First, Seattle’s Julia Robinson Math Festival is this Sunday. It’s a festival celebrating collaborative, beautiful, non-competitive mathematics. If you’re interested in signing your child up (4th grade or older), there’s still time! You can learn more about the festival here.
And third, I’ve been having a blast exploring the new website fractiontalks.com. This is a tremendous resource, a kind of one-stop-shopping experience to find fraction images to use with students of all ranges of experience. For example, what fraction of each shape below is green/yellow/blue/etc.?
More coming soon, including a new Kickstarter game for 3-8 year-old kids. Stay tuned!
I recently received this email from a teacher I work with:
“Dan, I have a question for you. I just introduced my [pre-algebra students] to slope and then to slope-intercept form of linear equations and wanted to explore with them some word problems which could be written in that form. (Ex: . For babysitting, Anna charges a flat fee of $10, plus $5 per hour. Write an equation for the cost, C, after h hours of babysitting. What do you think the slope and the y-intercept represent? How much money will she make if she baby-sits 5 hours?)
Do you have any ideas how to make this kind of lesson more fun, hands-on and exploratory for students?”
Here’s my response:
“This is a great question. Fortunately, there’s a large community online that’s working to solve it. I’ve got some ideas too 🙂
Strategies for making linear equations more relevant, more interesting, more exploratory:
Same problems, slicker delivery
An example might be the 100×100 cheeseburger problem. Same idea as the babysitting problem, but real life, involves a menu, and a compelling premise.
Slick delivery, unanticipated result
You can sometimes grab students attention with a problem that seems easy, but has a twist, like this stacking cups activity.
Pared down delivery
For example, visual patterns has the same info and the same question each time: here’s steps 1-4, and you can get step 43 as a hint. Write the equation. Purely visual, so students can begin immediately, and there’s actually more thinking work for them to do. It’s harder for them to just use a recipe approach.
Give two options to compare.
For example, a Would You Rather structure, as in, “Would you rather charge $5 base rate plus $7 per hour, or $15 base rate plus $5 per hour to babysit? Defend with algebra.” This is a harder question, and involves actually having to make an argument, which is a more compelling, more social reason to do something, and usually generates a deeper understanding.
I know a number of teachers who really like what they can do with this tool in the classroom. It basically allows kids to explore and solve problems with their computers or tablets, in some creative ways. There are some clever lessons in the teacher portal.
Raise the ceiling.
If you were to show the sequence of dominoes in a growth pattern below, there are natural questions that you (or the kids!) might ask:
-How many columns will there be in the nth stage?
-How many dominoes will there be in the nth stage (double n dominoes)?
-How many dominoes will be in the tallest column of this organization in the nth stage?
-how many dots total on all the dominoes in the nth stage?
Some of these questions go beyond linear equations, but provide a natural stretch questions, and can actually help kids understand how to model with equations even better.”
The teacher who wrote me closed by say “I have a few ideas, but would love to hear yours.” I feel exactly the same way. What else goes on this list?
My knowledge about the foundation history of irrational numbers was challenged today, and I’m pretty happy about it.
I had recently tweeted a Vi Hart video that gave a fun, geometric proof of the classic first proof of irrationality: is irrational. If it weren’t, that would mean you could build a square that had integer sides and an integer diagonal, and that would allow you to build a smaller square with the same process. To get a contradiction, repeat until you run out of integers.
After I tweeted the video, I got a response claiming that…
I was somewhat taken aback. In source after source, I’ve seen named as the first number ever proved irrational. Variations on the same proof abound. And here was a claim that , the golden ratio, actually holds the rightful place in history as humankind’s first brush with “the unnameable.” There seems to be a historical argument; but how complicated is the proof?
In fact, it’s so wonderfully simple that there’s a pedagogical argument to be made for teaching that is irrational before we even mention the Pythagorean Theorem or square roots. You need to know how to find angles in regular polygons and chase them around diagrams, and know how Isosceles triangles work. The thrust of the proof is the same as for , but it sidesteps the parity argument that can sometimes feel less tangible.
Let’s imagine, as the Pythagoreans might have, that every number is rational. An equivalent way to state this is there is always some scaling of any pair of lengths that allows them both to be positive integers. (To the Pythagoreans, the relationship between any two lengths was identical to the relationship between two whole numbers, axiomatically.) So suppose we have a regular pentagon with integer side length a and integer diagonal b.
The ratio of b to a, is precisely the golden ratio, by the way. But we don’t even need to know what it is. We’ll just try to show that a and b can’t both be integers.
First off, chase some angles around and you’ll see pretty much every angle is either 36, 72, or 108 degrees. This gives a bunch of Isosceles triangles. It follows quickly that
[I’ll leave that piece as an exercise. It’s pretty satisfying to chase angles around and have everything come out nicely.]
This implies that x and y are positive integers. But they are the side and diagonal of a regular pentagon again, so the argument repeats! And this is the crux of the problem: positive integers can get smaller for only so long before they run less than 1. (Just like the infinite regress that was hinted at in that error-laden but inspiring work, Donald in Mathemagic Land.)
Conclusion? The original pentagon couldn’t have been drawn with integer sides to start with. And that means it couldn’t have been drawn with two rational sides, or else we would have scaled them up to whole numbers. And that means the relationship between the side and diagonal of the regular pentagon is irrational.
And there we have it. Irrational numbers without actually dealing with numbers at all. Or evenness and oddness of numerators and denominators.
A delightful discovery. We’ll likely never know for sure what length the Pythagoreans proved irrational first, but that’s a strong claim for over .
Especially because, as Donald found out, the Pythagoreans were all over pentagons and the golden ratio.
I first played the game Aggression about five years ago. I had recently read Eric Solomon’s Games with Pencil and Paper, and tried the game out with a student I was working with. It was a hit. I heard later that the game was in regular rotation at his house, and had become a family favorite. “I should use this game more in schools,” I thought. And then I forgot about it.
Now, Gord Hamilton of Math Pickle has a beautiful explanation of the game up at his website, Math Pickle. Seeing it again, I’m reminded of just how good the game is. And happily, I don’t need to write it up anymore, because it’s hard to imagine improving on Gord’s description.
If you’re looking for a classroom game for 2nd or 3rd graders (or, honestly 1st or 4th or 5th), or just a great pencil and paper game to play at home, I highly recommend you check out A Little Bit of Aggression. Check out the PDF with different maps at the end of the slide deck. Also, check out challenges! Gord’s put some money on the line, and you might be able to claim it!
I recently collected a series of our favorite lessons for fifth graders on the topic of volume into one tidy booklet. I like these lessons a lot. They start with tangible activities involving squares and cubes, and build up from concrete to abstract until students are able to use an understanding of volume to solve truly complex problems.
I’d love some feedback, especially from 5th grade teachers. The lessons are available here.
In another experiment, the booklet of volume lessons is available by donation. This will hopefully allow everyone who wants it to get it for free, and for us to keep track of how many times it’s downloaded. At the same time, if people want to chip in for them, now there’s a way.
Recently, Dan gave a TEDx talk based on the blog post 5 Principles of Extraordinary Math Teaching. In the conversations we had with each other and with other educators in the run up to the talk, one principle came up repeatedly as the most nuanced of the five: Say Yes to Your Students’ Ideas. Perhaps the most challenging principle to implement but also the most rewarding, we’ve been thinking about this one a lot lately.
The ultimate outcomes of deciding to say yes to your students are rooted in the principles of creativity. We are living in a world of increasing complexity and innovation, the best of which requires us to offer our most dynamic and creative selves, and the worst of which hazards alienation and isolation. I recently encountered some work from Joi Ito, the director of MIT’s Media Lab, in which he offered nine variables of innovation and creativity (and, frankly, survival), principles which I thought perfectly captured the spirit of what excellent education can offer.
Joi Ito’s Nine Principles of Creativity
resilience over strength
pull over push
risk over safety
systems over objects
compass over maps
practice over theory
disobedience over compliance
emergence over authority
learning over education
In education we can think of these as follows:
We want students who can adapt their understanding to new contexts. Resilience is adaptive; strength is resistant.
We want students to find their own motivation to engage in work. The question should be designed to pull them in. Work should mostly come from inner drive (pull) rather than external coercion (push).
Staying in safety stifles intellectual curiosity. We want students to take creative risks. They will practice leaping, failing, and leaping again.
Objects exist in isolation; systems are objects in relationship. With the proliferation of objects in the world, the future likely belongs to those who understand the web of relationships, ie those with a systems perspective. This means developing an understanding of context: the reasons why something is true, and why it makes sense with everything else you know.
A map is a fixed picture of a landscape, but it becomes useless if the landscape is shifting under your feet. A map is difficult to use in a rapidly changing world. A compass gives direction even as the landscape is altered. It can be used to find the way through an uncharted (unmapped, unexplored) territory. It is the intuition of intellect, evolved through repeated exposure to rich, novel, changing contexts. It grows through mistakes, surprises, and direct experience with complexity. We want students to develop the abilities of intellectual intuition, this ‘compass’, so that whatever map they step on to, they are prepared to start navigating.
When a student has series of facts out of context and divorced from motivating reason, a student has theory. Theory tends to be tidy. Practice is messy because the work of learning is about breaking an accumulated body knowledge. The facts are associated, stress tested, fail, and the process is repeated with a yet stronger configuration. To stay in the realm of theory is to build a structure, adding yet more layers and stories, without ever testing it. It becomes precarious indeed. Better to build, test, break off the weak parts, rebuild, iterate. The best versions of practice aren’t necessarily ‘real world’ problems, but they are complex ones.
Disobedience breaks into the new. Compliance sustains the old. Disobedience explores the possible, which lies outside the known. Students who are intellectually disobedient are prepared to discover what we may not have imagined yet.
What emerges from a group of students working together is a reflection of what they are ready to know, what they are curious to know, and what they already know. This is the perfect medium of learning. If they are driven by the authority of the teacher, this ideal medium is trampled and it’s potential gains are lost. Teachers can meld their agendas with the students’ emergent understanding.
Learning is student-centered. Education is system-centered. Actually, in education, we want a blend of the two. As above, the position of greatest leverage brings together both the student and the system, a learning-education blend that meets the student’s own innate curiosities and interests with the goals of the education system at large. This is the domain of the principle of Saying Yes: by saying yes, we engineer this meeting of the student and the system, of the individual learning and the larger interests of education.
Saying Yes is a kind of guidance that uses the student’s inner world, their own mind and curiosities, to bring them through the educational goals we desire for them. It lies in the most productive intersection of child-centered learning and teacher-driven education. It takes the best of the Unschooling movement, the recognition that a student’s desires and interests are the best motivators for their learning, and the best of traditional education, which acknowledges that the outcomes of education prepare students for the future world that they (and we) may not have anticipated yet.
Saying Yes means teaching that the world of knowledge is rich, complex, unknown, and constantly changing. Saying Yes means teaching resilience (that the material can adapt to the student and the student can adapt to the material); it means working with the student’s internal motivation, constructing problems that pull them in and using where they are already pulled to direct teaching; it means practicing failure, courage, and risk-taking; it encourages a system’s perspective by taking advantage of the human impulse to make connections, to ask why, to look for meaning and make sense of things; it develops intellectual intuition, the compass of learning, by making exploration of the unknown a primary goal of the classroom; it emphasizes practice because getting our knowledge is inevitable when we follow our impulses; it means valuing intellectual disobedience; it makes use of the emergent knowledge of the classroom; it strikes the perfect balance between learning and education, between the individual and the system.
Our new lesson plan library is up in beta form. We’re not sharing it widely yet, but it’s getting close. Our goal is to have a collection of great problems for K-6th grade, easily filterable by topic, grade, Common Core, and keyword. We know that many teachers would love an easy place to find high-quality, easy-to-use complex problems and games. We’re hoping that Math for Love can be such a place, where each question, game, or lesson plan is a pleasure to use in the classroom and a hit with the students, as well as being rigorous and dynamic.
To that end, we want your feedback! Try these lessons out in your classroom or with your kids. Let us know what works and what doesn’t.
We’ll plan to highlight lessons from the new library weekly (or from time to time). This week: The Power of 37.
This lesson uses a surprising multiplication pattern to motivate multiplication practice for 4th and 5th graders. It’s highly arithmetic, and fits right in with the major work of 4th and 5th grade. It begins with a mystery.
Is it just a coincidence that (7 + 7 + 7) x 37 = 777?
The number 1,525,354,555,657,585,950 is, as it happens, evenly divisible by 99. Fix all the 5s digits where they are in the number, and rearrange the other digits randomly. What is the chance that a rearrangement of this form is still evenly divisible by 99?
I have mixed feelings about this puzzle. On the one hand, I love that it looks so hard, but can be unraveled so easily once you get past that, and attack it with the right tools (hints at Numberplay). On the other hand, I’m dissatisfied with its inability to generalize to other puzzles, or begin a larger investigation.
This week’s Sunday puzzle on NPR is a classic from Sam Loyd. Here’s Will Shortz:
This is one of the “lost” puzzles of Sam Loyd, the great American puzzlemaker from the 19th and early 20th centuries. It’s from an old magazine with a Sam Loyd puzzle column. The object is to arrange three 9s to make 20. There is no trick involved. Simply arrange three 9s, using any standard arithmetic signs and symbols, to total 20. How can it be done?
I played around with this puzzle this morning, and indeed, there is a way to solve it using only standard arithmetic (the four operations, parentheses, decimal points). However, as I was playing around, I found a second answer as well, using square roots and factorials.
A colleague of mine once remarked how strange it is that while the Greeks talked about 6-cornered shapes and 4-sided shapes, we talk about hexagons and quadrilaterals. Why is it, aside from the historical accident that it is, that we persist in making people learn Greek to talk about shapes they see everyday?
And quadrilaterals and hexagons are the easy ones. What’s a 7-sided polygon called? It’s either a heptagon or a septagon—I never remember (do you?). What about a 12-sided polygon? That’s a dodecagon. Thirteen sided? No idea—no one ever taught me that one.
Our vernacular around polygons is tied to an ancient system of numeration that not even experts know. We’ve created a system where we can speak properly about only a select subset of polygons: triangle, quadrilateral, pentagon, hexagon, octagon. Those prefixes denote the numbers 3, 4, 5, 6, and 8. And while there are certainly some of you who know more, I don’t think we ever bother teaching more than this. It’s like teaching inches and feet, and not bothering with miles.
We couldn’t convert to metric in the US, but we can do something even easier when it comes to polygons: name them by number. Forget the name of the icosikaitetragon? Just call it a 24-gon. Heptagons and decagons are 7-gons and 10-gons. We could even call hexagons 6-gons.
The advantages are immediate and enormous. First, every polygon now has an easy, instantly recognizable name. We’ve removed the barrier of Greek between ourselves and shapes. Second, we’re reminded of the defining trait of the thing when we name it. It’s why we call it a red-breasted robin instead of a Turdus migratorius.
Do you agree? Before you answer, let me add one more point: mathematicians use this nomenclature already. We even say n-gon instead of polygon, just so we can decide what n is later, or use the variable in equations.
We could have gone further. The prefix -gon is just Greek for -angle, as in triangle (3-gon). And while 5-angle and 9-angle have a certain poetry, I like the staccato of 5-gon and 9-gon. And it’s not so bad to keep a little Greek in there.
Sure, it’s nice for students to know the word triangle. And you can argue that learning vocabulary for the polygons is fun. But do we really need to add barriers around mathematical objects when we could just call them what they’re called? Do kids need to know quadrilateral or heptagon to relate to 4-gons and 7-gons?? If you’re a math teacher, you can start calling pentagons 5-gons and decagons 10-gons tomorrow.
And the beauty of it is, your students will know exactly what you mean.