To teach volume, I cut out a bunch of flattened cubes (a spreadsheet I made in Excel), built them, and taped them together. We recycled some of the boxes used for calculating surface area and filled them with cubes. Not only did it give her a feel for what volume is (how much space an object takes up), building with cubes spotlights the idea of cubic units being very different from square units. We carefully took the cubes out of the box and compared it to it, side by side. We separated each layer of the volume to enable Pamela to be certain of her count.
Pamela did free lance explorations in which I gave her 20 cubes and let her figure out how many ways to create a volume of 20 cubic inches. Another variation was to take a pile of blocks, create a rectangular prism, and count the dimensions and volume. Activities like this appealed to her orderly, patterned mind. She recorded everything she did in charts to allow her to discover the formula for volume.
By the fourth example of volume, Pamela spotted the formula for volume: length times width times depth. She quickly transition from doing hands-on work to picture (even trying her hand at drawing a 3-D box) to words and symbols. Once she solidified her ability to do volume problems, I introduced one variation: being given a problem with mixed units (1 foot by 10 inches by 14 inches). She is learning to be vigilant about checking the units before labeling her drawings.
Covering a Surface
Pamela struggled with coverage problems like,
Susan needs to buy paint to cover 2100 square feet. One gallon of paint covers 400 square feet. How many gallons of paint does she need?
I realized she did not quite understand what the problem was asking so we backed up and did some concrete activities like covering a 9" x 12" notebook with 3" x 3" sticky notes (pictured left) or covering books with index cards (or cut-up index cards of a standard size to make the numbers work). When we made the leap to word problems, I knew Pamela understood when she illustrated her problem with paint cans.
The next variation I plan to address are house problems including painting or covering sides with siding (you only hit the sides, unless you have triangular shaped attics) and shingling roofs. More variations will involve subtracting out the area of doors and windows or applying two coats of paint.