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Geometry Follow-Up: Proof in a Bag

The concept of proof-in-a-bag is simple.  Write out a two-column proof and then cut it up so that each statement or reason is by itself on a scrap of paper. Then put all the scraps in a bag (a small sandwich bag works well, though an opaque paper bag might have more of a dramatic effect) and have kids work on rearranging the scraps so that they form a coherent proof.  You can decide whether you want students to know ahead of time what it is they’re proving, or if you want them to figure it out by putting statements with “given:…”, “prove…” and a diagram in the bag as well.

Credit where credit is due: I got the idea for this from Laura Chihara while a student in her Algebraic coding class at the Carleton-St. Olaf Summer Math Program.

 It’s nice to have any activity where kids are physically doing something in a math class, of course, but I really like what kids get out of this activity.  It emphasizes the idea that you have to have enough evidence before you can conclude that triangles are congruent (otherwise, what are those “extra” statements doing in the bag?)  And it is very good for helping students understand what can be a statement vs. what can be a reason.  I often find that students want to use triangle congruence theorems like SAS when using properties of triangle congruence; the structure of this activity leads them to realize that they’ve already used SAS to justify the triangle congruence statement; they now need to use something else (CPCTC or the equivalent) to start using the congruence.

There are some times when I would definitely not use this activity.  If the proof is a particularly exciting one for kids to work out on their own, I wouldn’t rob them of the opportunity.  Proof-in-a-bag works best for simple, straightforward proofs, where the two-column proof format can be used without having to do a lot of extra explaining.  I generally use it for one day only, at a time when the class has had some practice writing proofs but has not yet reached a level of comfort with them.

Does anybody else have activities or techniques that they use to teach writing proofs?  I’d be especially interested in what people do who don’t insist on a strict two-column format all of the time.

A first post on geometry and proof

Euclidean geometry is for many students the first time they get a taste of what math is really about.  The problems don’t all fit the same pattern; it’s natural and expected that students will come up with their own arguments to prove something, rather than following a set of rules.  Ideally, geometry class also involves experimentation and conjecture.

I don’t think Park students are encountering these things for the first time in geometry.  Our students are used to investigating and asking their own questions.  And they are used to making careful arguments to support their claims.  Still, for Park students as much as students anywhere, geometry tends to be the first time that they are asked to write formal proofs.  Anyone who has taught geometry knows that writing proofs can feel to students like wearing a straightjacket.  For the first time, arguments that are correct but either vague or not axiomatic are inadmissible:

  • Opposite sides of a parallelogram have to be congruent because lines with the same slope stay the same distance apart.
  • Opposite sides of a parallelogram have to be congruent because there is no way to extend one of those sides without changing the angle of the side coming to meet it.
  • The base angles of an isosceles triangle have to be congruent because the triangle is symmetric.

I’ve stopped telling students that these arguments are not convincing.  Anyone who understands the terms they’re using would be convinced.  And I’ve even stopped telling students that they are incorrect.  They’re not incorrect; they’re just not arguments from first principles.  They appeal to intuition and common sense, as most arguments we’d make in daily life do.

Acknowledging those things, we still need to make rigorous arguments that appeal to specific principles we’ve studied in class, such as theorems about parallel lines, and theorems about congruent triangles.  For this reason, I stick to the “statement/reason” model of proofs taught in most geometry classes.  I find that if students don’t write proofs this way it is too easy for them to fall into arguments that are merely intuitive.  It’s also easy for them to fool themselves into thinking that they have enough evidence to conclude that triangles are congruent when, say, they’ve really only found two pairs of congruent sides.

I don’t, however, insist on the degree of rigor that most geometry books do.  Students in my classes do not write proofs that contain the sequence, “If angles form a linear pair, then they are supplementary.  If angles are supplementary, then their measures add up to 180 degrees.”  They can go right from linear pair to adding up to 180 degrees.  I don’t think that this level of following tiny steps in a chain serves the purpose of helping students to build new theorems out of the knowledge they already have.

Generally, my rule is that if students are using congruent triangles to prove something, they need to

  • Name the three pairs of sides/angles that they need to justify the congruence, providing a reason for each.
  • Name the pair of congruent triangles and say which theorem (SSS, ASA, etc) they are using.
  • Only after they’ve done all that, name the pair of sides or angles that they can now say are congruent.  To justify this, they will sometimes use the infamous “CPCTC,” or, since many students have trouble remembering what the acronym stands for, just say that they are using triangle congruence.

I think that, among Park faculty, I am one of the teachers who insists the most on some kind of standard template for proofs, even though I allow much more leeway in what can be used for a reason than most textbooks do.  I’d be interested in what other teachers ask of their students when writing geometry proofs.

Puzzles for 11.11.11

Some questions we are asking our classes today:

Anand: How many times this millenium will the date consist of a single digit?

Bill: Today, the day, month, and year are the same. In how many days will this happen again?

Angela: How can you get an answer of 0.0909090909… using only one number, but as many times as you want, and basic arithmetic?

What are you all doing?

The Cruel Irony of Algebra

You would think it would have occurred to me sooner, but it wasn’t until a few years ago that it really hit me what I think is the biggest problem most students have with Algebra:  they don’t actually think the letters represent numbers.   Here’s the kind of question I’ve asked that illustrates what I mean:

Chuck says that (xy)(wz) is always equal to (xyw)(xyz).  Is he right?

Over the years, I have found that if a student is unsure of whether or not they are supposed to “distribute the xy”, they often just guess.  When asked why, they say that they were unsure of the rule they were supposed to use, so they just took their best shot.  For many years, I tried to show them which rule to use in various situations and the principles involved, hoping that over time they would catch on to the logic of algebra.

But invariably, for many of my students, even a slightly changed question presented what seemed like a freshly baffling challenge.  After all, does the question below really seem all that different (other than to a math teacher)?:

Chloe says that (xy)(w+z)  is always equal to (xyw) + (xyz).  Is she right?

So what to do?  While it may seem like taking two steps backwards in the march towards abstraction and generalization, these days I ask my students how they could possibly figure out for themselves if the two expressions are always equal, and the ensuing discussions leads us to the question of what the heck those x’s and y’s and z’s and w’s represent—numbers!  So why not try out these equations with numbers?  The cruel irony of algebra is that what is intended to make generalization easier actually becomes so abstract for many kids that numbers are the last thing on their minds. They end up seeing algebra as a bunch of arbitrary rules that are hard to predict.

Of course, just because two expressions are equal with a given set of numbers doesn’t mean they always will be—and we discuss that eventually as well.  But as an entry point into algebraic identities, and as a gut check to see if my students “get” what algebra is about, I find this works because students believe what they can test and see for themselves.  Having them practice distributing multiplication over addition (and also practicing NOT distributing multiplication over multiplication), while it has its place, I find isn’t a good substitute for the intuition that develops by playing with the raw numbers.

Next, semi-related entry:  The joys and sorrows of “flip and multiply”!

The “Wow!” Factor

Sometimes you see things that are so amazing that it seems almost criminal not to share them with other people.  Being a teacher, I have a ready-made audience to share my enthusiasms, and I long ago concluded that anything that is extraordinary and mathematical, even if it is unrelated to the topic we are studying, is worth sharing.  There is nothing trivial about having students say “Wow!”  So here’s something that did that for me a year or so ago– Hans Rosling’s talk about the world’s demographics, using data-rich graphs:

In that talk, he uses a program that makes animations of data in beautiful and intuitive ways and that yield real insight.  Since he made that video, he has started a website where that program is available for use directly on the website using Adobe Flash. The site is   and it really is a wonder.  My son, in the 6th grade, couldn’t stop playing with it, and was immediately drawn in to the questions the graphs you create present.  Why was there a “Bangladesh Miracle”?  Why does China’s life expectancy suddenly drop 20 years around 1960, and then quickly rise again?  Why did the average number of children per women in the U.S. rise sharply between 1940 and 1970?  Why is the connection between average income and mortality so varied in different parts of the world?

I never was that interested in statistics and data analysis when I was in school.  With this program, it takes a herculean effort NOT to be interested.

But Who’s Counting? (Or: It’s Friday, Friday)

There are times when you can tell your class is a little fried and they need some short-term gratification rather than the usual drawn-out discussions.  Last period Friday, in particular, is a time when it is useful to have some math games ready to go.  So on this Friday I’ll describe a game that I stole from the wonderful Public Television show Square One.  (Remember that?)

As the “host”, I set up the digits of an arithmetic problem like _ _ _ + _ _.  (Actually, I make each digit a box rather than a dash, but that’s much easier to do on a whiteboard.)   Students copy this into their notebooks.  Then I use a calculator to randomly generate digits ranging from 0-9, repeats allowed.  As I read each digit, students place the number in one of the boxes, with the goal of making the largest sum possible.  The catch is that they are not allowed to move a digit once they place it.  The winner, at the end, is the student with the largest sum.  (You could also ask for the smallest sum, of course.)

Of course, this game is too easy for high school.  The strategy is clear to the students, and after that it’s all luck.  So after we play with the initial setup, to get the idea of the game, I begin to vary the game board, tailoring it to the level of the class.  _^_ + _ _ _ _ is an interesting one – is it best to use a nine, if it is chosen, as the base of the exponent, the exponent itself, or the first digit of the four-digit number?  Using fractions reinforces the concept that the bigger the denominator, the smaller the value of the fraction.  You can also involve “sin(_ _)” in your expression (specifying that you are using degrees), and students will need to think about what degree value will maximize the sine function.  For students thinking about limits, you can compare ratios of different types of functions so that students need to think about the shapes of their graphs and their end behavior.

I like this game for several reasons.  First of all, there’s an element of suspense when choosing the final few digits that always puts students on the edge of their seats.  Almost all students like this game and find it engaging.  Secondly, when the game is over (and they are allowed to talk about it), students want to argue about what the best possible result would be.  Because of this, they make all the mathematical arguments appropriate to the game board without my having to prompt them.  Third, it is extremely malleable to the needs of the class, and it can be anywhere from very easy to very hard.

One final note: I’ve been able to find only one web clip of this game, here.  They play a simple version in which the object is to make the largest five-digit number possible.  Watching it (if you can make it past the super-silly intro) might give you an idea of the principle of the game.