New Approach to the Product Rule

I tried a new approach today to teaching the product rule for my Calculus class.  I wanted to get my students thinking and trying to come up with what the rule is on their own.  I've looked at Think Thank Thunk by Shawn Cornally with his approach to teaching Calculus.  I like a lot of his ideas, but don't have the expertise he has in some areas so his approach to the product rule didn't real seem like something I could pull off.  But I wanted to do something.  I ended up just giving them a simple scenario:

"You are laying square tiles to form a rectangle.  Between each tile is a gap for mortar.  What happens to the total area covered when you increase the width of the mortar?"

I expected a bit of push back, but they all just jumped in creating variables for everything and setting to work.  I got a couple questions:

"Is there mortar on the outside?"  Yes
"Is the mortar the same width on all sides of the tiles?"  Yes

Mostly they just worked well.  I watched them work and offered little pointers on occasion, but for the most part they were all able to come up with good expressions for the area.

This was a great lesson for discussing conventions of writing math as well.  I was able to emphasize the importance of listing what the letters they use represent:

• Let x be the width of the mortar  (I hope I can get them to move away from always using x...)
• Let y be the number of tiles horizontally

I also got a chance to discuss conventions of which letters are often used as variables as opposed to constants:

• Early letters in alphabet are constants
• Late letters are variables
• n & m are integers

(Does this go directly against my previous point about moving away from always using x???)

In the end all groups ended up with something like:

Let x be the width of the mortar

Let n be the number of tiles horizontally

Let m be the number of tiles vertically

Let a be the width of each tile

Let A be the area of the region

$$A = [an+x(n+1)][am+x(m+1)]$$

Most groups multiplied this out to an ugly expression and a couple even got the derivative with respect to x.

For the last few minutes of class I took that area equation and discussed what it meant in context of the question.

Then I asked what they thought $$\frac{d}{dx} [f(x)*g(x)]$$ was.  Most guessed that it would be $$f'(x)*g'(x)$$.  Rather than telling them they were wrong I just calculated that derivative from the area equation getting:

$$A'=(n+1)(m+1)$$

They were quick to recognize that the change in area couldn't completely ignore the size of the tiles.

Next we tried to reason how the change would be affected.  If the width of the mortar increases along one strip, then the area will increase by the length of that strip.  There are $$n+1$$ vertical strips that are each $$am+x(m+1)$$ long.  Similar for the other direction, so the change in area will be:

$$dA = (n+1)(am+x(m+1))+(m+1)(an+x(n+1))$$

They were quick to notice that this is simply:

\[\frac{d}{dx} [f*g] = f'*g+f*g'\]

Tomorrow I will look more generically at the product rule simply as a rectangle (really, that is what multiplication is, a calculation of area!)  If the width and height both change, then the added area will be: 

• Let dW be the change in the width
• Let dH be the change in the height
• Let dA be the change in the area

$$dA=dW*H+W*dH+dH*dW$$

Overall I was surprised by how well this activity worked out.  The students really jumped in and did well manipulating lots of variables with very little direction.  I'm looking forward to more of this type of thing throughout the year.

Hopefully I will continue to share thoughts on here as I go!

Students hard at work on my whiteboards.