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/