# More Improvements to My Least Favorite Unit (Decimal Division)

3September 17, 2012 by ray_emily

Previously, I wrote about my least favorite unit, as well as my first attempt to improve it. While working with kids on division with decimals, recently, I attempted once again to step up my game (in other words, to shut up and give kids more opportunities to grapple with the material), and I was pretty happy with the results.

First we analyzed several situations where, when we multiplied a dividend and a divisor by the same power of ten, the quotient remained unchanged. For instance, we looked at this string of simple division problems – 8,000/4000 = 2; 800/400 = 2; 80/40 = 2; 8/4 = 2; .8/.4 = 2. I challenged kids to decipher what was going on, and to articulate their understanding of the pattern. (My kids still freak out a little when they are analyzing and not directly applying an algorithm. Must work on that.)

Next, I passed out this worksheet.

Here’s how it works – even though I bet you can figure it out, yourself. First, kids rewrite the problem so that it is both easier to solve and produces the correct (same) answer. (Their goal is to multiply the divisor by a power of ten – to turn it into a whole number – and then adjust the dividend in the same way.) Next, students use a calculator to check that they’ve altered the problem correctly. (Do the two problems produce the same solution?) The last step – and, really, the least important one – is to put away the calculator and solve the problem manually, ensuring that the answers match up, yet again.

The focus, as you can see, is on **setting up the problem** – which is really where my students have struggled, in the past. The set-up is also where the interesting, more challenging math is taking place. (A lot of my students weirdly enjoy long dividing – and do it almost thoughtlessly, on auto-pilot.)

The real reason I like this worksheet is because it is a preemptive strike against one particular error that I’ve seen far too often. In the past, I’ve noted that many students – in their efforts to change a messy problem into a simpler one – multiply the divisor and the dividend by *different* powers of ten. (So, *2.8467 / 0.3* becomes *28,467 / 3*, rather than *28.467 / 3*. Eek!) Previously, I would commend these students for turning icky decimals into nice pretty whole numbers – and then point out that they’d altered the problem *too much*: it was unrecognizable and had a different (incorrect) answer.

This worksheet helps students to arrive at this discovery on their own. Via either trial and error or consultation with another student, students can revise the problem until the old and the new produce the same solution. Additionally, if kids are having a hard time with the actual long division, there is plenty of time for me to circulate and help kids, one-on-one.

BTW: This worksheet was much more successful when I did a few of the problems with the class, before setting kids free. (There are many steps, and if kids don’t get the purpose of those steps, the worksheet feels endless.) Additionally, when demonstrating how to do the worksheet, I deliberately accepted (invited?) incorrect student answers, so that I could model the process of self-correction that I wanted my kids to adopt.

And, of course, if you have any thoughts on how to make this activity better still, please share!

For a topic that you admittedly dislike teaching, it’s impressive how much effort you’re putting into improving your teaching and the students’ learning. Way to go!! :-)

I like that you’re trying out several different approaches all with student understanding in mind. It’s great that you took the time to model some of the problems with the student so they would understand the steps. As you said, if they skipped those steps or didn’t understand them, then the activity was pretty useless. I also noticed you started with a problem string. Yay!

The only suggestion/thought/question is how better/worse you think students would understand the idea you’re going for if they saw the division in fraction form. From there, the idea you’re driving at that they have to multiply the dividend and divisor by the *same* power of ten can easily be related to the concept of creating equivalent fractions. I don’t know how to type the text that way here, but just imagine that problem 1 on the worksheet was written as the fraction 0.000144 over 1.2. Based on students’ prior work with equivalent fractions, they should understand either that you are multiplying by a fraction equal to 1 (10^1 over 10^1, for example) or that you have to multiply the numerator and denominator by the same factor (10^1) to create an equivalent fraction. The end result is the same. How they approach it will depend on how equivalent fractions was taught to them. This will yield the equivalent fraction 0.00144 over 12. Then it’s just a matter of dividing with long division as you asked them to.

Does that make sense? It sounds complicated to reread it, but I do see value in students being comfortable going between the two forms – division sentence and a fraction. Sometimes a division problem can look much easier if you write it as a fraction.

Again, it’s great to see you so dedicated to helping your students understand this topic, and it’s great that you’re making it a topic you’re happier teaching. Keep it up!

Ooooh! YES! This is such a great idea. The division symbol that I chose to use kinda gets phased out as kids move past middle school math, anyhow. Now that you’ve made this suggestion, it seems like the most obvious thing ever. Why didn’t I think of it? ;)

Lol. Believe me, I thought the exact same thing when I was first introduced to this idea at a math conference a couple of years ago. I definitely left feeling smarter than when I arrived. :-)