Volume: Decomposing Irregular Shapes

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SWBAT find the volume of irregular solids using the additive volumes of the decomposed shape.

Big Idea

Students use what they know (volume of rectangular prisms) to solve problems with irregular solids.


10 minutes

Students are given the dimensions of 2 different prisms.  Prism A and Prism B.  They use these dimensions to build each of the prisms and find the volume of each.

This is a variation of the launch I have used for the past 2 lessons.  Here, they are given the dimensions and asked to build the prism.  Previously they were given the volume and then asked to find the dimensions.  These variations help students develop a conceptual understanding because they are approaching the problem from different angles.  

After students build these prisms, I ask:

• What is the volume of prism A?  How do you know?

• What is the volume of prism B? How do you know?

When students answer, I emphasize the significance of using labels (cubic units).  I leverage their explanation to call attention to the formula for volume, as well as the benefit of counting the total cubes to check their thinking.

Next, students are instructed to slide both prisms together to make a new shape "C".  As a class, we discuss the attributes of this new shape to determine if it is a rectangular prism or not (it is not one singular prism).  

Students are reminded that we have learned how to find the volume of rectangular prisms, but not of other 3D shapes.  Then, immediately after, they are asked to tell me the volume of shape C.

Students turn and talk to determine if it is possible, using what they know at this time.  If they think it is possible, I ask them to attempt to find the volume. 

The students determine that the volume can be determined because we know the volume of the first prism and the volume of the second, so we can add these together. 

This launch is used to get students thinking about decomposing shapes to make finding the volume possible. In this lesson, students will use the mathematical practice (MP7) Look for and make use of structure.  Within these complex, irregular shapes, students need to find the rectangular prisms that create the structure of the shape.  Then, decompose the shape into smaller, manageable parts.  They must use what they know, to solve something they haven't learned yet.

Independent Discovery

20 minutes

Using the experience of combining two volumes to find the volume of an irregular solid from the launch, students then move on to solving problems of the opposite nature.  For the independent discovery, students work in pairs to decompose solids into two rectangular prisms in order to find the volume of the solid. 

I choose to have students work on this independently, rather than modeling it first.  This discovery time provides additional experiences that help students develop a stronger conceptual understanding. At this point, they have knowledge of how to solve these problems, but must grapple with choosing the right strategy and organizing their thoughts.  This tension of knowing how to find volume and decompose complex problems vs. finding the volume of these solids forces the students to push their thinking. 

Note: I thought these two problems would take the students just a few minutes (it would have if I'd modeled).  Although they were on task, it took much longer.  This shows me that they are making discoveries on their own.  While listening to the students' discussion, I found most of these discoveries focus on interpreting the drawing. I allow them to work through this as long as they need to, rather than pull them back together about five minutes in to model the rest of the problem.

Additional Practice

20 minutes

Students continue to solve problems involving the volume of solid figures.  They work in pairs and apply the strategy of decomposing the problem into rectangular prisms, finding the volumes, and then combining them. 

Interpreting the image and determining the dimensions is a challenge for many students.  Now that they have had the opportunity to make some discoveries on their own, I can see more guided practice is needed to help these students better understand diagrams when the shapes are irregular. 

To summarize the lesson, all students meet on the carpet, in front of the board, and we solve one example together. Prior to solving it, students turn and talk about the strategy they will use.