# Volume, From the Ground Up

### Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

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).

1. Oops, don’t you mean V = L x W x H? You have A = L x W x H.

• Thanks! Fixed it. 🙂

• I love the way you call them “floors,” I think that makes it very easy for the kids to relate to real life. I will have to use that with my 5th graders that are still struggling. BTW you still have what the previous poster said in your second-to-last paragraph.

• I have found the “floors” connection to be very helpful, Annie, and all mistakes cleared up now (I think…). Thanks for pointing that out. 🙂

• Donna,

Thank you so much for presenting this information in a clear way! I can’t wait to share this with my students and peers!

2. This reminds me a lot of the way Investigations addresses Volume 🙂 Love it! I have used the Muffles Truffles unit from Context for Learning Mathematics to teach multi-digit multiplication strategies using the array and open array within the context of making boxes of truffles… if you start to add “layers” like the Whitman’s boxes, you can explore volume as well. Now if only our “wellness policies” would allow us to give the kids truffles 🙂

• Ha ha, too true, Dawn! 🙂

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4. You can also get at partitive vs quotative thinking by looking at 15 ‘columns’ of 3, or 9 ‘beams’ of 5. It all depends on what you want to unitize as a standard referent.

Great post, Donna!