standard_description: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
problem_activity: Activity
In this activity, students make sense of three different methods for calculating the surface area of a figure. Three different methods are described to students and they are asked to determine which one they agree with (if any) (MP3). They then think about generalizing the methods to figure out if they would work for any prism. This activity connects to work they did with nets in a previous grade and builds upon strategies students might have to calculate surface area.
As students work on the task, monitor for students who understand the different methods and can explain if any of them will work for any other prisms.
Note: It is not important for students to learn the term “lateral area.”
Launch
Arrange students in groups of 2. Display the prism assembled previously in the warm-up for all to see. Ask students: “how might we find surface area of this prism?” Invite students to share their ideas. Give students 1 minute of quiet think to read Noah’s method for calculating surface area followed by time discuss whether they agree with Noah or not. Repeat this process for the remaining two methods. Once all three methods have been discussed give students 1–2 minutes of quiet work time to answer the rest of the questions in the task statement.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use different colors to represent each of Noah, Elena and Andre’s methods.
Supports accessibility for: Visual-spatial processing
Student Facing
Here is a picture of your teacher's prism:
[IMG]
Three students are trying to calculate the
surface area
of this prism.
Noah says, “This is going to be a lot of work. We have to find the areas of 14 different faces and add them up.”
Elena says, “It’s not so bad. All 12 rectangles are identical copies, so we can find the area for one of them, multiply that by 12 and then add on the areas of the 2 bases.”
Andre says, “Wait, I see another way! Imagine unfolding the prism into a net. We can use 1 large rectangle instead of 12 smaller ones.”
Do you agree with any of them? Explain your reasoning.
How big is the “1 large rectangle” Andre is talking about? Explain or show your reasoning. If you get stuck, consider drawing a net for the prism.
Will Noah’s method always work for finding the surface area of any prism? Elena’s method? Andre’s method? Be prepared to explain your reasoning.
Which method do you prefer? Why?
Student Response
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Anticipated Misconceptions
Students may think that Andre’s method will not work for all prisms, because it will not work for solids that have a hole in their base and therefore more lateral area on the inside. Technically, these solids are not prisms, because their base is not a polygon. However, students could adapt Andre’s method to find the surface area of a solid composed of a prism and a hole.
Activity Synthesis
Select previously identified students to share their reasoning. If not brought up in students' explanations, display the image for all to see and point out to students that the length of the “1 big rectangle” is equal to the perimeter of the base.
[IMG]
Students may have trouble generalizing which method would work for any prism. Here are some guiding questions:
“Which of the students’ methods will work for finding the surface area of this particular prism?” (all 3)
“Which of the students’ methods will work for finding the surface area of any prism?” (Noah’s and Andre’s)
“Which of the students’ methods will work for finding the surface area of other three-dimensional figures that are not prisms?” (only Noah’s)
If not mentioned by students, be sure students understand:
Noah’s method will always work, but it can be inefficient if there are a lot of faces.
Elena’s method will not always work because the rectangles will not always be the same size, but we can notice that some shapes are the same and not have to work them all out individually.
Andre’s method does always work even if the rectangles have different widths. The length of the rectangle will be the same as the perimeter of the base and the width of the rectangle will be the height of the prism.
Prisms can always be cut into three pieces: two bases and one rectangle whose length is the perimeter of a base and whose width is the height of the prism. This can be more efficient than the other methods because students only need to calculate two areas (since the two bases will be identical copies).
This method only works for prisms. For other shapes, such as pyramids, Noah’s method of finding all the faces individually or Elena’s method of combining those faces into identical copy groups will work. Solids with holes, such as the triangular prism with a square hole, can use a variation on Elena’s method: two congruent triangles with holes for the bases, one rectangle for the outside side faces, and another rectangle for the faces forming the hole.
Explain to students that they will have the opportunity in the next activity to practice using any of these strategies.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve a written responses to the question, “How big is the '1 large rectangle' Andre is talking about?” Give students time to meet with 2–3 partners, to share and get feedback on their responses.
Provide students with prompts for feedback that will help their partners strengthen their ideas and clarify their language (e.g., “Can you draw a picture to support your explanation?”, “You should expand on….”, “How does that match with Andre’s thinking?”, etc.). Invite students to go back and revise or refine their written explanation based on their peer feedback. These conversations will help students make sense of the different methods for calculating the surface area of a figure.
Design Principle(s): Cultivate conversation; Optimize output (for explanation)


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