Friday, February 26, 2016

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

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  

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?

Wednesday, January 27, 2016

Getting Rid of "No Calculator" Questions on Math Assessments

I have typically given assessments with both calculator questions and no-calculator questions.  I've reasoned that students ought to know how to do a few things without the use of a calculator.  (maybe we should say technology now instead of a calculator)  Now I've reevaluated my opinion.  I've changed over to the idea that all my questions on any assessment should be allowing the use of technology.  Why?  That is a question I hope to answer for you. 
1.  Change the question if you think they can solve it with technology too easily.  You will find it probably wasn't a good question in the first place.  I like to change my "too easy with technology" questions to be reversed. Example:  Change....Solve x^2-4x-12=0  to be this...Create a quadratic equation to have solutions of x = 6 and x = -2.

2.  Technology is a great way to reinforce answers.  My tests often have this as the directions.  Solve using algebra, and verify your answer using technology.  Justify your work.  Or even simpler, solve in two ways.
3.  No Calculator questions strip the students of their best and most visual resource:  THE GRAPH. What No Calculator questions do is get students to forget that GRAPHING is a GREAT way to solve almost anything.  Don't we want to encourage graphing?  I don't mean graphing by hand either.  I mean a really fast way to analyze a problem.   Are we discouraging graphing because that is too easy?    When we take that away from students that is a major loss for their problem solving skills.

4.   My assessments are changing.  I no longer have 30 questions on a test.  I given less computational questions and more conceptual questions.  I have fewer questions which will help my students focus in on the difficult ideas.  They often have to show answers in multiple ways.  I'm always asking them to explain WHY did you do something.  The questions are a little more challenging and need to be able to use technology to solve.

5.  Technology is NOT CHEATING!  Solve this problem.  3x + 5 = 23   How about graphing?  We have two lines and they are intersecting.  This is a system.  Maybe y= 3x + 5       and      y = 23.  Then find the intersection point.  Is that cheating?   I think the way to give this question is to ask students to solve this problem by graphing and algebra.

6.  A student without technology is often asked to find an answer a certain way.  It might even have only one way to solve it.  I think this takes the creativity away from a student.  It also causes them to be fearful of not making a mistake because they can't even check it with other methods.
7.  What about the argument that we need to have our students get better at computational math skills?  I AGREE.  However, I don't think that a student with good computational math skills equals a complex math thinker.  

8.  I'm going to give more messy number problems.  Technology allows us to give our students weird numbers.  We should be doing that on most of our questions.  Messy numbers actually will encourage them to understand what is happening.  (By the way, have you ever been asked by a student if the answer was wrong because it had a decimal in it?)

9.  Having students use or not use technology is not going to change their number sense during an assessment.  I think we have to promote number sense in many ways during our classes.  But I don't think we should force them into manual calculation during an assessment.  I also don't think that a bunch of calculations by hand will change their number sense too much.  It will probably just build the hate they already have for math.

How about you?  What do you think about having "no calculator" questions on your assessment?  I'd love to hear from you.

Thursday, December 31, 2015

Mental Math and More Than One Way to Solve a Problem

Try this for your opener next time you have class.  Or better yet, at a social event.  This is fascinating.

No calculator, no talking, and no writing...

What is 5 times 28?

Give your students (or friends) a minute to think through their answer.  You might even want to tell them to confirm their answer using another method if the have finished.

Group your students randomly into threes.  ( I do this by taking my class size divided by 3 and rounding up to the nearest whole number.... If I have 26 in the class that day, divided by 3 rounded up is 9.   Count up to 9 over and over until all have a number.  1's get together, 2's get together and so on.  The 9's group will have only two people in it. )

Have each person discuss their solution and how they arrived at it.   Once the group has finished talking about how they solved it, have them try to find other ways to solve the question.

Now go back as a big group and discuss the different ways of solving the problem.

 Here are some ways to solve this.  I'm sure there are many more too.

1.  5 times 8 is 40.  5 times 20 is 100.  Add 40 and 100 to be 140.  ( a great opportunity to talk about the distribution property)
2.  5 times 25 is 125.  5 times 3 is 15.  Add 125 and 15 to be 140.  (distribution property again)
3.  2 times 28 is 56.  Double that and get 112.  Add another 28 to get 140.
4.  10 times 28 is 280.  Take half of that to get 140.
5.  Take 28 and change it to be 14 times 2.  Now multiply the 2 and the 5 to be 10.  Then multiply the 10 and 14 to be 140.   ( I love the rearranging of factors to be helpful to multiply in other ways)
6.  Take 28 and change it to be 7 times 4.  Now multiply the 4 and the 5 to be 20.  Then multiply the 20 and 7 to be 140.
7.  I find many students go to doing this in their head.... See below.

Isn't that cool?  So many different strategies.  Did you find a different one?  Please comment below.

Once you have discussed all the possibilities.  Ask your students if they would change their original strategy?  Why would you change?  I think it is important for our students to see that other students have different approaches to the same problem AND THEY ARE CORRECT TOO.  

I think too often we as teachers give a certain method as THE answer and the students feel wrong if they have done it by another method.  This will help us all realize how important it is to justify what you do with words and explanations.  

This exercise will help your students realize the value of mental math.  It will also help your students appreciate the amazing variety of answers.  I like to do this at least every other week in my class to help promote mental math strategies and an appreciation of other peoples methods of solving.

What do you think?

Sunday, November 29, 2015

How Do You See the Shapes Growing?

I was at the ICTM Math Conference 2015 in Tinley Park IL and Jo Boaler @JoBoaler was the keynote.  It was fantastic.  Here is a question she presented to us.


What would this look like at the 100th step?  How did you figure it out?  These kinds of questions she told us were low floor and high ceiling questions.  They are easy enough to start but challenging to all as well.  

I have gone to the website and found a few problems.  This is the worksheet that we did in Precalculus just the other day.  I really liked that the students were engaged the whole period.  They did not know how the shapes would work out.  (linear, quadratic, exponential...) It was fun seeing them struggle.  They really were working it.  Here is the google doc if you would like to use it in any way.  For your information I randomly assigned students 3 to a group.  I thought 3 people working together created a nice dynamic.  

If you really want to take this a step further  (get it,  a STEP further) then you can try to work the digital drawing.

Here comes the fun part.  How about you create the next shape?  Can you show how it grows with color?  Click the link below and get the document in a Google Drawing.  Once you have done that get a screenshot and add it to the Padlet below.

Get a Screenshot of your creation and add it to the Padlet below. (Just double click anywhere in the Padlet and add the url or upload a screenshot or the actual image)

Saturday, October 31, 2015

What I Learned at ICTM 2015 in Tinley Park IL (Illinois Council of Teachers of Mathematics Conference)

I loved going to ICTM 2015 in Tinley Park IL this year.  I went with some colleagues from my school district and that always makes it more fun.  Thanks @tjgebbie @RachelFruin @smiller229 @PhelanHoward  and @1kjwilliams

Jo Boaler  @joboaler  Mathematics Educator, Author,  and Stanford Researcher

Eli Luberoff @eluberoff Creator of Desmos  

These two speakers were outstanding.  Here are just the highlights of some of the things I learned.  

Jo Boaler  

Questions that we should be asking our students…
Why does that work?
Can your ideas be represented in different ways?
Why do those methods work?  
How are they connect to others?
Why does the answer make sense?

Quotes that I loved…
To her students she would say….I’m giving you this feedback because i believe in you.
Math should never be associated with speed
Timed tests cause the early onset of math anxiety
When teachers ask me how this can be possible, I tell them that the best thinking we have on this now is that the brain sparks and grows when we make a mistake, even if we are not aware of it, because it is a time of struggle; the brain is challenged and the challenge results in growth.
Math is too much answer time and not enough learning time
Everyone can reach the highest mathematics levels
Growth mindset brains had enhanced brain responses to mistakes
The lowest achievers in the world are the memorizers

What you need to have Low Floor High Ceiling Tasks
Get Buy In
Students don’t always have to use the language of mathematics
Give Explore time
Devise their own strategies
Individual time BEFORE groups
Have Students listening to each other
No one stepped in to help.
Have your students use a good type of arguing

Eli Luberoff @Desmos @eluberoff

What a personable guy.  That was a day that was spent with someone who was really excited to see Math Teachers learning new things.  Here are just a few of my memorable moments.

Regression is a SNAP.  If you haven’t looked into DEMOS and regression you should go here to try it out.

In response to new features Eli would always respond enthusiastically.  
"Not Yet" but we are working on it.

Central Park helps students to gain an algebraic concept by discovery and decreasing fear of failure.

The Activity Builder is searchable and lessons can be copied.

The necessity principle
for student to learn what we intend to teach them they must have a need for it, where need means intellectual need not social or economic need
Guershon Harel

And of course getting a selfie with friends and Eli himself was a lot of fun.

Monday, September 28, 2015

World Record Distance for Throwing a Fish at a Stop Sign

What is the world record distance for throwing a fish at a stop sign?   Hey, I don't think this stuff up.  But one of my students did.  Try to imagine a competition where everyone was seeing how far they could throw a fish to actually hit a stop sign.  One of my students was searching the internet to see if there were any sites that would have some data on this.   Unfortunately he couldn't find any data on this.  He moved on to a new idea.

Letting my students choose their own rates of change brought a huge amount of buy in.  The used their Chromebooks to search for data.  It was a blast. The students came up with many different ideas. For instance one student picked McDonald's hamburgers.  Did you know that from 1990 to 2013 the price of a hamburger has gone up 14 cents every year?  They OWNED IT.  They also took the time to figure out exactly what the rate of change meant.  I have always had a hard time teaching this.  But because they tried to find the data themselves the could communicate the idea of rate of change and what what happening over time.

White Sox Tickets anyone?   The ticket prices went up $1.02 each year from 1991 to 2003.

One student chose this.  (1950 year,  0 people living in Antarctica)   (1990 year, 0 people living in Antarctica)   0 people over 40 years.   Rate of change is 0 people per year.  Very Clever.  However, I told him he needed to find some information that gave a rate of change OTHER THAN ZERO.

So what did this lesson do?
Differentiation?   Yep
Student Choice?  Yep
Engagement and buy in?  Yep
Crazy Ideas?  Yep
Messy Numbers?   Yep
Fun for the teacher?  Yep

 Here was the original assignment.

Rate of Change Ideas for class

Here are the other postings.  We posted them to Padlet.

Monday, August 31, 2015

Easy Digital Collaboration: Padlet

I have been using for a couple of years now.  This year it is especially nice because we just went to a Chromebook for each student.  The benefits are many for me. Accountability. The students see others giving correct form for their work, they will want to give the correct form too. It raises the bar for quality work by being public.  This is great for getting 100% participation. Another big benefit is the fact that ALL students will have a voice. You will hear from EVERYONE. Lastly, I love the fact that students really would rather type than write on paper. This will amaze you. They will fill a whole section on a Padlet post, but probably would not like to write much at all on paper. You can use a Padlet tomorrow in class if you like. Just take these easy steps.

1.  Go to and create an account as soon as you can.  It is free.

2.  Once you are logged in, then create a new Padlet.

3.  Click on the Settings Icon and name the Padlet and put any special instructions.

4.  Now set the Layout.  I like Grid the Best.

5.  Now the last thing I do is change the address and copy it.  This makes it easier for your students to get to your Padlet.  It is not necessary to take this step.  However, if you are just giving the students the URL by having them go to it, then you will want to change the name to something easy.  Now your ready for the students.

In class
1.  Have them sign up for their own account before you have them do anything.  When they have done this, they leave a cookie trail for any post they make. (this creates accountability)  I very rarely have them post things without their name.

2. Now you can give them the URL address that you copied above.  Just pose a question and see the Padlet fill up with student responses.

Check out the example Padlets


This Padlet is locked so you may not add to it.  There is one below if you would like to try to type into one.


Here is another one that asked to pick a city in the lower 48 states and guess the hours it would take to drive there.  Then to calculate how long it would take averaging 60 miles an hour.  Then to find the time Google said it would take to get there. Lastly answer the question of why the Google time is different than yours.

This Padlet is locked so you may not add to it.  There is one below if you would like to try to type into one.  

ADD TO THE PADLET BELOW.  Put your name, school, and location.  Please put an inspirational teaching quote for us all to benefit from.  It could be from a colleague or a former teacher or simply one you have always liked.  Please give credit to whoever gave the quote.  If you don't know, just put anonymous.