Trying to find math inside everything else

I bet you have all seen the following mistake:

There’s a problem here, but it’s certainly an understandable problem. It comes from, dare I say, a trick that we all teach. And it’s a trick we all think isn’t one – adding the same thing on both sides of the equation.

When I did my research project in grad school, I found that many students like the elimination method of solving a system of linear equations because of the way the elements lined up and made it very clear what to do – and this was true of students who used elimination well but could not solve a linear equation normally. It was then that I realized that elimination is actually the core of what we teach about solving equations – we just gloss over it.

It all comes down to two properties of equality: the reflexive property and the additive property.

Using elimination works because we have two different equations, but we add them together, like so:

3x + 2y = 20\\2x-2y=4\\ \line(1,0){75}\\5x\hspace{24 pt}=24

But the same thing is true when we normally solve a linear equation. It’s just that one of the equations is generated using the reflexive property.

-12 + 3x = 20\\+12 \hspace{24 pt}=+12\\ \line(1,0){75}\\\indent\indent3x=32

So actually a lot of things about solving equations become clear when I use elimination, which is why I try to introduce those ideas earlier. The goal here is that if we always have two equations that we are adding together (or subtracting, or dividing), then we can eliminate those mistakes where students add the same quantity twice on the same side.

(Basically, the additive property of equality is often formulated as a = b, therefore a + c = b + c. But I think it’s better formulated as a = b, and c = d, therefore a + c = b + d. And sometimes we use c=c instead of c=d.)

But it’d be great if they were introduced even earlier than when I do it – such as in middle school. Diving deep into the properties of equality, along with rate/ratio/proportion, are probably the two most important things for preparing for algebra.


While we’re talking about elimination, I want to bring up how it’s actually used. Today my kids were working on my Potato 3-Act problem. When solving that problem, you create the following system of equations:



So we solved by subtracting the two equations, giving r = $0.43 (the price of one red potato). Normally, at this point in the process, to solve for b we would use substitution, something like this:

2b + 3(\$0.43)=\$1.97

2b + \$1.29=\$1.97

And then you’d solve from there. But I realized that that’s not strictly necessary. Instead, we talked about what 3 red potatoes are worth, and wrote that as an equation, too. So now we had 2 equations, again, and we could use elimination.

2b+3r=\$1.97\\\indent3r=\$1.29\\\line(1,0){75}\\2b\hspace{24 pt}=\$0.68

No substitution necessary.


ETA: Additional examples of solving linear equations using elimination, at the request of Anna Hester:

CAM00486 CAM00485 CAM00484

What’s a routine?

At #TMC14, I made the following tweet,

Screen shot 2014-08-22 at 10.44.14 PMwhile I was in a session about warm-ups as review. I was reminded of what Jessica posted about her warm-ups, and how she does something different each day of the week.

I was thinking of doing something similar, but I also wanted to include things like Counting Circles. But…I’m having trouble with the disparity. Something like Counting Circles is very different from Estimation180 which is very different from a Throwback Thursday review problem. How can we get used to the norms of a counting circle if we only do it once a week?

I could just commit to one of these things, but I feel they are all important, so I didn’t know what to do. But now I had the thought…what if I did these routines but, instead of once a week, I did them for, saying, a marking period, then switched to another?

This could work because many of the routines match up with certain units – Counting Circles would be very helpful for linear functions, while Visual Patterns would be useful for functions in general (or perhaps for when we do quadratics). Does this sound like a good idea?

Have you ever listened to Pandora and wondered what method they used to determine what songs to play for you? I did and remember writing a research paper about it back in grad school.

Pandora makes use of something called the Music Genome Project. Professional musicians will actually listen to every song in their database and tag all of the songs along different dimensions – timbre of the instruments, vocal type, volume, bpm, etc. Each song then gets a vector associated with it where each dimension is one of those categories.

Then, when you put in a seed song, Pandora will calculate the closest songs to your seed, basically using the distance formula in hundreds of dimensions. (There’s some weighting and tweaking, of course, but that’s the core premise.)

At the Sunday My Favorites session of TMC14, Bob Lochel and Megan Schmidt show us how to find our closest buddies by filling out a survey about what movies they like. Then they calculated the correlation coefficient and the people who correlated the most were the best friends of the pair.

On Saturday, I was talking with someone (I think it was Matt Baker) about how to help people get into our community. He mentioned that while there are a lot of good ideas out there, the ideas that resonate the most with him are the ones that comes from they people he most identified with – whose teaching style was most like his. I had the idea that we could somehow make a survey that a new person could fill out and it would give them a personalized output of Twitter accounts and blogs to follow – a somewhat advanced version of the category lists we made here during TMC12.

I want to make this, but I need help. What are the dimensions that we should ask about? What are important aspects of your teacher identity, and what are some of the things that make you feel on the same wavelength as another teacher in the MTBoS? Please let me know so I can start compiling these dimensions and building this.

(Also, if there is anyone more skilled in programming who is willing to help me, ping me.)


I’m on the plane on my way back from Twitter Math Camp ’14, and it was, as it was the last two years, an amazing experience.

I’m trying to process everything – of course, a lot of that is looking through all of the resources I saw, which I can’t do on the plane. More of it is writing blog posts about specific things I want to talk about – those will come later.

But i want to write about, perhaps, not #whyMTBoS but #whyTMC. Maybe a few short vignettes:

- In my algebra morning session, we had a workshop where we created assessments/tasks for certain units (you can find those here) – when I pulled up the exam I wrote for functions last year, one person told me we could just use that as a product, they liked it so much. We didn’t – we made something even better than what I made myself.

- After Steve Leinwald’s keynote on Thursday full of spit and fire, I felt really energized, even though I had been tired just before.

- Thursday night a small group of people going to get BBQ snowballed to about 30 people, and no one was bothered by that – everyone was welcomed. The restaurant was super accomodating and even made a separate check for everyone (a theme during the trip) – though that wasn’t necessary, as the wonderful Jason covered all of those bills.

- On Friday Dan expanded all of our minds about the size of our community and how much more there is out here.

- Throughout the conference different people gave us “sneak peeks” on things they were working on, and we could get to see inside the process of making these cool things.

- On Friday night I was up until 230 having deep conversations and really connecting with people. It made me realize how much I’m affected by the negativity and positivity of others – TMC is so positive, my coworkers are sometimes negative, and I need to not accept it but work to change it, if I don’t want to absorb all that negativity.

- On Saturday I saw Mary Bourassa and Alex Overwijk present their spiraled task-based curriculum. I was amazed and wanted to be there, but I was scared about it. Alex said in the session that “When you try to make small incremental changes, it is so easy for the kids to pull you back down and flip back to what you’ve always done. But if you start with the huge change, even when you slide, some of that change remains.” I thought of people like Lisa who are worried about changing and how maybe those words might help.

- The last thing I did on Saturday was to take place in a body-scale number line exploration led by Max Ray and Malke Rosenfield I got to share my insights and experiences with number lines that others may not have had, I got to see it in other people’s eyes, and I experienced new revelations and am excited to dive into them deeply.

This last things leads me to my final thought. During our work with the number line, Malke constantly pushed back – what are we actually gaining my working with the number line using our bodies, instead of just paper and pen? It pushed us to keep developing new insights and sharing them until one moment I heard Malke make an involuntary gasp – there was a moment of breakthrough, one we never would have had without using our bodies.

So you could ask the same question – what do we gain from using our bodies to meet in person at TMC, instead of just writing to each other as we do in the MTBoS? There’s this energy that infuses all of it that you can’t feel remotely, these deep experiences and quiet moments that can’t be done publicly, this sense of connection that makes all the other work we do more powerful.

There’s a reason I am always following so many more people after TMC – I need that connection and once it’s there, I want to keep it going and make it grow. And even as there are more and more old friends I want to see at TMC and so little time, I still somehow make so many new friends. And that’s why.

Productive? Failure

The next chapter of Reality Is Broken starts off with this question: “No one likes to fail. So how is it that gamers can spend 80 percent of the time failing, and still love what they are doing?”

It’s an interesting question.  Many games, such as Demon’s Souls, as known for their fiendish difficulty – as that is often portrayed as a positive aspect, not a negative. Dr. McGonigal notes that in one bit of research from the M.I.N.D. Lab, the researchers found that players felt happy when they failed at playing Super Monkey Ball 2 – more so than even when they succeeded. Why would that be?

One thing they noted was that the failure itself was a kind of reward – when the players failed, the scene the played was usually funny. More importantly, though, players knew that their failure was a result of their own actions and symbolic of their own agency – they drove the ball off the course. Because everything was in their control, the players were motivated to give it “just one more try.” I know I’ve certainly had that feeling before – intense concentration on a hard task and then, “Aaaaah!” Coming just short of success, I immediately leap back into trying again.

Of course, that’s not true of every game. There are many games where failing makes me want to give up. There’s two main elements that differentiate the two – agency and hope. If failure is random and feels out of our control, it is demotivating. (Think Mario Kart when you get slammed with a slew of items right before the finish, when you were in 1st place.) But if we see that the failure was fully within our control – and another attempt shows us getting ever so slightly closer to that goal – then the hope of success can feel even better than success itself.

This feeds off of the idea that learning is inherently interesting. When you win at a game, you are successful – but then what do you do? But when you fail, you are learning how to play the game well, and that learning and the act of mastering the game’s mechanics is what is so motivating.


As math teachers, we often talk about Productive Failure – the idea that our students learn better by attempting something themselves, failing, and correcting, than by simply being instructed on the correct method ahead of time. The theory of this is matched by many of our observations (and by research) – but we often have the problem of people being shut down by failure. It ties in a lot with math anxiety and attitudes about math – if I think I am bad at math and that’s just the way it is, failure if just reinforcing that idea, not motivating me to try again.

In the book, Dr. McGonigal doesn’t talk about productive failure – she talks about fun failure. The key factors she mentions – a sense of agency and hope – are what’s so often missing from our math-phobic students. Math feels out of their control – and so any success is accidental, and any failure is predestined.

What can we do? Our main goal is to be a guide – because failure is productive for learning, we want to help the student overcome it themselves. And that means doing what we can to provide that sense of agency and hope.


For a gaming example, Rob was playing a game and was struggling against a particularly frustrating boss (Moldorm from A Link to the Past) – a single false move in the fight would knock him out of the room and he would have to start the whole thing over. Even though he had been having a lot of the fun with the game, this single frustrating experience was enough to make him consider giving up on the game altogether. I knew he would enjoy the rest of the game and wanted him to keep playing, so I stepped into action. One thing I did was provide him with the locations of some fairies – while they would not directly help him defeat the boss, they would lower the frustration of dying and having to repeat the dungeon. The second thing I did was just to watch his attempts.

After a while, when he was ready to give up, he said something to the effect of how he had tried over and over again but had gotten nowhere. But I told him that was not true – when he first tried, he would maybe get 1 hit, or perhaps none, off on the boss before being knocked off the ledge. But in later attempts, he was getting around 4 or 5. He had greatly improved in his tries – and so if he kept trying, he might succeeded. He conceded that might be true, but still took a break, frustrated and tired.

The next morning, I looked up some info and found that the boss only required 6 hits to be defeated – so that meant that in the last attempt, Rob had been very close to success! When I told him that, he was filled with hope (and well-rested), and upon loading up the game, proceeded to beat the boss on the first attempt of the day.


Our goal is productive failure, not frustration. When we are following the mantra of “be less helpful,” I think we still need to help in a different way – help dispel frustration and provide the tools for success, even if we are not telling the students the path they need to take. Be less helpful seems like a hands-off policy – but it’s quite the opposite; we need to devote even more attention to our students when we are letting them struggle on their own.


Satisfying Work

Earlier this year, Justin Aion wrote a post about how he tried to make his class boring on purpose by just giving silent independent work, to make them appreciate what he was normally doing, and how it backfired gloriously. At first, he wondered what he can do to break them of this preference for what they are used to and what is easy. About two months after that, we wrote about a similar situation, and wondered the following:

I’m beginning to wonder if my attempts to give them more engaging lessons and activities have burned them out.  I’m not giving up on the more involved activities.  I want them to be better at problem solving, but I think by trying to do it every day, I haven’t done a good job of meeting them where they are and helping to be where I want them to be.

As I read more of Reality Is Broken, though, I encountered an alternative explanation. In the book, Jane McGonigal wonders why so many people play games like World of Warcraft and other such MMORPGs where the gameplay is not, shall we say, the most thrilling. Many people find enjoyment in what other players call “grinding,” playing with the sole purpose of leveling up. In general, it’s a lot of work to level up in the game to get to what is considered the “good” part of the game, raiding in the end game.

But it’s work that people enjoy doing, and that’s because it is satisfying work. Dr. McGonigal defines satisfying work as work that has a clear goal and actionable next steps. She then goes on to say -

What if we have a clear goal, but we aren’t sure how to go about achieving it? Then it’s not work – it’s a problem. Now, there’s nothing wrong with having interesting problems to solve; it can be quite engaging. But it doesn’t necessarily lead to satisfaction. In the absence of actionable steps, our motivation to solve a problem might not be enough to make real progress. Well-designed work, on the other hand, leaves no doubt that progress will be made. There is a guarantee of productivity built in, and that’s what makes it so appealing.

Well, now, doesn’t that sound familiar? It kinda hit me in the gut when I read it. As math teachers, we are often preaching that we are trying to teach “problem solving” skills – but the thing is, people don’t like solving problems! It made make think of those poor grad students who are working towards their PhD – grad school burnout is a big issue, and one of the major contributing factors is that grad students are trying to solve problems, and so often feel like they are getting nowhere. Their work is inherently unsatisfying, which makes those that can finish a rare breed.

Our students, of course, are not all made of such stuff. But I’m not at all suggesting we drop our attempts at teaching problem solving and only give straight-forward work. Rather, I feel like we need to find a balance – for the past year, as I embraced a Problem-Based Curriculum, I may have pushed too far in the problem-solving direction, and found my students yearning for straight-forward worksheets, just as Justin did. But they also enjoyed tackling these problems, especially when they solved them, and I do think they had more independence and problem-solving skills by the end of the year.

So what should I do? Dr. McGonigal ends the chapter by noting that even high-powered CEOs take short breaks to play computer games like Solitaire or Bejeweled during the work day – it makes them less stressed and feel more productive, even if it doesn’t directly relate to what they are doing. (This reminds me of the recess debate in elementary school.) So even as I go forward with my problem-solving curriculum, I need to weave in more concrete work, and everyone will be more satisfied by it.

Parallel to a Parabola?

A while back, I was working on a lesson about average rate of change and wondered the following question: “Could you use the word ‘parallel ‘to describe two non-linear functions that have the same rate of change/don’t intersect?”

Screen shot 2014-06-18 at 8.53.48 AM

Jonathan’s response, though, made me think about what it actually means to be parallel. Often when you ask students, they will respond “two lines that never intersect,” which I usually push back against because 1) how do you know they never ever intersect? and b) skew lines never intersect, either. So when I explain parallel lines, I use the fact that they have the same slope/go in the same direction as the actual definition, which has the consequence of never intersecting. So I looked it up on Wikipedia.

Given straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean space:

  1. Every point on line m is located at exactly the same minimum distance from line l (equidistant lines).
  2. Line m is on the same plane as line l but does not intersect l (even assuming that lines extend to infinity in either direction).
  3. Lines m and l are both intersected by a third straight line (a transversal) in the same plane, and the corresponding angles of intersection with the transversal are congruent. (This is equivalent to Euclid‘s parallel postulate.)

I don’t think statement 3 was particularly useful to me, but the idea of being equidistant was interesting. A vertically shifted parabola is not equidistant from the original – though they never touch, the distance between them gets smaller and smaller.

So that raised the next question – how do I actually measure the distance between two parabolas at a given point? I asked my boyfriend and he responded, “Well, you definitely need calculus….” And who better to swoop in and help with that than Sam Shah.

Screen shot 2014-06-18 at 8.54.01 AM

So now that I know how to find the minimum distance between two functions, all I need to do is find a function that whose minimum distance to my original function is constant, and then I should have something that you could call parallel.

I made a little Desmos graphs with sliders, to help me visualize the process (click to access):

Screen shot 2014-06-21 at 1.27.13 PM

So I have the equation of the perpendicular line

y = \frac{-1}{f'(a)}(x-a)+f(a)

But that wasn’t really helping me see what the parallel function would actually look like. So then I turned to Geogebra. I needed to make a point on the perpendicular line that was a certain distance away from the function, say a distance of 1. So to figure out the coordinates of that point (x,y), I just used the distance formula, plugging in y from above.:

\sqrt{(x-a)^2+([\frac{-1}{f'(a)}(x-a)+f(a)] - f(a))^2} = 1

That gave me the coordinates of the point that is a distance of 1 away from f(x) at a:

(a + \frac{f'(a)}{\sqrt{1+(f'(a))^2}},\frac{-1}{\sqrt{1+(f'(a))^2}}+f(a))

So I made that point in Geogebra and activated the trace, which gave me this:

Screen shot 2014-06-21 at 2.20.38 PM

Lastly, I thought, well, what exactly is this function that I’ve traced? It looks kinda quartic, but that can’t be, because any quartic like this would intersect the parabola, right? So I tried to write the function for it, using parametric equations. Using f(t) = t^2, I made the parametric equation (t + \frac{2t}{\sqrt{1+4t^2}},\frac{-1}{\sqrt{1+4t^2}}+t^2).

I tried to plug that into Wolfram-Alpha to get the closed form, but it was a mess, so I still don’t really know what the closed form would look like. But who says a parametric form isn’t a solution?

(Here is a Desmos page with a summary of what I’ve done, and some sliders to play with, similar to the Geogebratube page, but with colors.)


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