PARENT TO PARENT

In the past two columns, I have explained how to introduce a student to multiplication as a calculating skill rather than a group of numbers to remember. The last two sets of multiplication facts are the elevens and twelves, which I will finish today.

First, don't forget to review. This includes the addition doubles through 24 + 24, the techniques for figuring answers to multiplication, and the 6, 7 and 8 combinations that need to be remembered.

To multiply a single digit number times 11, simply write the number twice.

I have an awesome trick to multiply two digit numbers times eleven. This is really fun for the students. You might want to grab a pencil (never pen for math, thank you) and try this on paper a couple times. To multiply a two-digit number times 11, write down the number with a small space between the digits, then add the digits and put that number in the middle. For instance, 11 x 23 = 2 3. Notice the space between the 2 and 3. Now, since 2 + 3 = 5, put the 5 between the 2 and 3, and you get 253, which is the correct answer. Again, to multiply 11 x 43, write down the 43, then add 4 + 3 = 7, and put the 7 in the middle. Your answer is 473. This works for all two digit numbers, through 99, but be very careful, since you do have to carry when necessary. For example, 11 x 58: Since 5 + 8 = 13, you would put the 3 in the middle and carry the 1 over, giving you 638. Like many of the techniques, this looks difficult in print but is unbelievably simple in practice. Here are two more examples: To multiply 11 x 54, write down 54; since 5 + 4 = 9, put the 9 in the middle and your answer is 594. To multiply 11 x 68, 6 + 8 = 14, so put the 4 in the middle and carry the one over. Your answer is 748. I hope you enjoy this as much as the students at the tutor center.

The twelves can be a bit of a challenge since there is no real technique or trick. However, by this time most students are comfortable choosing their own path so I add a couple of short cuts for them.

Remember, since a multiplication problem can be read either forward or backward, a student can choose his or her own technique to find the answer. Your student should have no difficulty figuring 12 x 1 through 12 x 5, using previously learned techniques. Also, we know that 12 x 10 = 120 by putting a zero on the end, and we can now figure 12 x 11 because 1 + 2 = 3, and that makes 132 (the eleven rule). That only leaves a few of the twelves to calculate.

In order to calculate answers to multiplication problems with twelve, it is necessary to digress. At this point, get out a clean sheet of paper (or clean off the board in the classroom), and start fresh. Begin by having your student start with 0, and count by tens to 100. As they count, write the numbers in a neat column. Make sure all the zeros at the end of the numbers line up and all the numbers in the tens column line up. This is ridiculously easy, so many students, especially older ones, look at me like I'm nuts and try to race through the numbers or skip the exercise altogether. Make sure your student counts carefully and slowly enough for you to write down their answers as they call them out. Now take a moment and point out the pattern they have created - the zero at the top repeats all the way down, and the number in front (in the tens place) increases by one each time. Once this pattern has been established start a new column next to this one, with 4 at the top. I always smile when I get here because it really surprises a lot of students. Have your child start with 4 and count by 10's to 104: 4, 14, 24, 34, 44, 54, 64, 74, 84, 94 and 104. Some students get this idea right away, some need a little help. You may have to ask your student to add 4 + 10, then add 10 again, etc., until the pattern is recognized. As always, take as long as necessary to ensure your student is comfortable with this technique. Continue creating new columns with other single digit numbers, and write the columns neatly so the pattern can be seen. Once your student is, have them count by 10 from a larger number, once. For example, start with 34 and count up 10, one time. Your student should be able to jump directly to 44, without calculating. The skill being learned is to add 10 to any number by "jumping" 10 to the next number, without actually counting. Practice this until your student can simply state the answer with little or no thought. To add 10 to 53, just count 53, 63. To add 10 to 87, count 87, 97. Take your time and practice this -a lot. This is an important skill that will come in handy from now on.

The second digression is to review counting by twos with your student. This should be simple and not take more than a few minutes.

The final digression involves the concept of Expanded Notation. This is the process of taking a number apart by digits. For example, 42 is 40 + 2, 586 = 500 + 80 + 6, etc. Most students learn how to do this in school, but make sure they are comfortable with the skill before moving on.

Once your student is comfortable with all three techniques, you can demonstrate how to calculate the twelves. Always look for a simple technique first. For instance, 12 x 1 = 12 (just write it down) and 12 x 2 = 24 (double the 12), etc. For the equations that have no quick, simple answers, simply count up the 12's tables. Since your student can now count "up 10" from anywhere, and count "up 2" from anywhere, counting "up 12" only takes two steps - simply jump 10, then 2. For example, since 2 x 12 = 24 (doubles), then 3 x 12 is just one more 12 and can be reached by starting at 24 and counting, 24, 34, 36 (jump 10 then 2). Your student can "walk up" the twelves tables this way, using only two steps each time. It is quick, simple and efficient, and most importantly accurate.

This seems like a lot of work just so your student can calculate a couple of multiplication facts, and it would probably be easier to help them memorize the answers. This is true, in the short run. In the long run, these are very important calculating techniques that I introduce at this time, that can (and should) be used regularly from this point forward.

Learning how to teach takes a lot longer than actually teaching. You, the teacher / tutor, will end up doing more work than the student, since you not only need to learn how to do the calculating, you then need to learn how to teach it to a student. The actual work that the student does is very little. As always, keep it fun. You should never work with numbers, you should play in them. Every technique introduced now will make learning future math skills easier.

Everything builds on previous skills, so if your child develops the habit of memorizing answers without skills, they will need to continue memorizing, since they will have no other base on which to build.

Enjoy the math, and enjoy your child.

Michele

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