Marriage Problem

Last week, we wrapped up our winter sessions with over 50 elementary school math teams. Part of these sessions are devoted to having teachers work together to solve problems. Having teachers “do the math” helps brings meaning to important topics in mathematics education. We gave the following problem, from Van de Walle:

In a particular small town, 2/3 of the men are married to 3/5 of the women. What fraction of the entire population are married?

This is a challenging problem, but only because traditional algorithms get in the way of sense-making methods. The gut reaction is to do something with common denominators. Time after time, with each group, primary and intermediate. Through questioning, the mistake can be recognized.

“In this context, what does the 15 over here represent?” [points to 10/15]
“The total number of men.”
“And over here?” [points to 9/15]
“The total number of wom–OOOOOh…”

Sometimes, it takes longer to reach an ‘OOOOOh’:

“What does the 10 represent?”
“The number of married men.”
“And the 9?”
“The number of married wom–OOOOOh…”

Once teachers realize that having 10 men married to 9 women is somewhat problematic, most model the problem using colour tiles. Two out of three men being married becomes four out of six and six out of nine. Three out of five women being married is equivalent to six out of ten. Six pairs of husbands and wives can be formed. We have 12 out of 19 people being married.

marriage (concretely)Others think logically to solve the problem. The number of husbands must equal the number of wives. The number of husbands and wives are represented by the numerators.  Therefore, the numerators must be made equal. With all due respect to Dr. Math, it just makes sense.

marriage (pictorially)The use of manipulatives to construct meaning continues to be a focus of teachers involved in the numeracy project, both for themselves and for their students. Long before I became involved in this project, my fellow Numeracy Helping Teachers (Marc Garneau, Selina Millar, Sandra Ball, and Shelagh Lim) worked tirelessly to set a climate in which teachers and students felt comfortable using a variety of manipulatives.

At these sessions, we present teachers with problems, not practice. It’s a pleasure to work with such an amazing group of educators so willing to explore, take risks, and persevere. But as much fun as these sessions with teachers have been, I’m looking forward to the real fun: problem-solving with their students.

The first step in adding fractions is to find a common numerator.

“Okay, listen up! Today’s lesson will be on adding fractions. Let’s start with an easy one like 1/3 + 1/6. The first step is to find a common numerator, which, in this example, we already have. This becomes the numerator of the sum so let’s write a 1 up there. The denominator is, of course, itself a fraction whose numerator is the product of the denominators and whose denominator is the sum of the denominators. This gives us 1/(18/9), or 1/2.

Let’s kick it up a notch and try 2/3 + 1/4. Remember, the first step is to find the lowest common numerator, or LCN. You guys look a little puzzled. You remember learning this in grade 7, right? Since the LCN is 2, we have 2/3 + 2/8. Write a 2 up top. To determine the denominator, simply multiply and add to get 24/11. We have 2/(24/11). This is a tricky one since 24/11 doesn’t reduce nicely. Multiplying the common numerator by the denominator of the denominator gives us 22/24. One more thing… if you don’t reduce to lowest terms, I’ll have to deduct half a mark. 22/24 should be written as 11/12. I’ve typed up some notes. Take one sheet and pass the rest back.”

Christopher Danielson over at OMT shared the method above with me earlier this year. Recently, I presented it to a group of secondary math teachers. Christopher’s algorithm brilliantly initiates conversation about what is important in teaching and learning mathematics. For example, one teacher said “It works. I can prove that it works. But, it doesn’t make sense.” Another asked “It’s quick and easy, but does that matter?”

I think Christopher (@Trianglemancsd) plays it straight when he shows his algorithm to pre-service teachers. I couldn’t pull this off – more of a tongue-in-cheek thing for me. This elicited some (nervous?) laughter as teachers put themselves in the role of their students learning about LCD’s.

This segued to activities that do build conceptual understanding of fraction operations. We looked at:

  • using an area model to represent multiplication,
  • using pattern blocks to explore quotative division, and
  • using a common denominator to divide fractions.

These last two are connected… more on this later.