Surface area is a such a fun topic for students to explore! To really understand what surface area means, students need to interact with three-dimensional actual objects. Before we talk about formulas or how to calculate, we spend time discovering how to find surface area in our own ways.
I give students every-day items to work with. Typically, we use product boxes (rectangular prisms) with different dimensions, and I ask the students to visualize and then draw what the boxes would look like if they were taken apart and laid flat. Most students take about 5 minutes to complete their drawings, depending on how detailed they choose to be, and for the most part, they do a very good job drawing the nets of the boxes. Next, I have them spend a few minutes comparing their nets with group members, deciding whether those nets are reasonable representations of the object (even if they are drawn a little differently), and determining whether anyone appeared to be missing anything (some students will draw only five sides, and their group members are able to help them figure out what's missing).
After drawing their nets, I assign the groups two tasks - to find the surface area of their particular box and to determine a formula for the surface area of rectangular prisms. At this point, we have already studied area, so the only thing we discuss before they set upon their tasks is the actual meaning of the term surface area....we brainstorm the possible meanings and agree on the defintion. Then they set off measuring and calculating.
Most groups determine the surface area of their by the end of the class period, but normally none of the groups are able to decide upon a formula (we only have 40 minutes). So, we continue for a second day. While some groups are able to write a formula that reflects a correct understanding of the concept (though not written correctly "variable-wise"), others often remain stumped. Even though they are stumped about writing a formula, the "stumped groups" are usually able to explain to me HOW they found their surface area. Most of them explain that they found the area of the front and multiplied by 2 because the back is the same; they found the area of the top and multiplied by 2 because the bottom was the same, and the same idea for the sides; then they added those three sums together. Other groups find the area of all six surfaces and add them all. Some groups will find the area of the 3 different sides, add them and then multiply by 2. Based on our conversations, I know when they have found a correct way to find the surface area (or can then guide them if they are off track - the conversations are so important!). When it comes to writing a formula, some groups get very close, but have to be guided toward naming the length, width, and height with different variables.
For those groups that are able to finish fairly quickly (correct formula and all!), I have them work on determining the surface area of a triangular prism (I keep a Toblerone box on hand to use:-)
In the end, we share and discuss the formulas as a class. The students really enjoy this activity - it's challenging but achievable:-) Giving the students the chance to explore the concept and to construct a formula based upon their understanding of surface area is a great use of class time!
Once the exploration is complete, we can move from the concrete to the more abstract notes. I created this Surface Area Math Wheel for students to keep in their notebooks. I included the nets on it so the visuals are always there for them, in addition to the formulas. They love it!