How to do a Low Floor High Ceiling makeover...

There is a loft at the top of my house.  Only the limber and sure footed can climb the rickety ladder to get up to it.  Once there, only the very small in stature can stand or sit up comfortably.  Pretty much, the only people who go up there are children between the ages of 5 and 14.  Every so often, I go up there with the vacuum hose, but I almost never climb all the way in...   

This space is a classic high floor, low ceiling space - only serving the needs of a narrow group.  Other areas of our house have low floors and high ceilings (relatively speaking).  Those are the rooms in which we spend most of our time together.

In math, there are high floor, low ceiling problems.  In fact, if we look with a critical eye, we might see that most of the math we do is high floor, low ceiling.  The issue with these problems is that very few of our students can approach them independently and those students finish them in no time.  So we have to have different kids working on different problems, and none of them are independent for long.  The result is a heck of a lot of management and more paper work than any teacher can reasonably keep up with.

Several years ago, I became very interested in low floor, high ceiling problems.  I noticed that certain problems just worked with everyone.  Rather than having a class fragmented by ability, they allowed me to have a class that was individualized but unified.  Students were independent enough that I could work with individuals or small groups based on formative assessment.  And these problems had legs... even my "speediest" students could be coaxed to see these problems as an opportunity to enter the sandbox and play.  Low floor, high ceiling problems encourage stamina and creativity.

Rather than having students work through to completion, I threaded these problems throughout my instruction for a period of a week or longer for about 15 - 20 minutes per class (skipping days here and there as needed).  They made differentiation so much easier.  I used them as formative assessment, individualized concept building and "sponge" activity (soaking up the last ten minutes of class when a group of students are "done").

I've been getting lots of questions about where to find low floor, high ceiling problems.  There are some good sources out there (search on "low floor high ceiling" at www.youcubed.org, for example), but sometimes it's hard to find the right problem for a particular concept.  So, I analyzed what made these questions so good, and starting giving low floor-high ceiling "makeovers" to the problems that I had on hand.

I compiled and use this set of criteria to makeover problems (linked here):

1. Able to be approached at different developmental levels (entry points for all);
2. Is open-ended and able to be approached multiple ways (possibly using multiple content domains);
3. Lends itself to satisfying the various types of learners (kinesthetic, audio, visual, etc.);
4. Incorporates mathematical practices that demand critical thinking, problem solving, perseverance, extensions and generalizations.
5. Connects to a real world application relevant to multiple developmental stages.
6. Leads to significant mathematics, higher-ordered thinking and discourse.
7. Is interesting and engaging to a wide range of students.
8. Has various solutions that require justification or comparison.
9. Allows opportunities to practice key skills.
10. Provides direction without limiting thinking and exploration.

Want to see a makeover?  Take this high floor, low ceiling third grade challenge problem that I encountered this week (I can't remember the exact wording, but here goes...):

At a farm, a lady looks out her window and sees a group of pigeons and donkeys passing by.  She counts up all of the legs as they go by and finds that there are 66. How many of each kind of animal are there if there are 24 animals?

 A few students quickly said that there were 12 of each kind of animal.  When I asked them to prove it, they made a number line showing that 12 plus 12 is 24.  Meanwhile, most of the students were circling the numbers and underlining phrases, but weren't sure how to begin.  Some had never even heard of a pigeon (gotta love Vermont kids, right?).

The problem has great potential, but as it is, the floor is too high, and the ceiling is too low for it to invite in and sustain the whole class.

What if instead we asked it this way:

Start out with a whole group tool building activity having students find all of the ways to make a length of 10 (or 20) using just the 2 (red) and 4 (purple) Cuisenaire rods.  Have them record how many of each color and how many rods total they used to make 10 pictorially.  Emphasize that the length is always ten.  So, one purple is 4 and three reds is 6 making the whole length 10?

Hold up the red (2) rod. Ask the group what the length is (check to see if they know where the length is). Ask them what number it represents (2). Ask students what else it could represent, what is something that comes in twos? Make a list of the various responses.  So, this could represent a pair of mittens?  The rod is a pair that contains two mittens?

Ask them if it could represent legs. What kinds of animals have exactly two legs? Maybe they come up with pigeon or maybe they come up with another animal that is more meaningful to them...  Sasquatch, an ostrich, an owl, whoooo knows?  Do the same thing with the purple rod to find a meaningful four legged animal for it to represent.

Then, you could present the problem like a Madlib:

At a __noun (place)___, a lady looks out her window __adverb__ and sees a group of ___2 legged animal___s and ___4 legged animal___s passing by.  She counts up all of the legs as they go by and finds that there are __a number between 20 and 100__. How many of each kind of animal could there be?

In this problem, students can use the red and purple Cuisenaire rods to help them visualize.  Instead of finding a "correct" answer, there are multiple possibilities (no pun intended).  The problem allows students to choose a total number of legs that they find accessible.

Notice that students may pick a number of legs that doesn't "work," but that's okay.  They may edit their madlib if they want to, and you can have a great discussion about how to predict which numbers will work and which won't.  And why?  How would things change if it was a 6 legged animal and a 4 legged animal?

Since the problem looks for how many of each kind of animal there "could" be, it has a high ceiling.  Some students could get into combinatorics.  The teacher does not have to figure out everywhere this problem could go ahead of time.  All the teacher needs to do is ask for concrete, visual and symbolic justification for the conjectures that students come up with.  Communication is the highest level of math knowing.

In the meantime, the teacher can be moving among the third grade class looking for which kids need a push to see repeated addition as multiplication, and which kids need to push through some additive reasoning blocks.

This problem could be scaled up to teach algebra.  Not that third graders would go there, but a 6th-10th grade teacher could.  I'd probably launch it in much the same way but just take it farther.  See below:

This problem could also be approached through inequalities, linear programming, solving systems of equations, recursive definition of a function, etc., etc...

Don't you just love math?!  If anyone tries this out (with or without the algebra), please let me know how it goes.