# Casting Off Nets.

Last week, I was working with my students on some word problems involving the multiplication of rational numbers.  We did the usual type of word problems related to recipes and I even had students write word problems to share with the class (a practice that is often as entertaining for the class as it is helpful in gaining practice with the ideas).  After taking our normal stretch break, I projected this image onto the board:

I asked students to recall the name of the shape and asked them what they normally did with prisms last year.  They remembered volume easily enough, but I steered them towards surface area.

When I said the phrase, a silence descended over the room.  I could tell it was a sore point.  I knew it was probably an ugly topic that my students didn’t like and I had a feeling it was due to one reason: they used nets all the time.

Me being me, I decided to have them face their fears a bit and walked through the method of unfolding the prism and writing down the dimensions on the net.  I sketched it on the whiteboard, but here’s an idea of what we ended up with by the end of 10 minutes:

Students were starting to catch onto the idea more and I asked the class if the process was coming back to them as we created the net.  Some students agreed, but many stayed silent.

Me: “How many of your learned surface area using this way in some form or fashion?”

Everyone raises their hand.

Me: “Who felt like this way was a pain in the neck and kind of hard to follow?”

A lot of students raise their hands.

Me: “Well, what if I told you surface area doesn’t have to be this complicated? Would you give me a chance to plead my case?”

About half of the class raises their hands.

Me: “Forget about nets for a minute and just look at the prism.”

At this point, I hold a Kleenex box and start spinning it around between my fingers.

Me: “If I’m at eye level with this Kleenex box and staring at up close [I move the box close to my face], how many sides of the box will I see?”

Someone says one.

Me: “Okay, so with the prism on the board.  If you were directly in front of this prism really close, how many faces would you see?”

One.

Me: “Which specific face is it?”

Someone says the front face.

Me: “What are the dimensions of this face?”

A student answers 4 by 2 1/2.

I direct students to sketch a rectangle with these dimensions and I do the same on the whiteboard. Here’s the pretty picture for your reference:

Me: “What face will have to have the same dimensions for this prism to be an even box?”

With some scaffolding, a student answers the back face.

Me: “Surface area is the areas of all of the faces added together.  We just got the areas of 2 of the 6 faces.  We’re a third of the way there.”

I place the Kleenex box on a table, lean over it, and get my face really close to the box.

Me: “When I’m staring at this Kleenex box so weirdly, what’s the only side of the box that I can see in detail?”

I proceed down the same road as I did with the front and back faces for the prism to eventually have students sketch a rectangle that represents the top and bottom faces of the prism. Cue pretty picture:

Rinse and repeat this process and we eventually ended up with a sketch for the left and right faces:

Me: “Now we just have to find the areas of the rectangles we sketched, double them, and add them to get the result.”

Student: “Could we just find the areas of the 3 rectangles, add them up, then double that answer?”

Me: “Bingo.  That’s a great idea.   Use either of those methods to find the surface area.”

After students worked on this problem for a while, I brought their attention back to compare answers and get their opinion about the process.

Me: “Who thinks they feel better about surface area after what we did today?”

Some students raise their hands.

Me: “I want your honest opinion. Who prefers the nets [I point to that work on the left side of the board]?”

No one raises their hand.

Me: “Who prefers thinking about what the faces look like when you change your point of view [I point to the rectangles we sketched on the right side of the board]?”

Everyone raises their hands.

Me; “Why do you like this way better?”

One student says they like it because there’s less numbers to remember.   Another student says they like the point of view way because it makes more sense.  Most curiously, one student said they liked it better because you don’t unfold boxes a lot in life, but you do look at different sides of boxes from time to time. YES!

I borrowed this idea from a teacher I saw at a conference the other year.  I wasn’t sure what he meant at the time about students understanding surface area more with the idea, but now I know.  Nets can make surface area (an idea that is really simple to state and remember) seem more complicated than is necessary. I think the idea of moving around something and looking at it is more intuitive for students.

What are your thoughts?  Is this “point of view” approach helpful?  Is it too abstract for middle schoolers?  Is it any less difficult than nets?