I  was hoping to have this post up a few days ago. However, I have been deeply disappointed with the lack of resources for students to truly gain an understanding of what multiplication IS. My least favorite involved rhymes for EVERY fact that had absolutely nothing to do with multiplication! (4 door x 6 chicks= denty floor, 24. Are you kidding me????)

There are a ton of “just fact” games: give the fact, type in the product, and so on. And these are fine for students who already know their facts. But if your child is struggling to learn their multiplication facts, that is the LAST place you want to go! So I am going to provide some alternatives for you to explore.

Before I do, I want you to truly ask yourself this question. Does your child REALLY know the facts, or is she merely skip counting quickly? This question could also be asked of your first grader with addition. (Does he really know his addition facts, or is he counting by ones really fast?) When you ask your child a multiplication fact, does she tap it out, count on her fingers, or (what my middle school students would do ) bounce through the multiples? Let me give you an example: 3 x 7. Does your child bounce through (3, 6, 9, 12, 15, 18, 21…21!)? If so, she was timed too early in learning her multiplication facts. Your child should not be timed until she can use a strategy or just knows the fact within three seconds. And that is one-one thousand, two-one thousand, three-one thousand. Once they can multiply that quickly, you can go to some of these fun blast ‘em multiplication games.

So how can you help get her to three seconds? Here are three games I used with my son. I used them in this order, as they get more difficult and begin to have a sort of time crunch to them. If you have others, especially great apps, please include them in the comments section. If you add a resource, please let us know if it is to review mastery or to build strategies towards mastery. Thanks!

1. Circles and Stars. I have seen this in a number of mathematics resources, but the one I love is from Marilyn Burns (see link below). This is slightly adapted. All you need are two dice, paper and a pencil (or whiteboard and marker, which is my personal choice).

Roll two dice. The first number tells you how many circles to draw. The second number tells you how many stars to draw in EACH circle. The total number of stars is your product. Play 5 rounds. The person with the most stars at the end of 5 rounds wins.

Example: Sam rolls a 3 and a 4. He draws 3 circles, with 4 stars in each circle. There are twelve stars in all. 3 x 4 = 12.

Take it one step further. Roll two dice. You choose which number represents the circles and which number represents the number of stars in EACH circle. At some point, you want to ask your child, “Does it matter?” “Will I get the same amount of stars in the total?” The answer is yes! However, 5 circles with 2 stars in each circle looks different than 2 circles with 5 stars in each circle. Yet they both give me 10 stars in all. This is a great place to discuss the commutative property. For multiplication (and addition), when I reverse the order of my factors (the numbers I am multiplying together), I will get the same result. So if your child knows 3 x 7, but thinks he doesn’t know 7 x 3, think again!

2. Math Boggle! I loved Boggle growing up. Shake up the letters, and try to find as many words as you could in the time allotted. This helped me a great deal with learning to spell. Well, we need to play with numbers just like we play with letters. We are just going to start with one fact, and try to figure out as many strategies in the time allotted to find the product (the answer to a multiplication problem). Your kids may know this as a “Number Talk” if their teacher uses them as warm-ups. (If not, encourage them to do so!)

Example: 5 x 6. You have five minutes to find as many different ways to get the value. Go!

1. 5 + 5 + 5 + 5 + 5 + 5= 30 (Still reliable and true, but not the fastest method.)
2. 6 + 6 + 6 + 6 + 6 = 30 (Great to have a discussion as to which method, a or b, is more efficient and discuss the commutative property yet again.)
3. 5 x (2 x 3) = 10 x 3 = 30 (I factored the 6 into 2 x 3. You know it as taking the prime factorization of 4. It’s actually a useful tool when working with multiplication and division.)
4. 5 x (3 + 3) = (5 x 3) + (5 + 3) = 15 + 15 = 30. This is using the distributive property. Please see prior post for details! Also notice it is 15 twice, which could lead to the next strategy.
5. 5 x (3 x 2) = 15 x 2 = 30. Similar to c, but using the commutative property. Also, similar to d, as I have 15 two times. This leads to a nice discussion about why we break apart the 6 instead of the 5. The number 6 is composite; it can be broken into the product of primes. 5 is prime, so it cannot be broken down into smaller whole numbers through multiplication.
6. 5 x (4 + 2) = (5 x 4) + (5 x 2) = 20 + 10 = 30. Again, using distributive property, but I broke up the 6 differently.
7. 5 x (5 + 1) = 25 + 5 = 30. Distributive property. Notice I stopped writing in the middle step. Once kids understand what is going on (in this case, I am multiplying the 5 times the 5 and then the 5 times the 1), they can omit this step. The goal is to get as many as you can. Further, we are eventually wanting them to do most of this in their head so they know their facts.
8. (2 + 3) x 6 = (2 x 6) + (3 x 6) = 12 + 18 = 30. You can use distributive property by breaking up the 5, but notice the partial products you get (12 and 18) are tougher to add together in your head. Great place to talk about why we naturally break apart the 6 instead of the 5.
9. There are many others, but I am hoping you get the idea.

You can play this all day long! Just use a whiteboard and work as a team. Take 5 minutes to work alone. Discuss all of the different strategies each of you used. Perhaps work together to find one more. Go to a different fact. Play in the car (Obviously the kids are playing, not you while driving!), while waiting for an appointment, at a restaurant. The more you play with numbers with your kids, the more fun they have, and the more important they see numbers playing a role in their lives.

3. The Product Game: http://illuminations.nctm.org/Lesson.aspx?id=5729

This is the first of four different multiplication activities your child can do with you to learn multiplication fluently. I highly recommend playing this after your child has really worked with the first two activities. The directions are included on the site as well as a pdf of the game you can print out. You are basically taking two numbers and multiplying them together to earn a spot on the grid. The first person to get four in a row (like tic-tac-toe), wins. Have a whiteboard or scratch paper handy in case your child wants to use some of their strategies to figure out the products. The other activities can be found through this link: http://illuminations.nctm.org/unit.aspx?id=6104

If you only have time or energy for one of these three activities, please use #2. It gets you the most bang for your buck. Happy number playing!!!

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Teachers: For a great resource for multiplication please check out Lessons for Introducing Multiplication, Grade 3 by Marilyn Burns (2001)

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This is the third in a series regarding fluency of basic facts. For the introduction to this series, please read Fluency and Memorization Are Not Synonyms!!!

I am a believer that all students can be successful in understanding mathematics. Maybe not memorizing erronerous facts, but truly understanding what the numbers mean and how they relate to one another. One of my favorite interviews to do with my sixth graders is on multiplication facts. You may want to ask your child this question as well…

If I know that 7 x 6 is 42, how does that help me figure out what 7 x 7 is?

Most students will say “49”, but will not find the relationship between the two facts. Here are the common answers for my sixth graders:

1. “I just know it.”

2. (Counting each one on his/her fingers) “7, 14, 21, 28, (usually begins counting by ones), 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 49…7 x 7 is 49.”

3. Gets the answer wrong, or says he/she does not know that one.

Now don’t get me wrong, I am appreciative when a student “just knows it”, because it usually makes her life easier. The middle school years are ALLLL about multiplicative thinking and proportional reasoning. If you do not know your multiplication facts fluently, you spend sooo much time figuring out the fact (such as student #2) you get lost in the actual work you are trying to do. This leads to frustration, feelings of failure, and a negative attitude towards mathematics.

The answer I am looking for is similar to the following.

“I can just add one more group of 7 to 42 and get 49.”

Why is this important? Because many children do not see multiplication of whole numbers as simply a certain number of equal groups. 7 x 6 means that I either have 7 groups of 6 OR 6 groups of 7 (Commutative Property. Please see my blog on fluency with addition facts.) So if I have 7 x 7, I have one more group of 7 than I did before.

Mathematically, it looks like this: 7 x 7 =(7 x 6) + (7 x 1) = 42 (the fact I was given) + 7 = 49.

This type of strategy, doing your multiplication in parts, is often called in elementary books the partial products. In middle school, this particular set of partial products is called the distributive property. It means you are going to distribute (pass out) a number, multiplying it by everything it has a relationship to. This is not new. We used this in Algebra back in the day. But to use it to help with multiplication facts is a novel idea for most, though it brings understanding to what they are learning.

Here is another example: 6 x 9. (One of the most commonly missed facts.)

Let’s say I do not know my 9’s. However, I do know my x 10’s and my x 1’s. I can think about multiplying 6 by 9 as multiplying 6 by one less than 10, since 10 – 1 = 9.

6 x 9 = 6 x (10 – 1) (true)

So now I need to know how much 6 x 10 is and 6 x 1 is. (You are distributing the 6.) Those are easy facts for me!

6 x (10 – 1) = (6 x 10) – (6 x 1) = 60 – 6 = 54.

I can do this with any of the facts. Say your student struggles with 7’s. However, they know their 2’s and 5’s. They can break the 7 into 2 + 5 and use the distributive property.

4 x 7 = 4 x (2 + 5)

Now multiply 4 by the 2 and 4 by the 5 (You are distributing the 4.):

4 x (2 + 5) = (4 x 2) + (4 x 5) = 8 + 20 =28. Trust me, it looks a lot freakier on paper than just doing the partial products in your head!

Here is one for you to try: 8 x 7. Try it on paper before sneaking a peak!

There are several options. Here are a few. If yours isn’t here, please add it in the comments for others to see!

Option 1 (what I do): Break apart the 7 into 2 and 5.

8 x 7 = 8 x (2 + 5) = 16 + 40 = 56.

Option 2: Break apart the 8 into 4 and 4.

 8 x 7 = (4 + 4) x 7 = 28 + 28 = 56.

Option 3 (what my son does): Break apart the 8 into 10 – 2.

8 x 7 = (10 – 2) x 7 = 70 – 14 = 56.

There is no counting on your fingers, no tapping of feet or bopping of heads (which is what your child does to keep count if they don’t have the facts fluent yet). They just break up one of the factors (one of the numbers you are multiplying together) and multiply by parts. I really only need to know my 1’s, 2’s, 5’s and 10’s. Every other factor can be broken into these parts. It takes the number of multiplication facts to memorize from 100 (if you count 10’s) to 36.

Here’s why you really want to use this strategy with your child. If you are reading this, I bet your child is struggling with memorizing the facts. This strategy isn’t drill and kill. It isn’t a slower method that makes your child feel less than intelligent. It is actually a more sophisticated way to do the math, as it shows where the relationships are among the numbers. It is algebraic thinking in an arithmetic expression. The more your child “plays around” with numbers, the more relationships he/she will see, and the easier algebraic thinking will become. And yes, most (after using the distributive property over and over) will find that the facts will become automatic over time.

Let’s show a problem with a multi-digit number. (Another favorite of mine to use when interviewing children.) 6 x 199. Try it out. How would you solve it?

I just did it in my head. And no, I am not brilliant! Just using relationships to make the problem easier!

6 x 199 is the same as 6 x (200 – 1) = 1200 – 6 = 1194. Sooo much easier than trying to carry and multiply a bunch of numbers by 9!!!

I have to end this with a final thought. I have two dear friends that blatantly state they do not have their multiplication facts memorized. “It’s a waste of necessary brain space when all I need to know are my simple facts and build up the others in my head”, claimed the first. “Why do I need them memorized when I can figure it out quickly using partial products???” stated the second. What to know what their occupations are? The first is a mathematician and professor. The second is a statistician and psychologist. Hmmmm….

Next Blog: On-line resources and games/activities to build multiplication fluency!!!