Every time I teach a unit on fractions, there are many students who insist that they’d rather use decimals, and I don’t blame them. The obvious parallels to the whole numbers make decimals a “friendly” extension from the integers into the rational numbers.1 Many of the things school math asks kids to do with rational numbers can be easily transferred into decimals with minimal stress on the algorithms. Such is not the case with fractions. Take addition for example.
When we add whole numbers, we combine collections of equal denomination until it is possible to collect, bind together, and count them alongside collections of a higher denomination (a process often called “carrying” when the process is abstracted to an algorithm, but I am still unsure of what, exactly, is being “carried”. I think I’d prefer to call this process, “re-grouping”). This process of “carrying” or “re-grouping” is preserved when working with rational numbers in decimal form. However, when asked to add fractions, students encounter problems above and beyond that of re-grouping collections. All of a sudden the process involves common denominators. Students quickly realize that they can always work with decimals at face value2, but need to massage fractions into form before their addition behaves nicely.
The same goes for ordering rational numbers. The bulk of the opportunities I provide to my students revolve around ordering fractions, because they always arrive in my room well-versed in ordering decimals. Most of our work with this involves a “dynamic number line“3. Usually, I check in with their understanding of ordering decimals quickly and then move on, but this semester I decided to provide a wrinkle to the usual question about decimals.
I wanted to accomplish two things:
- I wanted students to show they understand magnitude of decimals by placing them on the number line.
- I wanted to push their algorithmic thinking about what makes a decimal large or small.
The lesson went like this:
I asked each student to take out a blank sheet of paper and copy down the following structure:
I showed the class a 10-sided die with the digits 0-9 on the faces. The instructions were straightforward: I was going to roll the die four times. After each roll, they needed to choose a blank to place the result. They had to choose after each roll, and they were not permitted to switch locations later on. In the first round, students were tasked with assembling the largest possible decimal. In the second round, they were tasked with assembling the smallest decimal possible. As they placed the numbers, I asked questions to the class:
“Who dared to put that 6 in the hundreds place?”
“Does it matter where you put a zero?”
“If the next roll was a 9, would that be good news”
Often times, the last roll was met with equal parts pain and exhilaration, as half the class was hoping for a large number and the other half for a small number. All of these sounds signalled t0 me that the class had an understanding of what they wanted, and therefore an understanding of what made decimals large or small. After the two rounds, I asked if they could definitely create the largest possible decimal if I gave them the four digits beforehand. They guaranteed me, I rolled for four more digits, and each student came up with the same result. We were ready.
I asked them to write the following structure on their pages. (I added a spot because I wanted to have a lot of variety in the class).
I explained a new set of rules. They would get all five digits at the onset, and would still be required to place one in each spot. However, this time they were trying to create a decimal that would fall in the exact middle of the pack of created decimals. Notice that I did not ask them to create the middlemost decimal; I asked them to create a decimal where half of their classmate’s creations would fall above and half of the classmate’s creations would fall below.
Each student kept their answer to themselves and wrote it on a small, folded piece of paper we call a “tent”. (They put their name on the inside of the tent). One by one, they were asked to place their values on the clothesline across the front of the class.4 This allowed for conversations about magnitude to become public. I was able to assess if students understood not only the magnitude of the decimals but also the relative magnitude as they analyzed relationships between the created decimals.
After they all had gone, we pulled off the tents one by one to reveal the exact middle decimal.
To accomplish goal number 2, I asked the following question:
“How could we find out if this decimal is the “real” middle number?”
The conversation turned algorithmic quickly. To build the largest number, you place the largest digit first. To build the smallest number, you place the smallest digit first. So doesn’t it make sense that to build the middle number, you place the middle-est number first? The conversation was awesome!
Some questions I used to guide their thinking:
What if we tried it with three spots?
What if we forgot about the decimal point? Does that matter?
What if there were repeat digits?
We never arrived at a distinct algorithm on how to build the middle. We did, however, interrogate the algorithms for largest and smallest, and talk a lot about rational numbers, relative magnitude, and the base-10 number system in the process. In my book, these are all wins that resulted from a subtle shift in focus.
- I believe that a large part of this familiarity stems from the fact that students still see numbers represented with decimals as a single number that has a magnitude. Many students see a fraction and don’t have an adequate understanding of that fraction as a number. They see it as a relationship between two numbers.
- Of course, this nice behaviour does not extend into repeating decimals.
- Be sure to check out the amazing work Chris Shore has done with clothesline math.
- Again, see the links attached to footnote 3.