Lesson 5-4: Properties of Kites (Day 54)

Today I subbed in a special ed high school class. Three of the classes are math, and so it's worth doing "A Day in the Life" today.

But first, let me write in those programs that create Halloween Mocha music -- again, I post it today so that next year, I can just click "October 2018" and be led straight to the program. Our first one codes 18EDL again, but with an emphasis on open guitar strings:

http://www.haplessgenius.com/mocha/

10 CLS
20 N=8
30 FOR A=0 TO 6
40 B=4
50 X=A-INT(A/2)*2
60 D=20-RND(2)*5
70 PRINT D;
80 L=RND(B)
90 SOUND 261-N*D,4*L
100 IF L>1 THEN FOR I=1 TO L-1:PRINT "   ";:NEXT I
110 B=B-L
120 IF B>0 THEN D=19-RND(10):GOTO 70
130 PRINT
140 NEXT A
150 PRINT 18
160 SOUND 261-N*18,16

Here's how the new Line 60 works. The fundamental note of 18EDL is white D, and since this is Halloween music, we want the scale to be D minor. Let's look at the 18EDL scale again:

The 18EDL scale:

Degree     Ratio     Note

18            1/1         white D

17            18/17     17u D# (su D#)

16            9/8         white E

15            6/5         green F

14            9/7         red F#

13            18/13     thu G

12            3/2         white A

11            18/11     lavender B

10            9/5         green C

9              2/1         white D

The open notes of the guitar are E, A, D, G, B, E, while C is also an easy open chord to play. On the other hand, our one open B chord (B7) fits with E minor (or major), not D minor. I've decided that I don't want both B7 and C to appear together. C major is more closely related to D minor than B7 is, and so we choose to include C rather than B:

The 18EDL scale (guitar open strings):

Degree     Ratio     Note

18            1/1         white D

16            9/8         white E

13            18/13     thu G

12            3/2         white A

10            9/5         green C

Of course, there's no reason to exclude, say, green F, since as the minor third, green F sounds harmonious over a Dm chord. But for these simple songs, I want the first note of the measure to be played over a chord with that note as the root, to make it easier for me to remember.

The Degrees in this list are 18, 16, 13, 12, 10. This is so close to being an arithmetic sequence, which would be easier to include in Line 60. We notice that Degree 14 is red F#, while Degree 13 is thu G (formerly known as "ocher G"). Neither red F# nor thu G (which is almost G#) actually matches the string G on a guitar. But technically, white G is closer to red F# than thu G -- the former is 28/27 (about 63 cents) away, while the latter is 27/26 (about 65 cents) away.

So I feel justified in using Degree 14 in the program (which Line 60 codes for), but simply playing it as G on the guitar. In 12EDO, the 7-limit is only slightly better represented than the 13-limit -- and indeed, playing 14 as G makes 14/11 (about 418 cents) sound as G-B, a major third, rather than a perfect fourth. And besides, F# isn't really needed in a D minor song, since D-F# is a major third.

So this gives us:

The 18EDL scale (guitar open strings):

Degree     Ratio     Note

18            1/1         white D

16            9/8         white E

14            9/7         red F# (played as G on the guitar)

12            3/2         white A

10            9/5         green C

Since we're playing Degree 11 as B, the scale is more Dorian (with a major sixth) than minor. But we prefer B since G major, unlike G minor, is an open guitar chord.

If we write our songs in 12EDL (my originally intended scale, based on Degree 12 or A) rather than 18EDL and play Degree 7 (red F#) as G, then the scale resembles Kristen Lawrence's minor scale (minor seventh, no sixth). Earlier, I considered keeping red F# as F# and play the songs over two different chords, Am and Bm (or Bm7, which like B7 is an open guitar chord). But I didn't try out any 12EDL songs this year, so I don't know which way is better.

The randomizer program as written above creates songs with many whole and dotted half notes. The long notes make it more difficult to add lyrics. Over the summer, I wrote another program that includes more short notes, such as eighth and sixteenth notes, so that we can add lyrics. Let me include my latest version of these programs:

(18EDL)
NEW
10 CLS
20 N=8
30 FOR A=0 TO 13
40 B=2
50 L=RND(B)
60 BB=4
70 LL=RND(BB)
80 D=19-RND(10)
90 P=A*32+(4-B)*8+(4-BB)*L*2
100 D$=RIGHT$(STR$(D),2)
110 PRINT @ P,D$
120 SOUND 261-N*D,L*LL
130 BB=BB-LL
140 IF BB>0 THEN 70
150 B=B-L
160 IF B>0 THEN 50
170 NEXT A
180 PRINT @ 448, "";
190 SOUND 261-N*18,16

For 12EDL, make these changes:

80 D=13-RND(7)
190 SOUND 261-N*12,16

This program makes the rhythm fit 2/4 time (which can be interpreted as 4/4 time), with note lengths ranging from sixteenth to half notes (with a whole note at the end of the song). But unfortunately, this code no longer places guitar-friendly notes at the beginning of each measure.

Finally, I played Degree 18 (white D) on my computer, and it matches D on my musical keyboard at my home. I'm not sure why "white D" matched middle C on the piano in the classroom last week -- perhaps Mocha sounds on that laptop play as different tones.

OK, let's get to today's subbing and "A Day in the Life":

7:55 -- Second period (again, at high schools "first period" is more like zero period) is the first of the three math classes. It's Basic Math, and the students have a Pizzazz worksheet on factoring -- they must list all the factors of a number.

Anticipating a special ed math class, I bring my Number Talks book with me. But Cathy Humphreys writes more about multiplying than factoring. In fact, in her Chapter 3, one of the strategies for multiplying is "factor a factor," which presumes that students find factoring easy. Students are provided with a 25 * 25 multiplication table to assist with the factoring.

Without any Number Talks strategies available, I come up with my own strategies so that students can circle all the correct factors of a number from a given list:

1. Always circle 1 and the number itself.
2. If the number is even, circle 2. If the number is odd, don't circle 2.
3. Never circle 0.
4. Never circle any choice greater than the number.

These tips, of course, fit this particular Pizzazz worksheet, which often multiples of the number (including zero) as possible choices.

8:50 -- Second period leaves and third period arrives. This is another math class, except it's Algebra 1B, the second semester of Algebra I spread out to a year for special ed students.

These students are learning about exponents -- indeed, they have the same worksheet that the other special ed class was completing back on October 18th (at another school). And so I already know which problems I need to emphasize (including that infamous -2y^7/(14y^5) question).

9:50 -- Third period leaves for snack.

10:00 -- At this school, tutorial is after snack. A handful of students attend, including one student who stays just long enough to tell me that he's heading out for Cross Country -- my sport! His best time for three miles is just over 16 minutes -- he beats my best time of around 18 minutes (and that was almost exactly 20 years ago). I wish him luck at his League Finals.

10:30 -- Fourth period is the one science class. The class is labeled as Integrated Science. We know that California is encouraging middle school integrated science, but I still occasionally see high school integrated science from time to time (and recall that I as a young high school student -- back when I was still running XC races -- took integrated science). These students take a test on physical science, including the phases of solid, liquid, and gas (and plasma).

11:30 -- Fourth period leaves and fifth period arrives. This is a "Study Skills" class, but today they are watching Nightmare Before Christmas -- how festive for Halloween.

12:25 -- Fifth period leaves for lunch.

1:10 -- Sixth period arrives. This is the final math class, another Algebra 1B. This time, I have both third period and the October 18th to help me choose which problems to go over.

2:05 -- Sixth period leaves. Once again, this teacher has five straight classes, and so there's no need for me to stay for seventh period.

Since this is a special ed class, aides cover most of the classes. And so in this post, I'll focus on the two classes without aides, third and fourth period.

Third period has some of the hardest workers -- in fact, most students finish the first worksheet and a few finish the second as well. In fact, I name this class to be the best of the day.

Fourth period science, however, is a different story. It's always tough being a sub on a day when the students must take a test, since it's hard to make them be silent.

And today, two students take over the class. One student -- the lone senior in the class -- claims that he is a TA. Usually, TA's are clearly marked on the roster, but I've covered classes (often art or music) where the TA's aren't labeled. Since he's the only senior, it's plausible that he's a TA.

The other misbehaving student is a sophomore. I've subbed at this same school on October 19th -- this guy was one of the trio who leaves class early. He claims that he can't take today's test because he was absent all week -- instead, I should let him have a free day to study on his phone (as there are no extra science texts in the classroom).

I stop him from having a free phone day, but then both he and the senior continue to tell jokes, make noises, and generally make fun of me the entire period. Many students can't concentrate and are unable to take the test. The two troublemakers start throwing tennis balls at the test takers. I say nothing, since they'd consider yelling and getting upset at them to be a reward.

In the end, I inform the fifth period aide. She writes a referral for the senior, while I write up the younger student. But I'm worried, because to the other students during the actual class, it appears that I'm doing nothing about the pair of troublemakers. She also tells me that normally, the senior almost never attends class, yet he just happens to show up today.

And so I wonder what I could have done better in this class. Perhaps I should have just let the pair play on phones the whole time -- at least they might not have thrown objects at the others. But then when the teacher returns, they might claim "The sub told us we didn't have to take the test!" and so it makes me look bad.

I'd better figure something out quickly. This is the first day of a two-day assignment, and I worry about how I'm going to deal with fourth period tomorrow. Those students still have a test to finish, and the two troublemakers will be there.

Meanwhile, normally when I sub for science, I compare this class to what I should have taught two years ago at the old charter school -- but it's difficult to think about science because I'm so worried about these two troublemakers. But let's revisit the STEM (not science) project that I gave two years ago on Halloween:

On the October holiday two years ago, I tried to do another seventh grade Illinois State project, "Orienteering." I made the activity fit the holiday by dressing up as a pirate and having the students search for my treasure. I wrote on the blog that the activity didn't work as well as I wanted, because the public school (co-located with our charter) restricted us to a small area of the playground. It might have worked better this year now that we're no longer co-located -- if only I were still a teacher there.

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

The perimeter of this regular octagon is 20.4 cm. To the nearest [square] cm, what is its area?

(Yes, Pappas forgot to write square centimeters for the area.)

Notice that each side of the octagon has length 2.55 cm. The octagon can be divided into eight isosceles triangles with base 2.55 cm and vertex angle 360/8 = 45 degrees. Each triangle can be divided into two right triangles with leg 1.275 cm and opposite angle 22.5 degrees. These means that the other leg is 1.275/tan(22.5) = 3.078 cm, which is the apothem of the octagon. So the area is:

A = (1/2)ap = (1/2)(3.078)(20.4) = 31.3968 cm^2

To the nearest square cm the area is 31 cm^2 -- and of course, today's date is the 31st, Halloween.

(Notice that tan(22.5) is exactly equal to sqrt(2) - 1, but a proof requires half-angle formulas not often taught in general trig, much less Geometry.)

Lesson 5-4 of the U of Chicago text is called "Properties of Kites." In the modern Third Edition of the text, kites appear in Lesson 6-5.

This is what I wrote last year about today's topic:

Section 5-4 of the U of Chicago text covers kites. The kite is a relatively new quadrilateral classification. Not only did Euclid never define kite, but many texts made no mention of kites -- including my class geometry textbook from 20 years ago. Nowadays most texts define kite, but some include kites only in bonus questions, not in the main text.

Here's what John Conway wrote about the kite -- over 20 years ago, right around the time that I was taking my geometry class:

In fact it's not quite true, either, because "kite" is not
a very traditional name - it was obviously inserted because
this was a type of quadrilateral that SHOULD have received a
traditional name, but didn't, until recently.

Why do we include the kite - plainly because it represents
the one type of symmetry not otherwise mentioned. But this
reason suggests we should also EXCLUDE the non-isosceles
trapezoid.

David Joyce, meanwhile, doesn't find kites to be necessary at all:

Too much is included in this chapter. The sections on rhombuses, trapezoids, and kites are not important and should be omitted.

Notice that both Conway and Joyce want to exclude trapezoids -- but Joyce is the only writer I know who wants to omit rhombuses. Of course, for Joyce, the emphasis should be on triangles and parallel lines, not quadrilaterals like rhombuses. Also, notice that kites are not specifically mentioned in the Common Core Geometry Standards. But I find that students can identify a kite more readily than a rhombus or trapezoid.

Now every kite contains two special vertices, known as its "ends." As defined by the U of Chicago:

"The common vertices of the equal sides of a kite are the ends of the kite."

Notice that Michael Serra doesn't define ends, but does give a name to the two angles whose vertices are the ends of the kite -- the vertex angles, in analogy with the vertex angle of an isosceles triangle. I see that from a proof standpoint, this makes sense, since the first thing that we do in the proof of our main theorem is divide the kite into two isosceles triangles.

And now here is our main theorem: the Kite Symmetry Theorem. As I mentioned back in the lesson on isosceles triangles, we use symmetry to determine the properties of kites. (A pre-Common Core proof might divide the kite into two triangles to be proved congruent by SSS.) I will post a proof here of the Kite Symmetry Theorem. It is taken directly from the U of Chicago, except that I, as always, add a Given step to the beginning of the proof. Since the U of Chicago's proof has eight steps, mine has nine:

Kite Symmetry Theorem:

The line containing the ends of a kite is a symmetry line for the kite.

Given: ABCD is a kite with ends B and D.

Prove: Line BD is a symmetry line for ABCD.

Proof:

Statements                                           Reasons

1. ABCD is a kite with ends B and D  1. Given

2. AB = BC, AD = DC                         2. Definition of ends of kite (meaning)

3. Tri. ABC and ADC are isosceles      3. Definition of isosceles triangle (sufficient)

4. Let m be the perp. bis. of AC          4. A segment has exactly one perp. bisector

5. A' = C, C' = A                                  5. Definition of reflection (sufficient)

6. m contains B and D                         6. The perp. bis. of the base = angle bis. of the vertex angle

                                                                 (so it contains the vertex)

7. B' = B, D' = D                                 7. Definition of reflection (sufficient)

8. ABCD reflected over m is CBAD     8. Figure Reflection Theorem

9. m (Line BD) is a symmetry line      9. Definition of symmetry line (sufficient)

Notice that more than half of the reasons in this proof are definitions. This underlines how important definitions are to the study of quadrilaterals.

As for the other theorems in this lesson, the Kite Diagonal Theorem follows directly from lines 4 and 6 of the above proof. It makes the symmetry diagonal the perpendicular bisector of the other diagonal and so the diagonals of a kite are perpendicular, and the symmetry diagonal bisects the other one.

Finally, we have the Rhombus Symmetry Theorem. It states that a rhombus has two symmetry lines, as both of the lines containing its diagonals are symmetry lines. This follows directly from the classification of a rhombus as a kite. In texts that define kite exclusively, the theorems "The diagonals of a kite are perpendicular" and "the diagonals of a rhombus are perpendicular" are two separate theorems, often in two separate sections. But here we can easily see why the diagonals of both the kite and the rhombus are perpendicular -- because the rhombus is a kite! Also, since we are defining kite inclusively, we don't need any extra steps in the proof to ensure that our kite isn't a rhombus.

Lesson 5-3: Conjectures (Day 53)

Today I subbed in a special ed middle school class. It's one of those (mostly) self-contained classes, and so I become familiar with many of the students today. Aides run the classes all day, and so there's not much to say about classroom management today. Consequently there's no "Day in the Life" today.

The middle school rotation actually starts with first period today. Most students have an elective first period and P.E. sixth period, but for some students it's the other way around -- especially if their elective is music (band or choir). For all students, the other classes from second through fifth periods are social studies, English, math, and science.

As always, I do try to help out with math. In fourth period, the aides ask me to help out a pair of students (or "friends," as they like to call them) -- a boy and a girl -- who have a worksheet with four subtraction problems, with both numbers (the minuend and the subtrahend) less than 50. All four problems are set up for the standard algorithm with borrowing, with 34 - 17 as the first question.

This means that I couldn't use any ideas from the Number Talks book. Some strategies from Cathy Humphries might have been useful, such as "Decompose the Subtrahend." But this might have confused the friends since the worksheet was already set up for the standard algorithm. Still, I wish to remain on the lookout for opportunities to use Number Talks in class (especially special ed days).

As for the other classes, the only one in which any real work is done is social studies. The students are supposed to draw a picture that represents them. These pictures are drawn on puzzle pieces, so that all the friends' pieces are put together at the end.

In both English and science, the students watch videos. At first we try to watch Ice Age, but the DVD player doesn't work well. We're only slightly more successful with another DVD, Scooby-Doo and the Monster of Mexico. Notice that this movie takes place in Mexico on the Day of the Dead (Dia de Muertos) -- a holiday that falls slightly after Halloween (that is, at the end of this week). For science class the students watch Finding Nemo on YouTube -- and this we finish, since no disc is required.

Oh, and speaking of Halloween, at lunch the students have the opportunity to decorate pumpkins. A few volunteers from the Sheriff's Department come in to help the friends. Afterward everyone gets a slice or two of pizza to eat.

Today I finally get into Halloween mode and hand out some holiday pencils and erasers (which are orange just like their jack o'lanterns). Sixth period is the teacher's conference period. Most students are at P.E., and normally I'd join them on their walk, but today I don't. This is so that I could go out and purchase one last treat for them -- a tiny bag of Skittles candy.

Lesson 5-3 of the U of Chicago text is called "Conjectures." There is no real equivalent in the modern Third Edition, as Chapter 6 of that text goes directly from the equivalent of the old Lesson 5-2 to the equivalent of the old Lesson 5-4.

This is what I wrote last year about today's lesson:

Lesson 5-3 of the U of Chicago text discusses conjectures. The text defines a conjecture to be "an educated guess or opinion." It is a statement that has yet to be proved.

David Joyce has a low opinion of conjectures in a high school geometry class. In his scathing review of the Prentice-Hall text, Joyce writes:

(And this occurs in the section in which 'conjecture' is discussed. "Test your conjecture by graphing several equations of lines where the values of m are the same." What's the proper conclusion? That theorems may be justified by looking at a few examples?)

In summary, the material in chapter 2 should be postponed until after elementary geometry is developed.

That is to say, the statements should be postponed until they can be proved.

Now Michael Serra takes a diametrically opposite approach from Joyce. In Serra's Discovering Geometry text, a great many statements are given well before they are proved -- since the proofs don't occur until the final three chapters of the book, Chapters 14-16 (old version -- Chapter 13 in the new version). What would Joyce say about having so many of these unproved theorems in his text?

The theorems can be proven once a little actual geometry is presented, but that's not done until the last half of the book. A little honesty is needed here. Why not tell them that the proofs will be postponed until a later chapter? Or that we just don't have time to do the proofs for this chapter. Even better: don't label statements as theorems (like many other unproved statements in the chapter).

Well, Serra doesn't label statements as theorems -- he calls them conjectures! In all, 114 conjectures are in Chapters 3-13 of Serra's text -- Conjecture 1 is the Perpendicular Bisector Conjecture (which we've already proved on this blog) and Conjecture 114 is the Law of Cosines. Clearly, discovering and stating conjectures are the heart of Serra's learning philosophy.

As I've stated before, I want to lean towards Joyce's preference of proving all theorems as soon as they are mentioned. I've rearranged the order of the U of Chicago text in order to make sure that proofs precede applications of the theorems -- especially when I want to highlight the Common Core proofs, which may differ from traditional proofs.

But I'm also sympathetic to Serra's philosophy. I want to show some interesting results of geometry without being limited to what the students can prove. Also, one way to pique a student's interest is to show the result and ask, "Is this always true?" or "Why is this always true?" Proving many boring low-level theorems just to make sure that nothing is used before it is proved would result in students losing interest and wondering why they are forced to write endless proofs. And of course, Serra's text helps out students who may be weak at proof-writing and would easily forget how to write a proof once taught -- the main results are stated so the students can learn them, and then the proofs are given at the end of the book, right before the PARCC exams.

The point of this lesson is to get the students thinking about the properties of special quadrilaterals without worrying about how to prove them. In other words, I want to get the students engaged and thinking about the quadrilateral properties so that they can make the conjectures.

We begin by dividing the class into groups -- say of three or four students. Each group is assigned a worksheet -- or the members can write down answers on a common blank sheet. Then my usual set of ten questions are assigned -- but there are some differences between this and the usual individual worksheets that I post.

First of all, let's look at the first two questions:

1. What is the teacher's __________?

2. What is the teacher's __________?

Beforehand, the teacher fills in the blanks with words -- I'd fill them in with age and weight. I have no problem with giving this much information to the students -- but many people, especially women, are highly sensitive to revealing such personal data. This is why I left blanks in the questions -- so that the teachers fill in the blanks with words that they are comfortable revealing in class.

The teacher asks the question, "What is my age?" (or whatever is in the first blank). The groups signal when they want to answer. The teacher calls upon the group that signaled first to answer -- and since this answer will almost certainly be wrong, the teacher then calls upon another group. When a group finally gives the correct answer, the teacher awards this group a point. (In case you're as curious as the students are about my age, I am currently 37 years old.)

Notice several things about this game so far. The first team to give a correct answer -- and the answers in my version of this activity are numerical so far -- is the one to get the point. And after the first two questions, two groups have one point each -- or possibly one team already has two points -- and the rest have none.

Certainly the groups without points so far are eager to earn one. And so they are faced with the next question in the activity:

3. True or false: the diagonals of a rectangle are always equal in length.

Recall that this activity is all about conjectures. The students have already spent time making conjectures (that is, educated guesses) about the teacher's age and weight -- now it's time to make a conjecture about geometry!

This question serves several purposes. First, the students in groups that are trailing in points -- the same students who would have complained about doing math after the long exam -- now suddenly want to answer a math question because they want to catch up to the leaders. Second, this question is a true-or-false question, so students who might have tuned out if given an open-ended question will want to try this one at least since there are only two possible answers. The students are likely to guess at the answer -- and they're encouraged to do so, because a conjecture is a guess! Third, the conjecture in question involves rectangles -- and students who tend to forget what a rhombus or trapezoid is will still remember what a rectangle is. The only problem word that might be a barrier to participation is diagonal -- so the teacher reminds them that the two diagonals of a rectangle run from a corner to the opposite corner.

In my activity, every third question (that is, the third, sixth, and ninth) is a true-or-false question. I use these to give the students more opportunities to earn points. The teacher allows every group to give an answer of true or false before revealing the answer, and every group that gives the correct answer earns a point. In this way, groups can earn points without worrying about being the fastest group to get the answer.

Of course, the answer to Question 3 here is true. Hopefully, most, if not all, of the groups were able to guess that the diagonals of a rectangle are equal, so that every group is on the scoreboard. Now we move on to the next questions.

4. The diagonals of a square always divide the square into four triangles of __________ size.

5. The diagonals of a kite are always __________.

Now these questions are open-ended, just like the first two questions (but there are no more personal questions -- from now on, all are geometric). So we return to having the groups compete, and only one group will receive the point.

The students should test the fourth conjecture by drawing several squares -- by several, let's say one for each member of the group. So the first group to have drawn enough squares for the group as well as give the correct answer "equal" is the group to earn the point. Since these problems are increasing in difficulty, a teacher may choose to give two points, rather than one. (Notice that the four triangles are in fact congruent, but since congruence has not been taught yet, we instead say that they have equal "size" -- where the students can probably get an idea of what that might mean.)

The difficulty in the fifth question is that after having seen the diagonals in Questions 3 and 4 turn out to be equal, the students may jump to the conclusion that "equal" is correct yet again. The teacher should remind the students to draw the kites to make sure -- and to drive the point home, the teacher should draw a counterexample to the claim that the diagonals of a kite are equal. The students might not think to say that the diagonals are "perpendicular," which is the correct answer. So the teacher can give the hint that they should check the angle between the diagonals. By now, I'd award the point(s) to the group telling me that the angle is 90, even if the actual word perpendicular is not used.

Now we move on to our next true-or-false question:

6. True or false: consecutive angles in a parallelogram are always equal.

Of course, the teacher should define consecutive angles (or "adjacent angles"). If necessary, the teacher can draw a parallelogram on the board and point out where the consecutive angles are -- and naturally, that parallelogram should be a rectangle (or nearly so), in order to avoid giving away that the correct answer is false.

7. If ABCD is a parallelogram and angle A has measure 30, then angle B has measure _____.

8. Opposite angles in a parallelogram are always __________.

Notice that the seventh question is an extension of the sixth -- ABCD is a parallelogram, and the measure of angle A is 150 degrees. Because of question 6, the answer isn't 150 degrees. A point (or points) are awarded to the group correctly answers that the angle B measures 30 degrees. No conjecture is stated in this question, but the implied conjecture is that the consecutive angles in a parallelogram are supplementary. Since the answer to question 7 is numerical, I don't require the groups to draw a parallelogram for each student in the group.

 Of course, the answer to question 8 is that opposite angles in a parallelogram are always equal.

9. True or false: opposite sides in a parallelogram are always equal.

This one is self-explanatory -- the answer is also true.

10. A square has _____ lines of symmetry.

I've mentioned this one earlier on this blog. A square has four lines of symmetry -- if the sides of the square are parallel to the coordinate axes, then a square has one horizontal, one vertical, and two diagonal lines of symmetry. The diagonals are the ones that are often missed. Once again, this is a numerical answer, so I don't require a diagram for each member of the group. Of course, the students will want to draw at least one square in order to find its symmetry lines.

And now, as I often like to do, here's a Bonus Question. As I pointed out last week, I don't like it when students are eliminated from passing when there is plenty of time left in the semester, and in the same manner, I don't want students to be eliminated from winning this game too early. And so this question can be worth many points -- enough for the last (or maybe the next-to-last) place team to catch up. (In my game, I may deduct points for behavior -- so I might not want the last place team to be able to win if their behavior doesn't warrant it.)

Bonus Question: Take a quadrilateral and find the midpoints of its four sides. Join these four points to form a new quadrilateral, the Midpoint Quadrilateral. Midpoint Quadrilaterals are always what type of quadrilateral in the hierarchy?

Students will need time to figure out how to draw this Midpoint Quadrilateral. After each member of the group draws the Midpoint Quadrilateral, the winning team will be the one that correctly identifies the Midpoint Quadrilateral to be a "parallelogram."

For the rest of this post, I write about two memorable times that I played this game in class. One was as a sub, in fact a few days after Halloween. The other was as an actual teacher, last year on my birthday.

First, as a sub....

But yesterday, I had an opportunity to play my game. Most of the eighth grade math classes were assigned to take notes, but the Math/Computing class had a worksheet to finish. And so I began by asking the students to guess my age, and then my weight. So two of groups already had a point, while the other seven were scoreless.

Then the third question I asked was simply the first question from the worksheet -- namely to graph the equation x + y = 5 using intercepts. Just as I mentioned from my original Conjectures worksheet, every third question was a chance for each and every group to earn a point. I think that only about half of the groups earned the point. Some of the groups drew the graph incorrectly, while others had the correct graph but identified the slope as 1 instead of -1. My fourth question was the second question from the worksheet, x + 2y = 8 -- which, just as planned, allowed for only one group to earn the point.

I admit that graphing isn't necessarily the best sort of question for this game -- especially when the students had to do so much work to answer each question (finding both intercepts and the slope). The game worked out better on Monday, when the Computing class was working on the computers and the rest of the classes had equations to solve, so I played the game only in the other classes. When solving equations, nearly all the groups earned the point on the third question, which is what I want.

The worksheet consisted of about a dozen graphs, yet I only had time to play the game with six. So someone might point out that if I had simply passed out the worksheet and asked the students to work on it, they might have completed many more than six of the graphs. The game wastes so much time when the students can't work on the next problem until the class reaches it -- especially on every third question where I must check every group before proceeding.

But let's recall the situation the class was in. It was the second straight day the class had a sub. So the students, already knowing coming in that there was a sub, were already thinking about how much mischief they could get in -- things they'd never dream of doing with the regular teacher. And once they arrived, they were probably hoping that they could play around on the computer, only to find out that the teacher had locked the computers away and assigned the worksheet.

So we can see that the students weren't in much mood to work. And yet, I believe that there was a game, the allure of earning points motivates them. Many of the students might have just thrown the paper away, or worked on it at a snail's pace and still be on the first graph late into the period. Also, checking every third question keeps those who might have drawn every graph incorrectly or calculated every slope with the wrong sign.

Some people -- especially traditionalists -- dislike group work, and believe that students learn much, much more effectively when doing individual work. But I often find that as a sub, classroom management is easier when I only have to keep track of nine groups, not 35 students whose names I don't know. In order to earn a point, every member of the group must have drawn the graph on the paper, and so the students end up motivating each other at least to draw the graph on the paper (even if all they're doing is copying the other group members).

I remember that the group who debated with me about a slope of 1 vs. -1 failed to earn a point for that third question, and consequently fell behind the other groups. Even though by the end of the fourth question, I'd convinced them (using Slope Dude) that their slopes had the wrong sign, the group never caught up to the leaders. As I announced the winners, I overheard a member of that group saying to one of his teammates, "But we're the real winners because we did eleven problems and the others did only six." Yes, imagine that! The group kept on working on the graphs well ahead of my pace in an effort to catch up to the other groups' score. (This was my intent of including the Bonus Question on my original worksheet.) And these are students who would -- had I not played a game -- at best, have calculated all the slopes with the wrong sign, and at worst, have just thrown the papers away and not worked on the graphs at all!

And so I will continue to play this game in class. One difference between my original vision and the way the game has played out in the classroom is that I almost never make the questions for the entire class to answer true-or-false. After all, how often will the students be given a worksheet where every third question is true-or-false? One thing I might consider is, on every third question, have the groups work on a problem, and then present one group's answers to the class. Then all of the groups must determine whether that answer is correct (true) or incorrect (false). Indeed, even if I don't explicitly ask a true-or-false question, I might said to the group that insisted that the slope was 1, "Well, this other group says that the answer is -1. So let me take a point away from them and give it to you." So this threat would force the other group to defend their answer of -1 -- now the groups are debating the answer with each other, rather than the usual situation of me vs. the students.

Notice that I never actually give this game a name. I'm prone to name this game either Conjectures -- after the post in which it appears -- or Who Am I? -- which is the first part of the title of the post that appeared the day after I first posted the game. The only problem with either name is that they both refer only to the first two questions -- unless it's from my Conjectures worksheet -- since, for example, the fact that the slope is -1 isn't merely a "conjecture," but should be known. Of course, even on the Conjectures worksheet, "Who Am I?" only refers to the first two questions.

By the way, what exactly was the prize that I gave the winning groups? Actually, all I did was leave their names for the regular teacher with a positive note!

Second, as a teacher....

Now I decide to play this game today in all my classes. And you may ask, why today? Well, I actually played this game as a sub one year ago today -- and I did it for one very particular reason.

The answer to the first question "What is the teacher's age?" is 36. That's because today is -- you guessed it (or remembered from last year) -- my 36th birthday! And so I knew that if I was going to play a game which starts with my age, it might as well be on my birthday.

What lessons do I include in today's game? Well, just as in the version of the game I posted as a sub, I want to focus on geometry questions. As it turns out, the game fits the current seventh grade lesson like a glove. Yesterday, the students cut out triangles out of straw, and Illinois State even asks the students to make conjectures about the triangles they created. So it's easy to fit some of those right into the game.

Today is Wednesday -- always a scheduling adventure at our school. For once, we actually follow the same schedule as last week -- but again, it means that I don't see the seventh graders as much as the other grades. I try having them come up with Triangle Inequality as a conjecture. A few of them are able to get on the right track, especially after I give them the hint (or "lead them by the nose," much to David Joyce's dismay).

For eighth grade, I notice that the STEM project mentions the measures of angles that are vertical, adjacent, corresponding, and so on. So I play the game using these conjectures. One big problem is that some students can't use a protractor correctly, so many don't arrive at the conjecture that vertical angles have the same measure. (Actually, the seventh graders also had to conjecture Triangle Sum, but I don't even try to reach that conjecture, knowing that if the eighth graders won't use the protractor correctly, neither will the seventh graders.)

Meanwhile, for sixth grade, the animals project ultimately relates to guessing how much room animals need, so it fits into the game as well. They are learning about how to find the dimensions of a rectangle given its area -- that is, factoring.

I like this game as a sub because it gives the students something to do. But if I use it in the regular classroom, it might be better to do some preparation. Once again, I just took the STEM project and added my own "What is the teacher's age?" questions. But instead, I could have come up with some questions such as just measuring random given angles. If I award points in the game, then the students should be motivated to find them. Then after that I segue to finding specific angles such as vertical angles or those of a triangle. That should lead them to make the conjectures.

So far in 2018, I haven't played the Conjectures/"Who Am I?" game. This is mainly because I'm still focusing on classroom management techniques that I'd like to practice without playing a game. I came very close to playing it two weeks ago, when I subbed for another special ed class that happened to be studying geometric conjectures.