September 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!