Well, the new school year is officially underway. We touched division last year briefly, but fourth grade here is where we need to get our division facts down cold. Of course, the goal is working up to those long division worksheets but to get there, you really need to establish a solid grounding in the division facts first.
Unfortunately, division isn’t just multiplication in reverse. Oh yeah, we tell them that and it flies for a while, but then one ugly left over spoils the fun. Remainders. You only get so far into division before this remainder thing pops up, so if you arm your kids with only the “reverse the multiplication” strategy, division quickly develops a reputation as the nightmare operator. We relied heavily on the idea that addition and subtraction had “fact families” and you could always reverse them, but that clean relationship just isn’t there for multiplication and division. I mean, what’s the corresponding multiplication fact for 5 / 2 = 2 r 1 ? 2.5 x 2 ? We don’t get closure here until we’ve introduced fractions and decimals… Perhaps division’s reputation as a monster is a bit deserved. Either way, this post describes the various sets of division worksheets on the site to help you introduce division and remainders successfully.
There are a number of different ways to tackle division. The conventional RocketMath division problems work by progressively introducing a handful of division facts in a series of drills. We started with these worksheets here. The issue though is right at RocketMath Division Level G, division by divisors larger than the dividend is introduced and the answer (quotient) for all of these problems is given as zero (i.e., no wholes and the remainder is ignored.) This sweeps the whole remainder issue under the carpet, but when we circled back for actual division with remainders we struggled to get rid of this concept. I heard “Zero!” so many times when we were working steps in a long division problem here that I wish we’d never seen this treatment introduced. If your school is using either the RocketMath or Mastering Math Facts curriculum, you may find these division drills useful, but you may save yourself some “unlearning” by taking an alternate path.
Another approach is to introduce initially only the division facts that are perfect reversals of multiplication. This does ignore remainders initially, but it also doesn’t introduce a behavior that makes you think you can ignore them either. The SpaceShip Math Division Just Whole Facts series of worksheets does exactly this. In fact, once this series of worksheets has been completed, you could even step into the Long Division without Remainders worksheets to introduce the long division algorithm. We didn’t follow this route here, but the whole divide/multiply/subtract/bring-down process reinforces the relationship between digits in the quotient to each digit in the dividend, and I think that subtraction step might set up a notion of what the remainder thing is all about.
Whether you tackle long division earlier or not, after the core whole division facts are down, it’s time to introduce remainders in their full glory. The best success we had here over the summer was in simply drilling the division with remainder facts, just like any of the other basic operations.
The SpaceShip Math Division Focus on Remainders worksheets introduces division facts with remainders progressively by increasing the divisor as you go up levels. By starting with lower number divisors, the remainder is pretty easily to calculate and the concepts build up before you get to large remainders that aren’t quite as easy to recognize off the top of the head.
Another shorter series introduces remainders in a slightly different way. The Division with Progressive Remainders Worksheets introduce problems where the remainders increase (the whole part of the quotient may be larger, however). This is a similar approach to the SpaceShip math sheets mentioned above, but it does introduce the problems in a slightly different order that may help if the first series of worksheets seems to be a struggle.
The main division worksheet page also has practice worksheets for a variety of other division topics, including focuses on cubes and roots, factors of ten and powers of two. Of course, your real goal should be to get to the long division worksheets reasonably quickly, which will exercise all of those division facts (with and without remainders) probably much more than your little math fiends will care for. Dad’s supply of Pokemon reward packs is definitely dwindling.
Division has been a longer term project here, and I hope the different approaches are valuable to you. Let me know what worked and what didn’t in the comments below. And best of luck for the 2009/2010 school year!