Ceilings

Play With Your Math problems are designed to have low floors and high ceilings, but designing a task which has a high ceiling can be really complicated. Take, for instance, 18. X-Factor:

18-x-factor

Of all the possible numbers I could have chosen for this problem, why did I choose 12? Here are three reasons.

To make success attainable: Drawing a 12-factor graph is not easy, but it is very possible. Usually, it takes more than one try with some erasing and/or starting over. Some people try some smaller numbers first and build up to 12. It might take some work, but many people eventually succeed in drawing a 12-factor graph. And solving the problem is satisfying! By choosing a number that is possible, I create an opportunity for that feeling of satisfaction.
To make space for curiosity: After solving 12, there are many possible follow up questions. What about 13, 14, or 15? I like to ask students: What number do you think will be the next hardest? Like, just slightly harder than 12? Moments after we posted this problem, Benjamin Dickman tweeted out this great question: What is the smallest number that is impossible? What would make a number impossible? I chose 12 because I think it is just hard enough to pique curiosity about other numbers. I wanted to give people the space to ask their own follow up questions, which is its own fun and playful mathematical experience and adds fuel to their further exploration.
To shelter from inaccessible questions: Drawing a X-Factor graph is impossible for some numbers. I think justifying why a number is impossible is an enticing but daunting challenge without some background knowledge about graph theory. If I started by asking about numbers that were impossible, I would be setting most people up for failure. By asking about 12, I let people investigate the impossibility question without making it the sole definition of success.

In summary, when I present tasks with low floors and high ceilings, I try:

To make success attainable. In addition to a high ceiling, I also want a low ceiling, where students can feel a sense of accomplishment. There should be lots of ceilings.
To make space for curiosity. Just because there is a ceiling, doesn’t mean I have to show it to them.
To shelter from inaccessible questions. Some ceilings are just too high for some people, and that is fine. The high ceiling is not meant to intimidate.

The same logic went into our design of 19. Mountain Ranges.

Why did I ask for 4 units wide?

To make success attainable: It is not easy, but it is doable. Can you find all 14?
To make space for curiosity: How many are 5 units wide? How many are 6 units wide with 3 separate mountains? Is there a recursive strategy for finding the number of mountain ranges 1 unit wider? Is this similar to any other problems?
To shelter from inaccessible questions: The general question (“How many mountain ranges are n units wide?”) is interesting, but too hard to be the focus.

Some photos of people playing with these problems

I posed X-factor to some teachers at PROMYS for Teachers earlier this summer. It was a ton of fun. One participant had some graph theory background knowledge that he shared and helped us investigate which numbers were impossible and why.