The students will be looking at the faces, edges, and vertices of several prisms and pyramids to see if they can find a relationship between them to then make a generalization about any prism or pyramid. This is called Euler’s formula: F + V - E = 2 or F + V - 2 = E.
I’m going to give the students some background information about Euler before I let them start to work. There is a lot of information on the internet about Euler.
For example: ( taken from Interactive schooling)
Leonhard Euler was an eighteenth century Swiss mathematician and physicist, who made enormous contributions to a number of different fields. He is well remembered for his work in analytic geometry, trigonometry, geometry, calculus and number theory.
Arguably, Euler's greatest achievement is the formula he developed, which expresses the relationship between the number of faces, vertices and edges in a solid.
Students can work together to see if they can come up with the formula. Since it is not absolutely necessary for them to notice this, you can give them some time to work and then walk through with them how to get it. This is not meant to be an exercise that stresses them out. Just an exercise to get them thinking and noticing items.
Getting students to notice patterns and then make generalizations about their patterns by putting it into a formula, supports SMP 7 and SMP 8.
Tools: Chart with prisms and pyramids.
I’m going to walk the students through how to find the surface area of a rectangular prism. I will begin my discussion by telling the students that in order to find the surface area of a 3-D object, you have to add up the areas of all of the faces. I will also give them some real life examples of surface area. We find surface area when we want to wrap a gift. Surface area is used to find out how much paper is needed to make a label for a food item. Surface area is also used to create boxes that store food items. In order to make finding the surface area more conceptual and concrete, I will show them what they net looks like and how we can use the net to find the surface area of the 3-D object.
As we go through the example, I’m going to be asking the students where the numbers came from (SMP2: what do the numbers tell me). I want them to make the connections of where to place the numbers for their nets and how the numbers are combined to find the area.
Finally, we will find the areas of all the faces and add them together. I will be asking the students why we add the area of the faces together. I want them to make the connection between the definition of surface area (sum of all the faces) to adding. Students should know that sum means to add.
Tools: Rectangular prism example
Students are going to work on some example problems to find the surface area of other 3-D objects. They will work this out in their notes. Each example is already in its net form.
Square pyramid – As students are working out this problem, watch to see that they are using 12 cm for the height of the each triangle. Students may struggle with this because only one triangle is marked. If they do have problems, ask them what they know about the faces of this pyramid? Does it make sense to have triangles of different heights?
Students should add together:
Square: 15 x15 = 225
Triangles: 15 x 12/2 = 90. Since there are 4, they will need to multiply by 4 or add this in 4 times.
Total surface area: 585 cm²
Cube: Students will need to know that all sides of the cube are the same. This will be a great example of SMP 7 and 8 by asking students what they notice about the surface area of a cube and if they can find a way to show it (6s²).
Most students will do this:
7 x 7 = 49
7x 7 = 49
7 x 7 = 49
7 x 7 = 49
7x 7 = 49
7 x 7 = 49
Surface area: 294yd²
Expert level learners will see that they can take the side lengths and square them and then multiply by 6.
Students should see 2 triangles and 3 rectangles.
Triangles: 9 x 12/2 = 216
Triangle 9 x 12/2 = 216
Rectangle: 25 x12 = 300. Since there are 3 rectangles, they will need to do this 3 times or multiply by 3
Expert level learners will see that they only need to find the area of 1 triangle since there are 2 of them and that they only need to find the area of 1 rectangle and multiply it by 3.
As students work through these problems, they are implementing practices 4 and 5. They are using the visual of the net to help them model the math for finding surface area.
Tools: Student examples
For the closure, I want to see how the students will do with creating their own net and finding the surface area. The reason I want to do this is because they will be working on a project and will be using their own nets to create a robot. They will then need to find the surface area to cover the robot with tin foil. So, I will be watching for what students struggle with to work out these kinks before I let them loose on their project. I’m expecting them to struggle with how to measure out the net. I will have grid paper available for those that can’t get started. Grid paper will help with straight lines and will also help with measurement. I will also be giving them rulers to make exact measurements. (SMP 6: attending to precision)
I’m giving more time for this closure to allow the students time to think about how they want to start this problem (SMP 1: finding an entry point). For students that struggle, I will have them create a cube. This way they know that all sides need to be the same length. Other students can create any net they want. They should create the net, assign reasonable side lengths and then calculate the surface area using their net.
If time permits or if there is a need to use collaboration, have the students do a HUSUPU to find a partner to discuss strategies and/or solutions. (SMP3: justifying answers)
Tools: Grid paper and rulers