A Multiplication "Trick" To Know When You Are Multiplying by 11

Knowing how to do math does NOT require magic; although, sometimes working a problem can appear to be done magically.  This week I want to talk about multiplying by eleven. Before I demonstrate the "trick", I have to get on my soap box for just a moment. In my humble opinion, all students should know their times tables through 12 even though the Common Core Standard for third grade says through 10 x 10. Remember, Common Core is the minimum or base line of what is to be learned. In Algebra, I insist that my students know the doubles through 25 x 25 and the square roots of those answers up to 625. It saves so much time when we are working with polynomials.

Now to our our amazing mathematical "trick". Let's look at the problem below which is 231 x 11.

 

First we write the problem vertically. Next, we bring down the number in the ones place which in this case is a one. Now we add the digits in the ones and tens place which is 3 + 1 and get the sum of four which is brought down into the answer.


Moving over to the hundreds place, we add that digit with the digit in the tens place 2 + 3 and get an answer of five which we bring down. Finally, we bring down the digit in the hundreds place which is a two. The answer to 231 x 11 is 2,541.

Now try 452 x 11 in your head. Did you get 4,972? Let's try one more. This time multiply 614 by 11. I'm waiting...... Is your answer 6,754?

Now it is time to make this process a little more difficult. What happens if we have to regroup or carry in one of these multiplication problems?

We will multiply 784 by 11. Notice that we start as we did before by just bringing down the number in the ones place. Next, we add 8 + 4 and get a sum of 12. We write down the 2 but carry or regroup the one. We now add 7 + 8 which is 15 and then add in the 1 we are carrying. That makes 16. We bring down the 6 but carry the 1 over. We have a 7 in the hundreds place, but must add in the one we are carrying to get a sum of 8. Thus our answer is 8,624.

Let's see if you can do these without paper or pencil. 965 x 11 768 x 11 859 x 11 After working the problems in your head, write down your answers and check them with a calculator. Try making up some four and five digit problems because this is a non-threatening way to have your students practice their multiplication facts. Have fun!

Skip Counting and Learning How to Multiply Using Pattern Sticks

Most elementary teachers use a Hundreds Board in their classroom.  It can be used for introducing number patterns, sequencing, place value and more. Students can look for counting-by (multiplication) patterns. Colored disks, pinto beans or just coloring the squares with crayons or colored pencils will work for this. Mark the numbers you land on when you count by two. What pattern do they make? Mark the counting-by-3 pattern, or mark the 7's, etc. You may need to print several charts so your students can color in the patterns and compare them. I usually start with the 2's, 5's and 10's since most children have these memorized.

On the other hand, the Hundreds Board can also be confusing when skip counting because there are so many other numbers listed which easily create a distraction.  I have found that Pattern Sticks work much better because the number pattern the student is skip counting by can be isolated. Pattern Sticks are a visual way of showing students the many patterns that occur on a multiplication table.  Illustrated below is the pattern stick for three. As the student skip counts by three, s/he simply goes from one number to the next (left to right).

Martian Fingers

For fun, I purchase those scary, wearable fingers at Halloween time. (buy them in bulk from The Oriental Trading Company - click under the fingers for the link.) Each of my students wears one for skip counting activities. I call them the Awesome Fingers of Math! For some reason, when wearing the fingers, students tend to actually point and follow along when skip counting.

Most students enjoy skip counting when music is played. I have found several CD's on Amazon that lend themselves nicely to this activity.  I especially like Hap Palmer's Multiplication Mountain.  My grandchildren think his songs are catchy, maybe too catchy as sometimes I can't get the songs out of my mind!

$3.25

Think about this.  As teachers, if we would take the time to skip count daily, our students would know more than just the 2's, 5's and 10's.  They would know all of their multiplication facts by the end of third grade. And wouldn't the fourth grade teacher love you?!?

IMPORTANT:  If you like this finger idea, be sure that each student uses the same finger every time to avoid the spreading of germs. Keeping it in a zip lock bag with the child’s name on the bag works best. (Believe it or not, when I taught fourth grade, the students would paint and

decorate the fingernails!)

To help your students learn their multiplication facts, you might like the resource entitled Pattern Sticks. It is a a visual way of showing students the many patterns on a multiplication table. It also teaches how you to use the pattern sticks to recognize equivalent fractions, reduce fractions, and to change improper fractions to mixed numbers.

Looking For and Analyzing Mathematical Patterns

Mathematics is the science of patterns.  In this post, I want to target some mathematical problems in which we investigate developing patterns.

In the first example below, you will notice we begin by multiplying one by one; then 11 by 11, and so forth. Each time we multiply, the number of digits in the multiplier and the multiplicand increases. Do you see the pattern that progresses in the answer (product)? Notice how this multiplication pattern forms a triangle? Can you figure out what kind of triangle this would be if we added a "peak" or a row at the top?

Here is another interesting pattern. In this one, instead of multiplying by 1, then 11, then 111, the answer (product) looks like the multiplier in the pattern above. Do you notice anything else significant?

Yes, we are multiplying by 9 each time. Now look at the number being added, and count the number of ones you see in each answer. Surprised? Isn’t it amazing how math is ordered, methodical and precise? Maybe that is one reason I love to teach it! Encourage your students to look for and make sense of math patterns and structure in order to deepen their mathematical understanding and retain what they learn.

Dots Lots of Fun - Using Dominoes in Math

I am always looking for ordinary items that can be used in the classroom as manipulatives. I'm a firm believer in the Conceptual Development Model which advocates teaching the concrete (using manipulatives) prior to moving to the pictorial before even thinking about the abstract. When I was at the Dollar Tree (a great, inexpensive place to purchase school stuff) I saw sets of dominoes for $1.00 each. Since they were inexpensive and readily available, I decided to create several math activities and games to introduce, reinforce, or reteach math concepts.

The Number 52

Think about it; if you lay a domino horizontally, you have a two digit number. Put two dominoes side-by-side, and a four digit number is created. Now you can work with place value, estimation, or rounding.  How about lining up dominoes in a column, and working on addition (with or without regrouping) or subtraction (with or without renaming)? 

Another perfect domino activity is practicing addition or multiplication facts.  How about adding the two sides of the domino or multiplying the two sides together?

The Fraction 1/4

If a domino is placed vertically, you immediately have a fraction.  Placed one way it is a proper fraction, but rotated around, it is an improper fraction which can then be reduced.  A fraction can also be changed into a division problem, a ratio, a decimal, or a percent.

So think outside that box of dominoes and use them as an inexpensive math manipulative because Dots Lots of Fun!

Check out all my Domino Resources available on Teachers Pay Teachers.

The first two are absolutely FREE!

1. Dots Fun for Everyone - FREE  Three math activities and one game for the intermediate grades.
2. Dots Fun - FREE  Three math activities and one game for the primary grades.
3. Dots Fun   A 24 page resource for grades 1-3 that includes 13 math activities and four games.
4. Dots Fun for Everyone  A 29 page resource that features 15 math activities and three games for grades 3-6.
5. Dots Lots of Fun  Seven math games that use dominoes for grades 2-5.

Quick Times - Multiplication Tricks

I am always looking for different strategies when working with my remedial college students since many of the ways they were taught to do math aren't working for them.  I came across this "Quick Times" method and thought it would be another approach I could share with my mathphobics for multiplying.  They love anything that is different, quick and makes them look astute when doing mathematics.

Let's assume we have the multiplication problem of 41 x 12.  In the Quick Times method, first start by multiplying the first digit of 41 by the first digit of 12 to get the first digit of our answer.  We then multiply the second digit of 41 by the second digit of 12 as seen below to get the last digit of our answer (the ones place).

Now we need to find the middle digit of the product.  This is done by multiplying the outside digits, then the inside digits, and adding those two products together as shown below.

This quick method will only work when multiplying two digit numbers by two digit numbers, but it does cause the students to do mental math.  My students like the challenge of doing all of the computation in their heads.  Let's try another one that is a little different.  Let's do 63 x 41.  Again we multiply the first digit of each number and then the second digit of each number to get the first digits of the answer and the last digit of the answer.

As before, multiply the outside digits, then the inside digits, and add the two products together.

Now we must put the 18 into the middle spot, but there is only room for one digit in the tens place.  YIKES!!  What do we do now?  Very easy....because we can only have one digit where the question mark is, we must regroup (carry) the one in the tens place of the 18 and then add it to the 24.

Have you figured out the final answer?  It is.....

You are probably thinking the old method works so much better, but that is only because that is the method you are use to using.  Why not try the ones below using the Quick Times method and see if you get the correct answer.  Use the old method or a calculator to check your answers or go the the answer page above.

a)  36 x 21       b)  24 x  12      c)  48 x 29       d)  59 x 18       e)  63 x 13     

The Power of Math Tricks

Math tricks will never make you a great mathematician, but in the eyes of some, you can be a fantastic math-a-magician. My college students love it when I show them a trick they can then take home to amaze and impress their peers, parents, children or the best yet - their spouses. 

Remember when I demonstrated how to easily multiply by 11 in the post The Eleventh Hour?  Or how about the trick of multiplying by 12 in Quick Times?   Here is a new one I recently showed my students.

First, let's look at a problem where a number is divided by the decimal .25. 

The above example is really 9 x 4 which is 36, but why is this true?  Hopefully your students know that .25 is equivalent to 1/4; so this problem can be reworded as  9 divided by 1/4.  As seen below, when dividing a whole number by a fraction, we find the reciprocal of 1/4 and then multiply which gives us the answer of 36.

 

Based on the sample above, anytime a problem requires dividing by .25, simply multiply by four to get the correct answer.  Try these without using a calculator or paper and pencil.





Instead of using the reciprocal to divide fractions, I teach my students that this is the "cross" method. Simply look at the original problem and cross multiply as seen in the illustration below. 



Fractions for the
Confused
and Bewildered

First multiply the bottom right denominator with the top left numerator. (4 x 9)  Next multiply the bottom left denominator with the top right numerator, (1 x 1) and you get an answer of 36.  When doing the fractions this way, there is no confusion on the students' part about which fraction to invert. If you would like a more details on how to divide fractions this way, go to the post entitled: Don't Flip!

If you are interested in other alternative ways to teach the four operations of fractions, you can check out the resource on your right.


By the way, the answers to the above problems are  a) 24    b) 80    c) 72    d) 380.    How did you do?