Don’t forget Geometry when teaching Algebra

Right now, Michigan educators are trying to sort out the implications of the state switching to endorsing (and paying to provide) the SAT to all high school juniors statewide instead of the ACT as it had previously done. The SAT relies very heavily on measuring algebra and data analysis. This leaves plane and 3D geometry, Trig and transformational geometry under the label “additional topics in mathematics.” The new SAT includes 6 questions from this category. Compared to closed to 10 times that from more algebraic categories.

This comes up in every SAT Info session I lead. Should we just stop teaching geometry? All Algebra? Could geometry become a senior-level elective? If it’s not going to be on the SAT, then what?

These questions reflect a variety of misconceptions about the role of testing in curriculum decisions. In fact, these are the same misconceptions that are driving an awful lot of the decisions that are being made. And in this case, I think there’s more at risk than simply over-testing our students.

It would be a real shame to see Geometry become seen as an unnecessary math class. Because it’s not.

And to illustrate that, I’m going to tell you a story about a conversation I had with my daughter. She’s 6.

She wanted to try multiplication. She’d heard the older kids at school talking about it. So, I taught her about it and let her try some simple problems to see if she understood. And she did, for the most part.

So, I started to not only make the numbers bigger, but also reverse the numbers for some problems that she’d already written down. She had already computed 2 x 3, and 4 x 2, and 5 x 3, but what about 3 x 2, and 2 x 4, and 3 x 5. Were those going to get the same answers as their reversed counterparts?

She predicted no. So, I told her to figure them out to show me if her prediction was right or wrong. To her surprise, she found that switching the factors doesn’t change the product.

“Daddy… why does that happen? It should change, shouldn’t it?”

Translation: Daddy, how do you prove the Commutative Property of Multiplication?

How would you prove it?

The geometry teacher in me thinks about area when I see two numbers multiplied. We can model (and often do) multiplication as an array. It’s just a rectangle, right? A rectangle with an area that is calculated by it’s length and width being multiplied.

What happens if we rotate the rectangle 90 degrees about it’s center point? Now it’s length and width are switched, but it’s area isn’t. Because rotations are rigid motions. The preimage and image are congruent. Congruent figures have the same area.

If l x w = A, then w x l = A. 

Geometry helps prove this. Geometry also helps support a variety of other algebraic ideas like transformations of functions within the different function families. Connections between slope and parallel and perpendicular lines. There’s also the outstanding applications of algebraic concepts that geometric situations can provide. Right triangle trig, for example, is often a wonderful review of writing and solving three variable formulas involving division and multiplication. (A consistent sticking point for lots of math learners.)

As a teacher who spent years watching how Geometry presents such an environment for real, effective and powerful mathematical growth, eliminating it will leave a lot of holes that math departments are used to geometry content filling.

Coke vs. Sprite – One Class’ Response to Dan Meyer’s #wcydwt Video

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Almost three years ago, I highlighted the Coke vs. Sprite video that @ddmeyer made. It is a very intriguing question (which glass contains a greater amount of its original pop) without a clear answer.

Today, I let a class give it a go and here’s what they came up with.

First, they chose to model it with integers. Said one student: “I pretended it was a jar with 10,000 marbles.” 2014-04-03 11.07.36

Then they assigned “a dropper-full” to be 100 marbles. So, the first dropper took 100 red marbles and placed them into the jar with the 10,000 green marbles.

Now we assumed the stirring made the mixture homogeneous which meant that there was a consistent 100:1 green-to-red marble ratio in the right jar. So, when we pull another 100 marbles to put back, (with a little bit of rounding) we pull 99 green marbles and a single red marble back into the left jar.

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A little number crunching reveals (at least in this model) that there would be 9901 original-colored marbles and 99 other-colored marbles in each jar.

My original assumption matches their investigation. It seems (at least by this model) that at the end of the video, each glass is containing equal amounts of their original soda.

Another teacher Chris Hunter (@ChrisHunter36) also battled with this video and created an excellent read about his experiences. Check them out.

 

The Power of Network: Triangle Similarity

I want to share a story that shows the power of an effective PLN.

In a previous post, “proof and consequences: circular reasoning“, I begged for help solving a problem with students struggling to see their own logical crisis that was leading to predictable and consistent problems.

Several people reached out to me with suggestions. Thanks for that. I would like to highlight one specific suggestion that I tried to today and it worked just exactly as the designer predicted.

The suggestion was made by @nerdypoo.

From the comment:

“(i drew an example of this on the train home from work where i drew two isosceles triangles — the first had congruent legs 2 & 2 and the second had congruent legs 3 & 3, so scale factor of 1.5, but the first triangle was an isosceles right triangle and the second had an angle of a bit more than 90. i can send a jpeg if you want!)”

I loved this idea. And yes… I did. I did want that .jpg.

So, here’s a portion of what she sent back.

Dutch Triangle Idea Original

Today, I tried it in class. I began by putting up this image…

Dutch Triangle Idea Starter

… and asking the students to vote on whether the triangles were similar, not similar, or we don’t know. Overwhelmingly most students voted that they were similar. The thoughts they articulated were mainly that they could find the legnth of the missing side (which they claimed to be 3 cm long) and then could use SSS to show a consistent scale factor.

Then I showed them this image and asked them to vote again.
Dutch Triangle Idea

A lot of votes changed. Many changed from “similar” to “not similar”. A few others changed from “similar” to “don’t know”. An additional piece of information revealed an assumption. The assumption was that finding a consistent scale factor in two pairs implied the third. Perhaps an assumption that the angles were congruent.

It was essential that I made sure the students knew that I wasn’t changing the situation from the first question to the second. I was simply revealing information that was hidden. Those angles were never congruent. They simply didn’t know that, but most them assumed they were. But every person who voted that the first two sets of triangles were similar were making an assumption, an assumption that they didn’t recognize before. An assumption that shouldn’t be made because sometimes it’s incorrect.

Proof and Consequences: Circular Reasoning

I’m frustrating my students in ways that I don’t want to. I’m not sure exactly what to do about it. In geometry there’s proof. With proof comes a certain logical structure. Once you know this structure, it is terribly difficult to unknow.

Currently we are dealing with similarity, which involves using SSS, SAS, and AA postulates to prove whether or not two triangles are similar.

Suppose I gave this image to a student and asked them to find whether or not triangle FES and triangle GHS were similar.

Similarity4

Let’s suppose the student divides 54 by 24, and also 58.5 by 26. Both times the student gets 2.25 as a solution. The student assumes this is a scale factor and applies it to GH, finding that FE = 45. The student then divides 45 by 20 and gets 2.25 for a third time. That’s three pairs of proportional side lengths and BAM! Similarity proven by SSS.

Except…

Me, the teacher, is there is tell the student that he or she isn’t quite right. (You see the mistake, right?)

The student assumed similarity before it was proven. Then proceeded to use the assumed scale factor to find the missing side length, which ensured that the third quotient was going to be the same as the first two. This is circular reasoning. They are similar because FE = 45. FE = 45 because they are similar. I have seen this play out countless times.

I have addressed it with little success. I can’t seem to make sense to the students why that argument is weak. It sounds like “geometry teacher says we can’t, so do what geometry teacher says.”  (especially when the very next question asks for FE, which is 45… because the triangles ARE similar…). I can’t stand using my authority as a teacher to enforce a math idea that the students are perfectly capable of actually learning.

I’m trying to decide how picky to be with this. I have a hard time allowing that circular reasoning argument to be called correct, although it is clear that the student has learned a lot about similarity, proportionality, and the structure of a proof.

But the more I push the point, the more frustrated I get and the students don’t seem to be getting any significant gains. I just continue to enforce that “math teachers are just picky like that.”

I am hoping for some help on this one. I’ve tried a lot of things, but that thing you like that works really well for you… I haven’t tried that one. Toss it my way. I want to see how well it works.

For your students’ sake: Don’t stop being a learner

Yesterday, we designed an Algebra II lesson using 3D modeling to derive the factored formula for difference of cubes. As we began to finish up, Sheila (@mrssheilaorr), the math teacher sitting beside me made a passing reference to being frustrated trying to prove the sum of cubes formula. Me, being a geometry teacher by trade decided to give it a try perhaps hoping to offer a fresh perspective. I mean, I was curious. It looked like this:

2014-02-20 13.26.00

On the surface, it didn’t seem unapproachable. I quickly became frustrated as well. Most frustrating was the mutual feeling that we were so stinkin’ close to cracking the missing piece. Finally, Luann, a math teaching veteran sat down beside us, commented on her consistently getting stuck in the same spot we were stuck and then, as the three of us talked about it, the final piece fell in and it all made sense (it’s always how you group the terms, isn’t it?)

Then this morning, it happened again. Writing a quiz for Calculus, I needed a related rates problem. Getting irritated with the lousy selection of choices online, I decided that I needed to try to create my own. And I wanted to go #3Act and after some preliminary brain storming with John Golden (@mathhombre) (Dan Meyer’s Taco Cart? Nah… rates of walkers not really related…) we found some potential in Ferris Wheel (also by Dan Meyer)! Between my curiosity and my morning got mathy in a hurry.

First I tried to design and solve the problem relating the rotational speed in Act 1 to the height of the red car. That process looked like this:

2014-02-21 20.32.42

Meanwhile Dr. Golden found a video of a double Ferris wheel, which was pretty awesome. Seemed a little it out of my league, so kept plucking away at my original goal.

It clearly wasn’t out of Dr. Golden’s league, as he took to Geogebra and did things I didn’t know Geogebra was capable of. (You’re going to want to check that out.)

So, what is the product of all of this curiosity and random math problem-solving? As I see it, these past 48 hours have done two things: Reminded me of what makes me curious and reminded me what it’s like to be a learner.

I have a feeling my students will be the beneficiaries of both of those products. There’s a certain amount of refreshment that comes from never being too far removed from the stuff that drew us to math in the first place. The problem you want to solve just because you want to see what the answer looks like.

And this curiosity, the pursuit, it feeds itself. In the process of exploring that which you set out to explore, you get a taste of something else that you didn’t know you would be curious about until it fell down in front of you. (For example, Geogebra… have no idea what that program is capable of, which is a shame because it is loaded on all my students’ school-issued laptops…)

And this process breeds enthusiasm. Enthusiasm that comes with us into our classrooms and it spreads. I’m not trying to be cheesy, but much has been said about math functionality in the modern economy, how essential it is in college-readiness and the like with few tangible results. Let’s remember that there are kids who are moved by enthusiasm, who will respond to joy, who will pay attention better simply because the teacher is excited about what they are teaching. It won’t get them all, but neither will trying to convince them of any of the stuff on this poster.

Now, who’s going to teach me how to use Geogebra?

Thoughts on Proof… and showing your work.

Suppose I give a group of sophomores this image and asked them to find the value of the angle marked “x”.

G.CO.10 - #2

Consider for a moment what method that you would use to solve this problem. (x = 121, in case that helps.)

Now, suppose I asked you to write out your solution and to “show your work.” What do you suppose it would look like?

I was a little surprised to see what I saw from my tenth graders, which was a whole lot of long hand arithmetic. Like this…

2013-12-11 12.21.11

and this…
2013-12-11 12.21.42

One-in-three had a mistake, which, in the midst of grading about 90 started to become an entire class worth of young people who were making mistakes doing a process that seemed fairly easy to circumvent (and by tenth grade, seems fairly cheap and easy to circumvent without much consequence.)

So, I asked why they were so intent on doing longhand arithmetic. The responses were fairly consistent.

1. Our math teachers have asked us to show your work and that’s how you do it.

2. It’s easier than using a calculator.

I will admit I was not prepared for either answer. (In retrospect, I’m not sure what answer I was expecting.) When I was asking students why they resisted the calculators knowing that they lacked confidence with the longhand, they said multiple times that they could show me that they “really did the math” without demonstrating the longhand. Also, one girl wondered why I would be advocating for a method that, as she put it, “makes us think less.”

They knew that I expected them to provide proof of their answers. Most of them were perfectly willing to provide the proof.

This student is starting to suspect that proof means words. So, he used words to describe the process.

2013-12-11 12.21.31
The conversation was pretty engaging to the students. A variety of students chimed in, most of them willing to defend longhand arithmetic and the only “true” work to show. I had shown them a variety of different looks at the longhand (the ones picture here, among others… including some mistakes to illustrate the risk, as I see it.) Then I asked this question which quieted things down quite quickly:

“Okay… okay… you proved to me that you did the subtraction right. I’ll give you that. Which of them proved that subtracting that 146 from 180 is correct thing to do?”

At first, they weren’t sure what to do with that. Although, quickly enough they were willing to agree that none of the work got into explaining why 180-146 was chosen over, say 155+146 = x or something.

I tried to convey that by tenth grade, I’m really not looking for proof that students can do three-digit subtraction. I would very much prefer discussing why that is the correct operation. They didn’t seem prepared to hear this answer. Apparently we’re even…

To be fair, there was one example where a bit of the bigger picture made it into the work. Check it out:

2013-12-11 12.20.56

I learned a lot today. I feel like I got a window into the students who are coming to see me. I ask them to explain, to prove, to show their work. Many of them willingly oblige, they just see an effective mathematics explanation differently than I do. It might be time to help the students get a vision of what explaining the solution to a math problem really looks like.

I would very much like your thoughts on this.

Proof and Consequences

A conversation was taking place over at Dan Meyer’s Blog (http://blog.mrmeyer.com/?p=17964) about proofs, which is a topic that I find myself faced with about this time every year.

This isn’t a new conundrum for me. I’ve been working for while now trying to make this idea of proof, which, when compared to the typical form of textbook Algebra I should be an easier sell. But it just isn’t.

Here are some discussions of my previous attempts to sell it. Posts from Nov 2, 2012, Nov 16, 2012, Dec 7. 2012 are a few examples of my thoughts from around a year ago when geometry hit this place last year.

The problem I have is that the academic norms seem to prefer deductive reasoning to inductive and use of the theorem names. Those two things seem important to decide on before starting the journey of proof. If you are going to prefer deductive measurements, it rules out using measurements in proofs and it requires a lot more formal geometric language.

The problem that I see is that to rule out measurements (at least from the very beginning) and to strongly increase the formal geometric language in a way that makes deductive proofs possible from the very introduction of proofs creates… well… what Christopher Danielson is quoted as saying in Meyer’s post… “one of the most lifeless topics in all of mathematics.”

In order to breathe life into the topic, from the experience I’ve had, you need to let students engage in ways that make sense to them at first. The target to start the process is simply to get them comfortable with the idea of designing a functional persuasive argument about a mathematical situation. This requires recognizing that they need to start with a clearly stated claim (preferably something that is provable) and then start supporting it.

I find it helpful to let them pull measurements from pictures first and use those in the proof. The idea of comparing two things by length and NOT measuring them to get the length seems to a lot of kids like we are making the math difficult simply because we want it to be difficult. If they sense there is an easier way to solve a problem, then the explanation for why that method is against the rules had better be very strong, or else buy-in is going to suffer some pretty heavy causalities.

Once they get the hang of making an argument, then we can start by having discussions about what kinds of evidence are more compelling than others. This is usually where the students can figure out for themselves that each piece of information needs its own bit of mathematical support.

Next we can start deliberately exposing the students to different ways of proving similar situations. Triangle congruence seems to be a popular choice. We can have conversations about proving a rigid motion or proving pairs of sides and angles. Eventually certain kinds of explanations become more and more cumbersome. For example, using definition of congruent triangles to prove that two triangles are congruent as shown here:

Do we really need to keep going to find the three pairs of congruent angles?

Do we really need to keep going to find the three pairs of congruent angles?

Then, we can start pushing into shortcut methods. Mostly because those angles are going to be somewhat tricky to find (and why do more work than you need to… the students DEFINITELY identify with that.)

By using this method, I am trying to create what I’ve heard Meyer call “an intellectual need” for additional methods to prove this claim. (Keyword: trying… not sure how successful it is, but I’m trying.)

Then, that transitions fairly smoothly into stuff like this:

2013-10-28 08.28.32

… where we standardize the side lengths of two different triangles and see how many different triangles we can make and in what ways they are different.

Now, the tougher question is whether or not you allow the class consensus following the “Straw Triangle Activity” (which was a gem that came out of Holt Geometry, Chapter 3) to count as proof of the SSS theorem. In an academic sense, now we should “formally prove” SSS theorem. To most of the students, it’s settled. Three sides paired up means the triangles are congruent. What are we risking by avoiding the formal SSS proof? Do we risk giving the impression that straws and string are formal mathematical tools? But wait… aren’t they? What do we risk by doing the formal SSS proof? Do we risk our precious classroom energy by running them through an exercise there isn’t a lot of authentic need for right now?

Am I able to say that this is the definite recipe for breathing life into geometry proofs? Not even close. I am sure there are students who are completely uninspired by this. I can say using anecdotal evidence that engagement seems significantly and satisfyingly higher then when we used to run deductive two-column proofs at students from the very beginning.

But, we’ll have to see what the consequences are as we keep going.