PARENT TO PARENT

In the last column, I explained how to use patterning and "tricks" to compute answers to multiplication problems, through the fives tables. Let's continue working through the rest.

It is very important at this time that your student understand that multiplication problems can be turned around: 3 x 5 is the same as 5 x 3, etc. I suggest practicing this by reviewing some of the simple tables, like the twos, and alternating the style. For instance, the equations can written: 3 x 2 =; 2 x 4 =; 5 x 2 =, etc. No matter which way the equation is written, it can be calculated by using the two rule and doubling the 'other' number. Once this concept is mastered, practice it with a few other problems and you will be ready to move on.

I often tell my students that I have good news and bad news when it comes to multiplying by sixes, sevens and eights. The bad news is, there are no tricks. The good news is, you already know most of the equations, so you don't have to worry about it.

I take this time to show students exactly how quickly one can calculate answers using patterns, by "running up" the equations very quickly in one (overly exaggerated) deep breath. Try to read the following sentence as quickly as possible, and remember that this is all the time it takes to "calculate" answers to math problems:

(Deep breath, please)…"Since 6 x 1 is the ones rule then just write down the six, and 6 x 2 is the doubles so 6 + 6 = 12, and 6 x 3 is 6 + 6 which is 12, plus 6 more which is 18, and 6 x 4 is two doubles, so 6 + 6 = 12 and 12 more equals 24, and 6 x 5 is easy when you count by fives and that's 30, whew!" That's half the tables and the only thing anyone has to know is some very basic addition. There are also patterns and tricks with larger numbers, so the only equations your student needs to "memorize" are: 6 x 6 = 36, 6 x 7 = 42 and 6 x 8 = 48. That's right, there are just three equations. Now think about that for a minute. There are twelve multiplication facts for each table, so a child must memorize 72 separate equations just to get through the six tables. If your child learns how to calculate these answers with patterns and sequences, there are only three equations to memorize!

The same technique is applied to the sevens, and I include the fact that, since you've just learned that 6 x 7 = 42, why would you want to memorize it all over again, just because the numbers are turned around? Even if the equation is written 7 x 6, a young child can easily recognize that this is the same as 6 x 7, which you now know equals 42. So, you can skip that one, too.

The eights are the last difficult group, but - we already know how to calculate 8 x 1 through 8 x 5, and we've learned that 6 x 8 = 48 and 7 x 8 = 56, so all we need to learn is that 8 x 8 = 64.

At this point, all multiplication that has already been gone over should be reviewed as a preliminary warmup, not as part of a lesson. The actual lesson should concentrate on the six equations your student needs to learn - 6 x 6, 6 x 7, 6 x 8, 7 x 7, 7 x 8 and 8 x 8. I use the "Read, Write, Read" method. I have explained this in previous columns, but I will repeat it here, as it is imperative to moving through these six equations quickly and efficiently, as well as introducing a wonderful study technique your student can use in school.

On a sheet of paper, you write down the first equation, 6 x 6=. After your student gives the answer, write it down. Now you have written 6 x 6 = 36 on the paper. On the next line, below that, write the next equation, 6 x 7 = and write the answer after your student has given it to you. Continue with the six multiplication facts, through 8 x 8=64. Draw a vertical line down the paper, directly to the right of the column of equations you just made. Have your student read, very carefully, exactly what you wrote. Now have your student copy each equation, one at a time, to the right of the line. If your child makes a mistake and copies something incorrectly, do not let them erase the mistake and correct it. Erase the entire equation and rewrite it correctly. Also, no cheating! Children like to race through simple projects like this, and will often write in columns: all the numbers, then all the operational signs, etc. This is not acceptable. Make sure the equations are copied, with the answers, left to right and line by line. Now have your student read what they wrote, this time from the bottom line up. Do this only once in the lesson. Repeat this lesson a few times and your student should become very comfortable with these equations, with little effort on your part or theirs.

The nines are an amazing group of equations. The number itself is large and cumbersome, irritating because it is so close to 10 but not quite, and generally causes children a lot of grief. In truth, working with nine can be a lot of fun. It has quite a few interesting pattern sets and there are several ways to calculate the products of the nine tables. The one I teach is not the simplest to learn, but has the most power behind it, since the theories and patterns will carry over into division and fractions. This technique only works for single digit multiplicands (9 x 1 through 9 x 9), but that is enough right now. To multiply a number times 9, look at the number - not the nine, the other number - and write down the number "one less than" that one. Here are some examples: 9 x 4: since you are multiplying 4, one less than that is 3 - so write down the 3. Also, 9 x 7: since you are multiplying 7, one less than 7 is 6, so write down the 6. Again, 9 x 3: one less than 3 is 2, so write down the 2. This is true for all the nines, through 9 x 9 (one less than 9 is 8). That's the first digit of your answer. To find the second digit, subtract this first digit from (or add up to) 9: To figure 9 x 6: One less than 6 is 5, and 4 more makes nine, so the answer is 54. Notice that the digits (5 and 4) add up to 9 - that completes the rule: "To multiply a number times nine, write down one less than the number, then however many more it takes to add up to nine." This sounds confusing in print, but is actually very simple on paper. It is, of course, imperative that you actually try it a few times yourself so you can explain it to your students. Here are some examples: 9 x 4: One less than 4 is 3, and 6 more make 9, so the answer is 36 (check: 3 + 6 = 9). Also, 9 x 8: One less than 8 is 7, and 2 more make 9, so the answer is 72 (check: 7 + 2 = 9). When first introduced, this technique makes some children nervous - it looks rather complicated, but it really isn't. The idea that the digits of the product will add up to 9 will become very helpful in division and fractions, so it is worth taking a little time and practicing until you are comfortable with the pattern. Once it is established it is one of the quickest and simplest of the computing techniques.

Tens are as easy as the ones. To multiply a number times 10, write down the number and stick a zero on it. This is so simple many children already know it, but don't realize this is true no matter how large the number (which is one of the problems with teaching multiplication by tables). Make sure to throw in some large numbers to practice on. I would like to offer a caution before moving on. Some students get stuck here, because they think they need to "know" the answer before writing it down. For instance, some third grade children may not be comfortable figuring 387 x 10 in their heads, because they are just being introduced to numbers in the thousands and may not know how to say it. Make sure your student knows it is perfectly acceptable, practical and efficient to write the digits down and then read the answer. It is also more accurate, so I highly recommend making this the general practice here.

Most children have no problem with the elevens, but again I would like to offer a word of caution. Many students are taught to "just double the number", but double means to add. Make sure your student knows that, to multiply a single digit number by eleven, just write the number twice: 11 x 3 = 33, 11 x 6 = 66, 11 x 9 = 99, etc. Also remember that this only works on the single digits, one through nine. I often point out to younger children that there is no such number as "tenty ten", and 11 x 10 is not equal to 1010. Instead, we use the 10 rule, by writing down the 11 and sticking a 0 on it - 110.

This has taken a lot more print space than I thought, so I'll finish up (I promise) with a totally awesome trick for multiplying larger numbers times 11, the twelves and some general thoughts the next time. Remember, although it may seem simpler to just help your poor child memorize the facts right now, it isn't. A few extra minutes invested in learning these techniques will follow your child through the rest of school. This is part of the strong foundation in basic skills necessary for building more advanced techniques.

Hope this helps, Michele

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