There are several big ideas in teaching multiplication
a) Unitizing – The whole number is seen as a number of groups of a number of objects (four groups of six is 4×6). This is central for understanding place value and is the heart of learning exponents.
b) Distributive Property – Understanding the structure pf the part/whole relationship is key. Numbers can moved around and broken into smaller pieces (5×7 = 5×5 + 5×2, or with larger numbers 121×16 = 100×10 + 100×6 + 20×10 + 20×6 + 1×10 + 1×6). Think of it as a square, and you are trying to find the area of that square.
c) Associative Property – It doesn’t matter what numbers you associate together. Once they understand this, grouping becomes much more simple, and grouping is the key to multiplication (2x4x3; (2×4)x3 is the same as 2x(4×3). Get it?)
d) Commutative Property – changing the order of addends or factors does not change the product. 4×5 is the same as 5×4; though visually they would look different, mathematically they are the same. This is important for younger learners to grasp. If they see a 3×7 grid of stars and a 7×3 grid of stars, they will assume that one is bigger than the other without counting.
With this in mind, lets take a look at some specific strategies for teaching Multiplication.
This is the most basic of multiplication strategies, but a great place to start for younger learners. It can be used for small digits. It works by counting up by a number and keeping track with your fingers. 12×5 = 12, 24, 36, 48, 60!
This is very similar to counting up, the only difference is the student is taking the numbers and adding them together (9×8 = 9+9+9+9+9+9+9+9).
Doubling is a more advanced strategy that comes from repeated addition. Instead of doing 9×8 = 9+9+9+9+9+9+9+9, the student will do 18 + 18 + 18 + 18 and then 36 + 36. This is also a useful strategy to introduce a tree diagram as a way to organize the numbers in their heads. The diagram is below is for a similiar tyoe of question, but can you see how they used doubling to get to the answer?
This type of strategy is good for multiplying a single digit by a two digit multiplier (or for two digit by two digit, but for that particular strategy see the array below). This strategy shows a strong understanding of the concept of distributive property. Students will take a problem like 23×5 and break it down into smaller and more manageable parts, 20×5 and 3×5, and then add the sums together. It may even be used for three digit multipliers, 264×4 would turn into 200×4 + 60×6 + 4×4, and then add the sums together. As the numbers get bigger, the students have to store more information in their short term memory, and the problems become harder. This strategy does NOT work with a problem like 23×24, because when we have two digit by two digit, the picture gets more complicated.
For two digit by two digit problems, the students have to have an incredible sense of distributive property. Most of the time when you introduce the problem 16×16, the students will do this; 16×10 + 16×6, and then add the sums. If they can spot the problem, this is the beginning of algebraic thinking, and hopefully they will see that they answer is actually 16×10 + 16×6. Why? Lets look at our square from the problem above again, and try and think about the problem visually:
If we look at this type of problem, we can see that there are two tens columns, not one, and by breaking up the numbers into tens and ones (10×10 + 6×6), we are skipping many squares on the grid. Using this type of visual representation is key to understanding the concept, and once they have this idea down, they can begin to apply it mentally. If you listen closely to the link I posted at the beginning of the page – Arthur Benjamin doing Mathemagic – you will hear him using this exact some strategy to do 58343×58343 in his head.
This is a bit easier that the last strategy, however no less sophisticated. You simply overshoot by increasing the size of one multiplier and then compensate at the end. 17×70 is tough, so lets change it to 20×70, which is easy, it is 1400. Now, we need to take away the extra, which was 3×70 (3 more onto 17 equals 20) and then subtract that from 1400. Here is the complete string;
Much easier and completely possible to do in your head!
Partial Product Algorithm
Most parents are going to want their children to learn the traditional algorithm, because that is how they learned to do it. Also, in some situations, they will have already taught this algorithm to their children before they get to the third grade. Here is a way to take the traditional up and down method and make it based on an understanding of number theory and place value, rather than a just a rote memorization of steps. Instead of starting at the right and going left, have them start at the left and go right; this way we avoid the carrying over problem – which is the base of most multiplication errors. Essentially, what we are doing here is the same thing as a multiplication array, but in a vertical manner. Here is an example:
Notice how the student is getting the four different parts that need to be added together. They are partitioning the number into four easy to manage pieces. If the problem was a 3 digit by 2 digit problem, then they would 6 pieces. It it was 3 digit by 3 digit, they would need 9. Do you see the pattern? This pattern is key for them to get over the hump of using that traditional algorithm and moving onto more creative and quicker methods.