# What Does Twice as Big Mean?

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.

Well, since no one else has mentioned it yet, here is a link to the AIMS book that has hands-on activities on this topic http://store.aimsedu.org/item/1407DB/Effects-of-Changing-Lengths-6-8/1.html or if you need 2 units of professional development while learning more about this check out the online course http://ce.fresno.edu/cpd/courses/coursedetails.aspx?courseCode=MAT-959

I like the pizza problem. Kids might wonder what a fair price would be for a pizza twice the diameter of a personal size pizza. That would be a fun exploration!

I think an important piece in this exploration is labeling what is twice as big. If I say the diameter is twice as big, I think that is clear. The problem lies when a object has multiple things to measure. A cube has length, width, height, and volume. If you are talking about a cube being twice as big as another one, you would need to specify which aspect you are talking about. This is also a factor in 2D shapes like a rectangle. You could have the length, width, or area be twice as big. Sounds like an opportunity for precision.

Actually, I didn’t intend to do a follow-up, but now that you’ve asked, I might come back to explore these ideas a bit more. In the meant time I’m hoping others will chime in. One question to ask is about how a “scale factor of 2” is related to “twice as big”. I noticed today a couple of other blogs that have explored this idea.

Richard, yet again you’ve managed to look at something I take for granted and presented it to me in a way that has me second guessing my understanding. I suppose the answer to the question posed by your blog title is that “twice as big” means whatever I say it means. If I’m looking at perimeter, then I can’t also be talking about area since it may not have increased in size the same way. But whatever I mean, it sounds as though I can rarely be talking about the object or shape as a whole.

You kind of left us hanging at the end there with all the different possibilities for size increase. Is there another blog coming?