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 introduce long division 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!

Number patterns are a great way to revist basic addition and subtraction operations in a different format from the usual math drills. They also illustrate some interesting relationships like 15 minute intervals in hours, intervals of 25 in dollars and the ever-useful mechanics of multiples-of-five that seem to show up in every day life.

It seems like number patterns are increasingly appearing on things like AIMS tests and other placement exams. Include a healthy does of number patterns in your regular worksheet diet and pattern problems will be like striped candy. Or, so I keep telling my eldest.

New number pattern worksheets can be found at the link here…

Number Pattern Worksheets

…or under the ‘Worksheets’ menu to the right.

UPDATE! Additional worksheets involving negative numbers, including patterns that cross zero, have been added in their own section…

Negative Number Pattern Worksheets

Welcome to the Rocket Math Spaceship Math home page at DadsWorksheets.com.  Below you’ll find direct links to the Spaceship Math practice worksheets for all of the basic operations. Also, you’ll find a convenient web-based timer for administering practice tests.

Everything you need to rocket your kid’s math performance straight into orbit is below. If you find these materials useful, be sure to check out the rest of the math related content on the site. Good luck!

Please note that Spaceship Math is not the official RocketMath curriculum offered by R&D Instructional Solutions, and as per Dr. Crawford’s request in the discussion below, we’ve renamed the practice worksheets here as ‘Spaceship Math’ to avoid any confusion with the RocketMath commercial product.  Educators interested in using Rocket Math in their classrooms are encouraged to check out http://www.rocketmath.net for details.

Spaceship Math Strategies
Spaceship Math Addition Worksheets
Spaceship Math Subtraction Worksheets
Spaceship Math Multiplication Worksheets
Spaceship Math Division Worksheets
Spaceship Math Practice Timer

Printable Spaceship Math Progress Check Off Pages

Many grade schools now using various types of timed tests for basic arithmetic. This web site was originally created to provide practice worksheets for a time testing program used at a local school district. If your child’s school is using a similar program, these worksheets will provide several variations on the single practice sheet that typically comes home for each lesson.

The Rocket Math programs are typically divided into multiple levels usually identified by letter, where each level introduces a small number of basic facts. The problems on each level are built on the The tests are usually given daily, with each test lasting one minute. Practice on the problems is pretty critical to success, especially if your child isn’t one that works well under the pressure of the clock.

There are different Rocket Math levels for each of the four basic math operations. My kids started with Rocket Math addition in first grade and have generally worked up through the operations year-by-year, (subtraction in second grade, multiplication in third, and, still to come, division in fourth grade.)

For practice here, as part of home work two or three times a week I’ll run off five of the worksheets, usually two from the level my daughter is currently on and then a selection of three from the next two or three levels she’ll likely cover. I’ll then use the free web timer on the site and run a five minute test. Its very critical that you grade the tests and have your child correct anything that’s wrong. In fact, where an answer is consistently wrong, I’ll even resort to having my daughter write the correct math fact out eight or ten times. Memorizing a math fact wrong is simply deadly here and you need to catch this as early as possible when it comes up. Never (never, ever) skip grading a Rocket Math test. Child Protective Services has a special room for parents that let their kids memorize 6×7=56.

Anyway, apart from the raw practice these longer tests give, the logic here is that this is a much longer, harder and more stressful test than what she goes through at school each day. That makes the tests at school seem a whole lot easier by comparison. It does make create something of a motivation factor, and that’s really the key hurdle to overcome.

Motivation around here comes in many forms. It starts with having a positive attitude around the tests, no matter what happens. Reinforce that the practice is what counts (not getting 100% accuracy or completing it on time, even though that’s what you want ultimately.) Passing a particular five minute test under the timer usually merits a high-five and a serious break from whatever’s going on… Maybe a getting a little time on the Wii with Dad or a Godzilla movie or something else fun. We’ll set targets for levels and maybe working up to level ‘N’ means a $20 trip to Toys’R'Us or going out for a Quadruple-Venti Mocha at Starbucks (actually, cocoa for the kids… Dads need motivation, too.)

All the way around, the key here is making the activity fun and engaging. Making this enjoyable at home plus the success this will bring with testing at school will quickly turn Rocket Math into enormously positive experience. Honestly, how great is it to have your kids saying not only that “I’m good at math!” but also “I love math!” This Dad was never so proud.

Learning multiplication facts is a challenge because it’s the first math operation where your child needs to contend with relatively large numbers. Two digit addition and subtraction is squarely in the realm of numbers less than 20, which is familiar territory. There’s something concrete about 12 or 15 or similar numbers countable on fingers and toes, but 73 really is a big step out of the pond.

There’s two ways to approach this. One is just brute force memorization. I remember endless flash card drills after school, the timed tests in the classroom and the gradual accumulation of resentment towards anything with that little ‘x’ attached to it. While we love the Rocket Math program the schools use here, it is largely just memorization and could use something to back it up.

The other alternative is to make multiplication something of a game, with systems for some of the numbers. There still an inevitable amount of memorization that goes on, but by getting 90% of the multiplication table down to a few simple rules, the goal is suddenly within everyone’s reach. Split second, memorized results are still going to come, but having some means to reach incremental (albeit slower) success takes the fear and dread out of the process.

The place to start is understanding that multiplication is just repeated addition, and then using a few tricks to make the addition go faster. For example, if we need to multiply 4×6, an easy explanation is that this is just four copies of six added together, or six doubled twice. This “Double-Double” rule works for anything multiplied by four, and is easy to apply if basic addition has already been covered. A handful of these rules cuts 90% of the multiplication table away so that memorization is only required for ten facts.

Even if you ignore the ‘tricks’ in the rules below, the brute force way of multiplying by breaking it out into a bigger addition problem may seem like more work than just memorizing the facts, but it helps build some understanding of what multiplication actually means and provides a way to find the answers if they haven’t been memorized. Knowing what multiplication is, how multiplication works and (worst case, without any other tricks) how to solve a multiplication problem makes the whole process tangible. Also, this builds up some of the thinking processes used to multiply larger numbers where memorization isn’t possible.

Our focus for now is the core, so we’ll start with the 100 basic math facts (1×1 all the way through 10×10) and cut them down to size. The rules are ordered so that the easiest ones to memorize and use take the biggest chunks out of the table. If you learn them in this order, you cover the facts in the table in the fastest possible way.

Here are Dad’s eight simple rules for learning the multiplication tables.

Dad’s Eight Simple Rules for Mastering the Times Tables

This rule, more formally known as the reflexive property of multiplication, just means that A x B = B x A. If you can teach your child that 6×7=42, they should be able to remember that 7×6=42 as well. This should be the first question you ask if your child is stuck on a problem. If your child doesn’t know the answer to a multiplication math fact, swap the multiplicands and ask the question again. When you factor in the effect of perfect squares, this one rule cuts the number of facts we need to memorize almost in half to 55.

If multiplication is just instructions for addition, multiplying a number by one just means to add a single instance of that number up. The result is always that number. That takes 10 problems out of our remaining list of facts, dropping us already to 45. See how fast we’re moving?

Even if concepts about place value and shifting decimal places are new at this point, memorizing that multiplication by ten means just attaching a zero to the number is an easy rule to remember. The zero on the end of the ten should serve as a trigger, “Ten ends in zero. What do you attach to the other number?” Given the focus on reusing addition facts in our multiplication odyssey, I recommend avoiding the phrase “Add a zero” or you may garner some initial confusion. Multiplication by ten removes nine more problems from the grid and gives us 36.

This rule leverages facts learned during addition. 2×7 = 7+7 = 14. All of these facts should already be memorized, but even if they’re not they’re still in the range where counting on fingers and toes gets rapidly to a solution. Because we already crossed off 2×1 for Rule #2 and 2×10 for Rule #3, we only get to knock eight more off our list, but that still drops us to 28.

When my daughter pauses on a times-four problem, all I have to do is say “Double-Double” and the answer comes right back. 4×6 = 6+6+6+6 = 12 + 12 = 24. For numbers five and lower, the four double-double rule will work with addition math facts and should be performed in memory. If your child can do simple two digit addition without regrouping in memory, six and seven work as well. It’ll take a while, but eventually 4×8 and 4×9 aren’t too hard but you may find those facts get memorized before “carry the one” starts happening mentally. However you get there, we get to cross off seven more facts (skipping 4×1, 4×2 and 4×10 from the rules above), which puts us at 21 left!

Your child should already know how to count by fives by the time they’re in multiplication land, so a quick short-cut for solving a 5 times problem is just to skip-count by fives up to the number. There’s other more complex strategies for fives (if the number is even, divide it by two and add a zero, so 8×5 = (8/2) * 10 = 40) but these are typically a bit complex when making a first pass here. The “Count by Fives” rule drops us down to 16 remaining facts.

When you multiply a number by nine, the sum of the digits of the result is always a multiple of nine. For the basic math facts, the sum of the digits IS nine, and in fact it has some other interesting properties. The tens place value is always one less than the number being multiplied, and because of the nines rule the ones place is always the nine minus the value in the ten’s place. The basic script for learning this rule goes something like this: “Multiplying by nine? Okay, what’s one minus the other number? That’s the ten’s digit. Okay, what number plus that equals nine? That’s the one’s digit.” Again, this strategy just falls back on basic addition facts, and it cuts our total number of math facts to memorize down to 10.

The first seven rules cut our list of facts down from 100 to 10, so all we need to do is memorize the 10 multiplication facts to have the whole table down. We eliminated any number times 0, 1, 2, 4, 5 and 9. So here’s the multiplication facts that are left with a few rhymes to help remember them:

The first four facts are all from the three-times table, and they’re fairly easy to calculate using addition or find by skip-counting by threes. The remaining six are the nasty ones. If you really look back, you can probably remember struggling with one or more of the remaining ones as a kid. This is a link to practice worksheets specifically for the ‘Rule #8′ facts here.

So that’s it, multiplication in eight rules built on top of basic addition. If we count rule eight as ten facts, it really means the whole multiplication table is wrapped up in only seventeen pieces of knowledge. Easy!

You will still find conventional worksheets and flash cards to be a powerful way to reinforce the strategy presented here, but by quickly learning these few simple facts, your child will immediately have a solid and successful grasp of multiplication. Good luck and see you at Division!

A Summary of the Eight Simple Rules for Mastering Multiplication