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Math Games to Play at Home & NMF2017 Retrospective

I’m hosting a math night for parents and students tonight at a local elementary school tonight. These are always fun events, because

  1. I get to preach how games and puzzles are some of the best ways to support math understanding at home
  2. We get to play said games and puzzles and have a great time.

See the bottom of this post, where you can see my handout of puzzles and games to play at home, and other resources to check out.

In addition to my other favorites, I think I’m going to share the “Polka Loop Puzzle.” This is a classic puzzle we included as one of the puzzles to play in Tiny Polka Dot.

Last weekend I took this puzzle, among others, to the National Math Festival in DC. It was, not surprisingly, a blast. My only sadness is I was so busy sharing the puzzles and games I brought that I didn’t get to see all the other spectacular presentations.

I got to spend a lot of time with the folks from ThinkFun, from the Julia Robinson Festival nationwide, from Gathering for Gardner, and others, since we were all in the same section of the event. One big theme: the values of games in creating a math-positive culture at home. Everyone basically had their own variation on the sentiment that “What books are to reading, games are to math.” (One variant I was struck with: “What books are to writing, games are to math.”

Students would start by solving the puzzle from 1 to 5, and then add on a 6, then a 7, until they had it solved up to 10. One student really showed me something new when she concocted, and solved, a totally new variation on the puzzle I’d never seen before. Instead of putting one card on the bottom of the deck, you put one card on the bottom for each letter in the number that’s coming next.

Want more math games to play at home? Check out the handout I’m passing out at the Math Games Night tonight. And let me know what great games and puzzles I should add!

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Raymond Smullyan

The logician and puzzle-maker Raymond Smullyan died yesterday at the age of 97. After reports circulating on twitter, the news was confirmed, and articles in memorium have begun to appear.

Smullyan was a brilliant designer of puzzles, and his books, especially The Riddle of Scheherezade, had a big impact on me. His idea of coercive logic, in particular, impressed me deeply: with the right construction, and the stipulation that someone will answer you truthfully (or even, truly or falsely), you can compel them to do whatever you want.

The New York Times published this obituary today, which is worth the read.

The Times also published a small sampling of Smullyan’s puzzles here. It’s interactive, and a nice place to start remembering the contributions of a prolific and wonderful puzzle designer.

January Newsletter

Reposting our last email newsletter to the blog. If you’d like to sign up on our email list, click here.

January News from Math for Love

Loss: A farewell to Sid the goat

The year 2017 began on a sad note for us. After an abrupt decline, our beloved pet goat Sid passed away on January 3. Sid was a dear friend; he slept indoors, and has accompanied us, in his storied, 12-year existence, on hikes, cross-country car rides, adventures and abundant wonderful moments.

We will miss him, and remember him.

He is survived by us, and by his companion Myshkin. I wrote a blog post several years ago considering the two of them as problem-solvers, and it feels right to call back to it now.

If you’re interested to know about what we learned from Sid as a problem solver, you can read Goat River Crossing.

I highly recommend the videos.

Birth: Tiny Polka Dot arrives

In happier news, our new game, Tiny Polka Dot, has now been delivered to almost all of our Kickstarter backers, and is available on Amazon.

Photos and reviews are starting to roll in, and it’s fantastic to see the response. We’re hoping to see Tiny Polka Dot helping 3-8 year old kids fall in love with numbers in homes and classrooms around the world.

Julia Robinson Math Festival, Feb 25

Registration is open for Seattle’s 6th annual Julia Robinson Math Festival, February 25th. Get your spot now before it fills up!

Registration is open for 4th grade and up (i.e, 10 year old and up). Tickets are $10 – 15, sliding scale. Email dan@mathforlove.com for info about bringing groups.

If you’re someone who has a love of math to share, we’re looking for volunteers too!

Math for Love Classes start Jan 22 – register now!

There are still a few spots open for our Winter Math Classes, running at the PNA Sundays, Jan 22 – Feb 12:

For 1st & 2nd Graders
Mathematical Games and Puzzles
For 3rd – 5th Graders
Geometric Puzzlers

All classes take place Sundays at the Phinney Neighborhood Association.
Instructor: Paul Gafni

More info available here.

Problem of the Moment

Imagine walking in the following pattern:
1 block North
2 blocks East
3 blocks South
1 block West
2 blocks North
3 blocks East
1 block South,
and so on, repeating 1, 2, 3, 1, 2, 3 and N, E, S, W.

Do you ever get back to the spot you started? If so, how many blocks do you walk before you do? If not, can you prove it?

Try with other sets of numbers and directions.
What patterns do you find?

A Math Magic Trick

I’ve been meaning to write this lesson up for a long time. Finally, here it is, in all its glory. You can check it out below, or find it here on the lessons page.

This trick is an extraordinary introduction to the power of algebraic thinking, and a reminder of what makes algebra so awe-inspiring. When I was in middle school, I used to make up

A video Launch for this lesson is available in two parts. PDF of the lesson is below the videos.

Part 1. When using as a class launch, stop at 2:45 and let the class come up try to find their own counterexamples.

Part 2 – the solution to the magic trick.

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Warm Ups: Number Talks and More

In my last blog post, I wrote about creating Doing-Math and Thinking-Math classrooms. One small but important ingredient I’ve found helpful for both is a good warmup activity.

The goal of the warmup activity is to get students thinking and active right away. The barrier to entry should be extremely low, so that everyone can able to participate. It’s a chance for students to limber up their minds and possibly get an early success under their belts.

There are lots of great warmups out there, including games like Don’t Break the Bank and simple arithmetic challenges like Target Number. One teacher we work with has been leading some amazing Broken Calculator challenges as warmups, which has been fantastic to see.

But today, I want to talk about a classic warmup, and one that provides one of the best returns on investment in terms of time and energy required: Number Talks.

example-number-talk

Number Talks are one of the best bangs for the buck routines you can incorporate into your math classroom. They’re build number intuition and fluency while giving you insight into how your students think; they support the idea that math makes sense, and you can explain what you see to help it make sense to others.

If you’re not familiar with Number Talks, our how-to guide is here. In a nutshell, the idea is this: pose a simple arithmetic or counting question, along with the prompt to not merely answer the question, but to come up with as many different ways of answering the question as you can. That minor tweak makes the activity more challenging, interesting, and sparks conversations that the students actually get invested in. It takes a few days to help kids learn all the routine, but once they do the benefits are massive.

From teachers who are familiar with Number Talks, the number one request I get is where to find sequences of questions to pose. We’ve written up a bunch of these lately, starting with counting collections of dots, and moving into solving math expressions. If these would be useful for you, please take them for a spin.

There have also been some exciting developments on Number Talk technology recently. My favorites:

  • Fraction Talks (for upper elementary, middle school, and high school)
    This is a fantastic resource with tons of ideas for extending classic Number Talks into higher grades with fraction-based images.
  • Visual Patterns (for upper elementary, middle school, and high school)
    These tend to be accessible as warmups only after students have had some practice with how these growth patterns work. For upper middle and high school students, however, these are an effective warm up, and a place you can see the huge payoff in how the numeracy encouraged by doing Number Talks regularly at the lower grades connects to algebraic thinking in upper grades.

visual-patterns-4

  • Would You Rather
    Leading a Number Talk using a simple comparison question (“Would you rather have 16 dozen dollars or $150?”) can motivate estimation. Using questions like the ones at the link, where certain aspects may be more poorly defined, can be an interesting way to connect mathematical thinking, common sense, and logical argument.
  • Unit Chats
    Unit Chats are a new innovation of Christopher Danielson. These are in a nascent stage of development, as you’ll see if you click the link, but I’m very excited about them. The main idea is to show a picture that contains different choices for units (i.e., avocado halves or avocado wholes vs pits?. You’re not just giving an answer + your strategy; you’re giving a unit as well, which has the potential to change the question and the answer.

avocados

I recommend finding a warm up that works for you and then making it a habit. Keep it short—5 to 10 minutes is optimal, in my opinion—and make sure it gets students thinking immediately. Try to do it 2-3 times a week at least, or every day if you can. It’s a small adjustment to your teaching routine, and one that can pay off in a big way.

One last thought: most curricula start with an instruction for teachers to “show students” how to do such and such a problem. You can run these teacher demonstrations as Number Talks instead. By giving students a chance to think about the problem first, they’ll be that much more primed to learn when the answer—and the explanation—arrives.

The Doing/Thinking/Loving Math Classroom

Summer Staircase 1This summer, we had the opportunity to draw up and institute a wholesale program from scratch, using our own lesson plans and providing the PD and support. Following that program, we began some fascinating conversations about how to articulate our vision of excellent math classrooms in more detail. From 30,000 feet, the direction we want everyone to move feels clear: our goal is to help design experiences that give everyone a chance to fall in love with math. But I feel like we’re due for a new articulation of where we’re aiming.

So what does this loving-math classroom actually look like?

  1. The Loving-Math Classroom is a Doing-Math Classroom
    The central activity in the class must be the doing of mathematics. Our metaphor here is brain as muscle, and classroom as gym. The instructor needs to launch the activity so that everyone can get started (and won’t injure themselves), but fundamentally, the actual “working out” is done by the students. The students must encounter activities that are difficult enough to be challenging but not debilitating. Most of class time (50 – 80%) should be spent actually doing math, preferably individually or in groups. This might mean playing math games (i.e., Pig), working through complex math tasks (i.e. The Power of 37), and also story problems and more plebeian math worksheets, though it can be problematic to rely too heavily on these.
  2. The Loving-Math Classroom is a Thinking-Math Classroom
    In addition to the actual doing of math, which is primary, it’s vital to find moments to take a step back and reflect on the process of doing math. What type of tools are useful to solve problems? How should we organize our data to best find patterns? What are good problem-solving techniques, and which whens are most useful for which kinds of problems? What do we currently know, and what questions do we still have? These times for reflection and discussion are critical for encouraging a depth of thought and development of the habits of mind that are, arguably, the true goal of math education.

What does this look like in real life? Our Summer Staircase classroom had a four-part structure:

  1. Warm-up (5-10 min) — keeping with the gym metaphor, a quick game or exercise to get the mathematical thinking started. Number Talks are a prime example.
  2. Launch (5-10 min) — this is the only time in lesson structure that doesn’t explicitly include doing or thinking math: the time when the teacher is explaining something new. In general, the teacher should streamline all explanation to the minimum required to allow students to work on their own. Sometimes the launch is time for a mini-lesson, or to pull together ideas from previous lessons. Sometimes it’s simply to demonstrate a new game or activity. In general, the goal of the Launch is to get them to the starting line, not the finish line.
  3. Work (30 – 45 minutes) — this is the period when the students actually do the math. For the summer, we used a station model, since younger kids usually have trouble focusing on a single activity for a long period of time. Typically, stations included a math game, a new problem or task, and an activity aimed at helping kids practice to master a technique or get comfortable with a mathematical representation. Starting in 3rd or 4th grade, and certainly by middle and high school, it’s more reasonable for students to be able to focus on a single, more complex task for the duration of this time.
  4. Wrap Up (5 – 15 minutes) — this is when the doing-math class has the opportunity to become a thinking-math class. The Wrap Up allows students to articulate their conjectures, counterexamples, arguments, and questions from class. It’s also where the teacher can underline deeper lessons learned, in line with the Common Core Math Practices, for example.

We found that virtually everyone could create a doing-math classroom. All it takes is the teacher belief that student activity is the central function of the classroom, combined with good materials, which we provided. Creating a thinking-math classroom is trickier; it takes more artistry and more practice.

______________________________

Aside: Last year, we wrote an introduction for our 2015 Summer Staircase curriculum that outlined what we dubbed five principles of extraordinary math teaching that later become my TEDx Talk:

  1. Start with a question
  2. Give students time to struggle
  3. You are not the answer key
  4. Say yes to your students’ thinking and ideas
  5. Play!

It’s clear that these are all urging teachers toward a doing/thinking/loving math class as well. Principles 1-3 are designed to get kids doing and wondering as quickly as possible, and to avoid short-circuiting the doing/thinking process. Principle 4 is about expanding from doing to thinking. Principle 5 is about creating the atmosphere that actually makes it all jell.

______________________________

So here’s the goal for us now: we need to articulate and fully formulate a vision, first of the doing classroom that teachers can make their own by a series of small, manageable steps, and second, of a pathway from the doing to the thinking classroom. Here’s a small beginning to that process:

Doing-Math Classrooms ideally involve a Warm Up, as brief a teacher Launch as possible, and then most of the class time devoted to the students working on a deserving problem or task.

Thinking-Math Classrooms are Doing-Math Classrooms that additionally include Wrap Ups for reflection and development of higher order skills, in line with the Common Core Math Practices.

Loving-Math Classrooms are Thinking-Math Classrooms that, at least occasionally, blow kids’ minds.

We’ll try to go forward with sketching a fuller vision of all of this. In the meantime, I’d love to hear your feedback.

App Review: Exploring the Core

I just downloaded a new app that I think will be helpful reference! It’s called Exploring the Core. (At last check, it was $2.99.) It is, essentially, an easy-to-use encyclopedia of Common Core math content standards. It includes a list of the standards, by grade, and a glossary to help you understand what all the terms mean. So, handy, though not much added value there.

But what is really special is the skills section of the app, which gives a map by major content area of the skills students will need to know. Say we click on Operations and Algebraic Thinking.

Map in Exploring the Core

We’ve got an easy-to-scan map of skills by grade and type. And the types are easy to understand: word problems, fluency, equations, graphing, etc. Click on one and you get a page that shows you the target, the standards that match it, and mathematical images that explain it, along with the conceptual knowledge beneath it. skill in exploring the coreAnd here’s the real beauty of Exploring the Core: the examples are mathematically rigorous, clear, connected to a solid conceptual understanding, and feature useful diagrams/manipulative ideas to use when teaching. How easy would it have been to just say, know your sums and differences up to 10, and apply up to 20 with this standard? By applying the images, the app suggests the kind of thinking numerically fluent students should actually be doing to build fluency. If they don’t have everything memorized yet, you can use ten frames and manipulatives or pictures to help get your kids there.

And that’s what makes this app so useful. I think Exploring the Core will prove helpful for me and the teachers I know, as a reference for the CC standards that fills in what’s between the lines with mathematically accurate images.

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Breaking and Remaking Lesson Plans

I just had an interesting lab session with a school district we’re working with this year. The teachers (K-2) want to extend the scope and dynamics of their lesson plans, and make sure they challenge their students. But the teachers also don’t have much free time, are, in many cases, new to the curriculum, and also feel that they need to be in alignment with the curriculum, for various administrative reasons. (Although the expectation from admin is that the curriculum will be supplemented. But still.)

The implications for me were that rather than demo one of my favorite Math for Love games or lessons, I wanted to model how one might break and remake a lesson plan. I wanted a process that would be quick and dirty, so teachers could implement ideas quickly; after all, they might have only a short time to review a lesson plan from the curriculum before teaching it.

The lesson that was given to me to demo was flawed in some pretty intense ways. For one thing, it was a lesson for 1st graders that was meant to be teacher-led instruction for a full hour. (Let that sink in.) The goal of the lesson was to have students connect story problems and subtraction equations. It included, in the middle, almost as an afterthought, the instruction to let students come up with a few of their own ideas for story problems, which the teacher would transcribe on the board and solve altogether.

Here’s the process I wrote down, which I think does a pretty good job of imitating the sieve I pass lesson plans through in my own mind.

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I thought this was a pretty good draft, and a good introduction to breaking and refashioning lesson plans to fit your classroom better. I’d appreciate feedback, especially if you have some favorite strategies my list is missing.

For this particular lesson, I launched with a related warmup (Target Number), and built an example story problem with the class. In fifteen minutes, we had accomplished what the lesson had said should take forty minutes, and then I released the kids for what we decided should be the main event: writing and solving their own story problems, and challenging each other with the ones they’d written. By launching quickly and letting the students generate their own work, we’d also succeeded in raising the ceiling of the task, and many of the students showed that they were comfortable with much larger numbers than the class had explored up to that point, adhering, as it had, to the curriculum.

And more than that, it was fun. The students were proud of their work, and stayed engaged, even though the material, without their input, wouldn’t necessarily have held their interest. Here are some of the student’s problems.

student-crafted-story-problems1 student-crafted-story-problems2 student-crafted-story-problems3

After 15-20 minutes of students dreaming up and solving their own problems, we had plenty of time to wrap up and pose even more difficult problems, involving one-digit numbers everyone was comfortable with, but more complicated stories, involving addition AND subtraction.

Many teachers don’t have the time, expertise, or permission to go too far outside their curriculum. But the good news is that most lesson plans in your curriculum are built around some good idea, even if it’s only a kernel at the center. But there’s a filter you can put these lessons through that accentuates the good while straining out the bad.

You’ve seen my draft above. Try it out, and let me know how it works for you, and if there are changes that should be made to it.

Thoughts on story problems

Story problems! They are the great bugaboo of math class, the problems everyone remembers hating, On the other hand, when educators think of “real-world” math, useful math, or motivated math, story problems are where they want to go. And this instinct makes sense. Story problems should be a fantastic resource in the classroom—a chance for reading comprehension, making sense of problems, modeling, and more—but instead they’re dreaded and ineffective.

The seedy underbelly of story problems was stunningly revealed recently in this video by Robert Kaplinsky.

Let’s take a minute and consider what happened here. Three quarters of the students who saw this question attacked it with a kitchen-sink strategy: just do some arithmetic with the numbers and maybe you’ll get it right. Why don’t they do what the other quarter did and say, “this doesn’t make sense”?

It’s possible that they don’t want to disappoint the questioner, and that they figure doing something is better than doing nothing. It’s possible they didn’t feel comfortable asking questions or expressing confusion. But look at that last girl describing why she decided to divide; this strategy doesn’t come from nowhere. These kids are doing story problems as they’ve been taught to do them.

In school, many teachers teach kids how to solve story problems as a sort of code. There’s a protocol:

  • Step 1. Underline the numbers
  • Step 2. Circle the important words, such as plus, minus, sum, product, difference, quotient, together, and, more, less, etc.
  • Step 3. Create an equation using the numbers and the operations corresponding to those words. (If the operation is subtraction or division, we’ll subtract the smaller number from the larger, or divide the smaller number into the larger)
  • Step 4. Solve the equation. That’s probably your answer.

This kind of rubric for solving story problems is self-defeating. We’re basically turning the intuitively sense-making project of reading a story into another kind of encoded math project, devoid of meaning. There’s a subtle line here, because teachers want to help, and underlining or noticing key words isn’t inherently a bad thing. But to begin by sweeping aside all pretense of meaning in favor of a mechanical process is bad mathematics.

Not surprisingly, it ends up being self-defeating as well. Story problems get trickier as students get older, and when they do, these kinds of mechanistic strategies backfire big time. The fact that most of the eighth graders in the video still seem to approach problems in this way spells trouble for their future in math.

So what’s to be done? Story problems have been around for millennia, and though they’ve often felt a little contrived, I don’t think they’re going anywhere. And truly, they are a relatively untapped resource. How do we use them to better effect?

Here are some ideas. I’d love to hear yours too.

Idea 1. Use story problems, but don’t teach a rubric to solve them.
Drop the story-problem “strategy” and focus on helping kids draw pictures or models of the situation instead.

Idea 2. Create story problems that resist basic strategies.
Interestingly, it’s not that hard to write story problems that can’t be cracked with the rubric-based story-problem-solving strategy. I’ve seen these pop up on high-stakes tests at the end of the year, leaving teachers to feel cheated that the test was gamed with questions designed to fool their students. But what if all the questions always resisted the basic hack of getting the answer without understanding the problem?

There are a number of ways to create hack-resistant story problems. The simplest is to create problems that don’t follow the same basic structure, but can be better solved by understanding the situation or drawing a picture. Here’s an example of this kind of problem from our Summer Staircase curriculum. This is a sheet for 2nd graders. Simply adding more steps to the problem makes these virtually impossible to solve without understanding what’s happening. On the other hand, if you draw a picture or build a model, they’re quite straightforward.

a-trip-to-the-bakeryGet the full worksheet here.
Idea 3. Create story problems with real interest as well as complexity.

The name “story problem” suggests a story. Why not tell a real story? Folks like Marilyn Burns and Greg Tang have been writing math story books for some time now (see a long list at living math.net). What’s nice is that these aren’t too hard to write, they get huge buy-in from the students, and they combine the fun of story time with a much deeper thoughtfulness regarding the math. Here’s one I wrote for our Summer Staircase curriculum, suitable for 3rd/4th graders.

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Notice that the engagement created by a good story allows for a much greater complexity in the mathematical modeling. What’s the relevant info for each question? The meaning can’t be lost, and kids are tuned into the meaning because it’s a real story. Some teachers gave kids a chance to draw their own version of the monster, creating an opportunity for an interdisciplinary lesson, involving reading, math, and art.

the-monster

Even if we skip the pictures, we get reading comprehension combined with mathematical meaning. The downside is that creating these kinds of stories is more work. But I can imagine a collectively-produced library of them.

Here are a few more examples of these long-form story problems.
Story Problem – The Ant and the Grasshopper
Story Problem – The Kite

Idea 4. Switch to Video
The 3-act math lesson is another way to grab attention and focus on meaning-making. There’s a lot to be said for this format (and a lot has already been said by others). Check out a lovely example here: The Cookie Monster.

Using video or rich images as a launch can be great. Really, though, this is a different animal than story problems, so I won’t focus on it here.

Idea 5. Use story problems as launches to complex tasks
This is another idea that stretches the very idea of a story problem. Consider a problem like the indecisive director problem.

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This is a great project, but it resists a straightforward solution, and requires less of the modeling and simple arithmetic of the story problems above, and is more about digging in to a much deeper problem. I’m a huge fan of complex tasks, and personally think they should be much more represented in math class, but this feels like a different animal to me too.

Idea 6. Have students write their own story problems
There’s nothing like standing on the other end of a process to understand its inner workings. A student who finished all the problems in The Kite asked what she should do next, and I suggested she write her own question. Here’s what she came up with—I liked it so much I converted it into a challenge problem for the other kids. (Check out the lesson to see.)

Having kids write story problems is usually a great idea. The dangers are that it becomes another unmotivated exercise, and that their story problems may not be appropriate for others to solve—it can be tough to write a story problem of the appropriate difficulty! That said, there are great opportunities when students are involved in the back end of producing problems as well as solving them.

What’s your take on story problems? It seems that story problems are an untapped resource, and with the right approach, they could be leveraged in all sorts of powerful ways. I’m still hopeful about producing a free library of good story problems in the style of The Monster. I like the prospect of combining the meaning-making involved in reading comprehension and mathematics.

Thoughts?

In Praise of the Open Middle with Pyramid Puzzles

Sometimes you hear the perfect word to describe something you were already doing, but didn’t quite realize you were doing. My most favorite recent case of this came from openmiddle.com.

We’ve all heard of open-ended problems, the genuine, intriguing problems that can take us on journeys to unexpected answers. A lesson based around an open-ended problem can be a beautiful thing, and done right, there’s hardly a more exciting educational experience you can have.

The trouble is, it doesn’t always go so well. Guiding a class through a discussion and exploration of an open-ended problem requires tremendous insight and understanding on the part of the teacher, and even the most experienced teachers sometimes have them go awry. Will this student’s comments lead in a productive direction, or just muddle the issue, and leave us all frustrated? Is this problem so hard that we can’t hope for more than a partial understanding of it, or is there some key idea that we’re all missing? When you have to make assessments on the fly, things don’t always go well.

One trick to coax these explorations in productive directions is to game them just a little bit. The teacher can go in with a plan, or an arc of where they expect the lesson to go. So they can “discover” something like the fact that the midpoints of quadrilaterals seem to form parallelograms and ask, “that doesn’t always happen, does it?”, all the while knowing that it is, in fact, true, and with a couple of good ideas about how to prove it. This little bit of lesson planning can greatly improve the chances of an individual lesson feeling like a success, even if it gives up a little bit of the total free of open ended lessons.

Enter the Open Middle. The idea of Open Middle problems is that the beginning (the problem) and the end (the answer) are both clearly well-defined, but the middle of the problem—the part where you look for the solution—is wide open. The middle is where you cast around for ideas, structures, tools, or just blind guesses about what’s going on. Because Open Middle problems have such a well-defined structure, there’s much less of a chance of them going off the rails. This makes them a fantastic tool for teachers who might be taking their first steps into less regimented math teaching, or who don’t feel like taking a chance on an exploration yielding a broad discussion that doesn’t go anywhere. (Read more about Open Middle problems here.)

I produced a number of these Pyramid Puzzles recently, and I think they’re great examples of Open Middle problems. it’s relatively easy to adjust their difficulty as well.

Pyramid Puzzle 1.

Each number must be the sum of the two directly below it in the pyramid. Fill in the blanks.

pyramid-puzzle-1

This is a puzzle that seems easy, but the solution is just out of reach. We can put a 7 above the 2 and 5, but what then? There’s no obvious next step, and our best guess is to take a stab at it, and see what happens. Some people like to work from top to bottom, and try taking a guess of what two numbers might go below the 30 (say, 12 and 18?), while some prefer to work up from the bottom, putting a number in the blank and going from there.

Suppose I put a 10 in the bottom blank. Then I’d have this:

pyramid-puzzle-2

A 10 on the bottom leads to a 51 at the top, which is a problem. But it’s also a useful mistake, since now I know the number in the bottom must be smaller than 10. One wild guess gives me traction, and it’s only a matter of time before I solve this problem.

Pyramid Puzzle 2.

Each number must be the sum of the two directly below it in the pyramid AND no number can appear more than once. Fill in the blanks with positive integers so that the top of the Pyramid is 20.

Bonus: Could the 20 at the top of the pyramid be replaced with a smaller number and the pyramid still be solved? Show how if it’s possible, or show why it’s impossible.

pyramid-puzzle-3

This is a much subtler puzzle, and solving the bonus problem in particular requires you to delve into the workings of these puzzles. Is there some kind of theory that can tell us what the maximum number at the top of the pyramid can be, given the rules that we’re filling them only with positive integers, and never the same one twice? What about differently-sized pyramids?

pyramid-puzzle-4

 

This is clearly the smallest number that can be atop a pyramid 2 stories high.

 

 

pyramid-puzzle-5

 

And this seems to be the smallest number that can be on top of a pyramid three stories high.

 

 

 

So what’s the smallest for four stories high, or more?

We’re in the territory of open-ended problems now, which is where I always seem to end up. But that’s the exciting thing about mathematical exploration: there are always questions pointing in directions you’ve never gone before.

Find more of our Pyramid Puzzles here, at our lessons page.

Check out openmiddle.com for more examples of Open Middle problems.