What is “Rounding Up”?

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

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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…

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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..

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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:

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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:

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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

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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.

PEMDAS? Is order of operations the only way?

This year I have had quite a few students who are pushing themselves mathematically and doing a lot of great thinking.  This also equates to making me think as well, and I really need that.  Now a student has me thinking about PEMDAS, how it is taught, and how it is a coveted norm of mathematics.

One thing I have always struggled with when talking with students about PEMDAS is how to make it meaningful for students without pulling out the “This is the way it is” card.  I don’t want it to become a memorization practice only, I really want students to understand why we do operations in a particular order.

One thing I have used and students understand (not surprising because I adopted this because of a student) is that multiplication implies “groups of.”  I’m sure many pure mathematicians might frown upon this, or the fact that it also leads to repeated addition, but students really grasp the concept and start visualizing order of operations.

3×4+2

3 groups of 4 plus 2

4+4+4+2

14

When students think of it this way, they don’t want to add 4 and 2 and then multiply that by 3.  This is also true when they see 2+3×4, the “groups of” thinking prevents them from adding 2 and 3.  Is this just replacing a trick with a trick?  It took me a few years to even accept this type of thinking for my students- I don’t want to provide a new crutch for students.

This all leads me to the other day, when a student was solving the equation 3X + 2 = 8.  He was unsure of what to do, so I asked him what step he would try.  He repled: “I would divide by 3 since I want to solve for X.”  I immediately balked, that answer went against all of my mathematical fiber, but I tried to not show it visibly.  I told him to go ahead and try his idea.  This is what was produced:

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Wait now!  Hold on, that is the answer I expected, but not attained in the way I assumed it would be.  I can’t tell you how many times I have used the last in- first out type of thinking with students to solve equations, and I expected seeing something like this:

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But all of these provide the same solution!  Is that a fluke?  Some weird oddity that can’t be reproduced?  It had me thinking of possible counter-examples and what else it could imply.  Does order of operation last in-first out thinking absolute in solving equations?  Or is it something like the standard addition algorithm that we accept as the best method and ignore others?  My student gave me another example to consider later that hour.

This problem involved the area of a trapezoid.  They wanted him to algebraically solve it for b2.  I was very curious what he was going to do, and instead of starting with the parenthesis, so subtracting b1 (which is what I assumed he would do because of his previous reply), he told me he needed to distribute before solving.  His work was this:

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Which again produced the correct solution after he cleaned up the compound fraction.  Once again his work produced the right answer without doing the expected procedure:

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So currently I am rethinking approaches to solving equations.  How will not using the standard approach effect him later on?  I can see how trying some sort of variation on this with powers will be messy.  I’m totally open to suggestions and comments on this, as it has my brain working on overload ATM.