## Thursday, December 17, 2015

### Chapter 6 Test (Day 74)

Today I post the Chapter 6 Test. My original intention was for this test to be mostly unchanged from the test that I posted last year, except adding a few Frosty graphing questions based on the worksheet that I posted yesterday.

But there's just one tiny problem -- I never posted a Chapter 6 Test last year! I looked back at the post from last year in which I was supposed to have given the Chapter 6 Test, and I found out that it was posted incorrectly. Ironically, I'd written in the correct answers to the test that I should have posted, but not the test itself. It didn't help, of course, that last year I finished Chapter 6 just before Thanksgiving break, when I was having trouble with my computer and getting ready to replace it.

I actually did post a Chapter 6 Quiz last year -- this was just before a four-day weekend. With the changes to this year's calendar, there was no four-day weekend, so I didn't post it. I could have posted that Chapter 6 Quiz today, but I didn't because it doesn't correspond to the Chapter 6 Review page that I posted yesterday. The last thing I want is for the students to freak out after seeing a quiz that has nothing to do with the review.

And so I finally posted the correct Chapter 6 Test today. Yes, it could have been lost on the old computer that I discarded, but fortunately I still have a copy of it. I was originally planning on adding the four Frosty questions, but I don't really want to change a test that has never been posted. So I decided to keep the old test as it is -- but it now means that the students might freak out after seeing four test questions that have nothing to do with the review.

Let's look at these final four questions in more detail. Questions 17 and 18 are graphing questions, except that one is transforming triangles, not snowmen. One of them is a glide reflection, while the other is a translation. I just hope that students won't be thrown off by seeing rules for each of these transformations, such as T(x, y) = (x + 5, -y).

Question 19 is about the cardinality of a set, N(S), which is mentioned briefly in Lesson 6-1 of the U of Chicago text, but I only discussed it briefly this year. Here's what I wrote about N(S) last year:

I decided that the only real reason that the U of Chicago introduces the N(S) notation for cardinality (number of elements in a set, previous question) is to prepare the students for function notation, so I might as well use it here. There's only one other place where I see n(A) used for number of elements in set A -- the Singapore Secondary Two standards!

Last year, I also wrote: (A Thanksgiving reference! These are the seven dates in November which could be turkey day!) Again, I originally wrote this test last year at Thanksgiving. Okay, this year we can pretend that the seven elements of S correspond to Christmas Day and the three days before and after Christmas.

The final question shows one more transformation -- which happens to be a dilation. Neither last year nor this year did I formally cover dilations. This is supposed to be a think-outside-the-box question where students should try to reason out what's going on. But think about it for a moment -- the graph makes it appear that (3, 3) is the image of (1, 1). So all students have to do is plug in x = y = 1 into each of the four choices and see that only choice (d) gives (3, 3) as the answer. (Of course, this year, the students have seen dilations because of Tom Turkey and Thanksgiving again.)

Students might consider the last four questions to be unfair. But even if they get all four wrong, it's still possible to get 80% -- the lowest possible B. So strong students who completed the review worksheet yesterday should still earn at least a C on this test.

Here are the answers to today's test -- the same answers I posted last year to the invisible test:

1. a translation 2 inches to the left

2. a translation 2 inches to the right

3. a rotation with center O and magnitude 180 degrees

4. a translation 8 centimeters to the right

5. true

6. angles D and G

7. triangle DEF, triangle GHI

8. Reflexive Property of Congruence

9. definition of congruence

10. Isometries preserve distance.

11. translation

12. translation

13. glide reflection

14. glide reflection

15.-16. The trick is to reflect the hole H twice, over the walls in reverse order, and then aim the golf ball G towards the image point H". In #15, notice that y and w are parallel, so reflecting in both of them is equivalent to a translation twice the length of the course. In #16, notice that x and y are perpendicular, so reflecting in both of them is equivalent to a 180-degree rotation.

17. glide reflection (changing the sign of y is the reflection part, adding to x is the translation part)

18. translation

19. 7

20. d (for dilation, of course!)

Now today's a test day, and it's been a while since I posted a topic about traditionalists, so let's make this our traditionalist post. I've been thinking about the Presidential Birthday plan -- and no, I know that it won't be implemented since I'm not running for president and the real successor to No Child Left Behind (the Every Student Succeeds Act) has already been signed, but still, I'd like to discuss what the Common Core would look like if I were in charge.

In particular, I've been thinking about what Geometry would look like under that plan. I mentioned in that birthday post how I wouldn't mind setting up a new Geometry test -- one that can assess students at three different levels of Geometry classes. These Geometry classes correspond to the three levels that are offered at the school the First Daughters attend, Sidwell Friends (the "presidential" part of the Presidential Birthday Plan) -- Discovering Geometry for the low track, Plane Geometry for the middle track. and Integrated Math I for the high track.

On this blog, my lessons are based on the U of Chicago text. We don't know what text two of the three tracks use, but we definitely know that the low track uses Michael Serra's Discovering Geometry (hence the name of the course). So we should use the Serra text as a base. We begin by writing down the chapters of the Serra text and then insert new units to represent the more rigorous material to be taught on the higher two tracks.

The following list is based on the modern version of the Serra text. My version contains an extra chapter between the modern Chapters 3 and 4, and has three chapters at the end on logic and proofs while the modern text has only one, Chapter 13. But let's focus on the main Chapters 1-12 first:

1. Inductive Reasoning
2. Introducing Geometry
3. Using Tools of Geometry
4. Triangle Properties
5. Polygon Properties
6. Circles
7. Transformations and Tessellations
8. Area
9. Pythagorean Theorem
10. Volume
11. Similarity
12. Trigonometry

This list will look a bit cluttered when we start adding units to this, so let's make it shorter by combining every two chapters into a unit:

-- Inductive Reasoning and Introducing Geometry
-- Using Tools of Geometry and Triangle Properties
-- Polygon Properties and Circles
-- Transformations, Tessellations, and Area
-- Pythagorean Theorem and Volume
-- Similarity and Trigonometry

Some of these chapters already go together -- for example, similarity and trig, since the trig ratios are defined using similar right triangles. But others don't really fit together -- for example, polygons and circles, since a circle is not a polygon.

Now we wish to insert new units in between these, as follows -- for the middle track, we want to add some basic postulates that will allow us to prove all the conjectures presented in Chapters 1 to 12 of the Serra text. And on the high track, we will add back the Common Core transformations that will allow us to prove the middle track postulates as theorems.

Here's my attempt at accomplishing this:

1. Inductive Reasoning and Introducing Geometry
2. Reflections
3. Triangle Congruence
4. Using Tools of Geometry and Triangle Properties
5. Rotations
6. Parallel Lines
7. Polygon Properties and Circles
8. Translations
9. Area Formulas
10. Transformations, Tessellations, and Area
11. Transvections
12. Volume Formulas
13. Pythagorean Theorem and Volume
14. Dilations
15. Triangle Similarity
16. Similarity and Trigonometry

Here's how this list works: all three tracks begin with Unit 1. Then the remaining units are divided into sets of three: 2-4, 5-7, 8-10, 11-13, and 14-16. The first unit in each set is intended for the high track only and focuses on transformations. The second unit is appropriate for both the middle and high tracks and focuses on the properties that traditionalist Geometry teaches, such as SSS, SAS, ASA, the Corresponding Angles Test, and so on. The last unit is appropriate for all three tracks, and each unit corresponds to the original two chapters from Serra. In other words, different tracks cover some or all of the units listed above, as follows:

High Track: all 16 units
Middle Track: Units 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16
Low Track: Units 1, 4, 7, 10, 13, 16

Here's an example: in Unit 2, students learn about reflections and use them to derive the congruence theorems SSS, SAS, and ASA. In Unit 3, SSS, SAS, and ASA are already assumed -- to those who covered Unit 2 they are proved, to those who skipped it they are postulates -- and these are used to prove interesting theorems such as the Isosceles Triangle Theorem. In Unit 4 the Isosceles Triangle Theorem is assumed -- to those who covered Unit 3 it's a theorem, to those who skipped it it's a conjecture -- and the lesson follows Serra.

I inserted the various transformations where they make the most sense. For example, it's a no-brainer to place dilations in Unit 14, because it precedes similarity in Unit 16. But then we wonder why transvections are being placed in the curriculum at all, much less in Unit 11. Well, I looked ahead to Unit 13, which corresponds to Serra's Chapters 9 and 10, and I asked myself, what transformation links the Pythagorean Theorem and Volume? After thinking about it for a while, I realized that they actually have transvections in common -- Euclid used transvections to prove Pythagoras and the volumes of certain solids, such as right and oblique prisms, are related via 3D transvections.

On the other hand, notice that transvections appear after area in Unit 10 --  but then we may ask, don't we use transvections to define area? After all, a transvection is a transformation that preserves area in the same way that an isometry is a transformation that preserves congruence.

Actually, transvections are not used to define area in this manner. For example, the unit circle has the same area as a square of side-length sqrt(pi), but there is clearly no transvection, isometry, or any other affine transformation mapping the circle to the square. On the other hand, if two figures are congruent then there must be an isometry mapping one to the other.

And besides, it may be a transvection that shows us that a parallelogram has the same area as a rectangle of the same base and height. But we can determine the area of a parallelogram simply by dividing it into triangles, just as the U of Chicago text does, without any need for transvections.

Yet it's not as obvious how to divide an oblique prism or pyramid into right prisms or pyramids so we can use their volume formulas. It's easy, though, to show that a transvection maps the oblique solid to the corresponding right solid. This eliminates the need to use Cavalieri's Principle to find the volumes (although I still like the derivation of the volume of a sphere from those of the cylinder and cone using Cavalieri).

In Unit 2, reflections are the only transformations that are taught. This means that only reflections are used to prove the SSS, SAS, and ASA theorems. We see that AAS is traditionally proved using parallel lines, but these don't appear until Units 5 and 6. Actually here I have no problem using Dr. Franklin Mason's TEAI to prove AAS in Unit 2, since this is a unit that only high trackers will see, and I'd rather keep all of the triangle congruence theorems in Unit 2.

Notice that rotations are in Unit 5, ahead of parallel lines in Unit 6. This means that we're using rotations to derive parallel line properties, returning to Dr. Hung-Hsi Wu's 180-degree rotations. We can use rotations to prove the Alternate Interior Lines Test in Unit 5 and introduce a Parallel Postulate to give us the AIA Consequence. Then the proofs of Corresponding and Same-Side Interior Angles from AIA can be given in Unit 6, since we expect middle-trackers to be able to prove these. The polygons mentioned in Unit 7 are mostly quadrilaterals (especially parallelograms), so these follow directly from the parallel line properties taught in Units 5 and 6. The circle properties in Unit 7 (corresponding to Serra's Chapter 6) are mostly about angles (central, inscribed, etc.) and these can be proved using theorems demonstrated in Units 1 through 6 -- the power of a point theorems (given in Chapter 15 of the U of Chicago text) don't appear in Serra.

Unit 8 and its translations do lead to tessellations for Unit 9. Notice that tessellations and area are linked somewhat -- think about Lesson 8-3 of the U of Chicago text, which uses tessellations of squares to motivate a definition of area.

Finally, I like to think ahead to Pi Day. Since Pi Day is about 2/3 to 3/4 of the way through the year (depending on whether it's a Early Start or Labor Day Start calendar), we expect to be around Units 11 to 13 on Pi Day. These units involve volume, and of course many volume formulas use pi.

The different tracks use various types of proofs. We can reinforce this by teaching paragraph proofs in Unit 2 (since many transformation proofs are written in paragraphs), two-column proofs in Unit 3 (as we expect middle-trackers to understand two-column proofs), and flow proofs in Unit 4 (as Serra often uses this type of proof). Let's rename the units to remind us that the middle-track focuses on traditionalist two-column proofs of the various properties while the low-track focuses on conjectures and their applications (as in Serra) as follows:

1. Inductive Reasoning and Introducing Geometry
2. Reflections
3. Triangle Congruence Properties
4. Triangle Congruence Applications
5. Rotations
6. Parallel Line and Polygon Properties
7. Parallel Line and Polygon Applications
8. Translations
9. Tessellations and Area Formulas
10. Tessellations and Area Applications
11. Transvections
12. Pythagorean Theorem and Volume Formulas
13. Pythagorean Theorem and Volume Applications
14. Dilations
15. Similarity and Trig Properties
16. Similarity and Trig Applications

With all of this detail, you may wonder why the blog posts aren't organized this way. The reason is that the blog follows the U of Chicago text and Common Core. All of this is a fantasy about what I'd do if I were in charge, but as long as California is a Common Core state, I must organize blog lessons such that they prepare students for the Common Core tests.

My dream is that a single test can assess students at all three levels. We can accomplish this by having a Geometry test where the first few questions focus on low-track material, the next few on middle-track material, and then the final questions on high-track material.

An advantage of this is that some low-track students might be able to figure out a few of the questions intended for middle-trackers, and likewise some middle-track students could answer a few of the questions for high-trackers. Serra's Chapter 13, to be taught at the end of the year, can serve as a brief introduction to proofs for low-trackers, and similarity, we can place an introduction to transformations (say Chapter 9 of the Glencoe text I discussed last year, or Chapter 12 in other traditionalist texts such as HRW) at the end of the middle-track year.

And with regards to my deep concern that schools may use tracking to enforce demographic segregation, students can challenge their placement into lower tracks simply by studying the lessons on their own (or with a tutor) and answering questions intended for the higher tracks -- this simply isn't possible if each track had its own exam.

Notice that in this "traditionalists" topic, I haven't actually mentioned any traditionalists. Well, I do want to point out another feature of the three-track system -- the middle track at Sidwell is said to lead to "non-AP Calculus." It's often pointed out by traditionalists that in other countries such as Japan, all students take Calculus (and no, it's not just the ASTC mnemonic to remember trig quadrants).

I'm trying to imagine what a non-AP Calculus class would entail -- since again, I'm not quite sure what's taught at either Sidwell or Japan. The easiest functions to differentiate and integrate are of course the polynomial functions, so I can envision a non-AP Calculus course where the entire year is devoted to the derivatives and integrals of polynomial functions and their applications. OK, I'd throw in the exponential function e^x, as that's really the easiest function to differentiate and integrate!

OK, here's the Chapter 6 test that you've been waiting more than a year for: