Unit 1, Lesson 15
More Nets, More Surface Area
Grade 6
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This is the first Math Talk activity in the course. See the launch for extended instructions for facilitating this activity successfully.
This Math Talk focuses on multiplication of two whole numbers. It encourages students to observe the impact of adjusting a factor and to rely on the structure of base-ten numbers and the properties of operations to find products (MP7).
Each expression is designed to elicit slightly different reasoning. In explaining their strategies, students need to be precise in their word choice and use of language (MP6). While many ways of reasoning may emerge, it may not be feasible to discuss every strategy. Consider gathering only 2–3 different strategies per expression. As students explain their strategies, ask them how the factors impacted their approach.
This is the first time students do the Math Talk instructional routine in this course, so it is important to explain how it works before starting.
Explain that a Math Talk has four problems, revealed one at a time. For each problem, students have a minute to quietly think and are to give a signal when they have an answer and a strategy. The teacher then selects students to share different strategies (likely 2 or 3, given limited time), and might ask questions such as "Who thought about it in a different way?" The teacher then records the responses for all to see, and might ask clarification questions about the strategies before revealing the next problem.
Consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. This signal could be a thumbs-up, a certain number of fingers that tells the number of responses they have, or another subtle signal. This is a quick way to see if the students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Keep all previous problems and work displayed throughout the talk.
Find the value of each product mentally.
To involve more students in the conversation, consider asking:
MLR7: Compare and Connect
To Copy (from Blackline Masters)
Building Prisms and Pyramids Cards
Previously, students used a given net of a polyhedron to find its surface area. Here they use a given polyhedron to draw a net and then calculate its surface area.
Use the provided polyhedra to differentiate the work for students with varying degrees of visualization skills. Rectangular prisms (A and C), triangular prisms (B and D), and square pyramids (F and G) can be managed by most students. Triangular Prism E requires a little more interpretive work (in that the measurements of some sides may not be immediately apparent to students). Trapezoidal Prism H and Polyhedron I (a composite of a cube and a square pyramid) require additional interpretation and reasoning. The work here offers opportunities for students to make sense of problems and persevere in solving them (MP1).
As students work, remind them of the organizational strategies discussed in previous lessons, such as labeling polygons, showing measurements on the net, and so on.
Arrange students in groups of 2–3. Give each student in the group a different polyhedron from the blackline master and access to their geometry toolkits. Students need graph paper and a straightedge from their toolkits.
Explain to students that they will draw a net, find its surface area, and have their work reviewed by a peer. Give students 4–5 minutes of quiet time to draw their net on graph paper and then 2–3 minutes to share their net with their group and get feedback. When the group is sure that each net makes sense and all polygons of each polyhedron are accounted for, students can proceed and use the net to help calculate surface area.
If time permits, prompt students to cut and assemble their net into a polyhedron. Demonstrate how to add flaps to their net to accommodate gluing or taping. There should be as many flaps as there are edges in the polyhedron. (Remind students that this is different from the number of edges in the polygons of the net.)
Your teacher will give you a drawing of a polyhedron. You will draw its net and calculate its surface area.
What polyhedron do you have?
Study your polyhedron. Then draw its net on graph paper. Use the side length of a grid square as the unit.
Label each polygon on the net with a name or number.
Find the surface area of your polyhedron. Show your thinking in an organized manner so that it can be followed by others.
Ask students who finish their calculation to find another person in the class who has the same polyhedron and discuss the following questions (displayed for all to see):
If time is limited, consider having the answer key posted somewhere in the classroom so students can quickly check their surface area calculations.
Reconvene briefly for a whole-class discussion. Invite students to reflect on the process of drawing a net and finding surface area based on a picture of a polyhedron. Ask questions such as:
Optional
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In this activity, students compare the surface areas and volumes of three rectangular prisms given nets that are not on a grid. To do this, they need to be able to visualize the three-dimensional forms that the two-dimensional nets would take when folded.
In grade 5, students had learned to distinguish area and volume as measuring different attributes. This activity clarifies and reinforces that distinction.
Keep students in the same groups of 2–3. Tell students that this activity involves working with both volume and surface area. To refresh students' understanding of volume from grade 5, ask students:
Reiterate that volume measures the number of unit cubes that can be packed into a three-dimensional shape and that we can find the number of unit cubes in a rectangular prism by multiplying the side lengths (length, width, and height) of a prism.
Give students 1–2 minutes to read the task statement and questions. Ask them to think about how they might go about answering each question and to be prepared to share their ideas. Give students a minute to discuss their ideas with their group. Then, ask groups to collaborate: Each member should perform the calculations for one prism (A, B, or C). Give students 5–7 minutes of quiet time to find the surface area and volume for their prism and then additional time to compare their results and answer the questions.
Here are the nets of three cardboard boxes that are all rectangular prisms. The boxes will be packed with 1-centimeter cubes. All lengths are in centimeters.
students who worked on the same prism if they agree or disagree about the calcualation. Record and display the results for all to see.
Invite students to share a few quick observations about the relationship between the surface areas and volumes for these three prisms, or between the amounts of material needed to build the boxes and the number of cubes that they can contain. Discuss questions such as:
Students will gain more insights into these ideas as they explore squares, cubes, and exponents in upcoming lessons. If students could benefit from additional work on distinguishing area and volume as different measures, do the optional lesson "Distinguishing Between Surface Area and Volume."
A net can help us find the surface area of a polyhedron that has different polygons for its faces. We can find the areas of all polygons in the net and add them.
A square pyramid has a square and 4 triangles for its faces. Its surface area is the sum of the areas of the square base and the 4 triangular faces:
The surface area of this square pyramid is 96 square units.
Help us improve by sharing suggestions or reporting issues.
Students may know what polygons make up the net of a polyhedron but arrange them incorrectly on the net (for instance, allowing the faces to overlap instead of meeting at shared edges, orienting the faces incorrectly, or placing them in the wrong places). Suggest that students label some faces of the polyhedron drawing and transfer the adjacencies they see to the net. If needed, demonstrate the reasoning, for instance: “Face 1 and Face 5 both share the edge that is 7 units long, so I can draw them as two attached rectangles sharing a side that is 7 units long.”
It may not occur to students to draw each face of the polyhedron to scale. Remind them to use the grid squares on their graph paper as units of measurement.
If a net is inaccurate, this becomes more evident when it is being folded. This may help students see which parts need to be adjusted and decide the best locations for the flaps. Reassure students that a few drafts of a net may be necessary before all the details are worked out, and encourage them to persevere.
Students should have little trouble finding areas of rectangles but may have trouble keeping track of pairs of measurements to multiply and end up making calculation errors. Suggest that they label each polygon in the net and the corresponding written work and double-check their calculations to minimize such errors.
If students struggle to find the volume of their prism using information on a net, suggest that they sketch the prism that can be assembled from the net and label the edges of the prism.
Students may need a reminder that area is measured in square units and that volume is measured in cubic units.