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.
Launch
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:
Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Find the value of each product mentally.
Student Response
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Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
Math Community
At the end of the Warm-up, display the Math Community Chart. Tell students that norms are expectations that help everyone in the room feel safe, comfortable, and productive doing math together. Using the Math Community Chart, offer an example of how the “Doing Math” actions can be used to create norms. For example, "In the last exercise, many of you said that our math community sounds like ‘sharing ideas.’ A norm that supports that is 'We listen as others share their ideas.’ For a teacher norm, ‘questioning vs telling’ is very important to me, so a norm to support that is ‘Ask questions first to make sure I understand how someone is thinking.’”
Invite students to reflect on both individual and group actions. Ask, “As we work together in our mathematical community, what norms, or expectations, should we keep in mind?” Give 1–2 minutes of quiet think time and then invite as many students as time allows to share either their own norm suggestion or to “+1” another student’s suggestion. Record student thinking in the student and teacher “Norms” sections on the Math Community Chart.
Conclude the discussion by telling students that what they made today is only a first draft of math community norms and that they can suggest other additions during the Cool-down. Throughout the year, students will revise, add, or remove norms based on those that are and are not supporting the community.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
12.2
Activity
Standards Alignment
Building On
Addressing
6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
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.
Adjust the timing of this activity to 25 minutes. Students may not have time to assemble their net into a polyhedron during class. Also, consider posting an answer key so students can quickly check their surface area calculations.
Launch
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.)
Activity
None
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.
Student Response
Activity Synthesis
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):
“Do your calculations match? Should they?”
“Do your nets result in the same polyhedra? Should they?”
“Do your models match the picture you were given? Why or why not?”
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:
“How did you know that your net shows all the faces of your polyhedron?”
“How did you know where to put each polygon or how to arrange all polygons so that, if folded, they can be assembled into the polyhedron in the drawing?”
“How did the net help you find surface area?”
MLR7 Compare and Connect. Invite students to prepare a visual display that shows their net drawings and surface area calculations. Encourage students to include details that will help others interpret their thinking. Examples might include using specific language, different colors, shading, arrows, labels, notes, diagrams, or drawings. Give students time to investigate each others’ work. During the whole-class discussion, ask students:
“What did the nets and calculations have in common? How were they different?”
“What kinds of additional details or language helped you understand the displays?”
Advances: Representing, Conversing
12.3
Activity
Optional
Standards Alignment
Building On
5.MD.C.5
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
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.
Launch
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:
“When we find the volume of a prism, what are we measuring?”
“How is volume different from surface area?”
“How might we find the volume of a rectangular prism?”
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.
Representation: Internalize Comprehension. Provide students with a graphic organizer such as a two-column table to record measurements and calculations of surface area and volume. A table with space for students to record their calculations for volume and surface area for Boxes A, B, and C will help them keep their thinking organized. Supports accessibility for: Visual-Spatial Processing, Organization
Activity
None
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.
Compare the surface areas of the boxes. Which box will use the least cardboard? Show your reasoning.
Now compare the volumes of these boxes in cubic centimeters. Which box will hold the most 1-centimeter cubes? Show your reasoning.
Student Response
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Building on Student Thinking
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.
Activity Synthesis
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:
“If these prisms are boxes, which prism—B or C—would take more material to build? Which can fit more unit cubes?” (Prisms B and C would likely take the same amount of material to build because their surface areas are the same. Prism C has a greater volume than does Prism B, so it can fit more unit cubes.)
“Which prism—A or B—would take more material to build? Which can fit more unit cubes?” (A and B can fit the same number of unit cubes but, B would require more material to build.)
“If two prisms have the same surface area, would they also have the same volume? How do you know?” (No, Prisms A, B, and C are examples of how two figures with the same volume may not have the same surface area, and vice versa.)
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."
Lesson Synthesis
To highlight some key points from the lesson, display a picture of a prism or a pyramid and a drawing of its net. Discuss these questions:
“Can you find the surface area of a simple prism or pyramid from a picture, if all the necessary measurements are given?”
“Can you find the surface area from a net, if all the measurements are given?”
“Which might be more helpful for calculating surface area—a picture of a polyhedron or a net?” (If the polyhedron is simple—such as a cube or a square pyramid—and does not involve hidden faces with different measurements or require a lot of visualizing, either a picture or a net can work. Otherwise, a net may be more helpful because we can see all of the faces at once and can find the area of each polygon more easily. A net may also help us keep track of our calculations and notice missing or extra areas.)
Student Lesson Summary
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.
Standards Alignment
Building On
4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas and to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
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.
