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!