One of those principles is:

Create an intellectual need for new mathematical skills.

Nowhere is that principle more necessary, in our view, than in the instruction of algebraic expressions. Three of my least favorite words in the English language are “write an expression” because they *so often* mean we’re asking the student to do the difficult work of variable manipulation without experiencing any of the *fruit* of that work.

In both of the questions below, students are likely to experience the work of writing an expression as punishment, not power.

Given the width of the lawnmower (W) and the length of the rope (L), write an expression for the pole radius (R) that will make the lawnmower cut the lawn in a perfect spiral.

Given the width of the pool in tiles (n), write an expression for the number of tiles that will fit around the pool border.

We recognize that one reason variables give us power is that they let us complete lots of versions of the same task quickly and reliably. So in our version of both of the above problems, we asked students first to work *numerically*, both to acclimate them to the task, but especially to establish the feeling that, “Okay, doing a lot of these could get tedious.”

And then we use their expression to power ten pool borders.

And ten lawnmowers.

Those activities are Pool Border Problem and Lawnmower Math. In Picture Perfect, for another example of this principle, we give students the option of either a) filling in a table with 24 rows, or b) writing an algebraic expression *once*.

In each case, students are more likely to see algebra as power than punishment.

]]>As I approached B and R, I misread them as disengaged. In fact, they were thinking really, really hard about this beast:

B suggested they multiply by two as a “fraction buster.”

(One small pleasure of guest teaching is trying to identify and decode the vernacular of each new class. I heard “fraction buster” more than once.)

R asked, “But do we multiply *this* by two or the whole *thing* by two?”

If you’ve taught math for a single day, you know the choice here.

You can tell them, “You multiply the whole *thing* by two.” That’d be helpful by the definition of “helpful” that includes “completing as many math problems as correctly and quickly as possible.” But in terms of classroom management, I’ll be doing myself no favors, having trained B and R to call me over whenever they have any similar questions. More importantly, I’ll have done their relationship with mathematics no favors either, having trained them to think of math as something that can’t be made sense of without an adult around.

“**Variables like x and y behave just like numbers like -2 and 3.**” I said. I wrote this down and said, “Try out both of your ideas on this version and see what happens.”

After some quick arithmetic, they experienced a moment of clarity.

In the next class, students were helping me solve 2x – 14 = 4 – 2x at the board. M told me to add 2x to both sides. One advantage of my recent sabbatical from classroom teaching is that I am more empathetic towards students who don’t understand what we’re doing here and who think adding 2x to both sides is some kind of magical incantation that only weird or privileged kids understand.

So at the board I was asking myself, “Why *are* we adding 2x to both sides? What if we added a different thing?”

Then I asked the students, “What would happen if we added 5x to both sides? What would break?”

Nothing. We decided that nothing would break if we added 5x to both sides. It wouldn’t be as *helpful* as adding 2x, but math isn’t fragile. **You can’t break math.**

**BTW**

- I haven’t found a way to generate these kinds of insights about math without surrounding myself with people learning math for the first time.
- One of my most enduring shortcomings as a teacher is how much I plan and revise those plans, even if the lesson I have on file will suffice. I’ll chase a scintilla of an improvement for
*hours*, which was never sustainable. I spent most of the previous day prepping this Desmos activity. We used 10% of it. - Language from the day that I’m still pondering: “We
*cancel*the 2x’s because we want to*get x by itself*.” I’m trying to decide if those italicized expressions contribute to a student’s understanding of*large*ideas about mathematics or of*small*ideas about solving a particular kind of equation.

Here is an awesome sequence of comments in which people *savage* the term “cancel,” then temper themselves a bit, and then realize that their *replacement* terms are similarly limited:

I have a huge problem with ‘cancel.’ It’s mathematical slang, and I’m OK with its use among people who really understand the mathematics. But among learners it obscures the mathematics and leads to things like “cancelling terms” in rational expressions.

I think the word “cancel” is misused in math when teaching students. We are not canceling anything we are making ones and zeros.

I never say cancel. I’ve worked hard to eliminate it from any teaching I do. Same with cross multiply, never say it. Instead I say “add to make zero” or “divide by or multiply by the reciprocal of to make one”.

We use “cancel” to mean too many things and so they use the term anytime they want to get rid of something or slash something out. The basis for my concern: when I ask kids “why can you do that?” they often can’t explain.

However, when it comes to squaring a square root, what is most accurate to say? I don’t correct the kids in that case, and I tell them that cancel means the same thing as “undo”.

If the point is to be rigorous, “apply the inverse” is more rigorous and technical than “cancel”.

More miscellaneous wisdom on language in mathematics:

In the topic of “get x by itself”, I’ve started saying “We want to say what x is. What would that look like?” They usually say “It would look like x = something”. Then they’ve chosen what the final equation ought to look like for themselves.

I wonder what would happen if we had an equation and then asked them to find out what, say, 2x+1 was.

Even in my Advanced Algebra 2 classes I have started using the phrases “legal” and “useful”. In the original post adding 5x to both sides was definitely “legal” just not as “useful”.

]]>One of my refrains is that an algebraic step is “correct, but not useful.” Inspired by my dance teacher, I also talk about how a particular procedure is lovely, just not in today’s choreography (which is geared towards solving for an unknown/simplifying a trig expression/ finding intercepts, etc).

**Poll**

What mathematical skill is the textbook trying to teach with this image?

Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.(If you’re reading via email or RSS, you’ll need to click through to vote. Also, you’ll need to check that link tomorrow for the answer.)

**Current Scoreboard**

*Team Me*: 6

*Team Commenters*: 4

**Rules**

Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that *isn’t* pseudocontext, collect a personal point.

(See the rationale for this exercise.)

Mega-Grandma!

Later the text goes on to ask students to construct and graph the formula for the average cost of producing the Mega-Grandma, which turns out to be a rational function given the constraints.

But the text might have been better served by asking students to solve for generic widgets, or tennis balls, or something a little less gonzo-bananas than the Mega-Grandma exoskeleton, which is all I’m going to remember from this unit in the textbook.

**BTW**. Thanks to Jasper LaFortune for the submission.

First, we made substantial upgrades to our entire activity pool. Second, we released ten new activities in the same amount of time it took us to release *one* activity two years ago. This is all due to major improvements to our technology and our pedagogy.

*Technologically*, our engineers created a powerful scripting language that hums beneath our activities, enabling us to set up more meaningful interactions between teachers, students, and mathematics.

*Pedagogically*, my teaching team has spent the last year refining our digital mathematics pedagogy through daily conversations, lesson pitches, lesson critiques, summary blog posts, occasional lunch chats with guests like the Khan Academy research team, and frequent consultation with our Desmos Fellows.

The result: we cut an activity pool that once comprised 300 pretty good activities down to 127 great ones, and we gave each one of those 127 a serious upgrade, making sure they took advantage of our best technology and pedagogy. Then we added ten more.

I don’t think I’ve learned as much or worked as hard in a three-month span since grad school, and I owe a debt of gratitude to my team – Shelley Carranza, Christopher Danielson, and Michael Fenton – for committing the same energy. Also, it goes without saying that none of our activity ideas would have been possible without support from our engineers and designers.

In future posts, I’ll excerpt those lessons to illustrate our digital pedagogy. For today, I’ll just introduce the activities themselves.

Hang loads of pictures precisely and quickly using arithmetic sequences.

Your shipment of tennis balls has been contaminated. Use exponential functions to find the bad ones.

Graphing Stories comes to Desmos just in time for its tenth birthday.

One of the oldest and best problems for exploring algebraic equivalence. We wouldn’t have touched it if we didn’t think we had something to add.

Use your intuition for angle measure to bounce lasers off mirrors and through targets.

Use Algebra and the properties of circles to help you mow ten lawns automatically and quickly.

Use linear equations to land airplanes safely and precisely.

Practice circle equations by completing artistic patterns.

Develop your understanding of the behavior of polynomial graphs by creating them piece by piece, factor by factor.

This is my favorite introduction to the concept of a transformation. “Actually, there’s really only one parabola in the world – we just move it around to make new ones.”

We are still testing these activities. They are complete, but not complete complete, if you know what I mean. You won’t find all of them in our search index yet. We welcome your feedback.

]]>**Poll**

What mathematical skill is the textbook trying to teach with this image?

Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.(If you’re reading via email or RSS, you’ll need to click through to vote. Also, you’ll need to check that link tomorrow for the answer.)

**Current Scoreboard**

*Team Me*: 5

*Team Commenters*: 4

**Pseudocontext Submissions**

*William Carey* has offered two additional genres of pseudocontext that are worth your attention:

One motif in pseudocontextual questions seems to be treating as a variable things that, you know,

don’t vary.

The car question follows a fascinating pattern that shows up in lots of physicsy work: it begs the question. Physicists like to measure things. Sometimes measuring something directly is tricky (or impossible), so we measure other things, and then calculate the thing we actually want.

Questions like that have as their givens the thing we

can’tmeasure and ask us to calculate the thing that wecanmeasure. It’s absolutely backwards.

**Rules**

Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that *isn’t* pseudocontext, collect a personal point.

(See the rationale for this exercise.)

The commenters bit down hard on the lure this time, folks. The correct answer – “calculating area of parallelograms” – was selected *least*.

Delicious pseudocontext, right? The judges all suffered massive strokes when they saw this problem so I couldn’t get their official ruling, but I don’t think it matters. This context fails the “Come on, really?” test for pseudocontext.

“This unpredictable force of nature is threatening a precisely-bounded parallelogram? Come on, really?”

How could we neutralize the pseudocontext? I would be thrilled to see a task that invited students to select and *approximate* important regions with various quadrilaterals, but let’s not *lie* about where our tools are useful.

Hey history teacher-friends!

Lately, I’ve been interested in the math teaching opportunities that arise when we delete and then progressively reveal details of a task. Digital media offers us that luxury while paper denies it.

I saw an opportunity to apply the same approach in history and geography. I took the license plates from all fifty United States and removed explicit references to the state or its outline. Then Evan Weinberg turned it into an online quiz.

Feel free to send your students to that quiz, or to use the images themselves [full, deleted, animated] in any way you want. If you’re feeling obliging, stop by and let us know how it went in the comments.

]]>I said that I wanted to stay skilled as a math teacher. As much as I’d like to pretend *I’ve still got it* in spite of my years outside of the classroom, I know there aren’t any shortcuts here: I need to do more math and I need to do more teaching. I have plans for both halves of that goal.

In support of “doing more math,” I’ll periodically post about my recreational mathematics. Please a) critique my work, and b) shoot me any interesting mathematics you’re working on.

**When Should You Bet Your Coffee?**

Ken Templeton sent me an image from his local coffee shop.

Should you bet your free coffee or not? Under what circumstances?

This question offers *such* a ticklish application of the Intermediate Value Theorem:

If the bowl only has *one* other “Free Coffee” card in it, you’d want to bet your own card on the possibility of a year of free coffee. But if the bowl had *one million* cards in it, you’d want to hold onto your card. So *somewhere* in between one and one million, there is a number of cards where your decision *switches*. How do you figure out that number? (PS. I realize the IVT doesn’t hold for *discrete* functions like this one. Definitions offer us a lot of insight when we *stretch* them, though.)

I asked some of my fellow New Year’s Eve partygoers this question and one person offered a concise and intuitive explanation for her number, a number I personally had to calculate using algebraic manipulation. Someone else then did his best to translate some logistical and psychological considerations into mathematics. (eg. “Even if I win it all, I won’t likely go get a drink every day. Plus I’m risk averse.”) It was such an interesting conversation. Plus what great friends right?

Here’s my work and the 3-Act Task for download.

**How Many Bottles of Coca-Cola Are in That Pool?**

When I watched this video, I had to wonder, “How many bottles of Coca Cola did they have to buy to fill that pool?”

I tweeted the video’s creator and asked him for the dimensions of the pool.

“12 feet across by 30 inches high,” he responded.

Even though the *frame* of the pool is a dodecagon, the pool lining itself seems roughly cylindrical. So I calculated the volume of the pool and performed some unit conversions to figure out an estimate of the number of 2-liter bottles of Coca Cola he and his collaborators would have to buy.

Here’s my work and the 3-Act Task for download.

**How Do You Solve Zukei Puzzles?**

Many thanks to Sarah Carter who collected all of these Japanese logic puzzles into one handout.

Carter describes the puzzles as useful for vocabulary practice, but I found myself doing a lot of other interesting work too. For instance, *justification*. The rhombus was a challenging puzzle for me, and this answer was tempting.

So it’s important for me to know the definition of a rhombus – every side congruent – but also to be able to *argue* from that definition.

And the challenge that tickled my brain *most* was pushing myself away from an unsystematic visual search towards a systematic process, and then to write that process down in a way a computer might understand.

For instance, with squares, I’d say:

Computer: pick one of the points. Then pick any other point. Take the distance between those two points and check if you find another point when you venture that distance out on a perpendicular line. If so, see if you can complete the square that matches those three points. If you can’t, move to the next pair.

**Commenter-friends:**

- What are your professional resolutions for 2017?
- What recreational mathematics are you working on lately?
- If any of you enterprising programmers want to make a Zukei puzzle solver, I’d
*love*to see it.

**2017 Jan 2**. Ask and ye shall receive! I have Zukei solvers from Matthew Fahrenbacher, Jed, and Dan Anderson. They’re all rather different, each with its own set of strengths and weaknesses.

**2017 Jan 2**. Shaun Carter is another contender.

First, Wilson’s hypothesis:

Wilson fears that students who depend on technology [calculators, specifically –

dm] will fail to understand the importance of mathematical algorithms.

Next, Wilson’s experiment:

Wilson says he has some evidence for his claims. He gave his Calculus 3 college students a 10-question calculator-free arithmetic test (can you multiply 5.78 by 0.39 without pulling out your smartphone?) and divided the them into two groups: those who scored an eight or above on the test and those who didn’t. By the end of the course, Wilson compared the two groups with their performance on the final exam. Most students who scored in the top 25th percentile on the final also received an eight or above on the arithmetic test. Students at the bottom 25th percentile were twice as likely to score less than eight points on the arithmetic test, demonstrating much weaker computation skills when compared to other quartiles.

I trust my readers will supply the answer key in the comments.

**BTW**. I’m not saying there *isn’t* evidence that calculator use will inhibit a student’s understanding of mathematical algorithms, or that no such evidence will ever be found. I’m just saying this study *isn’t* that evidence.

**Featured Tweet**

Got one!

@ddmeyer I think you just found me a new example for chapter 4 (experimental design)...

— David Griswold (@DavidGriswoldHH) December 23, 2016

**Featured Comment**

]]>The most clarifying thing that I can recall being told about testing in mathematics came from a friend in that business: you’ll find a positive correlation between student performance on almost any two math tests. So don’t get too excited when it happens, and beware of using evidence of correlation on two tests as evidence for much.

Some of you knew what kind of problem this was before you had finished the first sentence. You could blur your eyes and without reading the words you saw that there were two unknown quantities and two facts about them and you *knew* this was a problem about solving a system of equations.

Whoever wrote this problem knows that students struggle to learn how to *solve* systems and struggle to *remain awake* while solving systems. I presume that’s why they added a context to the system and it’s why they scaffolded the problem all the way to the finish line.

How could we improve this problem – and other problems *like* this problem?

I asked that question on Twitter and I received responses from, roughly speaking, two camps.

One group recommended we **change the adjectives and nouns**. That we make the problem more *real* or more *relevant* by changing the *objects* in the problem. For example, instead of analyzing an *animated* movie, we could first survey our classes for the movie genres they like most and use *those* in the problem.

This makeover is common, in my experience. I don’t doubt it’s effective for some students, particularly those students already adept at the formal, operational work of solving a system of equations through elimination. The work is already easy for those students, so they’re happy to see a more familiar context. But I question how much that strategy interests students who *aren’t* already adept at that work.

Another strategy is to ignore the adjectives and nouns and **change the verbs**, to change the *work* students do, to ask students to do informal, relational work first, and use it as a resource for the formal, operational work later.

This makeover is *hard*, in my experience. It’s especially hard if you long ago became adept at the formal, operational work of solving a system of equations through elimination. This makeover requires asking yourself, “What is the core concept here and what are early ways of understanding it?”

No adjectives or nouns were harmed during this makeover. Only verbs.

The theater you run charges $4 for child tickets and $12 for adult tickets.

- What’s a large amount of money you could make?
- What’s a small amount of money you could make?
- Okay, your no-good kid brother is working the cash register. He told you he made:

- $2,550 on Friday
- $2,126 on Saturday
- $1,968 on Sunday
He’s lying about at least one of those. Which ones? How do you know?

This makeover claims that the core concept of systems is that they’re about *relationships* between quantities. Sometimes we know so many relationships between those quantities that they’re only satisfied and solved by *one set* of those quantities. Other times, *lots* of sets solve those relationships and other times those relationships are so constrained that they’re *never* solved.

So we’ve deleted one of the relationships here. Then we’ve ask students to find solutions to the remaining relationship by asking them for a small and large amount of money. There are lots of possible solutions. Then we’ve asked students to encounter the fact that not every amount of money can be a solution to the relationship. (See: Kristin Gray, Kevin Hall, and Julie Reulbach for more on this approach.)

From there, I’m inclined to take Sunday’s sum (one he *wasn’t* lying about) and ask students how they know it might be legitimate. They’ll offer different pairs of child and adult tickets. “My no-good kid brother says he sold 342 tickets. Can you tell me if that’s possible?”

Slowly they’ll systematize their guessing-and-checking. It might be appropriate here to *visualize* their guessing-and-checking on a graph, and later to help students understand how they could have used algebraic notation to form that visualization *quickly*, at which point the relationships start to make even more sense.

If we only understand math as formal, operational work, then our only hope for helping a student *learn that work* is lots and lots of scaffolding and our only hope for helping her *remain awake* through that work is a desperate search for a context that will send a strong enough jolt of familiarity through her cerebral cortex.

That path is wide. The narrow path asks us to understand that formal, operational ideas exists first as informal, relational ideas in the mind of the student, that our job is devise experiences that help students access those ideas and build on them.

**BTW**. Shout out to Marian Small and other elementary educators for helping me see the value in questions that ask about “big” and “small” answers. The question is purposefully imprecise and invites students to start poking at the edges of the relationship.

**Poll**

What mathematical skill is the textbook trying to teach with this image?

Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.(If you’re reading via email or RSS, you’ll need to click through to vote. Also, you’ll need to check that link tomorrow for the answer.)

**Current Scoreboard**

*Team Me*: 5

*Team Commenters*: 3

**Pseudocontext Submissions**

**Rules**

Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that *isn’t* pseudocontext, collect a personal point.

(See the rationale for this exercise.)

This was a nail-biter between Team Commenters and Team Me this week, with Team Commenters narrowly tipping the scales in their favor.

The judges rule that this satisfies the second rule of pseudocontext:

Given a question, the assigned method isn’t a method most human beings would use to find it.

Reasonable people might wonder about the dimensions of a water tank. The judges rule that most human beings would use a tape or a stick or any other kind of measuring device to answer it, not a cubic polynomial.

I can’t think of any way to neutralize this pseudocontext. The number of actual contexts for cubic polynomials with non-zero quadratic and linear terms is *vanishingly small*.

Here is an activity I would much prefer to use to teach the construction of polynomials. It doesn’t involve the real world but it does ask students to do real work.

**Featured Comment**

]]>One motif in pseudocontextual questions seems to be treating as a variable things that, you know,

don’t vary. I have a funny video playing in my mind of some surprised fish watching the volume of their tank become negative. But happily the volume of that tank is not varying, inasmuch as it’s sides are made of glass.