 # How to Teach Volume of a Right Rectangular Prism ## Manipulatives and Volume: Volume of a Right Rectangular Prism

I have noticed that students often leave out the unit label when doing word problems or measuring.  For the longest time, I didn’t understand why this was the case. However, one day I had an epiphany.  My students didn’t understand what the label actually meant!  When teaching Volume of a Right Rectangular Prism, we draw 3D figures on the board and students see them in their math books.  We try to help them connect labels to what they see,  but that’s often not concrete enough to really help students understand a unit like a square centimeter.

## How I Tested My Hypothesis

To test my hypothesis that they don’t understand what a square centimeter is, I held up a single blue centimeter square place value block.  When I asked them if they knew what it was, I got every answer in the book other than a cubic centimeter.  “It’s a blue box.”  “A cube.” When I told them it is a cubic centimeter there was a chorus of, “Ohhhhhhh, that’s what you’re talking about?”

Yes.  This is the unit label we have been talking about for days.  Now, I keep a set of place value blocks out at all times when exploring volume.  It’s an easy reference to give students a concrete understanding of what they are measuring.  This hands-on lesson for exploring volume is what I use to make sure students start out with a concrete understanding of a cubic centimeter and can visualize what they’re actually measuring when they use the formula for volume.

## Set up for the Lesson: Volume of a Right Rectangular Prism

• Each student needs their math notebook or a piece of paper
• A pack of six small, 2D squares for every 2 students
• Place value blocks separated into containers for individual use or larger containers for groups to share.
• A small rectangular prism brought in by students – for example, a toothpaste box, raisin box, etc.
• One copy of the Volume Worksheet for each student

## How to Launch the Lesson

Before you begin, review with students how to find the area of a rectangle.  Draw a 3 ft. x2 ft.  rectangle on the board. Write 3 ft. x 2 ft underneath.  This will obviously not be to scale.  Walk through what area means; the number of squares that cover a given 2D space.  Hand out packs of six small, 2D squares to each set of partners. You can also have the squares on their desk when they enter for class to save time.  Ask students how you can figure out how much carpet you need to cover this space and ask them to show you how they can cover the space with their squares.  The students should arrange their squares so that there are three rows of two squares.  On the board, you can have precut squares with tape on the back so you can model the arrangement of squares.

Have them turn to the neighbor and explain their thinking.  Why did they arrange their squares the way they did?  Are there other ways to arrange them? Do they mean the same thing?

Ask the students, What is the formula we use to find the area?  Write it on the board.  Ask the students why that works. They may say it’s multiplication. Ask the students to explain.  If you look at your arrangement,it is repeated addition, in this case.  Know that irregular shapes will not always follow this rule.  You would need to break it down into more than one section to get to this rule.

## The Lesson: Volume of a Right Rectangular Prism

Now hold up a single place value block next to one of the flat squares used above.  Ask students how they are different.  Have them write their answers in their notebooks.  Let them know that the place value blocks in front of them are for their use, and that they may need to share.  If they need more blocks they should raise their hand and wait quietly for you to call on them.  You may also want to set up a station where students can access more cubes on their own.

### Split Up the Students

Split the students into groups of four.  Each student should have their own Volume of a Right Rectangular Prism Worksheet.  This ensures that they all have the opportunity to do the work, but can help each other if needed.  They should now take out the rectangular prism they brought from home.  I usually tell them to bring a  box that is smaller than a Rubik’s Cube.  Good choices for this exercise are:

• raisin boxes
• toothpaste boxes

You should also have some extras on hand for students who do not bring one into class.  You do not want to leave a student out of a learning activity because they do not have the materials at home.  About a week before I do this activity I will send an email to all faculty asking them to drop some boxes off in my room, if they have them.  I get all kinds of boxes and usually more than I need for my classes.

### Share Their Boxes

Have students share their box with their group.  Each person in the group should estimate how many cubic centimeter blocks it will take to fill each box in the group.  This gives them the chance to reflect on what a cubic centimeter actually is and how much space it takes up.  They can share their estimates and explain their reasoning.

### Fill the Right Rectangular Prisms

Next, have each student use their blocks to fill the rectangular prism and find the actual number that fills the space. Write it on their worksheet.  If they can manage, have them try to remove the rectangular prism so that blocks are left intact.  This way they can see the multiple levels of their blocks.  This last step will help them see the third dimension of volume, which is height. The last section on the worksheet asks them to find the formula for volume and use it to find the volume of their shape. Many will be able to come up with the formula and some may not.  As you circulate, ask guiding questions to get them thinking about what elements need to be in the formula.  Some you may ask are:

• Do you see the area formula here?
• What do you see that is not part of the area formula?
• How is your arrangement of blocks different from the squares on your desk?

## Share

Then have students share out the formulas they designed and explain how they chose which elements should be included in the formula. Help students to focus on the difference between area and volume.  The difference here is that you are moving from 2D to 3D.  You transition from a flat shape to a shape that exists in three planes, instead of two.  This is why it is a cube and why we measure it with cubic centimeters.  There are three sides we measure and multiply – length x width x height.  Area is length x width.  In volume that third plane of the cube, height, is multiplied.  That is why we use a superscript of three on the abbreviation cm.  Believe it or not, this is an aha moment for many students.  We often assume this is understood by students, when it is not.

## Length x Width x Height

Once students are clear that the formula is length x width x height, have them jot it on their worksheet and then solve for the area of their rectangular prism, counting out the number of blocks for the length, width, and height. Once they solve, they should check that their answer matches the volume they got when they counted the total number of cubes they packed into their rectangular prism.

## Reflecting on the Activity: Volume of a Right Rectangular Prism

It is interesting talking about this experience with students.  They are often surprised by how many blocks it takes to fill a space. You will want to talk about their estimates and how close they were to guessing the number of blocks. Of course, some students have great spatial reasoning skills and will be super close in their estimates, others not so much.  I always let students know that this is not my strong suit. I often find my estimating to be way off, but it’s something that can improve with practice.

It is also a great opportunity to help students be metacognitive about their learning. Talk to students about how much deeper their understanding is of the volume formula after doing this hands on work with the cubes. The cubes helped them to visualize what they were actually measuring when plugging numbers into the formula. Encourage students to be aware of the importance of being able to visualize when working with formulas.

## Extensions/Next Steps

There are more unit measurements than cubic centimeters.  You could have a few prepared cubic foot cubes or cubic yard cubes to share with students. We often use cubic centimeters or cubic inches in the classroom because they are manageable. However, we buy large amounts of soil, mulch or compost in cubic feet or yards. Calculator Soup is a great resource to extend your students’ thinking or have them work on a problem based project for a garden.

## Resources:

https://www.calculatorsoup.com/calculators/construction/cubic-yards-calculator.php

https://www.calculatorsoup.com/calculators/construction/cubic-yards-calculator.php

## FREE Volume of a Right Rectangular Prism Worksheets and Resources

These are all PDF Files. They will open and print easily. The Student Edition Files are labeled SE and the Teacher Editions Files are labeled TE. Click the links below to download the different resources.

## Volume of a Right Rectangular Prism Worksheets and Resources

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## This Lesson is From Unit 5 – Multiplying and Dividing Fractions

5-1  Fractions as Division

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5-2  Multiplying Fractions and Whole Number

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5-3  Multiplying Fractions

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5-4  Interpreting Multiplication

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CCSS.MATH.CONTENT.5.NF.B.5.A

CCSS.MATH.CONTENT.5.NF.B.5.B

5-5  Multiplying Mixed Fractions

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CCSS.MATH.CONTENT.5.NF.B.4.A

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5-6  Dividing Unit Fractions by Whole Numbers

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5-7  Dividing Whole Numbers by Unit Fractions

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CCSS.MATH.CONTENT.5.NF.B.7.A

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5-8 Fractions in Real World (Multiplication and Division)

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## Don’t Forget to Pin this Lesson on Teaching the Volume of a Right Rectangular Prism… 