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/ 

Walking Meetings with Students

I was in Stockholm and Copenhagen in March of this year.  I was at a Move and Learn Conference in Stockhom and our host Martin Lossman gave an amazing idea.  He told me about his walking meetings with his students.  He talks about goals and how the class is going on a quarterly basis with his students on a 1 on 1 basis.  So simple.  So I tried it.  It worked like a charm.  A student had come in for help in the afternoon and we simply went for a walk to discuss how things were going.  We just went for a 2-3 ;minute walk.  We talked about what he had been working on and where he hoped to be. It took away all those uncomfortable pauses and eye contact issues some students have.  The other thing that walking does is it puts you on the same level as your student.  Often we are on physically different levels than our students which creates a "I'm Helping You" attitude.  Walking beside students creates a "Let us do this TOGETHER" attitude.   We got back to the math office and we went to work on what he wanted to cover.   Brilliant.  Thanks Martin! twitter @marlos71

Paul and Tina Zientarski, Martin Lossman, Linda and David Sladkey in Stockholm

You are Not Defined by a Grade

This is a note to self...

You are not defined by a grade, a number, a letter, a percentage, or a category.  These are labels.  They are not giving the whole picture of who you are.   Do you sometimes dwell on the high category you were rated at.  Other times do you wallow in the low percentage you earned.  Both are like grabbing the wind.  These are labels.   Remember that you are not defined by a label like a grade.  You are so much more than a grade.   Grades are temporary.  A percentage, a number or a grade can never give the whole picture of who you are.  It is fleeting and it will pass very soon.  So ACKNOWLEDGE your label.  This just means that you need to recognize that a label was given to you.  Secondly, REFLECT and LEARN from your label.  Maybe you need to be more humble, or maybe you need to try a little harder next time.  Lastly, you need to MOVE ON.  Labels are quickly in the past and they don't define who we are.

One other thing I want to tell you.  You are important.  You impact this world.  This world needs you.   So give a little back today.  Give a smile.  Engage in conversion and actually listen to the response.  Give a compliment to someone difficult in your life.  Write a note of thanks to someone who would never expect it.  Give your best even though you don't feel like it.  Go the extra mile for someone or something even though no one is looking.  Put a little extra time in your schedule today to just notice others and their needs.   When you do this you will smile and be at peace.  So go and make a difference.

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?

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.

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?