Pyramid Puzzle

This Pyramid Puzzle was recently featured in a post in Forbes about our Tiny Polka Dot campaign.

Take two suits–that’s 22 cards, with 0 – 10 each occurring twice. The puzzle is to make a pyramid using 10 cards of those 22 cards, so that each number in the pyramid is the sum of the two below it. Here’s a near-solution: every card is the sum of the two dots below it; the only problem is that there’s no third 1 to go in the last space on the bottom. (Excess cards are on the right.)

I just received this email from a friend who’s been play-testing Tiny Polka Dot with her kids.

“[my daughter] couldn’t make it work [with 10] with two sets then decided “OK let’s try 9.” Off to verify 10 really doesn’t work”

Here’s the photo she attached with the email (note: spoiler below!)

I actually convinced myself that 10 couldn’t work at the top of the pyramid… for a while. Turns out, I was wrong! More ends up being possible with this puzzle than meets the eye.

But I love this puzzle for precisely the reason that it worked so well for my friend’s daughter: 10 doesn’t seem to work, so she takes a leap of faith and tries 9 at the top of the pyramid; the puzzle rewards the courageous step of trying an even harder puzzle!

Is it possible to put an even smaller number at the top of the Pyramid?

[Sidenote: you can now get Prime Climb and Tiny Polka Dot together at a big discount if you support our campaign. Pledge here!]
1

What are the aims and goals of math education, K-2?

As part of the Math for Future Elementary School Teachers class we’re teaching at the UW, we regularly ask our students to reflect on what they’re learning in the class. This particular student reflection felt so dense and comprehensive that I thought it would be worth sharing here (with her permission, and her name removed).

The prompt: What do you think are the aims and goals of math education for grades K-2? Include topics, values, and outcomes.

What do you think?

So you want your kid to know math? Of course you do. Math is an important tool, used widely in many disciplines, and helps us make sense of our world. It’s also beautiful, fun, and interesting, especially for young children.

Kids are just entering the world of patterns and numbers, and their love of math is ready to bloom. They are ready and excited to count, classify, name, and look for patterns. But too often, parents unintentionally do damage, either by neglecting their kids’ mathematical development, or by pushing them too hard, too quickly. It’s important to find the right balance, but many parents aren’t sure how.

This is an issue now, as more attention is being paid to the development of a human intellect. Pre-Kindergarten education, Head Start, Common Core, and the achievement gap are all looming large in public discourse right now because we care so much about educating our children. Research has shown the best education begins at home, with a thoughtful approach to nurturing your child’s natural math instincts.

The following guide is adapted from our new math game deck for three to eight year olds, Tiny Polka Dot. These ideas are, essentially, the keys to nurturing the natural mathematical instinct that is growing in every child.

Guide for Grown-ups

1. Play! Play is the engine of learning for young children. Provide your kids with a rich environment to play in, and let them take the lead. What makes for a rich math environment?

• Blocks – pattern blocks, legos, tangrams, and other blocks for building, sorting, and playing are the best.
• Games – Classics like Uno, concentration, war, dots and boxes, and Tiny Polka Dot are great as soon as kids are ready for them. Winning is irrelevant at first, and many games can be played collaboratively. Keep it light, and have fun yourself!
• A mathematical perspective – a walk in the woods, a pile of buttons, an old egg carton… these all hold rich mathematical structure if you look at them with the right eyes. Any pile of assorted object can become an opportunity to group by color, by number of holes, or by size. Walks become a chance for counting steps or physical challenges that involve gut estimations and intuitions.

EXAMPLE
You: How many steps do you think it will take to get to that tree?
You: Let’s find out!
[You walk and count. The answer, it turns out, was 15.] You: How many steps to that next tree?
You: Let’s find out!

Math teachers in elementary, middle and high school know how hard it is to get kids to intuitively understand whether their answers make sense. Connecting numbers up to the world with fun guess-and-check challenges pays off big time later.

As for egg cartons, check out the patterns emerging from kids playing with colored plastic eggs and egg cartons at Math-on-a-Stick at the Minnesota State fair. (These taken from Christopher Danielson’s aptly titled blog post, Let the Children Play.)
You might not recognize this as mathematical play, but sorting, symmetry, and grouping by equal numbers is precisely the foundation of the more abstract mathematics that will come later in school. And most importantly, it’s fun!

2. Learning takes time! The process of learning and mastering a new skill can be slow and complex. This is not a test. Do not to rush your child or expect them to know something after they have seen it once or twice. You may find a child makes the same mistake over and over again. As long as they are having fun, trust that your child is learning! It’s easy to over-help. Take a breath, and make sure you are letting your child take as active a role in the game as possible. It doesn’t matter if they get the answer right. Because it isn’t a test.

• WRONG QUESTION: “Is my child smart?”
• RIGHT QUESTION   “How is my child thinking?”

3. Think out loud. Your child is imitating everything you do, and the more you can narrate your thoughts, the better a model of thinking you can be. Slow down, find reasons to count slowly and clearly, and use mathematical language to describe what’s happening in your head. Never say, “Mommy (or Daddy) is terrible at math!” If you don’t have the ideal relationship to math, now’s your chance take a fresh look.

The happy truth about doing math with your kids is that it’s way more fun than you’re expecting it to be. It’s not about right answers, and it’s not about speed. It’s about playing, counting, building, sorting, and studying the wonderful, colorful world around us.

If you’re interested in more ways to explore math with your three- to eight-year-olds, our new math game, Tiny Polka Dot, is on Kickstarter now. It provides multitudes of ways to play with number and pattern in a colorful and fun family game.

Because really, we should all be having way more fun.

Tiny Polka Dot – the colorful math game for young kids

We’re happy to announce that our newest math game, Tiny Polka Dot, is now on Kickstarter!

Tiny Polka Dot is a math game deck for 3- to 8-year-old children and their families. Over the past several years we’ve designed series of games through our work with pre-K and K-2 teachers. Tiny Polka Dot is the culmination of that work: a deck of colorful cards that pulls together the best of those games.

If you or anyone you know would like this game to exist, please support the campaign now. Learn more here.

Why Tiny Polka Dot?

The most common question we get asked about Prime Climb is: “can my 4 (or 5 or 6 or 7) year old play this game?” And although some people have reported having fun playing the game with young children, Prime Climb was designed with older kids in mind.

We had some great card games to recommend, but the most mathematically relevant ones usually involved removing the face cards or a bunch of elaborate rules. (Cribbage jumps to mind.) And the design of standard playing cards makes simple games like memory or war or their variations less mathematical than they ought to be.

Why is there no deck of cards that’s expressly designed to bring out the mathematical nature of these games? Three- and four-year-old kids often need practice counting different arrangements of objects, while Kindergarten-aged kids transition from counting to adding and subtracting; the right series of games and a thoughtfully-designed deck of cards could back up the entire sequence of early numeracy.

And, in the pivotal moment that often occurs in these stories, we suddenly realized that we should build it.

And so we did!

If you have a 3- to 8-year-old in your life, please consider backing the campaign, and spread the word to folks you know who would be interested! We’re really excited about this game, and with your help we can get it produced before the year is out.

Math Festival, TEDx Talk, Fraction Talks

So much is coming up!

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.

Second, my TEDx Talk, Five Principles of Extraordinary Math Teaching, if finally out. I’d love to know what you think of it.

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!

4

Thoughts on linear equations

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:

1. 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.
2. 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.
3. 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.
4. 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.
5. Use Desmos.
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.
6. 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?

Phi is the new root 2

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: $\sqrt{2}$ 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 $\sqrt{2}$ named as the first number ever proved irrational. Variations on the same proof abound. And here was a claim that $\phi$, 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 $\phi$ 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 $\sqrt{2}$, 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

$x = 2a - b$ $y = b - a$ [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 $/phi$ over $/sqrt{2}$.

Especially because, as Donald found out, the Pythagoreans were all over pentagons and the golden ratio.

2

Aggression: the amazing arithmetic game no one knows about

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!

Lessons on Volume

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.

Saying Yes: Joi Ito’s 9 Principles

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

1. resilience over strength
2. pull over push
3. risk over safety
4. systems over objects
5. compass over maps
6. practice over theory
7. disobedience over compliance
8. emergence over authority
9. learning over education

In education we can think of these as follows:

1. We want students who can adapt their understanding to new contexts. Resilience is adaptive; strength is resistant.
2. 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).
3. Staying in safety stifles intellectual curiosity. We want students to take creative risks. They will practice leaping, failing, and leaping again.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.