Volume as Multiplication

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The students will understand volume as a multiplication of layers.

Big Idea

Layer it up!


15 minutes

In this lesson students will be using their knowledge from investigating volume thus far in order to view volume as property of multiplication of layers.  This will need us to the multiplicative rule of volume.  They will be using the one of the box dimensions in the previous lesson’s investigation to look at the prism in layers.  The lesson is set up in a similar way to the previous lesson of examining volume as addition of layers.

To begin this lesson I have a Number Talk with students.  I pose a simple problem to the students and once we discuss the first problem, we move on to the next step.  I want the students to visualize how multiplication can be broken down into pieces.  The topics for the Number Talk are as follows.

6 x 4

(6 x 2) + (6 x 2)

6 + 6 + 6 + 6


30 minutes

For this part of the lesson I invite the students to self-discover how to look at volume as multiplication of layers.  Each partnership recreates the 6 x 3 x 4 prism from the previous lesson using base ten cubes.  I then pose a simple problem to the students.

We need to create a number a different number sentence for determining the volume of this prism.  I would like you to use the multiplication symbol and the addition symbol in one side of your sentence.  There may be more than one way to write this new sentence.  See what you can come up with as you work with your model.  Feel free to discuss with your group as you investigate.  As you investigate look for connections between the addition sentences we created yesterday and multiplication sentences you are creating today.  Make quick sketches of your ideas.

The goal is that students are able to see that the volume of the prism can represented in three distinct ways using multiplication in layers.  This will help lead them to the self-discovery of the l x w x h formula associated with volume.

(6 x 4) + (6 x 4) + (6 x 4) = 72                                             (24 + 24 + 24 = 72)

(4 x 3) + (4 x 3) + (4 x 3) + (4 x 3) + (4 x 3) + (4 x 3)  = 72      (12 + 12+ 12+ 12+ 12+ 12 = 72)

(6 x 3) + (6 x 3) + (6 x 3) + (6 x 3)  = 72                                (18 + 18+ 18+ 18 = 72)


15 minutes

To wrap up this lesson I facilitate a student led whole group discussion around the conclusions of our investigation.  I ask students to think about what they discovered in the lesson and then let the students share and continue the conversation as a class.

I close this activity by having students draw the 6 x 3 x 4 prism three times on isometric dot paper.  I then have them label the three ways they divided the shape into multiplication layers.  For each layer I have them use a different color so the layers can be seen easily.