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!!!