36 Students on a Bus

Reading my post entitled "MET Building Futures Workshop" I alluded to a problem we were asked to solve.

"There are 36 students on a bus. There are 8 more boys than girls on the bus. How many boys and girls are there on the bus?"

Here are at least 6 different ways to solve this problem.

PROCEDURE 1 (Keeping the sum constant-36)

( STEP 1) Choose two numbers that add to 36. (e.g. 20 + 16)
(STEP 2) Find the difference between the numbers. (e.g. 20 - 16= 4)
If the difference is less than 8, then add one to the larger number and subtract one from the smaller number. Return to (STEP 2) (e.g. 20+1 = 21 16-1=15 >>> 21-15=6 and so on)
If the difference is greater than 8, then subtract one from the larger number and add one to the smaller number. Return to (STEP 2)

PROCEDURE 2 (Keeping the difference constant-8)

(STEP 1) Choose a number equal to or less than 36. (e.g. 25)
(STEP 2) Add or subtract 8 from this number to obtain your second number. (e.g. 25-8=17)
(STEP 3) If the sum of these two numbers is less than 36, then add one to each number and return to (STEP 3) OR If the sum of these two numbers is greater than 36, then subtract one from each number and return to (STEP 3) (e.g. 25+17=42. 25-1=24. 17-1=16. 16+24=40 and so on)

PROCEDURE 3 (My strategy)

Divide 36 by 2. >>36/2=18. Divide 8 by 2. >> 8/2=4. Add 4 to 18 to find the number of boys. Subtract 4 from 18 to find the number of girls. 18+4=22 boys. 18-4=14 girls. Verify (22+14=36 / 22-14=8)

PROCEDURE 4 (solving algerbraically)

Let X=number of boys.
Let Y=number of girls.

X+Y=36
X-Y=8

Solve for X in X+Y=36. X=36-Y
Substitute X=36-Y in the other equation X-Y=8 >>>
(36-Y)-Y=8
36-2Y=8
-2Y=8-36
-2Y=-28
Y=-28/-2
Y=14
Substitute Y=14 in either equation.>>> X+Y=36 >>> X+14=36 >> X=36-14 >> X=22

PROCEDURE 5 (graphing)

Let X=number of boys
Let Y=number of girls

X+Y=36
X-Y=8

Solve for Y in both equations

X+Y=36 >>>
Y=36-X

X-Y=8 >>>
Y=X-8

Plot Y=36-X and Y=X-8 on a graph. Where the two lines intersect is the solution.

PROCEDURE 6

Add 8 to the number of girls so that the number of girls and the number of boys is now the same.
Adding 8 girls brings the new total to 44 (36+8 added girls)
Divide 44/2=22.
Subtract 8 from 22 to find the number of girls. (22-8=14)

THERE ARE PROBABLY MANY OTHER WAYS TO SOLVE THIS.
If fact, I can think of at least 3 other ways right now.

The point?...

Don't tell your students formulas to solve problems. Let them discover them on their own. And the ones they do not discover, you need not worry because if your classroom is collaborative and cooperative, they will learn these other ways from their peers.

What do they even need teachers for anyways? :)

MET Building Futures Workshop

After attending the literacy, numeracy and assessment workshops I had some time to think on the long drive home. I was very impressed with the numeracy presentation put on by the literacy and numeracy secretariat of Ontario. I became familiar with this organization during my obsessive drive to find as many videos of teachers teaching and found one of the best online resources. http://www.curriculum.org/secretariat/literacy_en.html The other really good one I found was eworkshop.ca. http://www.eworkshop.on.ca/cfmx/edu/core.cfm?p=home.cfm&L=1

The speaker made a wise decision. Rather than talk briefly about the hundred and fifty thousand topics relevant to mathematics instruction, she choose to spend most of her time discussing one of the most important. She presented the class a problem to solve.

"There are 36 students on a bus. There are 8 more boys than girls on the bus. How many girls and boys are there on the bus?"

She then had us share the different methods we used to solve the problem. This is a key point. She didn't show us "her" way. She let us find our own ways and through each of us sharing our ideas, we all discovered not only that there were many ways to solve this problem, but that our peers can be valuable assets in exposing us to ideas and perspectives that we would otherwise never be aware of. The problems of our society are complex to say the least. No longer can we rely on rote application of set procedures to solve them. We need creative, new ways to solve problems and we need collaborative and cooperative means to do this. Gone are the days when teachers tell us to multiple 47 x 38 by carrying the one and adding a zero on the second line. The true spirit of inclusiveness demands teachers provide opportunities for their students to be exposed to the ideas and strategies of their peers. It's wonderful to think that inclusiveness can be fostered while solving a math problem.

Waiting To Exhale

3 months to go. Only a month and a half until our second placement and I can't wait. Not that I do not love the 3 hour daily commute to Hamilton or the 2.5 hour breaks in between classes on Tuesdays and Thursdays (after all, I can finish my 200 assignments during these breaks...in theory) but I can't wait to be back with the kids.
One of our counsellors said, "after this program, you will be ready to learn how to teach." Well I may not be ready as of yet but as long as the school boards, faculty, and parents don't mind us learning with actual children I'd just as soon prefer to practice on/with them.