Monday, August 29, 2016

10 Days of Number Talks for High Schoolers

This summer I read a book called "Making Number Talks Matter" by Cathy Humphreys and Ruth Parker.  It was outstanding.  It really called me to action.  I highly recommend it.  

Also, I read a great blog post regarding Number Talks by Sara VanDerWerf @saravdwerf called "Secondary Number Talks"  Sara challenged me to give Number Talks 30 times in a year and also for 10 days in a row (I'm trying it for the first 10 days of school)  

Number Talks are short mental math problems given to students to work out individually and then discuss as a whole group.  They are fascinating and fun.  Mostly it reverses the typical way a math class is run.  You give the problem with no introduction and no explanation and no hints.  Then you discuss the many different ways students solved the problem.  It is amazing the different thought processes that happen in with this method.  The students are set  free from the boundaries of whether they did it the same way as the teacher.  They relish in the coolest and fastest way.  They oooh and aaahh at the different methods.  The most amazing part is that the students start to see the connections in math that we have been trying to beat over their head for so long.    

Typically this is how the Number Talks have gone.

  1. You will be given a problem and time to work out the problem mentally.
  2. You will be given some time to share your idea with someone near you and get feedback.
  3. As a whole class we will share out potential solutions.
  4. Then as a whole class we will share out methods for those solutions.
  5. Depending on time, we will use one or more methods on a new problem.

Here are my Ground Rules

Everyone's voice matters.  Be respectful when someone is giving their opinion.
Everyone will be asked to take risks and be uncomfortable. Being uncomfortable is OK.  It is a part of growth and learning.
If you find one way to figure out the problem, then see if you can find a different way.
Try to think of a visual method of solving and explaining your solution.

Here is the one I gave DAY 1 (I got this problem from @joboaler ICTM 2015 keynote)

Simplify




POSSIBLE DISCUSSIONS and SOLUTIONS

 



Here is another thing that I think would be great to start doing on day 1.  EQUATIONS

5(18)=90
5(10+8)=90

5(18)=90
5(20-2)=90

5(18)=90
5(2)(9)=90

5(18)=90
2(5)(18)=2(90)

5(18)=90
18(2+2+1)=90

10 DAYS OF NUMBER TALKS
DAY 1 TOPIC: Multiplication
Simplify

DAY 2 TOPIC: Subtraction
Simplify
DAY 3 TOPIC: Visual Pattern
How is this growing?
What does step four look like?  How many small squares are in step four?
What does step 43 look like?  How many small squares are in it?  
What does the xth step look like?  What is the equation for it?

DAY 4  TOPIC: Fractions
Simplify

DAY 5 TOPIC: Visual Pattern
What does step 10 look like and how many mini squares are in it?  What does the xth step look like?  What is the equation for it?

DAY 6  TOPIC: Division

DAY 7 TOPIC: Visual Pattern
What does step 43 look like and how many squares are in it?  What does the xth step look like?  What is the equation for it?


DAY 8  TOPIC: Multiplication
Simplify

DAY 9 TOPIC: Visual Pattern
What does step 43 look like and how many squares are in it?  What does the xth step look like?  What is the equation for it?

DAY 10 TOPIC: Percents
Simplify
25% of $200





Here is what I give to the students
Day Topic URL (to copy to your drive)




Friday, July 22, 2016

Visual Patterns: Concrete Results

I would encourage you to use visual patterns in your class as soon as possible.  There is an excellent website to get yourself started at http://www.visualpatterns.org/ 
I gave this problem to my students in the middle of the quadratics unit for Algebra 1.  Any student has a chance at this.  That is what makes this so wonderful.  The playing field is leveled.  This particular problem is very cool because there are so many ways to view it.  So the beauty of visual patterns is that each student can look at the problem differently and still get the same answer.  The key is having them JUSTIFY their work.  You might want to try it yourself before looking below.
Here is my original problem 
Some things that I try to do when using this kind of problem.
1.  Have students work on their own first.  Then after some alone time, give time for collaboration.
2.  Have students try to work multiple solutions if they finish with one.
3.  Push students to visualize the problem in some way.  (manipulatives or drawing or sketch or computer based image...)
4.  Push students to give some meaning to the problem with algebraic symbols.
5.  I try to remember that this problem might take 20 or more minutes to work out.

Here are a few examples of what my students created with this problem.

This one saw the perfect squares involved and then just subtracted the two missing pieces off at the end.  


This group saw the smaller perfect square in the pattern.  Then dealt with the rest in a linear way.
  

I love this one because it incorporated a graph to make sure of their answer

This one is detailed.  The recognized the perfect square in the middle and then dealt with the other stuff as linear.  
I made these visualizations for the problem.  However, many of the students had these types of things on their papers before they wrote up the equations.  You can also see by the video below how the students were visualizing this problem.  Also, next year I'm planning on having the students do this type of visualization with a spreadsheet that @alicekeeler made.  http://alicekeeler.com/pixelart I found it in her blog post:  http://alicekeeler.com/2016/07/17/modeling-division-brownies-joboaler/ 

(x+2)(x+2)-2







(x+1)(x+1)+2x+1

(x)(x)+4x+2

More questions were asked for this problem.
Are the algebraic results all the same?
How do we know that the algebraic results are all the same?
Can we use DESMOS to see if they are the same?
Can we simplify to see if the algebraic results are the same?

Can you imagine your students wanting to know these questions?  It was so much fun.  


Here is the reward of the day.  One student who has trouble with the algebraic concepts and who almost never wants to talk about it got up and did this....magical.

Lastly, I'm thinking about taking the @saravwerf Number Talks challenge.  See her blog post at https://saravanderwerf.com/2016/06/27/secondary-number-talks-ill-convince-you-with-ducks/