Drawing Rectangles Instead of Writing Equations

I don't see dead people but I see rectangles all the time. 

The first time I saw a multiplication fact, like 3 x 5, as a rectangular array was after I'd graduated from college. 



And because I was so very late to this game, I thought drawing rectangles must be how everyone else — at least math teachers — solved math problems. 

But apparently not so. I started going to math workshops and often the teachers sitting next to me would look at my drawings and ask, "What are you doing?" I glanced over at their papers and saw mostly numbers and equations and thought, What are you doing?

So this is how I've always solved these rather mundane but classic problems.


Problem: In a town, 3/7 of the men are married to 2/3 of the women. What fraction of the people in the town are married? 

My mind sees these two rectangles.



I notice that the blues have to match the pinks, so I make them match.



The answer is 12/23 of the people in town are married. (So fun to add fractions straight across.)


Problem: Danielle and Jennifer can do a job in 2 hours working together. Danielle could do it in 3 hours alone. How many hours would it take Jennifer to do the job alone?

I see a 2 x 3 rectangle as the "job," and I choose 2 and 3 because other dimensions just get messy. 



It's easy then to see how much Jennifer can do.



The answer is it would take Jennifer 6 hours to do the job alone. I mean the answer is looking at me.


Problem
: [mathforum.com] 
Dad bakes some cookies. He eats one hot out of the oven and leaves the rest on the counter to cool. He goes outside to read. Dave comes into the kitchen and finds the cookies. Since he is hungry, he eats half a dozen of them. Then Kate wanders by, feeling rather hungry as well. She eats half as many as Dave did. Jim and Eileen walk through next, and each of them eats one third of the remaining cookies. Hollis comes into the kitchen and eats half of the cookies that are left on the counter. Last of all, Mom eats just one cookie. Dad comes back inside, ready to pig out. “Hey!” he exclaims. “There is only one cookie left!” How many cookies did Dad bake in all?

Dad eats 1. Dave eats 6. Kate eats 3. So, 10 cookies are gone already.

Now, I use the rectangle to help me figure out where the rest of the cookies went. I'm reading "thirds" in the problem so I make sure I draw something that's easy to divide into thirds. (I'm smart like that.)



Now, I read that Mom eats 1 cookie and there's only 1 left for Dad. So the last three white squares in the rectangle represent 2 cookies. 



I see 12 cookies. These 12 plus the 10 from the beginning equals 22 cookies in total that Dad baked.


Problem: A class has 5/9 girls. If the number of boys were doubled and 12 girls were added, there would be an equal number of boys and girls. How many students were in the class at the outset?


Naturally, I see this rectangle.



Then I do what the problem says.



The 5 pink boxes equal the 5 white boxes, so I cancel them out, leaving me with this.



At the outset there were 9 boxes, each box worth 4 students, so the answer is there were 36 students in the class.


So this is how I teach it to my students too. No equations. I remember a few years back one of my 6th graders' tutors was not happy with me that I'd assigned a PS involving "systems of equations." He said it was an algebra topic and how was a 6th grader supposed to solve it except for using guess-and-check. Really? Stuff people say that makes me fart.

Anyway, I've always looked at a square as two interlocking staircases, each a sum of consecutive integers. (Right, Chris Hunter?) I also see a non-square rectangle as two interlocking staircases, one is the sum of consecutive odds and the other sum of consecutive evens.

   

Finally, my love for rectangles proved helpful — if not the key — in solving this problem that Josh Zucker had presented at last summer's Math Teachers' Circle.

On a blackboard are written the numbers 1 through 100. At every stage, two are selected, erased from the board, and their sum plus product is added to the list on the board. At any stage, you're free to choose any two numbers. When the board is reduced to a single number, what possible values can it have?

Hooray for rectangles in problem solving!



 
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Comments

  • December 11, 2012 10:10 PM Robert Kaplinsky wrote:
    It similarly amazes me how many ways of thinking about math I did not learn until after college. I recall how I felt silly yet empowered by being challenged to explain what it meant to divide by a fraction. That helped me understand the significance of having conceptual understanding in addition to procedural understanding.

    The strategies you use certainly make these problems easy to understand, and it is lots of fun asking students to show where their numerical answers are represented in solutions like yours.

    Reply to this
    1. December 12, 2012 10:31 PM fawnnguyen wrote:
      That's why I wish I had some of my tweeps as my math teachers growing up; we are all striving to do a better job teaching conceptually and making math as exciting and beautiful and relevant as it truly is. Thank you, Robert!
      Reply to this
  • December 13, 2012 4:12 AM Mary Dooms wrote:
    How timely! My sixth graders are reviewing fractions and I'm going to use your strategies. I did have them model fractions using one of the GeoGebra applets http://www.geogebra.org/en/upload/files/english/Megan/Fractions/Multiplying%20and%20Dividing/Mult_Div_Fractions.html but you are taking it much farther. Thanks!
    Reply to this
    1. December 17, 2012 11:02 AM fawnnguyen wrote:
      Hey, last year I just learned how to make sliders in GSP/Geogebra! So far I've been pretty happy teaching multiplication and division of fractions with pattern blocks. We'll see how it goes this year -- really low group, about 1/4 are at least 2 grades below! Yikes. Thanks, Mary.
      Reply to this
  • December 13, 2012 8:39 AM Denise wrote:
    I love thinking with rectangles. I didn't learn it in school, but picked it up as I used the Singapore Primary Mathematics books to homeschool my kids. Such a powerful way of looking at number relationships!
    Reply to this
    1. December 17, 2012 11:05 AM fawnnguyen wrote:
      Thanks for dropping in, Denise. I love Let's Play Math!
      Reply to this
  • December 16, 2012 1:38 AM Steve Grossberg wrote:
    Great stuff, Fawn! I plan on thinking further about some or perhaps all of the problems you've solved here, in the spirit (I believe) of what Robert K. wrote above, trying to see the connection between your rectangles and the equations I would have used (and taught my students to use). I stress thinking about multiplication (of two numbers anyway) as a rectangle area problem all the time to my students, but you take it much further.

    I very recently ran into the issue of a problem where one step was to, as you say, "add fractions straight across." A flag went up in my brain, asking myself why this was the right thing to do, but I didn't pursue it at the time. (I was teaching, and made the split-second decision that I couldn't afford the time to follow this path-of-indeterminate-length.) I will look at your first problem and consider that point further. I think that you are not "really" adding the fractions, in that adding their decimal equivalents is completely incorrect. But if "adding straight across" gets to the right answer (and not accidentally, to be sure), then what *are* you doing there?

    Also, I am intrigued by the use of rectangles to solve a combined-rates problem, for at least two reasons. One, they love this kind of thing on the SAT, and two, students are nearly always dying to do these incorrectly by adding (or in this case) subtracting the rates. So their answers are almost always of the it-takes-longer-when-we-work-together variety, which has some intersection with reality but is clearly not the issue with this question. (It doesn't reference the girls' personalities or relationship, for example.)

    And even after I teach the students how and why the rates need to be denominators and not numerators (implicit in your drawings), it would seem to be much easier to make a mistake setting up the problem than it might be with your method--given some practice with it, of course.

    I think I will throw this at my Math Club kids and see what insight they bring to it. And I may try it out with my own children, in 6th grade math and Algebra 2 currently.

    Thanks so much for sharing this intriguing idea!

    Reply to this
    1. December 17, 2012 12:54 PM fawnnguyen wrote:
      Right, for the married couples problem, adding straight across -- based on the drawings -- shows that I'm just adding people who are married (numerator) and adding the total men and women in town (denominator). Some of the kids will realize too that since this is just a ratio, so the town population can be any size that guarantees whole-number married couples. It's very common for kids to set up a proportion to do work problems only to realize that it doesn't make sense, that's why I show the visual model first. Then we look at the rational equation after. 

      Thanks so much, Steve, for your comment.

      Reply to this
  • May 21, 2013 2:31 PM Tami redus wrote:
    I have to say, I am very confused at how you arrived at the married couple answer. And what you mean by making the blue and pink match by changing 3/7 to 6/14 and 2/3 to 6/9. What matches there? I just can't seem to wrap my brain around it for some reason. What am I not seeing?
    Reply to this
    1. May 21, 2013 8:31 PM fawnnguyen wrote:
      Because we assume that marriage is one-to-one, and the numerators represent the people who are married, therefore I need the numerators to match. Thus changing both original fractions of 3/7 and 2/3 to become 6/14 and 6/9, respectively, allow the "six" men (out of 14 men) and "six" women (out of 9 women) to marry.

      Hope that helps, Tami. 

      Reply to this
  • May 22, 2013 2:29 PM Deborah Boden wrote:
    Tami - Since there has to be the same number of men as women, the numerator needs to be the same number. If you change the 3/7 to an equivalent 6/14, and the 2/3 to an equivalent 6/9 you now have 6 men and 6 women. Therefore, there are 12 people married (the numerators) and 23 people in total (the denominators). It isn't the usual way to add fractions, but we're not really "adding" them, we're finding equal numbers. I'm sure Fawn could explain the last part beter!
    Reply to this
    1. May 22, 2013 7:29 PM fawnnguyen wrote:
      No, Fawn could not have explained it any better. Thanks, Debbie.
      Reply to this
  • May 22, 2013 2:30 PM Deborah Boden wrote:
    OOPS! I didn't see that Fawn had answered Tami already!
    Reply to this
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