<p>In this lesson, students learn how to break compound two- and three-dimensional figures into smaller parts and then find the area, volume, and/or surface area of those parts. Students will:<br>- determine the basic shapes and solids that make up compound figures.&nbsp;<br>- calculate the volume of compound figures by determining the volume of the basic shapes and solids that comprise those figures.<br>&nbsp;</p>

<p>- How do spatial relationships, such as shape and dimension, help to create, construct, model, and represent real-world scenarios or solve problems?<br>- How may geometric properties and theorems used to describe, model, and analyze problems?<br>- How may patterns be used to describe mathematical relationships?<br>- How may detecting repetition or regularity assist in solving problems more efficiently?&nbsp;<br>- How may using geometric shape features help with mathematical reasoning and problem solving?&nbsp;</p>

<p>- Circumference: The distance around a circle. (C = 2π<i>r</i>)</p>

<p>- one copy of the Templates sheet (M-7-4-3_Templates) for each student&nbsp;<br>- one copy of the Two Prisms sheet (M-7-4-3_Two Prisms and KEY) for each student&nbsp;<br>- one copy of the Three Prisms sheet (M-7-4-3_Three Prisms and KEY) for each student plus a few extra copies&nbsp;<br>- scissors for each student (unless the shapes on each copy of the Template sheet are cut out prior to the lesson)</p>

<p>- Assess students' comprehension by examining their finished two-dimensional figures from Activity 2.&nbsp;<br>- The Two Prisms exercise results (M-7-4-3_Two Prisms and KEY) will be used to assess student mastery.&nbsp;<br>- Students' performance will be evaluated using the findings from multiple, group-completed copies of the Three Prisms sheet (M-7-4-3_Three Prisms and KEY).<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, and Explicit Instruction<br>W: Students will learn to break down complex two- and three-dimensional figures into smaller, simpler figures. Students will find the areas/volumes of the simpler figures and use that information to calculate the areas/volumes of the complex figures. Students will use various deconstructions of the same complex figure to demonstrate that the area/volume remains constant regardless of how the figure is broken up.&nbsp;<br>H: A simple example will pique students' interest. Students will rapidly learn to use patterns to create their own composite figures. Students will enjoy presenting "puzzles" to their classmates and enjoy solving their classmates' "puzzles" as well.<br>E: Students will get lots of practice with key ideas by creating their own two-dimensional figures and practicing multiple methods of breaking down composite solids. They will experience the ideas in an abstract sense and then in a concrete, hands-on sense.&nbsp;<br>R: Students are given opportunities to improve their thinking through complex examples. Students can rehearse their skills by creating their own figures and experimenting with different methods of solutions for finding the volume of three-dimensional solids.<br>E: Students will evaluate their performance by comparing their answers to those of their classmates in Activity 2. They can also compare their answers in the first part of Activity 3 and collaborate with their classmates in the second part of Activity 3.&nbsp;<br>T: Use the Extension section to customize the lesson to match the needs of the students. The Routine section includes ideas for reviewing lesson concepts throughout the year. The Small Group section provides suggestions for students who may benefit from additional training or practice. The Expansion section is intended for students who are ready for a challenge that goes beyond the requirements of the standard.<br>O: The lesson starts with a common two-dimensional example of a composite figure. Activity 2 begins with students solving each other's simpler composite creations before progressing to more complicated creations. Activity 3 begins with a teacher-led investigation and progresses to individual practice. Students will collaborate on the final part of Activity 3, which expands the concept to a composite solid made up of 3 separate rectangular solids.&nbsp;<br>&nbsp;</p>

<p>Prior to this lesson, students should understand how to calculate the area of triangles and special quadrilaterals. Students should understand how to calculate the volume and surface area of right rectangular prisms, including cubes and triangular prisms.<br><br><strong>Activity 1</strong><br><br>Start the activity by drawing a parallelogram with a base of 12 and a height of 8.</p><figure class="image"><img style="aspect-ratio:266/129;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_43.png" width="266" height="129"></figure><p>Ask students to find the area. (<i>96 square units</i>) Then, draw a vertical line to separate one of the triangle sections at the end:</p><figure class="image"><img style="aspect-ratio:265/130;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_44.png" width="265" height="130"></figure><p><strong>"We may divide our parallelogram into smaller shapes, such as a triangle, by cutting off the end. We can cut off the other end to make another triangle."</strong> Show by separating the opposite end of the parallelogram. <strong>"So in this case, what shapes can we break our parallelogram into?"</strong> (<i>Two triangles and one rectangle</i>)&nbsp;<br><br><strong>"If we knew more about the parallelogram, we could use this method to calculate its area by breaking it down into smaller shapes with areas that are easy to find. For this example, now we already knew how to find the area of a parallelogram, so there's no need to divide it down into smaller shapes. Consider this shape."</strong> Draw the shape shown below on the board.</p><figure class="image"><img style="aspect-ratio:161/342;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_45.png" width="161" height="342"></figure><p><strong>"This is a pentagon, as it has five sides. Do we have a formula for finding the area of a pentagon?"</strong> (<i>No</i>) <strong>"So, as it looks now there is no way to find the area. Can we break this shape into smaller, 'easier' shapes?"</strong> Students should understand that it can be divided into a triangle at the top and a rectangle. If not, suggest that with a single "cut," it can be divided into two basic shapes. Once students have determined how to separate it into rectangles and triangles, draw a line parallel to the base to demonstrate the separation.&nbsp;<br><br><strong>"How can we find the area of the whole pentagon?"</strong> Students should understand that we can calculate the areas of two simpler figures and then add them together.<br><br>Ask students to calculate the area of the rectangle. They should calculate 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 only reaches to the height of the rectangle. On the board, write "Area of Rectangle = 80 sq. units". Explain to students that while calculating the area of compound figures, they should make a habit of writing down the area of each part of the figure and labeling it clearly.<br><br><strong>"Now, how about the area of the triangle that's on top of our rectangle?"</strong> Students may suggest that the area is 30 square units; if so, they have multiplied the base of the triangle by the height but not divided by 2 (multiplied by \(1 \over 2\)). Remind them about the formula for finding the area of a triangle. When at least a few students have determined that the area of the triangle is 15 square units, ask them how they did it. Make sure they explain how they calculated the height of the triangle (by subtracting the height of the rectangle from the height of the entire figure). Also, emphasize that the height makes a right angle with the base, so we don't need to know the lengths of the triangle's two unlabeled parts of the triangle. Write "Area of Triangle = 15 sq. units."&nbsp;<br><br><strong>"So, what is the area of the entire figure?"</strong> (<i>95 square units</i>)&nbsp;<br><br><strong>Activity 2&nbsp;</strong><br><br>Give students a copy of the Templates sheet (M-7-4-3_Templates) and have them to cut out the individual shapes (or give each student a copy of each individual shape if they have already been cut out).<br><br><strong>"You each have 5 basic shapes and know how to calculate the area of each. Your task is to create four compound figures using these shapes as stencils. Trace some of the shapes on a new sheet of paper, placing them side by side to make compound figures. You don't want to 'give away' the shapes you've used, so just trace the outer border of the compound figure without drawing the lines inside to indicate how it can be divided into separate shapes."</strong><br><br>Draw three to five shapes with increasing difficulty and label them as <i>1, 2, 3</i>, and so on, with 1 being the easiest. For the first shape, have them mix just 2 of the stencil shapes. For the second, have them mix 3 or 4 shapes, and for the last, have them combine 5 or 6 (repeating shapes as needed). Students should write their names at the top of each sheet and note the areas of each of their compound figures, as well as list what shapes were used to make it. (For example, a big triangle and a rectangle have a total area of 150 square units.) This will serve as the answer key.&nbsp;<br><br>After each student has drawn at least three shapes, make stacks at the front of the class 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 page of the stack without looking at it and place it face down. Once all students have received a 1, they should turn it over and write their name on the bottom. They will then decide which shapes make up the figure and calculate the total area, writing it on the bottom of the 1 in the same way they did for their own answer keys.&nbsp;<br><br>When students finish a 1, they should keep it at their desk before moving on to a 2 and repeating the process and so on. When students have completed one of each type, they should wait until all of the rest of the figures are done being "decomposed." You can monitor and clarify any confusion.&nbsp;<br><br>After calculating the areas of the compound figures, students should return each sheet to its creator. Students will review their work for each figure. All the completed figures should be gathered and submitted for teacher evaluation.&nbsp;<br><br><strong>Activity 3&nbsp;</strong><br><br><strong>"We can apply the same concepts from the last exercise to three-dimensional figures as well. For example, suppose we need to determine the volume of this figure."</strong> Draw the following figure.</p><figure class="image"><img style="aspect-ratio:294/240;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_46.png" width="294" height="240"></figure><p><strong>"Remember that we know how to calculate the volume of a rectangular prism—a box—but this is not a rectangular prism. Is there some way to divide it up to make two rectangular prisms?"</strong> Students should identify at least one of the two ways it can be divided up. Explain the two ways (as indicated below).</p><figure class="image"><img style="aspect-ratio:496/198;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_47.png" width="496" height="198"></figure><p><strong>"With many solids, there are several ways to divide them up; in most cases, either way will work. Let's examine this solid both ways and make sure we get the same result."</strong> On the left-hand diagram (the one with the vertical division), label the left-most prims A and the right-most prism B.<br><br><strong>"What are the dimensions—length, width, and height—of prism A?"</strong> (<i>4 × 6 × 12</i>) Make sure students recognize 12 as one of the dimensions (rather than 18). <strong>"What is the volume of prism A?"</strong> (<i>288 cubic units</i>). Write "Volume of A = 288 cubic units" on the board, again emphasizing careful labeling.&nbsp;<br><br><strong>"What are the dimensions of prism B?"</strong> This is the most challenging prism in this example. Point out that the 6-unit lengths are the same throughout, and the height is given as 14 units. <strong>"How can we determine the last dimension?"</strong> Help students realize that the missing dimension is 18 - 12 = 6 units. <strong>"So what is the volume of prism B?"</strong> (<i>14 × 6 × 6 = 504 cubic units</i>) Write "Volume of B = 504 cubic units" on the board.<br><br><strong>"So what is the volume of the entire rectangular solid?"</strong> (<i>288 + 504 = 792 cubic units</i>) Write this on the board.&nbsp;<br><br><strong>"Now, if we evaluate the volume by dividing the solid horizontally instead of vertically, should we get the same volume?"</strong> (<i>Yes</i>) Label the lower prism C and the upper prism D.&nbsp;<br><br><strong>"Do we know the dimensions of prism C based on the measurements given in the diagram?"</strong> (<i>Yes, they are 4 × 6 × 18</i>). Have students calculate the volume of C and write "Volume of C = 432 cubic units" on the board.<br><br>Assign students the task of calculating 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.<br><br><strong>"So, to find the total volume, we just sum the two separate volumes and what do we get?"</strong> (<i>792 cubic units</i>) <strong>"We can calculate the volume of this rectangular prism in two different ways. And, actually, we're going to find it in a third way, but not by dividing it into two different prisms. What we're going to do is imagine it's an entire rectangular prism."</strong> Modify the prism to look like this:</p><figure class="image"><img style="aspect-ratio:392/306;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_48.png" width="392" height="306"></figure><p><strong>"Imagine it as a complete prism. What are the dimensions?" </strong>(<i>6 × 18 × 14</i>) <strong>"So what would be its volume if it were a complete prism?"</strong> (<i>1512 cubic units</i>).<br><br><strong>"But, is it a complete prism?"</strong> (<i>No</i>) <strong>"That's correct, because we cut out the dashed part and got rid of it. So, let's calculate the volume of the part we cut out and got rid of. What are the dimensions?"</strong> (<i>6 × 12 × 10</i>) <strong>"So what is the volume of the part we got rid of?"</strong> (<i>720 cubic units</i>)&nbsp;<br><br><strong>"How should we find the volume of the part of the prism we're interested in?"</strong> (Subtract.) Explain that we are subtracting because we are removing part of the prism. <strong>"So, we find the difference of 1,512 and 720, which is…"</strong> (<i>792 cubic units</i>)<br><br><strong>"So there are actually three different ways to calculate the volume of the prism. Does it make a difference which method you choose?"</strong> (<i>No</i>) <strong>"Sometimes a particular method is easier than another, but ultimately, it's up to you which way you choose. Choose the option that makes the most sense to you."</strong>&nbsp;<br><br>Give each student a copy of the Two Prisms worksheet (M-7-4-3_Two Prisms and KEY). Students should work individually. Each student should come up with two different ways to calculate the volume of the solid presented. Students should indicate the method they used (by drawing dashed lines) and calculate the volume using each method. Remind students that they should get the same volume with each method.&nbsp;<br><br>While students are working, recreate the diagram on the board to use in explanation.<br><br>After students have completed, ask one of them to come up and explain how he or she found the volume. Then, have another student who used a different method come up and describe how he or she determined the volume. Finally, have a student who used the third method come forward and explain how he or she determined the volume. (Review any methods that were not chosen by the student.)<br><br>Give each student a copy of the Three Prisms worksheet (M-7-4-3_Three Prisms and KEY). Assign students to work in pairs or trios. Each group should have three copies of the worksheet, while pairs will need an additional copy.<br><br><strong>"Work together, using one copy of the worksheet, calculate the volume of the solid displayed. You will need to separate it into more than two prisms. Once you've determined the volume of the solid one way, try again on an additional copy of the worksheet to see if you can divide the solid differently and calculate the volume another method. On a third copy, try to calculate the volume by imagining the prism as a complete rectangular solid and then subtracting the volumes of the missing parts."&nbsp;</strong><br><br>[Note: calculating the volume using subtraction is more complicated.]<br><br>Students should turn in their completed Two Prisms and Three Prisms sheets and compare them to their corresponding keys for evaluation.<br><br><strong>Extension:</strong><br><br>Use the strategies listed below to adjust the lesson to your students' needs throughout the year.&nbsp;<br><br><strong>Routine:</strong> Providing students with varied shapes throughout the year, even if only to draw lines demonstrating how the shape can be broken down into simpler shapes, will help students in developing their "instinct" to see how shapes can be decomposed into simpler figures.&nbsp;<br><br><strong>Small Group:</strong> Students who could benefit from more practice can be divided into smaller groups. Each student can create a complex rectangular prism and label the side lengths. Have students trade papers to determine the volume of each other's complex prisms. Make sure there is enough information (i.e., the dimensions) to calculate the volume of such solids. Students can practice and learn more about finding the volume of complex rectangular prisms by visiting the following Web site:&nbsp;<br><a href="http://learnzillion.com/lessons/1809-find-the-volume">http://learnzillion.com/lessons/1809-find-the-volume</a>.&nbsp;<br><br><strong>Expansion:</strong> Students who are ready for a challenge beyond the requirements of the standards may use blocks or design their own stencils to make different / more complicated figures, and then calculate the volume.&nbsp;<br>Students can experiment with creating convex figures by beginning with a larger figure (such as a large rectangle) and removing other figures from the rectangle's area, requiring the use of subtraction rather than addition to calculate the figure's area. Allow students to sketch the figure, indicating the removed portion.<br><br>The three-dimensional exercises can be expanded by incorporating formulas for spheres, cylinders, pyramids, and cones.<br><br>Students may take the challenging quiz at the following Web site: <a href="http://www.sophia.org/finding-the-volume-of-odd-solids-with-composite-figures/finding-the-volume-of-odd-solids-with-composite-fi--5-tutorial">http://www.sophia.org/finding-the-volume-of-odd-solids-with-composite-figures/finding-the-volume-of-odd-solids-with-composite-fi--5-tutorial</a>&nbsp;</p>