr. Katharine Beals, an opponent of Common Core:

http://oilf.blogspot.com/2014/11/math-problems-of-week-common-core_14.html

Beals begins by showing a question from New York State -- recall that NY state has one of the most developed Common Core curricula in the country. This question involves Cavalieri's principle. Then after stating the same three Common Core Standards that I did above, Beale asks her readers the following six "extra credit questions":

1. Will a student who has never heard the phrase "Cavalieri's principle" know how to proceed on this problem? 

2. Should a student who has never heard the phrase "Cavalieri's principle" end up with fewer points on this problem than one who has? 

3. Should a student who explains without reference to "Cavalieri's principle" why the two volumes are equal get full credit for this problem? 

4. Is it acceptable to argue that the volumes are equal because they contain the same number of equal-volume disks? 

5. To what extent does knowledge of "object permanence," typically attained in infancy, suffice for grasping why the two stacks built from the same number of equal-sized building blocks have equal volume?

6. To what degree does this problem test knowledge of labels rather than mastery of concepts?

The question involves two stacks of 23 quarters -- one arranged as a cylinder, the other arranged not so neatly -- and asks the students to use Cavalieri to explain why they have the same volume.

Notice that in four of the "extra credit questions," Beals criticizes the use of the phrase "Cavalieri's principle," arguing that the label makes a simple problem needlessly complicated. But Cavalieri's Principle is no less a rigorous theorem than the Pythagorean Theorem. Imagine a test question which asks a student to use the Pythagorean Theorem to find the hypotenuse of a right triangle, and someone responding with these "extra credit questions":

1. Will a student who has never heard the phrase "Pythagorean Theorem" know how to proceed on this problem? 

2. Should a student who has never heard the phrase "Pythagorean Theorem" end up with fewer points on this problem than one who has? 

3. Should a student who explains without reference to "Pythagorean Theorem" what the hypotenuse is get full credit for this problem? 

6. To what degree does this problem test knowledge of labels rather than mastery of concepts?

As for the other two "extra credit questions," yes, the volumes are equal because they contain the same number of equal-volume disks. We don't need Cavalieri's Principle to prove this, since the Additive Property -- part (d) of the Volume Postulate from yesterday's lesson -- tells us so.

The true power of Cavalieri's Principle is not when we divide a solid into finitely (in this case 23) many pieces, but only when we divide it into infinitely many pieces. In higher mathematics, we can't simply extract from finite cases to an infinite case without a rigorous theorem or postulate telling us that doing so is allowed. I doubt that the infant mentioned in "extra credit question 5" above is intuitive enough to apply object permanence to infinitely divided objects.

I wonder how Beals would have responded had the question been, "These two cylinders have the same radius and height, but one is oblique, the other right. The right cylinder has volume pi r^2 h. Use Cavalieri's Principle to explain why the oblique cylinder also has volume pi r^2 h." Nothing less rigorous than Cavalieri gives us a full proof of the oblique cylinder volume formula.

Furthermore, in another post a few weeks before this one, Beals tells the story of a math teacher who would only allow those who successfully derive the Quadratic Formula to date his daughter:

http://oilf.blogspot.com/2014/11/my-daughter-can-now-date-barrys-daughter.html

I claim that deriving the sphere volume formula from Cavalieri's Principle in Geometry is just as elegant as deriving the Quadratic Formula from completing the square in Algebra I. If a math teacher would let anyone who derives the Quadratic Formula date his daughter, then let anyone who derives the sphere volume formula date his niece.