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How to help every student understand volume in elementary math

How to let every 5th grader find out why length × width × height works, with a 3D box they fill themselves and a question after every wrong count.

Jemel Fanfan

Jemel Fanfan | Founding Member at Flint

Title card reading 'How to help every student understand volume in elementary math', next to three cards from a fish tank volume activity: a student finding 12 cubes in one layer, the same student counting only the 47 cubes he can see on the outside, and his explanation that 5 layers of 12 is 4 × 3 × 5 = 60 cubic units.

Why this matters

In one 5th grade class, most students can recite length × width × height and can't say why it gives the number of cubes in a box. A student who looks at a drawing of a box counts the cubes on the front, the side and the top, adds them up and gets 47 instead of 60. The teacher has to wait for the quiz to find out who understands volume and who memorized a formula.

What understanding volume looks like

Volume is the number of unit cubes that fill a solid with no gaps. 5th graders find it by packing a box with cubes, then see that one layer holds length × width cubes and the box holds that many layers. Only then do they use length × width × height, and they finish by adding the volumes of two boxes joined together (CCSS 5.MD.C.3 to 5.MD.C.5).

  1. Let the student fill the box one layer at a time
  2. Ask why the multiplication works before the formula has a name
  3. Ask a question instead of correcting a wrong count

A teacher with 25 students can't build a box of cubes for each one, or stop each wrong count with the right question. Sparky, Flint's AI tutor, makes a 3D box for each student to fill and turn, and asks the next question in each student's chat.

One student fills a fish tank, then two tanks joined together

Theo is in the volume activity. He picks a fish tank, so every box Sparky gives him is a fish tank. Sparky can't see what Theo does inside a model, so Theo tells Sparky what he counts. Every student in the class has their own chat like this one, with the box they picked.

Sparky uses the box Theo pickedSparky asks a question instead of giving the answerSparky builds on what Theo already found
T

Theo

Box: Fish tankStarting point: Adds the edges, counts only the outside

Sparky

Hi! Today you'll fill boxes with cubes to find out how much space is inside them. First, what box do you want to fill: a fish tank, a shoebox, a toy chest, or something else?

Theo

A fish tank

Sparky

Your fish tank is ready! Drag to turn it, and add one layer of cubes at a time. How many unit cubes cover the bottom of the tank? How did you count them?
A 3D model Sparky made in the chat: a glass fish tank with its edges labeled 4, 3 and 5, and two layers of blue unit cubes inside. Below it are an Add a layer button, a Remove a layer button, and the text Layers filled: 2 of 5.
InteractiveSparky's fish tank, with 2 of 5 layers filled

Theo

7. I counted 4 along the front and 3 along the side.

Sparky

Good thinking on how you counted. If there are 4 cubes in a row, and 3 rows, how many cubes is that in one layer?

Theo

Oh, 3 rows of 4. That's 12 cubes on the bottom.

Sparky

Exactly right! How many layers does it take to fill the whole fish tank?

Theo

I filled it up. I counted the cubes I can see: 20 on the front, 15 on the side and 12 on top. So 47.

Sparky

Good noticing that the tank is full! But you counted only the cubes you can see on the outside. You already know there are 12 cubes in one layer. How many layers did you add?

Theo

5 layers. 5 × 12 = 60 cubes.

Sparky

Yes! 60 unit cubes. Why does layers × cubes in one layer give the same answer as counting every cube?

Theo

Every layer has 12 cubes and there are 5 layers, so it's like counting 12 five times.

Sparky

Perfect explanation! Can you write the volume as length × width × height? What are the units called?

Theo

4 × 3 × 5 = 60. Cubic units.

Sparky

Spot on! Now try one with no model. A new fish tank is 6 cubes long, 4 wide and 3 tall. What's its volume?

Theo

6 × 4 × 3 = 72 cubic units.

Sparky

Correct! Last challenge: two fish tanks joined into an L shape. Select Pull Apart to see them separately. What's the volume of Box A?
A 3D model Sparky made in the chat, titled Two Boxes, One L-Shape: a low blue Box A labeled 5 cubes, 3 cubes and 2 cubes, joined to a taller green Box B labeled 2 cubes, 3 cubes and 4 cubes. Below it are Pull Apart and Join Together buttons.
InteractiveTwo fish tanks joined into an L shape

Theo

Box A is 5 × 3 × 2 = 30 cubic units. Box B is 2 × 3 × 4 = 24.

Sparky

Nice! So what's the total volume of the whole L-shaped tank?

Theo

The whole thing is 7 long, 3 wide and 4 tall, so 7 × 3 × 4 = 84.

Sparky

Good thinking, but the L shape isn't one plain box. It has two different heights! You found 30 for Box A and 24 for Box B. What do you get when you add them?

Theo

Oh. 30 + 24 = 54 cubic units. The 84 counted space on top of Box A that isn't there.

Sparky

Exactly right! That space above Box A isn't part of the tank, so you can't count it.

Explained why 5 layers of 12 cubes is 60, and found 54 cubic units for the joined tanks

Theo got three counts wrong: he added the edges, counted only the outside cubes, and multiplied the outside of the L shape. Each time, Sparky asked a question about what he had already found, and Theo fixed the count himself. Both 3D models are Sparky's own, from a test run of the library activity in which a test student typed the same answers. Theo is an example, and we shortened his chat for this page.

See who can explain volume

Sparky grades each chat against the activity's rubric. The activity page shows who needs help right now, what the class understands, and what to practice next.

Flint's activity page for an activity called Volume with Unit Cubes: Fill the Box. Needs attention shows a live alert that one student has counted only the cubes on the outside of the box twice. Activity analytics lists what the class does well, including 18 students who explained why layers times cubes in one layer gives the volume, an area of improvement where 9 students multiplied the outside measurements of the L shape, and a follow-up with a button to create a follow-up activity. The Sessions table shows two submitted students graded Proficient and Developing and one still in progress.

Example activity page

After a 5th grade class finishes the volume activity

The school and students are made up for this page. In this example, most of the class can explain why the formula works, and the next thing to practice is splitting a joined shape into boxes before multiplying.

Set up the activity for free

The volume activity is in Flint's public library, with Sparky's role, guidelines, greeting and a rubric already written. Open it, select Try this activity to copy it into a free Flint account, and move it to a class. Each 3D model opens beside the chat, where students drag to turn it. In the activity settings, select Try session myself to test it as a student before you assign it.

To practice with real measurements, change the boxes in Guidelines for Sparky. For example:

Example guideline

After step 3, give a real box measured in centimeters, like a cereal box 20 cm long, 6 cm wide and 30 cm tall. Ask how many 1 cm cubes would fit in one layer before asking for the volume.

For more math activities, see Flint for math.

Start the activity for free

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