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.  

Friday, July 31, 2015


The First Three Days of Class

  1. Parable of the two golfers
                    Click Here

  1. Student Partner Interviews
Place students with the new Seating Chart.     Have students interview their new partner
Share out the interviews by introducing your partner to the rest of the class and answering 2 or 3 of the questions.

  1. Partner List

This list will be used when are working in cooperative learning groups.  Click here to access it.  

  1. GRIT Discussion

Take the GRIT SCALE Quiz:  or    If you want to look at the questions again click here.  Or go to

  1. Brain Break

  1. Estimation 180
How many Almonds in the cup?

  1. Desmos Sign Up

Make a design using desmos.  There is a lot of math involved here.  Use the help button for tutorials.  Here are some examples
Spend at least 20 minutes trying to make a design.   Save and name your design. Get the url.  (use the share button) Post the url at

Period 1:

  1. Google Drive Setup
Create your folders like this in your main drive…..
__School Misc
_Home Misc
_P1 Algebra 15-16
_P2 French 15-16
_P3 Anthropology 15-16
Pictures and Videos
Z_Catch All

Color Them as you wish

_P1 Algebra 15-16
_Assignments and Projects
_Class Notes
_Important Class Stuff        passwords might be in here
Pictures and Videos

  1. Brain Break

Get in pairs.  Decide who is person A and who is person B.

Person A starts and asks Person B a question from below.  After the discussion, then person B asks Person A THE NEXT QUESTION and so on.  
  • Who is responsible for the device – what does that entail?
  • Discuss the idea of passwords and password literacies
  • Decorating of devices?
  • When can they be on social media and communicating with others?
  • Consequences of off-task behavior in class
  • Limits on personal work on device
  • Charging of devices

Pick a free space in the appropriate document below to give your thoughts on one or more of the above topics.

  1. All are True Except One
Go to the Google Slides document below and make a Slide that represents you.  Make all the things you say to be true except one.
Period 1     Period 2      Period 4     Period 6

  1. Differentiated Instruction
Dollar Bill Activity:  Invite someone to get the $1 from taped to the high part of the wall.  Then ask a person who can’t reach it to try to get it, then ask a taller person to get it.  Discuss the implications for this as we try to work together to learn more math.

Learning Style Quiz

Meet with your favorite food partner and discuss the following quote.

Quote 1:  ” ‘differentiated instruction’ refers to a systematic approach to planning curriculum and instruction for academically diverse learners.  It is a way of thinking about the classroom with the dual goals of honoring each student’s learning needs and maximizing each student’s learning capacity.”
-Carol Ann Tomlinson and Cindy Strickland

Quote 2:  
  1. Brain Break

  1. Mindset Quotes  
Read to yourself the Mindset Quotes.  3 minutes.  Which one do you identify the most with.  Why?  Now meet with your group of 4 and have each person discuss the quote they most identify with.

  1. Importance Order Smartboard Activity

  1. SMARTBoard Rules Activity

  1. Homework Expectations
Use Socrative to get opinions on what homework expectations should be.  We will build the homework framework off of these.

  1. Goals Google Form

Click here to get to the Google Form to submit your Goals.

Below is what it looks like.