# Making Math Visible

by Ruth Crum

Ruth Crum earned a M.Ed. degree from Seattle Pacific University. She taught for 25 years in the state of Washington (grades 1-9) and 12 years overseas in Peru, Yemen Arab Republic, and Taiwan. She also served as a volunteer helper at the Global TCK Care & Education office in Dallas.

Did my determination to help kids understand math begin when my eighth grade teacher sarcastically berated Donald for making a wild guess when asked a math question? I remember cringing and grumbling to myself: “Why doesn’t she explain it so he can understand?”

## All from a Box of Straws!

Or was it during my early days of teaching when I dropped a box of straws on the floor? My Depression-days frugality rebelled at throwing away all those unused straws.

My first-grade pupils were struggling with place value, so I cut the straws in half and bundled them in groups of ten with rubber bands.

After that, when I asked what the “3” meant in the number 35, I consistently got “three bundles of ten.” When they were asked to write the number 95, I no longer got a few “59s.” Soon I splurged on another box of straws and put ten bundles of ten in large rubber bands to make bundles of one hundred.

The children could now select the correct number of straws to place under any three-digit number I wrote on the board. Or if I held up the straws, they could write the correct numeral on their papers. (Straws are also great for teaching regrouping in subtraction.)

Another year, my fourth graders wanted to expand to a thousand straws. “OK,” I agreed, “rinse your straws at lunch and we’ll save them.” When the great day came in the beginning of the third month of school, they were jubilant!

“Let’s save a million!” they shouted. “Well,” I pointed out, “if it took us two months to save a thousand, when would we reach ten thousand?” They pondered this question. “And one hundred thousand?” I prompted. At lunch, a dejected bunch tossed their straws in the wastebasket.

## On to Toothpicks

Laughingly, I shared the incident with the staff in the lounge. The next week, a sixth grade teacher reported his class wanted to buy a million toothpicks. After counting the picks in one box, they discovered it would cost them \$130 and would consume much of their cupboard space to store all the boxes.

After laying out toothpicks side by side and calculating the area, my class found that a million toothpicks would “wallpaper” two and one-third walls of our classroom.

I wonder if any of the legislators who so glibly toss around billions (and now trillions) really know how many a million is? Bundles of straws are also useful for teaching other number bases. At first I used coins to demonstrate base five | quarter | nickel | penny |. Five pennies = one bundle of five = 10 in base five, or . Five bundles of five = . It’s easy to rebundle the straws to make any number base you wish. Try it!

Comprehension comes quickly when students can see the meaning. Soon they can add, subtract, and even multiply in other number bases. This brief practice makes the base two of computers easier to understand.

## Area and Volume

Understanding the concepts of area and volume is difficult for some children. Empty a two-liter bottle of water into a gallon container. Empty a half-gallon carton or bottle of water into another gallon container. The slight visible difference will make “one liter = 1.057 quarts” easier to comprehend and remember.

A collection of empty cans of varying sizes used to pour beans, rice, or popcorn will soon demonstrate how a small increase in diameter will produce a large increase in volume. (If your corn is popped, you had better have a backup for a treat, or you may lose part of your volume!)

A collection of cosmetic and other odd-shaped jars can show how thick walls and curved bottoms deceive the consumer as to the true volume of the contents. Read the label to know how much there really is!

## It’s Fun to Compare

A ruler with both inches and centimeters helps establish the relationship of centimeters and decimeters to inches and feet. A yardstick and a meter stick show that a meter is only 3.37 inches longer than a yard.

Draw a square yard on a board or floor with chalk, or in the dirt with a stick. Then enclose that in a square meter. It’s easy to see the square meter is greater than the square yard. What would a cubic meter look like in relation to a cubic yard? Is “one cubic meter = 1.31 cubic yards” more comprehensible now?

A kilometer is more than half a mile, but a square kilometer is less than half a square mile. If that is hard to understand, make it visible.

I thought I understood area and volume until I ordered a yard of sand to fill my daughter’s sandbox, which was 3 x 4 feet and 8 inches deep. Fortunately, the excess was useful to amend our clay soil! No wonder my students so often confused square units and cubic units.

Remedy? We saved corrugated cartons, cutting out one-foot squares until we had nine. Already we could see that three square feet   was not the same as three feet square.

With eighteen squares and plenty of masking tape, we could build three cubic feet, using six squares for each cube. Persistence brought us to our first layer of nine cubic feet, a second layer of nine, and finally all 27 cubes produced a quite visible cubic yard.

Wanting to share our triumph, we arranged a visit with a second grade class. Twenty-seven children carrying 27 boxes waited at the door. I went to the blackboard. Using a ruler, I drew a 12-inch line which the second graders could identify as a foot.

When I added three more lines, they said “square” and agreed it was a square foot. While asking what would happen if we brought that square out into the room a distance of one foot, I signaled my first student to come forward and place his cubic foot on the chalk square. From the back of the room came the other teacher’s voice: “Is that what cubic means!”

Quickly placing four yardsticks on the floor to identify a square yard, I had the rest of my class place their cubes to build one cubic yard. Yes, squares and cubes are quite different! When the carpet man and the gravel man speak of a yard, they are not talking about the same thing!

To make an acre visible, we measured enough twine to mark the perimeter. Then four students, using the knots we had put in the twine to mark the four corners, walked onto the playground and we saw how large an acre really was. You might want to do the same with a hectare.

In another article, Ruth shares more suggestions for helping children understand mathematical concepts under the title, “Geometry to Touch.”

Permission to copy, but not for commercial use.