Some people wonder why kids of different ages have difficulties multiplying numbers. I think one of the explanations is coming from the way multiplication is being taught in school.
If teachers could teach multiplication in different ways besides the traditional method, kids will find Math class more interesting and be able to understand why multiplication is so important. Most of the teachers prefer the traditional method because it is the most used and described in literature.
Even so, there still are kids that struggle with multiplication tables and have a difficult time understanding the concept of multiplication that was taught to them in elementary school. We can also find kids in middle school that often use addition to solve a multiplication problem. It is interesting to see that some children like to try and use other methods besides the traditional method because they only require the basic skill of counting that was mastered at an early age. I think it is a great idea to expose children to different multiplication methods until they become confident in knowing the multiplication facts and understand the concept of multiplication. I think when teachers use different methods of teaching multiplication kids could deviate from memorizing the multiplication facts and make sense of numbers.
Last year, one of my students got very excited because he found out how Asian kids multiply whole numbers with multiple digits The method may not be for everyone but I am sure there will be some kids that will be willing to give it a try at least for fun!
Multiplying Asian Style
36 x 45
Step 1: Draw three lines grouped together for “3”, or the tens place value, and six lines grouped together for the “6”, or the ones place value at a distance from each other (green lines)
Step 2: Draw four lines grouped together for “4”, or the tens place value, and six lines grouped together for “5”, or the ones place value a distance from each other (purple lines). THEY MUST “OVERLAP” with the green lines for number 36.
Step 3: Look at the last digits at the ones place value for both numbers: “6” and “5”. You will see 6 GREEN lines that overlap 5 PURPLE lines. Draw a curved line to show the grouping.
Step 4: Count the points or “dots” at the intersection of all these lines and then write their total on the side. The “last digit” of this sum will represent the last digit of our product.
Step 5: Draw another curved line in the “opposite side” of our first group of dots because that total will represent “the first digits” of our product.
Step 6: Now you have to worry about the other TWO groups of dots that are located in opposite side from each other. Count the dots of the first group, ADD the number of dots from the second group, AND add the number at the tens place value from the intersection of the 6 “green” lines and 5 “purple” lines which is “3”. “Zero” will be the last digit of the product of the problem and “3” will be “carried over” to the total of dots from the middle sections located opposite from each other. Count ALL dots together from the middle sections including “3” that was carried over (“orange dots”): 15 + 24 + 3 = 42. Keep “2” and carry over “4” clockwise to the last group that has a curved line.
Step 7: Count the dots from the section with the curved line AND “add” 4 from “42”. You will get “16”.
Step 8: You will write “16”, “2”, and “0” as “1620”, place comma for place value and get in the end the product of 36 x 45 = 1,620!
Step 9: If you don’t believe the answer verify it using the traditional method or a calculator!
Let’s try another example: 23 x 44 (23 “green” and 44 “pink”)
I am sure that you are wondering: what about numbers with more digits than 2? The method works in the same way. Now you will be able to notice some patterns: the dots are going to be combined “in diagonal”, you will keep the last digit of the total and carry the number for tens place value to the next set of dots clockwise that are also placed in diagonal. Here’s an example:
231 x 425
Multiplying Using the”Lattice”
“Lattice multiplication” first was introduced to Europe by Fibonacci (Leondardo of Pisa), whose 1202 treatise Liber Abacii (Book of the Abacus) was the most sophisticated work on arithmetic and number theory written in medieval Europe (Google search of “who invented lattice multiplication”). “Lattice multiplication“, is also known as gelusia multiplication, sieve multiplication, shabakh, Venetian squares, or the Hindu lattice (Wikipedia). Some children in the United States are currently using this method . It is usually taught in elementary school. I teach it to my middle school students that struggle with the multiplication tables and the traditional method especially when they have to align the digits of the numbers that are being multiplied. Once they noticed how easy it was to multiply large numbers using the “lattice”, they would not want to use the traditional method anymore. The advantage of this method is that no matter how many digits the numbers have the technique remains the same. Let’s look at the following problem:
245 x 57
Step 1: Draw a grid with 3×2 squares according to the number of digits in the problem. I shade the “ends” of the grid because I do not want kids to get confused about the placement of the numbers.
Step 2: Write the large number at the top and the smaller number on the right side. Some people place them on the left side but the technique is the same. “Split” each square with a “diagonal” line towards right.
Step 3: Multiply 2x 5 = 10 and write 1 in the left half of the square under 2 and o in the right half of the same square. Continue multiplying 2×7=14. Write 1 in the left half of the square and 4 in the right half of the same square.
Step 4: Continue multiplying in the same pattern. The tens of the answer will go in the left side of the square and the ones will go in the right part of the square.
Step 5: Color the diagonal lines of the entire square (orange lines) to show what numbers are found in each diagonal and add the numbers in each diagonal. “Carry over” if the sum is greater than 10.
Step 6: Read the digits from left to right: 1 3 9 6 5 and place comma for place value. Your answer is 13,965!
Multiplication With “Napier’s Rods”
“Napier’s Bones”, also called “Napier’s Rods”, are numbered rods which can be used to perform multiplication of any number by a number from 2 to 9 (http://mathworld.wolfram.com/). This method allows kids to easily multiply large numbers and enjoy multiplying! They need to have “strips of paper” already made up for each number from 2 to 9 that are written in the top square, with the multiplication table underneath. This method is similar to the “Lattice Method”. Each strip shows squares that are were divided by a “diagonal” inclined towards left side. Some people draw the diagonal towards right side. In our example, each square will have the tens in the left half and the ones in the right half of the square.
Let’s try a problem:
478 x 23
Step 1: Arrange the strips as follows: multiplier, 4, 7, and 8.
Step 2: Break down 23 in 20 + 3
Step 3: You will have two problems to solve: 478 x 20 and 478 x 3. Let’s multiply 478 x 2. You will “add a zero” at the end of your product. This problem will look like this (blue lines). Just like in the lattice method, you will add in “diagonal” from right to left because the diagonals are leaning towards left. The last number will be 6, then move towards left with a total of 4 + 1 = 5, 8 + 1 =9, and 0. You will read the number from left to write: 956 and place a zero next to it for multiplication by 10. So, 478 x 20 = 9,560.
Step 4: Now, you will multiply 478 x 3. You will add in diagonal just like in the lattice method from right to left. Your last digit will be 4, then 2+ 1 = 3, 2 + 2 = 4, and 1. Read the number from left to right: 1, 4 34 (orange lines).
Step 5: Add the two products to get the final answer: 9,560 + 1,434 = 10, 994
Is Anybody Still Using the Multiplication Used In Ancient Egypt?
This method is still used in many rural communities in Ethiopia, Russia, the Arab World, and the Near East (http://www.atozteacherstuff.com/). Some teachers in the United States are teaching this method to their students to show them that there is another way that can help them with problems multiplying numbers (“Adam”, Ninth Grade Teacher – http://www.atozteacherstuff.com/).
First, you arrange the problem with the larger number listed first so you will do less work. The first column will show how you get from 1 to 2 , from 2 to 4, from 4 to 8 and so on using addition. The second column shows multiples of 29 also calculated using addition: 29+29=58, 58+58=116, 116+116=232, 232+232=464, 464+464=928. By looking at the table you will notice that 32+8+4+2=46, which is one of the two numbers. In order to find out the product you will use the corresponding value of each of these numbers according to the table. For 32 we will have 928, for 8 we will have 232, for 4 we will have 116, and for 2 we will have 58. The sum of all these numbers will give us the answer.
***Note: I wrote that each number will “double” its value to get to the next number up by showing “multiplication by 2” but in reality it is addition since the ancient Egyptians did not know the concept of multiplication.
There are a lot of multiplication methods out there besides the traditional method. Some have been around for hundreds of years and other represent variations of other methods. I chose the ones that I thought kids can use on a regular basis, like “Lattice Multiplication”, help them practice multiplication in different ways, and get them motivated to learn Math. Some of my students use the traditional method for multiplication of small numbers and the “lattice” for large numbers.”Napier’s Rods” is a good method but it requires “rods” usually made of Popsicle sticks or strips of paper already cut out. I enjoy when my students “play” games using these methods and compete with each other to find out who is the fastest AND the most accurate. I hope you will like them, too!