# What is “Rounding Up”?

So, Christopher (@trianglemancsd) posed this question to twitter over the weekend:

Now, this weekend I was out in the woods hunting deer, so I got this notification while trudging through a cedar swamp.  I quickly jotted down a reply that first hit me, put the phone in my pocket and trudged on, but of course that is never enough with Christopher…

DOH! Yes Christopher, it is.  I’m in the WOODS, leave me alone!  So when I came in for dinner I tried to construct a better response..

It’s not the most mathematical definition, so I was wondering what my students would say… This was awaiting them when they entered the room today:

I haven’t had a lot of time with the group I currently have to work on constructing good thinking responses so I didn’t know what I would get, but here is what I got.

The -0.13 Camp:

Overall, this group considered “rounding up” as only “making the digit bigger”, and had their mind blew when I asked how rounding up was making it negative 0.13?  There was a long moment of pause; ideas flashing across their features as they struggled with this concept.  Many became unsure of their answer.

The -0.12 Camp

Overall, this group was confused by the rounding “rules”.  Many explanations would not lead to correct rounding for positive numbers and these students need a quick refresher.  There was one student who understood what to do and took that into consideration when rounding, going to -0.12 because to round up was to make a bigger number.

The “Other” camp:

I am not going to post a picture of these responses, but these students had answers other than -0.12 and -0.13, and had major errors in their mathematical thinking about rounding or just guessed.

The result:

It was really had to discuss this question without imposing my idea of what the answer should be.  I had many students ask me what the correct answer was throughout the discussion.  I told them that was what we were trying to discover, and would not tell them my answer until they all agreed upon their way to round this number.  They were confused with this concept at first because it following their rules did not produce what they expected- but only when that was implicitly pointed out to them.  Many did the mechanical procedure for rounding and didn’t examine the number or it’s implications.

Once we finished our discussions, each group came to the same conclusion.  That while they want math to be consistent- this did not appear to be until you considered the concept of negative.  They initially wanted the procedure to be the same, by using the terminology of rounding up they wanted the number to be larger.  Then they moved into the number line and comparing the distance from specific numbers.  Since this was a half number- that caused a little more discussion about which way to go.  They decided to round it to -0.13 because it would remain consistent with their concept of rounding, but with reflection around 0.  Since a number would be rounded up in the positive, it would “round up to more negative”.

I challenge those of you who read this blog to introduce the question and discussion to your students, and blog about it.  There was a lot of great mathematical thinking that happened today.