If you asked a student to look at this figure and tell you how many cubes it’s made of, he might tell you 16. Can you figure out why? Well, if you think about it, he can only see 16 cubes. He’s not thinking about the ones tucked away underneath. Here in Texas, 4th grade students must “use models of standard cubic units to measure volume”, and in 5th grade they need to “connect models for perimeter, area, and volume with their respective formulas.” Notice that both standards feature the use of models.
The danger is in jumping to the formula, v=l x w x h, before students have an understanding of what the formula means. In other words, jumping to an abstract idea without concrete experiences.
Give students concrete experiences with volume by comparing the process to building a building. Look at the picture below as you read the following steps. Have the kiddos build the bottom floor, a 5 by 3 array, using the unit cubes from your base-10 block set. They’ve already worked with tiling to find area, so this should be a familiar task. Ask them how many cubes they needed to make this floor of the building (15). Now have them build another layer, or floor, on top of the first one. Ask how many cubes they used on that floor (15) and how many cubes they have used so far (30). Engage in a number talk about how they knew it was 30. Some students may have added 15 + 15 while others might have multiplied 15 x 2. Now have them build the top floor. Ask how many cubes they used on the top floor (15) and how many they used for the whole building (45). Again, have students share their strategies for determining is was 45.
For my fourth graders who don’t need to know the formula, they now have a strategy for determining the volume using a picture. They can see the top floor, and they know it’s 15. They can see there are 3 floors with 15 on each, so they can add or multiply to arrive at 45 cubic units. I have the kids label their models just like I showed in my picture.
For the fifth graders who need to connect the formula to the model, they now have an understanding of why v=l x w x h works. What do you do to find the cubes on one floor? (multiply length times width) And why do you multiply that by the height? (Because there are 3 floors and each has 15).
Ready to try it with your kiddos? Click here to grab a couple of sheets with volume models. As always, I love to read your comments!