Lesson 26

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), and select students who correctly name each prism to share during the whole-group discussion.  

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 each net with a partner. 

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 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.

This activity reviews the work students did previously drawing 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 measure 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.

Provide access to geometry toolkits and compasses.

Students may try to multiply two side lengths of the triangle to calculate the area. Remind them that the height must be perpendicular to the base. If necessary, demonstrate using an index card to draw in the height of the triangle.

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 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 students have calculated the area of their selected triangle correctly, because this will affect their volume and surface area calculations in the next activity.

In this activity, students take the triangle they selected in the previous activity and use it as the base of their triangular prism. 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.

Demonstrate the positions 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 \(A\) and \(C\). 
• 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.

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.

Students combine their solid with a partner’s and examine the new solid’s properties.

If there is time, this activity can be extended to review relationships between angles as well.

• Take two of the prisms and put them together so that their \(45^\circ\) 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.

Arrange students in groups of 2. 

Student might struggle trying to measure and calculate the area of the base of their new prism if it is an irregular quadrilateral. Prompt them that they can just add the areas of the bases of their individual prisms together.

Discuss 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 various ways.

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?