Taking 20 Minutes for One Problem? A look at a Low Floor High Ceiling Problem

Starting.  Seems so easy.  But for many students this is very difficult.  When all of your students can start on a question, then you as a teacher have accomplished a lot.  So I give a lot of low floor high ceiling questions like this one.  Low floor questions are ones that ANYONE can begin. The high ceiling part is where it is really challenging for EVERYONE to get parts of the question.  This combination is very important.  It's called DIFFERENTIATION.   Now the beauty of low floor high ceiling questions is that you don't tell the students how to do it.  You let them get struggle.  You let them come to you with questions and observations.  You try to HOOK them.  Here is an example.  I got this problem from @davidwees and @rachelfruin.  There are many problems like this at http://www.visualpatterns.org/

This was so fun.  The students really went after the problem.  I had them working in pairs.  I choose 5 or 6 pairs to do all their work on the white board wall.  I have taken pictures of a few examles. 

   

I loved the way this group separated the colors without using colors.  

This group was very methodical and to the point.  I thought it was interesting that they did not have any figure for their 4th step.  

This pair was so proud of their work.  Amazing detail here.  

This group showed the next step correctly.  They did not get the variable expression correct, but we discussed their thinking.  I love the legend on this one.  

My favorite part of this one is the check at the bottom to see if their expression was correct.

Most interesting was the fact that none of the groups used this idea.  We had done some problems previously that were not so colored centered.  See http://tinyurl.com/visualpatternspractice  

So we asked the students to think of this problem in terms of area.  If the step was 1, then the area was 2 times 5.  If the step was 2 the area was 3 times 6 and so on to make....  

However, it was a great teaching moment because we asked the students to show that the variable expression that they found was the same as the variable expression we all found.  It was an ah ha moment.  

Justify that these are the same

It was terrific.  One student did this.

Another student did this...

  The cool thing was that many of the students did this to justify

This took 20 minutes to do this one question.  An it was worth every minute.   I think this is what Micheal Fenton @mjfenton calls Slow Learning Math.  If you are on Twitter you can use the hastag #slowmathchat.    We thoroughly took a look at this problem.  We let the students choose how they wanted to represent the problem.  They didn't have a set way of doing it because we haven't showed them the way to do these.  The were engaged completely because they owned how they did it.  They were open to new ideas because they were confident in how they did it.  

I love being a math teacher.  I can't wait for a new question for the next lesson.

Where do you get your Low Floor and High Ceiling questions?

Do you show how to do things before or after your students have tried to them?  Is there a balance with this approach?

Find all the Conics that you can in this 1 minute Video

Conics are EVERYWHERE.  I just took a flip video and taped of a few clips of some conic sections around town and then put them in a 1 minute video.  I asked the students to remember as many of the images that they could.  They were to try to identify the shape (parabola, ellipse, hyperbola, or circle) and what the object was.  It was very interesting to hear the discussion.

Some observations the students made.

The Taco-Bell clip looks like the bottom of the bell is an ellipse, whereas the top part looked like a parabola. There were some arches that looked like a semi-ellipse.
Someone pointed out the circle that is the steering wheel and the tires on cars.
The tower was a parabolic dish.
McDonald's had two parabolas.
The gas station had an elliptical sign.
There was a big argument about the wires on the sidewalk. The only thing it was not called was circular.
The stone arrangement looked like a semi-circle. Since the video is from an angle it almost looks like an semi-ellipse.
The bridge had some discussion as well between a hyperbola and a parabola.
The last clip was simply a nuclear plant shaped like a hyperbola.

Most of all, it created some great MATH discussions. Give it a try.
Dave