This lesson in meant to help students transition from thinking of area as a two dimensional measure to being able to apply it to the concept of surface area of a 3 dimensional object. I don't mention the vocabulary of "surface area" in this lesson because I want students to create their own definition in a later lesson (Making Sense of Surface Area). When I introduce surface areaI want students to have a model for representing a 3 dimensional object in a two dimensional way. If students have learned area in a traditional algorithmic way they may have the misconception that area IS multiplying, which gets in the way of surface area calculations which include addition. Exploring flat patterns helps students focus on the "faces" of an object and encourages them to think conceptually and visually rather than trying to apply a memorized (or not so memorized) formula. Working with visual models is really helpful for ELL students especially when they are trying to sort out all the geometric vocabulary like side, area, measurement, dimension, face, edge, etc.
This warm up asks student which of the two given flat patterns will fold into the given box. In my classroom I don't like to set out different supplies like scissors when I introduce a task. I feel that influences and in some cases stunts the method students choose to investigate the problem. I keep as many supplies out and visible and available to students at all times. I expect students to figure out for themselves how they want to proceed. If they want scissors they know where they are in the room and they are free to go get them. Not all students, however, choose the same method of investigation. Some cut out the patterns and fold them up, some use other ways of visualizing the box, and others compare the flat patterns. The key is to give students autonomy over their tools.
However they choose to explore, students discover that both patterns will work. My intention is twofold. I don't want students getting stuck on the idea of one right answer and I want them to get into the habit of double checking their assumptions. If they see that both of these will work the natural next question I want to raise is what other flat patterns will work. The intent of this lesson is not actually to find a flat pattern that works but to focus student attention on the flat, two-dimensional surfaces of a 3 dimensional object. Multiple solutions helps to cement this idea.
In their math family groups I ask students to come up with as many different additional flat patterns that will fold into the same box from the warm up. Asking them to create additional solutions will not only help them focus on the flat outer surfaces, but will help them visualize those surfaces. In order to make flat patterns that will work students must be able to figure out the number and dimensions of each face as well as how they fit together. When students are then asked to calculate the surface area of a 3 dimensional figure they can better "unfold" it in their minds.
There are a number of Possible solutions and several common mistakes. Some of the solutions are reflections or reversals of others and this can be an interesting conversation to have with students by asking "how many different patterns do we have?" or "Are some of these really the same thing?" This just provides additional experience that will help them visualize. It can also help ELL students practice the language of comparison that is so important in math. Here are some examples of some Incorrect patterns that students may come up with. The first one includes all the faces with the correct dimensions, but they are not connected correctly. This may or may not pose a problem with surface area, however it does indicate that a student is having trouble visualizing the figure. It is definitely worth pointing out and can help engage students in argumentation if you pose the question for discussion within the "math family" group. Trouble visualizing can lead to one of the other mistakes that do interfere with calculations. In the second example the "tabs" or sides of the box are represented with the wrong dimensions and will lead to the wrong answer. The mistake found in tricky tabs will lead to a correct response, but also can give rise to the last mistake.
Students have an opportunity to compare and discuss the patterns in the next section, but ELL students would benefit from getting a "head start" in their small groups first.
At the end of class I want to spend some time to flush out some conclusions about 2 dimensional nets that fold into 3 dimensional boxes. This is when I may highlight some of the Incorrect patterns that will not fold into a box and ask what conclusions students might make about what must be true for a flat pattern to "work". I really want them to focus on visually imagining the 3D box unfolding so that when we start working with surface area they can picture all the faces.
One idea that I want to come out in the conversation is the attention to which faces/sides connect or attached to which other faces/sides. This really seems to help them transition mentally between two and three dimensional representations. Another tricky part seems to be the size and placement of the "tabs" or the closing sides tricky tabs. Students may have all the faces correctly represented but have the "tabs" placed incorrectly. When I see some of these mistakes I know that this student may have trouble visualizing the faces of 3 dimensional object and they may have trouble with surface area. However, if they are representing the correct number and sizes of the faces I am less concerned, because it shouldn't interfere with surface area calculations.