• Evaluate students’ comprehension using their completed two-dimensional figures from Activity 2.
• Student mastery will be gauged using the outcome of the Two Prisms exercise (M-7-4-3_Two Prisms and KEY.docx).
• Students will be assessed using their results from the multiple, group-completed copies of the Three Prisms sheet (M-7-4-3_Three Prisms and KEY.docx).

Prior to this lesson, students should be familiar with how to find the area of triangles and special quadrilaterals (M06.C-G.1.1.1). Students should also be familiar with how to find the volume and surface area of right rectangular prisms (including cubes) and triangular prisms
(M06-C-G.1.1.6).

Activity 1

Begin the activity by drawing a parallelogram with a base of 12 and a height of 8.

Ask students to find the area. (96 square units) Then draw a vertical line to separate one of the triangular portions at the end:

“It’s possible to break our parallelogram up into smaller shapes—for instance, we can cut off the end to make a triangle. We can cut off the other end to make another triangle.” Demonstrate by separating the other end of the parallelogram. “So in this case, what shapes can we break our parallelogram into?” (Two triangles and a rectangle)

“If we knew more about the parallelogram, we could find the area of it using this method—breaking it down into smaller shapes with areas that are easy to find. Now, for this example, we already knew how to find the area of a parallelogram, so there isn’t much need to break it down into smaller shapes. Consider this shape.” Draw the shape shown below on the board.

“This is a pentagon, since it has five sides. Do we have a formula to find the area of a pentagon?” (No) “So, as it looks now, there’s no way to find the area. Can we break this shape down into smaller, ‘easier’ shapes?” Students should recognize that it can be broken down into a triangle on the top and a rectangle. If not, hint that with one “cut,” it can be made into two basic shapes. Once students have identified how to separate it into a rectangle and triangle, draw the line parallel to the base to show the separation.

“How can we find the area of the whole pentagon?” Students should recognize that we can find the areas of the two simpler figures and then add them together.

Ask students to find the area of the rectangle. They should determine that it is 80 square units. If they suggest 110 square units, point out that the 22-unit label goes the entire height of the whole figure, whereas the 16-unit label just extends to the height of the rectangle. Write “Area of Rectangle = 80 sq. units” on the board. Explain to students that when finding the area of compound figures, they should get in the habit of writing down the area of each part of the figure and labeling it clearly.

“Now, how about the area of the triangle that’s on top of our rectangle?” Students may suggest that the area is 30 square units; if that’s the case, they’ve multiplied the base of the triangle by the height but not divided by 2 (multiplied by ). Remind them of the formula to find the area of a triangle. When at least some students have determined that the area of the triangle is 15 square units, ask some of them how they determined it. Make sure to have them elaborate on how they determined the height of the triangle (i.e., subtracting the height of the rectangle from the height of the entire figure). Also, reinforce that the height makes a right angle with the base, so we don’t need to know the lengths of the two unlabeled parts of the triangle. Write “Area of Triangle = 15 sq. units.”

“So, what is the area of the entire figure?” (95 square units)

Activity 2

Give students a copy of the Templates sheet (M-7-4-3_Templates.docx) and have them cut out the individual shapes (or give each student a copy of each individual shape if the shapes have been cut out ahead of time).

“You each have 5 basic shapes, and you know how to find the area of each. Your task is to make four compound figures using these shapes as stencils. On a new sheet of paper, trace some of the shapes, placed side-by-side, to make compound figures. You don’t want to ‘give away’ which shapes you’ve used, so just trace the outside edge of the compound figure—don’t draw the lines inside to show how it can be divided into separate shapes.”

Have students draw three to five shapes of increasing difficulty, labeling them 1, 2, 3, and so on¸ with 1 being the easiest. For the first shape, have them combine just 2 of the stencil shapes. For the second, have them combine 3 or 4 shapes, and for the last, have them combine 5 or 6 (repeating shapes if necessary). Students should put their name on the top of each sheet and they are responsible for recording the areas of each of their compound figures and listing what shapes were used to make it. (For example, 1 big triangle and 1 rectangle: total area = 150 square units.) This will serve as an answer key.

Once every student has drawn at least three shapes, create stacks at the front of the room for each number of diagram (i.e., a stack of 1s, a stack of 2s, etc.) Then pick up the stack of 1s, mix them up, and have each student take the top sheet of the stack without looking at it and put it facedown. Once all students have a 1, they should flip it over and write their name on the bottom. They will then determine which shapes make up the figure and find the total area, writing it on the bottom of the 1 in the same way they recorded their own answer keys.

When students complete a 1, they should keep it at their desk and then get a 2 and do the same thing and so on. When students have completed one of each type, they should wait until all the rest of the figures are done being “decomposed.” You can monitor and clarify any confusion.

Once the areas of the compound figures have been calculated, students should return each sheet to its creator. Students will check the work for each of their figures. All the completed figures should be collected and turned in for teacher evaluation.

Activity 3

“We can apply the same ideas that we used in the previous activity to three-dimensional figures as well. For instance, suppose we need to find the volume of this figure.” Sketch the following figure.

“Remember, we know how to find the volume of a rectangular prism—a box—but this isn’t a rectangular prism. Is there some way we can divide it up to make two rectangular prisms?” Students should identify at least one of the two ways it can be divided up. Illustrate the two ways (as shown below).

“With many solids, there are multiple ways to divide them up; in most cases, either way will work. Let’s examine this solid using both ways and make sure we get the same answer in either case.” On the left-hand diagram (the one with the vertical division), label the left-most prism A and the right-most prism B.

“What are the dimensions—length, width, and height—of prism A?” (4 × 6 × 12) Make sure students identify 12 as one of the dimensions (as opposed to 18). “What is the volume of prism A?” (288 cubic units) Write “Volume of A = 288 cubic units” on the board, again emphasizing careful labeling.

“What are the dimensions of prism B?” This is the more difficult prism in this example. Point out that the 6-unit lengths are the same throughout, and the height is given as 14 units. “How can we determine the last dimension?” Guide students toward the realization that the missing dimension is 18 – 12 = 6 units. “So what is the volume of prism B?” (14 × 6 × 6 = 504 cubic units) Write “Volume of B = 504 cubic units” on the board.

“So what is the volume of the entire rectangular solid?” (288 + 504 = 792 cubic units) Write this on the board.

“Now, if we evaluate the volume by dividing the solid horizontally instead of vertically, should we get the same volume?” (Yes) Label the lower prism C and the upper prism D.

“Do we know the dimensions of prism C based on the measurements given in the diagram?” (Yes; they are 4 × 6 × 18.) Have students determine the volume of C and write “Volume of C = 432 cubic units” on the board.

Guide students towards determining the dimensions of prism D (6 × 6 × 10; the height is 14 – 4) and have them find the volume of D. Write “Volume of D = 360 cubic units” on the board.

“So, to find the total volume, we just sum the two separate volumes and what do we get?” (792 cubic units) “So, we can find the volume of this rectangular prism two different ways. And, actually, we’re going to find it a third way, although not by dividing it up into two different prisms. What we’re going to do is imagine that it is an entire rectangular prism.” Modify the prism so it looks like this:

“So, imagine it is a complete prism. What are its dimensions?” (6 × 18 × 14) “So what would be its volume if it were a complete prism?” (1512 cubic units)

“But, is it a complete prism?” (No) “Right, because we cut out the dashed part and got rid of it. So let’s find the volume of the part we cut out and got rid of. What are its dimensions?” (6 × 12 × 10) “So what is the volume of the part we got rid of?” (720 cubic units)

“How should we find the volume of the part of the prism we’re interested in?” (Subtract.) Explain that here, we’re subtracting because we are getting rid of part of the prism. “So, we find the difference of 1,512 and 720, which is…” (792 cubic units)

“So there are actually three different ways to find the volume of the prism. Does it make a difference which method you choose?” (No) “Sometimes, a particular method will be easier, but ultimately, it’s up to you which way you choose. Pick the way that makes the most sense to you.”

Give each student a copy of the Two Prisms sheet (M-7-4-3_Two Prisms and KEY.docx). Have students work individually. Each student should come up with two different ways to find the volume of the solid shown. Students should indicate which method they used (by drawing dashed lines) and find the volume using each chosen method. Remind students that they should get the same volume by each method.

While students are working, recreate the diagram on the board for use in explanation.

After students have finished, have one student come up and explain how s/he found the volume. Then have another student who used a different method come up and explain how s/he found the volume. Finally, have a student who used the third method come up and explain how s/he found the volume. (Go over any of the methods that weren’t chosen by a student.)

Give each student a copy of the Three Prisms sheet (M-7-4-3_Three Prisms and KEY.docx). Have students work in pairs or trios. Each group should have three copies of the worksheet; pairs will need an extra copy.

“Work together, using one copy of the worksheet, to find the volume of the solid shown. You will need to divide it up into more than two prisms. Once you’ve found the volume of the solid one way, on another copy of the worksheet, see if you can divide the solid up differently and find the volume a second way. On a third copy, see if you can find the volume by imagining the prism is a complete rectangular solid and then subtracting the volumes of the missing parts.”

[Note: determining the volume using subtraction is more difficult.]

Have students turn in their completed Two Prisms and Three Prisms sheets and compare these to their corresponding keys for evaluation.

Extension:

Use the following strategies to tailor the lesson to meet the needs of your students throughout the year.

• Routine: Providing students with various shapes throughout the year, even if only to draw lines showing how the shape could be broken into simpler shapes, will help students continue to develop their “instinct” to see how shapes can be decomposed into simpler figures.
• Small Group: Students who may benefit from additional practice can be pulled into small groups. Each student can draw a complex rectangular prism and label side lengths. Have students trade papers to find the volume of each others’ complex prisms. Make sure there is sufficient information (i.e., the dimensions) to be able to determine the volume of such solids. For additional practice and instruction, students may use the following Web site:

http://learnzillion.com/lessons/1809-find-the-volume-of-complex-rectangular-prisms

• Expansion: Students who are ready for a challenge beyond the requirements of the standard may use blocks or create their own stencils to make different / more complex figures, and then calculate the volume.

Students can experiment with making convex figures by starting with a larger figure (a large rectangle, for example), and removing other figures from the area of the rectangle, requiring the use of subtraction rather than addition to find the area of the figure. Have students sketch the figure, indicating the removed portion.

The three-dimensional activities can be expanded through the introduction of formulas for spheres, cylinders, pyramids, and cones.

Or students may take the challenging quiz at the following Web site:

http://www.sophia.org/finding-the-volume-of-odd-solids-with-composite-figures/finding-the-volume-of-odd-solids-with-composite-fi--5-tutorial