What Are Cubic Units Anyway?
When I was a kid, I learned math the "old fashioned way" with the teacher in front of the classroom telling us how to do the problems and showing us on the chalkboard. Then we had to do about 100 similar problems from the book to show that we "learned" it. For measurement, I remember learning the formulas for calculating perimeter, area, and volume. But I never understood why those formulas worked. I just know that when we were asked to figure out the volume of a rectangular prism, we did L x W x H to get the answer. Then we had to put a little number 3 after our unit (i.e., cm3) or it was marked as incorrect.
FOCUS math consultants and teachers are using manipulatives to help students understand the concepts behind the procedures. At Te Tsu Geh Oweenge Day School, Sky City Community School and Isleta Elementary School, teachers and I work collaboratively with the 4th - 6th grade students on linear measurement, area and volume, with manipulatives. Students measure the lengths of each side of a rectangle with rulers. They find the areas of the same rectangles using color tiles (square units) and discover the shortcut (Area = L x W). They find out how many blocks it will take to make different sized cubes, like one that is 5 by 5 by 5. They then discuss how many blocks are completely hidden ("If I spray paint all around the outside of this cube, how many blocks won't get any paint on them?" It's 27, which is 33). Listening to students discuss their thinking and model their ideas with the blocks provides into what students do and don't understand. They are pretty excited when they figure out that the reason you can multiply L x W x H is that there are layers you can actually see! They see the "5 x 5" part, and that's 25! And there are 5 of those! Cool! So I can actually multiply that 25 times 5 and I get 125. I need 125 blocks to build a 5-inch cube. That's a lot because a 4-inch cube only needs 64 blocks to build it, but adding one inch in every direction makes it a lot bigger, almost twice as big! The light bulbs are so bright we can hardly see!
Now that's real learning!