The purpose of this Warm-up is for students to reason about prisms formed from various nets. During the partner and whole-group discussions, listen for how students name each prism: pentagonal prism, triangular prism, square prism (but not a cube). Select students who correctly name each prism, and ask them to share during the whole-class discussion.
Launch
Arrange students in groups of 2. Give students 30 seconds of quiet think time to look at the nets, followed by 2 minutes to describe, with a partner, each net.
Activity
None
Here are some nets for various prisms.
What would each net look like when folded?
What do you notice about the nets?
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is to identify the prisms from their nets and to share key features of nets of prisms. Ask selected students to describe the object formed by each net. Record and display their responses for all to see. If a student's description does not include the name of the prism, ask other students to name the object and to explain how they know.
Ask students to share what they notice about all of the nets. Record and display their responses for all to see. While students may notice many things, important ideas to highlight during the discussion are:
They all have a long rectangle in the middle.
The bases on top and bottom are upside down.
There’s one base on each side of the rectangle.
22.2
Activity
Standards Alignment
Building On
Addressing
7.G.A.2
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
This activity reviews the work that students did previously when they drew shapes with given conditions. Students draw as many different triangles as they can that could be the base of the triangular prism, given two side lengths and one angle measurement for the triangle.
In preparation for calculating surface area and volume in the next activity, students select one of their triangles and find its area. This will require them to draw and measure the height of the triangle. As needed, remind students that the height must be perpendicular to whichever side they are using as the base of their triangle. Also, prompt students to measure the height as precisely as possible, because it will influence the accuracy of their later calculations.
Launch
Provide access to geometry toolkits and compasses.
Activity
None
The base of a triangular prism has one side that is 7 cm long, one side that is 5.5 cm long, and one angle that measures .
Draw as many different triangles as you can with these given measurements.
Select one of the triangles you have drawn. Measure and calculate to approximate its area. Explain or show your reasoning.
Student Response
Activity Synthesis
The goal is to show the four possible triangles and make sure each student has calculated the area of their triangle correctly because this will affect their calculations in the next activity.
Ask students to share triangles they drew so that everyone has an opportunity to see all four triangles. If any of the four triangles are not presented by students, demonstrate how to construct it. Ensure that the class agrees that 4 unique triangles have the given measurements. The third side length of the triangle could be 5.0 cm, 7.3 cm, 2.6 cm, or 9.7 cm.
Make sure that students have calculated the area of their selected triangle correctly, because this will affect their volume and surface area calculations in the next activity.
If the third side of the triangle is
then the area of the triangle should be about
possible strategies
5.0 cm
13.7 cm2
or
7.3 cm
18.2 cm2
or or
2.6 cm
6.3 cm2
or
or
9.7
18.9 cm2
or
or
22.3
Activity
Standards Alignment
Building On
Addressing
7.G.B.6
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
In this activity, students take the triangle they selected in the previous activity and use it as the base of their triangular prism.
Launch
Demonstrate the positions at which triangles should attach to the rectangle to form a net, by displaying an example and describing the important parts:
The triangles must have a vertex at and .
The triangles must be identical copies with one “upside down” from the other.
Corresponding sides of each triangle must be along the side of the rectangle.
After students have drawn their net and before they cut it out and assemble it, make sure they have correctly positioned their bases—opposite from each other on the top and bottom of the rectangle and reflected. It will also make assembling the net easier for students if they draw lines subdividing the large rectangle into the individual rectangular faces and draw tabs where the faces will be glued or taped together.
Activity
None
Your teacher will give you an incomplete net. Follow these instructions to complete the net and assemble the triangular prism:
Draw an identical copy of the triangle you selected in the previous activity along the top of the rectangle, with one vertex on point .
Draw another copy of your triangle, flipped upside down, along the bottom of the rectangle, with one vertex on point .
Determine how long the rectangle needs to be to wrap all the way around your triangular bases. Pause here so your teacher can review your work.
Cut out and assemble your net.
After you finish assembling your triangular prism, answer these questions. Explain or show your reasoning.
What is the volume of your prism?
What is the surface area of your prism?
Stand your prism up so it is sitting on its triangular base.
If you were to cut your prism in half horizontally, what shape would the cross-section be?
If you were to cut your prism in half vertically, what shape would the cross-section be?
Student Response
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Building on Student Thinking
Activity Synthesis
The goal is to make sure that students understand how the changes in the base triangle affect the prism’s surface area and volume. Select students to share their answers for the cross-sections. For cross-sections taken in these two ways, all triangular prisms should have the same shapes as answers, although the actual size of the cross-section will differ based on the size of the base triangle and the height of the prism.
The volume and surface areas will depend on the triangle they have chosen to use as their base.
Select students to share their methods for computing volume and surface area. The base area is important in the calculation of each, so students should use the values they computed in the previous activity.
22.4
Activity
Optional
Standards Alignment
Building On
Addressing
7.G.B.6
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Students combine their solid with a partner’s and examine the new solid’s properties.
Launch
Arrange students in groups of 2.
Engagement: Develop Effort and Persistence. Provide prompts or checklists that focus on increasing the length of on-task orientation in the face of distractions. For example, provide students with the printed Student Task Statement to use as a checklist for task completion. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
None
Compare your prism with your partner’s prism. What is the same? What is different?
Find a way you can put your prism and your partner’s prism together to make one new, larger prism. Describe your new prism.
Draw the base of your new prism and label the lengths of the sides.
As you answer these questions about your new prism, look for ways that you can use your calculations from the previous activity to help you. Explain or show your reasoning.
What is the area of its base?
What is its height?
What is its volume?
What is its surface area?
Student Response
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Building on Student Thinking
If students struggle with calculating the area of the base of the new prism, consider asking:
“What shapes is your new base made of?”
“How can you use what you know about the original bases to help you calculate the area of the new base?”
Activity Synthesis
The purpose of this discussion is to explore which attributes of the larger prisms were easiest to determine based on the original prisms and which were hardest.
Using two prisms that are identical, demonstrate putting them together against a matching side in various ways.
For each configuration, discuss with students:
“How does the area of the new prism compare with the area of the original prisms?”
“How does the height of the new prism compare with the height of the original prisms?”
“How does the volume of the new prism compare with the volume of the original prisms?”
“How does the surface area of the new prism compare with the surface area of the original prisms?”
In the first configuration,
The area of the base is the same as in the original prism.
The height is twice the height of the original prism.
The volume is twice the volume of the original prism.
The surface area is less than twice the surface area of the original prism (because of the sides that are put together).
In the second and third configurations,
The area of the base is twice the original.
The height is the same as the original.
The volume is twice the original.
The surface area is less than twice the original.
How could you put these two prisms together to make the largest surface area possible for the new prism? The smallest surface area possible?
If time allows, this activity can be extended to review relationships between angles as well.
Take two of the prisms and put them together so that their 45 angles are adjacent.
Ask students what type of angle is created and what the relationship between the two angles creating that angle must be. (a right angle, complementary angles)
Take three identical prisms and put them together so that a different angle from each prism is adjacent. Ask students what type of angle is created. (a straight angle). Ask students to identify pairs of angles that are supplementary.
MLR8 Discussion Supports. Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement “Putting the small sides together makes the biggest prism” as “Putting the triangular faces together makes the longest prism.” Advances: Speaking, Representing
Lesson Synthesis
Ask students to reflect on what they have learned in this unit, either in writing or by talking to a partner. Here are some suggested prompts:
“What is something you learned in this unit that surprised you?”
“What is a new mathematical word you learned in this unit, and what does it mean?”
“What is an idea that you learned about in this unit that is useful in the real world?”
“Describe something that you were confused about at first, but that you understand now.”
“Describe something that you found challenging, but that you understood with some effort.”
“What is something from this unit that you are still wondering about?”
“What was your favorite activity, and what did you learn from it?”
Standards Alignment
Building On
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.
