"Working Together" Problem

Heya, back-to-back post about a problem from Five Triangles mathematics.

When I tweeted how much I love this problem, a few people did not feel the same at all. Here are my reasons for appreciating this problem:

1. It's a notched up "working together" problem that I have not seen before.

2. It has percentages and fractions.

3. I can use rectangles to solve this. (I was asked on Twitter how I would solve this using rectangles, hence this post.)

4. I had to work on this problem. This is a big reason for me. We should assume that if we're teaching a particular math subject — Geometry, Statistics, or Calculus — that we're able to easily do all the exercises in the textbook. A set of exercises allows us to practice a particular skill. But a problem should require us to think. I hope I've encouraged problem-solving enough with my students that they value a problem more when they have to struggle with it, when they don't know immediately how to start it, when they get stuck and become frustrated, when they seek others for help, when they can leave the problem and come back to it another day.

While I'm at it, I also love the site Five Triangles in general for a couple of reasons:

1. The Geometry problems are simply stated and interesting. They make me pause and think, very few have been automatic gimmes.

2. The solutions are not posted. I really appreciate this because if they were, we might be tempted (mainly due to lack of time) to check the answers too early before we allow ourselves a chance to work through the problem and perhaps struggle with it. "Anticipating" is the first of 5 Practices that gives us insight on how students might solve the problem.

I did, however, retype the question above so it's easier to read and track information. I also numbered the paragraphs for quicker reference.

How we worked through this problem. Colors and all.

Draw a rectangle to represent the task. It has an area of 80 square units because that's the LCD of the three fractions in the problem.

Because this grid represents the task, we use it to fill in the amount of work done. Paragraph [3] is the first concrete piece of information that allows us to do this.

We continue to fill in the work done as described in paragraph [4].

Paragraph [5] is the first piece of information that allows us to figure out C's rate. Knowing that C can do 16 boxes in 8 hours means C can do the task — 80 boxes — in 40 hours.

With C's rate, we can now take on paragraph [2]. We know from the last step that C's hourly rate working alone is 2 boxes per hour or 10 boxes in 5 hours. But when working with A, C's rate is 40% faster, therefore instead of getting just 10 boxes done, C can get 14 boxes done in 5 hours when working with A.

From picture above in green, we know A and C did 24 boxes in 5 hours, and since C was responsible for 14 of those, the remaining 10 boxes were done by A.

Then A's hourly rate when working with C is 2 boxes per hour. Because this hourly rate represents a 20% increase than if A were to work alone, the math we need to do is 2 boxes divided by 1.2 to get 5/3 boxes. Solving for x in the proportion below gives us the answer that A completes the task in 48 hours.

Lastly we use paragraph [1] to figure out rate for B. We know A's alone rate is 5/3 boxes per hour, but when working with B, A's rate is 40% faster. Thus we multiply 5/3 by 1.4 to get 7/3. If A can do 7/3 in 1 hour, then A can do 35/3 in 5 hours when working with B.

The yellow boxes show that A and B can do 25 boxes in 5 hours, so subtracting 35/3 from 25, we see that B did 40/3.

To get B's alone rate, we divide 40/3 by 1.2 (because B is 20% faster when working with A) to get 100/9. Solving the proportion below gives us the answer of B completing the task in 36 hours.

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: 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.

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. 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.