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Donna Boucher
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Take a look at the figure below. How many cubes is it made of?
A student might answer 16. Can you figure out why? If you look closely, there are 16 cubes that can actually be seen. What the student may not yet understand is that there are additional cubes hidden underneath the ones on top.
This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.
When teaching volume, this kind of misconception shows exactly why concrete experiences are so important. A two-dimensional drawing requires students to visualize cubes they can’t see. Before we expect them to do that—or introduce the formula V=l×w×hV = l \times w \times h—students need opportunities to actually build rectangular prisms with unit cubes, take them apart, and reason about how the cubes are arranged.
That’s the Concrete-Representational-Abstract (CRA) sequence at work. Students begin by building prisms with cubes, move to reasoning about drawings and other visual representations of those prisms, and eventually connect what they’ve experienced to the abstract volume formula. The danger is jumping straight to V=l×w×hV = l \times w \times h before students understand what the formula actually represents.
Give students concrete experiences with volume by having them build a rectangular prism one layer at a time. I like to compare the process to constructing a building, with each layer representing a floor.
Begin by having students build the bottom floor—a 5 by 3 array—using unit cubes. Because students have already used square units to tile rectangles and find area, this should feel familiar. Ask how many cubes they used to build the first floor. 15.
Next, have students build a second identical layer on top of the first. How many cubes are on this floor? 15. How many cubes are in the building so far? 30. This is a great opportunity for a quick number talk. Some students might reason 15+15=3015 + 15 = 30, while others might see two groups of 15 and think 15×2=3015 \times 2 = 30.
Finally, add the third layer. Again, ask students how many cubes are in the new layer and how many are now in the entire prism. Encourage students to share how they determined there are 45 cubic units altogether. They might add 15+15+1515 + 15 + 15, think 30+1530 + 15, or multiply 15×315 \times 3.
The important thing at this stage isn’t getting students to use the volume formula. It’s helping them see the structure that will eventually give the formula meaning: 15 cubes in each layer and 3 equal layers. When students later encounter V=l×w×hV = l \times w \times h, those numbers can represent something they have actually built and understood.
Once students have built rectangular prisms with unit cubes, they can begin reasoning from pictures and visual models. Now they no longer have to physically build every layer—they can visualize the cubes they can’t see.
In the model above, students can see that the top layer contains 15 cubes. They can also see that the prism is 3 layers high. If each layer contains 15 cubes, they can determine the total volume by adding 15+15+1515 + 15 + 15 or multiplying 15×315 \times 3. I like to have students label each layer on their models, just as I’ve done in the picture.
At this point, students aren’t simply counting the cubes they can see. They’re reasoning about the structure of the prism and using what they know about equal groups to account for the cubes hidden from view.
For students who are ready to connect their understanding to the volume formula, the transition to V=l×w×hV = l \times w \times h now has meaning.
Ask: How can we determine the number of cubes in one layer? Multiplying the length by the width tells us there are 15 cubes in each layer. Then ask: Why do we multiply by the height? Because there are 3 equal layers, each containing 15 cubes.
Now V=l×w×hV = l \times w \times h isn’t just a formula to memorize. The length and width tell us how many cubes are in each layer, and the height tells us how many of those layers there are.
Volume makes much more sense when students have experiences that give the formula meaning. By building rectangular prisms with unit cubes, reasoning about visual models, and eventually connecting those models to V=l×w×hV = l \times w \times h, students can understand why the formula works instead of simply memorizing it.
I’ve put together a free set of volume models you can use to help students make that concrete-to-representational connection. Download the free sheet and try the models with your students. Have them visualize the hidden cubes, reason about the layers, and explain how they determined the volume.
And if you’re looking for meaningful practice for finding the volume of rectangular prisms, check out my Finding the Volume of a Rectangular Prism Scoot Game.
The post Volume, From the Ground Up appeared first on Math Coach's Corner.
A student might answer 16. Can you figure out why? If you look closely, there are 16 cubes that can actually be seen. What the student may not yet understand is that there are additional cubes hidden underneath the ones on top.
This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.
When teaching volume, this kind of misconception shows exactly why concrete experiences are so important. A two-dimensional drawing requires students to visualize cubes they can’t see. Before we expect them to do that—or introduce the formula V=l×w×hV = l \times w \times h—students need opportunities to actually build rectangular prisms with unit cubes, take them apart, and reason about how the cubes are arranged.
That’s the Concrete-Representational-Abstract (CRA) sequence at work. Students begin by building prisms with cubes, move to reasoning about drawings and other visual representations of those prisms, and eventually connect what they’ve experienced to the abstract volume formula. The danger is jumping straight to V=l×w×hV = l \times w \times h before students understand what the formula actually represents.
Start with concrete models
Give students concrete experiences with volume by having them build a rectangular prism one layer at a time. I like to compare the process to constructing a building, with each layer representing a floor.
Begin by having students build the bottom floor—a 5 by 3 array—using unit cubes. Because students have already used square units to tile rectangles and find area, this should feel familiar. Ask how many cubes they used to build the first floor. 15.
Next, have students build a second identical layer on top of the first. How many cubes are on this floor? 15. How many cubes are in the building so far? 30. This is a great opportunity for a quick number talk. Some students might reason 15+15=3015 + 15 = 30, while others might see two groups of 15 and think 15×2=3015 \times 2 = 30.
Finally, add the third layer. Again, ask students how many cubes are in the new layer and how many are now in the entire prism. Encourage students to share how they determined there are 45 cubic units altogether. They might add 15+15+1515 + 15 + 15, think 30+1530 + 15, or multiply 15×315 \times 3.
The important thing at this stage isn’t getting students to use the volume formula. It’s helping them see the structure that will eventually give the formula meaning: 15 cubes in each layer and 3 equal layers. When students later encounter V=l×w×hV = l \times w \times h, those numbers can represent something they have actually built and understood.
Connect to visual models
Once students have built rectangular prisms with unit cubes, they can begin reasoning from pictures and visual models. Now they no longer have to physically build every layer—they can visualize the cubes they can’t see.
In the model above, students can see that the top layer contains 15 cubes. They can also see that the prism is 3 layers high. If each layer contains 15 cubes, they can determine the total volume by adding 15+15+1515 + 15 + 15 or multiplying 15×315 \times 3. I like to have students label each layer on their models, just as I’ve done in the picture.
At this point, students aren’t simply counting the cubes they can see. They’re reasoning about the structure of the prism and using what they know about equal groups to account for the cubes hidden from view.
Connect the Model to the Formula
For students who are ready to connect their understanding to the volume formula, the transition to V=l×w×hV = l \times w \times h now has meaning.
Ask: How can we determine the number of cubes in one layer? Multiplying the length by the width tells us there are 15 cubes in each layer. Then ask: Why do we multiply by the height? Because there are 3 equal layers, each containing 15 cubes.
Now V=l×w×hV = l \times w \times h isn’t just a formula to memorize. The length and width tell us how many cubes are in each layer, and the height tells us how many of those layers there are.
Build Meaning Before the Formula
Volume makes much more sense when students have experiences that give the formula meaning. By building rectangular prisms with unit cubes, reasoning about visual models, and eventually connecting those models to V=l×w×hV = l \times w \times h, students can understand why the formula works instead of simply memorizing it.
I’ve put together a free set of volume models you can use to help students make that concrete-to-representational connection. Download the free sheet and try the models with your students. Have them visualize the hidden cubes, reason about the layers, and explain how they determined the volume.
And if you’re looking for meaningful practice for finding the volume of rectangular prisms, check out my Finding the Volume of a Rectangular Prism Scoot Game.
The post Volume, From the Ground Up appeared first on Math Coach's Corner.