Wednesday, January 20, 2016

The MTBoS, Week 2 (Day 87)

Before I get to Lecture 13, it's time for Week 2 of the 2016 Blogging Initiative. This week's idea comes from Julie Reulbach, a North Carolina high school teacher:

Called a “My Favorite,” it can be something that makes teaching a specific math topic work really well.  It does not have to be a lesson, but can be anything in teaching that you love!  It can also be something that you have blogged or tweeted about before.  Some ideas of favorites that have been shared are:
  • A lesson (or part of one) that went great
  • A game your students love to play
Reulbach lists more options here, but this is the "My Favorite" that I wish to post. I've mentioned My Favorite Game here on the blog a few times before. In the following description, this game is set up for a Geometry lesson on quadrilaterals, but it can be adapted to any lesson.

Oh, and I've noticed that based on the blogs I've glanced at so far, many teachers' "My Favorite" posts involve computer programs. I seem to be behind the times as my activity uses pencil and paper. But remember that I'm just a sub who isn't guaranteed a classroom with computers for student use, much less ones with the programs mentioned on the other blogs.

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 35 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.

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

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

And the game continues in this fashion. At the end of this post is a worksheet containing all ten questions plus a Bonus Question.

I'll let the teachers decide what prizes to award the winning team -- or teams, since I prefer to give the reward to the top two groups.

Now returning to the present, let me say that when I first posted this activity last year, I was unsure how well it would work in the classroom. At the time, it was just before my first day in the classroom as a substitute teacher. Now I say that this game works out well enough that I try to play it every time that I sub in a math classroom.

Of course, I can't play the game every time I sub, but only when it fits the teacher's lesson plan. For example, just before Halloween, some of the sixth graders were finishing a test that they had started the previous day, while the rest of the students had a Pizzazz worksheet to complete. I actually tried to start asking the students who had finished the test about my age and weight, but then it disturbed the students who were still working. So I didn't play my game in any subsequent period.

But in early November, 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 want to name this game either Conjectures -- the title of my 2014 blog post in which it appears -- or Who Am I? -- which is the first part of the title of the post the day after I first posted the game in 2014. 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 (just like the day described in my MTBoS Week 1 post)!

I try to play the Conjectures/"Who Am I?" game every time I sub in a math class -- so it definitely counts as "My Favorite" game -- but I don't always have the opportunity. The day I described in my MTBoS Week 1 post was a quiz day (just like before Halloween), with only 20 minutes to review before the quiz -- and besides, it was the second time I'd subbed in that room, so a few of the students already remembered my age and weight! The last time that I played the game was back on December 7th, and this is how the game went:

The Integrated Math I students were also in Module 6 of their respective text, "Forms of Linear Equations," and they were working on Lesson 6.3, "Forms of Linear Equations." In this class, I played the usual game where I start by asking the students, "What's my age?" And of course, I had fun telling them that I was 35 years old, because it's my birthday. Unfortunately, the students in sixth period must have heard from the ones in fifth period that it was my birthday, because many groups were trying to shout out 35 at the same time.

Speaking of my birthday, I'm now halfway through watching my gift, the DVD Mind-Bending Math: Riddles and Paradoxes, that I've been discussing on the blog everyday. Normally I'd be describing Dave Kung's next lecture, Lecture 13, called "Games With Strange Loops," right now.

But this post -- like so many on my blog -- is much too long already. I was hoping to shorten this post by talking about some of the games Kung describes (since this post really is about games), but I want to reserve MTBoS posts for things that I've actually done in the classroom. Instead, I'll just link to each of the games mentioned in Kung's lecture (and why each one is paradoxical):

Pirate Game
Two Envelopes Paradox
Newcomb's Paradox
Puzzle of the Blue-Eyed Islanders
    (Solution to Blue-Eyed Islanders Puzzle)
Prisoner's Dilemma

(The Quick Conundrum for the lecture is, when water boils, what's in the bubbles? It's not air, hydrogen, or oxygen, but water vapor.)

If I were in a classroom and wanted to play one of these games, I might choose the Puzzle of the Blue-Eyed Islanders, since it leads directly to mathematical induction. In a way, the Pirate Game also involves some sort of induction. On the other hand, I claim that the Prisoner's Dilemma is one of the most commonly played games by the teenagers in our classes -- except it's the Lover's Dilemma:

-- If I like my friend, and my friend likes me back, then we can start a relationship.
-- If I don't like my friend that way, nor does my friend like me, then we don't start a relationship, but at least we can remain friends.
-- If I like my friend, but my friend doesn't like me back, then our friendship is ruined. My (former) friend gets to laugh at me -- the best of all situations for my (ex-)friend, the worst for me.

Then the Nash equilibrium is for neither lover to confess his/her true feelings. This explains why two teens may like each other, but so often neither one admits it. As Kung points out at the end of the lecture, these simple games tell us a little more about how the world works.

Let's wrap up this post with a few more links. Here's the blog of Julie Reulbach, the teacher who came up with the Week 2 Question:

And here are the blogs of the three teachers to whose blogs I will respond. I prefer responding on Blogspot websites as I'm already a member, so I chose the two blogs just above mine than skipped up to the next Blogspot site. (And yes, I know that I've posted at the first link last time!)

Finally, here are the two sides of the worksheet for the Conjectures/"Who Am I?" game:


  1. I don't know that I've ever met a sub as committed as you! I appreciate your efforts to create an activity like this for the students you're working with. You've got some fun competition, some chances to earn points for teams that might be struggling, and a fun way to engage with concepts and skills. Thanks for sharing!

    1. The thing is that the students usually enjoy this game -- the week after I posted this, I subbed for one teacher for three days and we played it the second day, but then they were disappointed when they couldn't play it the third day!

      But sometimes this game doesn't work well. It works best with groups of four -- I tried it later last week with groups of six and the class became too noisy.

      Thanks for reading and participating in the MTBoS.