Let’s face it – we all love magic tricks. There is something mysterious about the unknown that intrigues us. And combining mathematics with magic to make magic math tricks is even more mysterious. But how is multiplying by 11 a magic trick?
What starts out as a common multiplication problem, slowly turns into a rule that becomes evident when you do a few of them. A pattern starts to unfold that students, if guided a little, can figure it out. As a result, there is no need to do the working for multiplying by 11.
Let’s Start Multiplying by 11
First, give a multiplication problem to students by writing it on the board. Let’s use 43 x 11. Set up the problem in the normal say of multiplying whole numbers and draw the line underneath.
Next, go through the procedure with the students asking them each step of the way what to do. Most students will be able to follow the routine of starting with the ones column and multiplying it by each number above it. As they move to the tens digit, they will place the ‘zero’ as always at the right and then multiply by the ones and the tens.
Finally, they will add up the numbers and get the answer. This may have taken us a minute or two depending on what grade level you do this with. The answer is 473. Bravo!
Do It Again
Once you praise the students for getting the right answer, tell them since math is so easy and so much fun, we are going to do another problem with another two digit number that we will multiply by 11. Students will find this easy, because at each step, they are multiplying the digits above it by 1, resulting in the same answer.
Let’s try now the problem 27 x 11. Set up the problem in the normal say of multiplying whole numbers and draw the line underneath. Repeat the same procedure asking students what to do each step of the way and write the digits in place.
The answer to this problem is 297. Well Done! Repeat a few times with other problems like 31×11, 45×11, 52×11.
(the two digit number you multiply by 11 has to be two digits such that their sum is less than 10).
Analyze the Questions and the Answers
After you do a few problems, stop and ask the students for each question to look at the digits in the question and look at the digits in the answer. As them what patterns they see forming in each of the question and answers. Use the word ‘pattern’ as a part of the discussion so that they hear these mathematical words being linked to numbers.
They should immediately see a few things:
1. The first digit in the question is also the first digit in the answer.
2. The last digit in the question is the also the last digit in the answer.
3. The answer at each step of the question is repeated and written under each other, the second row has the same digits shifted to the left.
They may take a little more time to figure out where the middle number comes from. But after a few moments, there will be some student that realizes that the middle number in the answer is simply the SUM of the two digits in the question.
Like the sounds when you see a magic trick done by a magician or at a party, you will hear a lot of “ohhhhhhh” around the room as if some magic has been done with numbers.
Make a Rule For Multiplying By 11
Now ask students to come up with a RULE for multiplying by 11 and write it as a class. Write the rule together and have students take it down in their notebooks for later. Once they understand that there is no need to do any working, but rather:
- 1. Keep the First Digit
- 2. Sum the first and second digits and place it next
- 3. Keep the Last Digit
The answer can now be done in their heads.
Have Students Test Each Other
Now that students have the rule, they should be able to wow each other with this new magic trick they have learned.
- Group students into pairs
- Have Student A write down and work out on their own piece of paper the following problems: 33×11, 24×11 and 63×11.
- Have student A ask student B the first problem
- Student B should answer the question in the their heads without doing any working, but using the rule that they just learned.
- Once completed, Have Student B write down and work out on their own piece of paper the following problems: 35×11, 15×11 and 42×11.
- Have student B ask student A the first problem
- Student A should answer the question in the their heads without doing any working, but using the rule that they just learned.
- If a student stumbles along the way, they can write down the problem on the paper to assist them but shouldn’t do the working.
This will create the excitement that is needed in the classroom to cause students to enjoy mathematics. Who would have though that multiplying by 11 could be so much fun.
Ask students to try this out with their parents and friends when they get home. You will hear the stories about how much they impressed their parents throughout the term.
Get More Resources
Many more math ideas and resources are available at the proteacherresources home page along with other worksheets, reviews and videos. Many exciting articles and researched strategies for making math fun in the classroom can be found in Jo Boaler’s awesome book entitled Mathematical Mindsets.