Tag Archives: Proportional Reasoning
I’m often puzzled by the way we use phrases like twice as big. What does that mean? For example, I understand that if my debt 5 years ago was $10,000 and today it’s $20,000, then my debt is twice as big today as it was 5 years ago. I also understand that if one two-by-four is 8 feet long and a second one is 16 feet long, then the second two-by-four is twice as long as the first. I could also say the second two-by-four is twice as big, since not only did the length double, so did the volume and most likely the weight. In these two examples, the meaning is clear.
But what about two squares, where one has a side of length 2 and the other a side of length 4? Is the square with a side of length 4 twice as big as the one with a side of length 2? Clearly, while the length of a side is twice as big, the area is 4 times as big. So, is the square with side of length 4, twice as big or is it 4 times as big? In fact, we could also ask about the perimeter of the square with side of 4: is the perimeter twice as big? Of course, the answer is yes; the perimeter is twice as big. It seems to me that this question opens the door to some important measurement conversations with students, beginning in third grade, where they are first introduced to area and perimeter.
Let’s back up for a second. What if we started with that smaller square with side of two? That means both length and width of the square are 2. Suppose now that we double the length but leave the width unchanged. Is the resulting rectangle twice as big as the square with a side of two? One pair of sides is twice as big, but the other pair stayed the same. What about the area? Is it twice as big? What about the perimeter, is it twice as big? Actually, the area is twice as big, but the perimeter is just one and a half times as big. Why would that be?
Here’s another example from the real world. Suppose you have the choice of ordering a personal size pizza that has an 8-inch diameter, or ordering one that has a diameter of 16 inches. Is the 16-inch pizza twice as big as the 8-inch pizza?
Here is just one more example. When I ask the copy machine to enlarge a drawing by 200%, will the copy of the drawing be twice as big as the original? What does enlarging by 200% mean to the copy machine?
As we teach measurement concepts, especially as we teach area of rectangles in third grade and volume of rectangular prisms in fifth grade, we have a great opportunity to explore questions like the ones I’ve been asking. What does twice as big, or three times as big, mean for objects that are two-dimensional or three-dimensional?
Just some thoughts.