Uncovering Coherence Using Area Models

The challenge of adult numeracy encourages us to persist to find ways to communicate concepts to learners.  I have found that the best work comes from collaboration.  In that spirit, Connie Rivera and I have created Uncovering Coherence Using Area Models, a webcast now available on the LINCS YouTube Channel.  Please watch the webcast and then come back here to share your reflections!

Which webcast activity was the most useful to you?  Why?

What insights did you gain about coherence, arrays, or area models?

Comments

The webcast contains hands-on activities and discussion questions.  (Plan for about an hour to complete the activities.)  

Who can you team up with to go through the webcast?  Share your webcast viewing plan here to encourage others.

Amy and Connie,

Thank you so much for the fantastic webcast. I love the approach, which encourages us to stay active and work with a one-way medium. It’s a nice model for this technology.

I was lucky enough to meet both of you at COABE last year and participated in Amy’s workshop on area models. It was great to be reacquainted with the approach, and I loved the additional details you added (quantities that can’t be made into rectangles with sides larger than 1, decimals, more fraction practice). I’m looking forward to discussion of polynomials as well, since I have found the area model to be really helpful in my own math explorations. No need to rush into that yet, though.

The decimal area model is really interesting and still confusing for me. I think there is a lot to unpack there. The columns and the rows are 1/10. The squares are 1/100. The model forces you to think about what the decimal point means, in the same way that the area model for two digit multiplication forces you to keep place value in mind instead of just following steps.

I’ve been thinking about square numbers a lot lately, and how to calculate them mentally. I’ve been using a mental area model (can I say that?) when thinking about something like 17 squared (100 + 70 + 70 + 49 = 289). This is all for my own entertainment for now, but I’m starting to think about how I might bring this to students. I’m curious to know what other people think about connecting conceptual understanding and mental math, but don’t want to pull this away from the intuitive-concrete-abstract progression you’re following.

Best,
Eric

I agree that there's a lot to unpack.  I can tell you that most of my students wouldn't intuitively get the idea that "okay, now the whole grid is one!!"   We have to work quite a bit with the idea that multiplying by something smaller than one shrinks things.   (This comes up a lot in our math literacy course where they learn about exponential growth and decay.)  

 

I totally get that it may not be intuitive for students to automatically understand that the whole grid is one, if it is a new idea for them.  As I think about decimals, pennies, and percentages, though.  I'm thinking that it would be useful to have the 10x10 square available as a go-to model, one that I could use perhaps as much as the number line.  In what other ways could we make use of the 10x10 square that would help to build understanding and connections?

Thank you for sharing your thoughts, Eric!

I’m glad you appreciate the variety we are able to explore because of the format of this offering.  Those other ideas (rectangles with ‘extras’, etc) are also one of the bonuses of collaboration and building on each others' ideas.  In fact, you’ll see if you hang in with us, that a resource we’ll share has your work in the cycle of building on each other’s ideas!  I believe that our students, too, benefit when we spend time talking about math ideas and sharing our thinking and reasoning.

I’m finding your idea about using a mental area model, which could look like the partial product method but easier to keep in your head, to find squares intriguing.  There’s something about picturing it occupying a space that makes it easier for my working memory to handle as I calculate the other pieces.  I like it!  I don’t find that to be too far off the pictorial level, even if it is mental.  How would we make that thinking visible to students?  Well I’m thinking… go back to the concrete level first and build to this.

Thanks,

:-)Connie

Hi Eric,

Good to see you here!  I remember talking with you at COABE, and as Connie alluded to, we will be sharing some of your curriculum work in our webinar. 

For me, using area models for squaring is a way to make larger numbers more accessible and have more meaning.  Also, it gets my attention that the middle 2 products match, just like in squaring a polynomial.  That may be a useful pattern for students to notice and build on.

Yeah, I agree. The application of an area model to multiplying binomials is fascinating. I was in a meeting with teachers on Tuesday looking at area models and there were a couple interesting connections made. This is one of the things that came out.

(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2

a2 ab ab b2

The area model becomes an easier way (at least for me) to do FOIL. Top left is First, then top right and bottom left are Outside and Inside, then bottom right is Last.

I've heard teachers say they prefer FOIL because the analogy of an area model doesn't work with negative numbers {(x - 2)2, for example}, but I'm sold. And maybe there is a way to capture negative quantities?

Here's a question that occurred to me when I was showing teachers: Is it important that we have students divide into the quantities that preserve place value 13 broken into 10 and 3, for example) or should we let students trying breaking into whatever works for them (maybe 8 and 5, if that's useful). A teacher noticed that 132 = 82 + 52 + (8*5) + (5*8). Interesting, right? But probably not helpful for students to discover?

Eric

Thanks for begin as interested in this stuff as we are!  Amy and I talked about the idea of preserving place value that you bring up, Eric.  We come from different perspectives, though we don’t really disagree.  I’ll share mine.

As you point out, the examples we chose to use in the webcast started with two digits and preserved place value.  We did this in order to connect it to the standard and partial products method because we wanted to make connections to the algorithms our students come to us recognizing and remembering pieces of.  Adult students are never ‘clean slates.’

With my students, I do not start by limiting them to place value when we are working with area models.  In fact, I begin with single digit multiplication and ask them to break down the numbers however they see fit, then write the equation for it that shows the distributive property, for as many ways as they can think of.  Some students are able to break down something like 8x9 to show 8x3 + 8x3 + 8x3 = 8(3 + 3 + 3) and others break down one of the factors into just two numbers.

Then we move into using double digit by single digit multiplication.  I don’t show them how since they’ve had the single digit experience; we play with it and see what they can figure out.  They might see that when multiplying, for example, 23 by 5, the distributive property works even if we use 5(12+11) = 5x12 + 5x11.  For me, accepting this answer and others helps students eliminate anxiety about “the” right answer, as well as helps them develop the idea that math can be flexible and numbers can be manipulated in different ways.  After we’ve shared ideas, I do find that they begin to settle into breaking down into either 5(10+10+3) or 5(20+3) pretty quickly.  I want them to reach the conclusion on their own that it’s more efficient and more relevant to consider place value.

What a great question to consider, Eric!  I really like Connie's emphasis on flexibility in her response.  MP 7:  Look for and make use of structure shows students that underlying mathematical concepts can be discovered and tested.  Encouraging students to explore different ways to construct an area model will reinforce the structure of the workings of area models as well as each student's trust in the model and confidence in their own mathematical reasoning. 

A point to add to the intentional use of place value in constructing area models is that place value is such a key math concept that it would be useful to reinforce it as often as possible.  Certain concepts and models are so helpful through all levels and contexts of math:  the number line, the 10x10 grid, place value, operations, etc.  In taking with Connie, I described those as the steel beams in skyscrapers.  Linking back to those fundamental ideas as often as possible will help students feel more steady in their learning and help them build and strengthen their own mathematical schema.

Eric,

I've also had teachers in workshops comment that the area model doesn't work with negative numbers. I guess I always thought of abstract area models as being able to represent both positive and negative numbers in the same way that (x-2)2  is an abstract representation. Using an abstract area model to multiply binomials with negative numbers seems to be just the next step after using an abstract area model to multiply binomials with positive numbers. I think it's a way to "look for and make use of structure".

 https://drive.google.com/file/d/0B1jlyb7MgOxkQXpOX3dhNTlmUlU/view?usp=sharing

You got me thinking, though, about how negative quantities could be captured using an area model and I found this youtube video. What do you think?

https://www.youtube.com/watch?v=wQ_Y98ovkMk

Patricia


 

 

 

Patricia, great video!  During a workshop I gave, a participant showed us that method, though not as clearly and colorfully as the woman in your video describes it.  This video demonstrates a way of drawing the area model that is less abstract than the image in Drive (and like the images we will share in the webinar).  I can see exactly what is happening with the subtraction the way she describes it. 

I also can see how taking a shortcut like we do to represent the negative binomials in the same way as the positive binomials doesn’t show everything.  My general philosophy about shortcuts is that I don’t want to take shortcuts FOR my students… shortcuts are for them to take once they discover a more efficient path to doing something for themselves.  Yet somehow, I am still ‘drawn’ to the shortcut simply because it doesn’t require a new way of drawing.  Would there be an easier connection to algebra tiles if we used the method shown in the video?

Thanks for sharing this example!

:-)Connie

Thank you Patricia for this valuable resource! In the past, I had used algebra tiles that you could flip for positive or negative values. However, it didn't really explain as well as the youtube video did which clearly showed removing parts of the square and then replacing a part that was removed twice. I LOVE IT!!!!! It reminded me of the proof for a^2 - b^2 = (a+b)(a-b) using paper, literally cutting away b^2 and rearranging what was left into a rectangle whose dimensions were (a+b)(a-b).

Pam,

Lynda talked about that paper cutting activity and it sounds interesting. I remember when she first mentioned it to me I did a youtube search but didn't come up with anything. I can't quite picture in my mind how it works so hoping that you can show me sometime. Maybe at our next ANN Board meeting?

Patricia

Thank you for the video, Patricia!  It totally makes all of the pieces fit!  I used her picture to find the solution in a different way, just to see.  I added the area of each of the 4 rectangles as defined in the video.  The point is not to teach this as a method, but to show connections.  x^2 - (2(x-3) -3(x-2)) + 6 

This could also be used to show the order of operations and precision with positive and negative signs as being important because the symbols and terms are describing a very specific situation.  Moving the parentheses or changing a sign would tell a different story!

Whenever I've used the Solar Panels pic it always gets students up from their seats and up to the SMARTboard to count the squares. Of course, they intuitively do not count every square because there's just too many! I like the idea of exploring different equations to represent what they are seeing. I don't think I ever explored that aspect of the pic as fully as it's addressed in the resource, so thanks for ways I can get better use of the pic.

Also,  I really like the "1/2 problem" that that the heart pic presents, so interested in seeing how my students respond to that. I haven't used that pic in a long while.Thanks for posting this great resource!

And thank you, Patricia, for all the field testing and sharing of materials that we’ve had over the years.  Amy and I discussed several times while preparing these activities how collaboration makes us grow and makes us all better teachers, both in the classroom with our students and with other teachers.  Let me encourage anyone reading who is looking for like-minded colleagues to join the Adult Numeracy Network, for which Patricia is the awesome editor of The Math Practitioner – a publication for members.

During our Uncovering Coherence Using Area Models webinar, chatter in the chat box turned to manipulatives for understanding math concepts.  I recently came across this Progression of Multiplication video, that makes it easy to see how we would move to algebra tiles.  In his demonstration, you can also see a move from pictorial to abstract (squares, bars, and dots... then actual numbers). 

As a side note, check out the rest of his Making Sense Series!