Fluently multiply multi-digit whole numbers using the standard algorithm.
The standard method of multiplication is based around the idea of partial products.
Let's look at the related addition problem. If you are adding 73 four times, you will be adding 3 four times, which can be represented by 3 x 4. You are also adding 7 four times, but since the 7 is in the tens' place, you are really adding 70 four times, which can be represented by 70 x 4. If you add these two partial products together, you arrive at the answer 292.
In the same way, the standard method of multiplication relies on the idea of partial products.
First, we will multiply 3 x 4, carrying over the tens' digit, as we did in the addition problem.
Then, we will multiply 7 by 4.
The same general principle applies to 2 digit x 2 digit multiplication. In this addition example, we find 3 x 12, then 10 x 12.
First, we find 12 x 3 = 36.
Then, find 12 x 10. The "placeholder zero" that we write beneath the 6 in 36 signifies that we are multiplying by 10, not 1.
Finally, we add our two partial products together.
3 digit by 2 digit multiplication follows a similar pattern.
If we look at the matching addition problem, you will see that we need to figure out 3 x 12, 20 x 12, and 100 x 12 in order to find the sum.
When we set up the multiplication problem, however, we are first finding 123 x 2, then 123 x 10. Instead of breaking up 123 into 100 + 20 + 3, we instead are breaking up 12 into 10 + 2. The first step is to multiply 2 x 3, then 2 x 2, then 2 x 1, as indicated by the arrows. The reason why we multiply in this order is the same reason why we add in that order: we start with ones, then go to tens, then hundreds, etc.
Next, we multiply 123 x 10. Since we are multiplying by 10, not 1, we need to write a placeholder zero below the 6. Then, multiply 1 x 3, 1 x 2, and 1 x1. Just like in the previous step, we start with the lowest place value, and work our way to the highest place value, in order.
Finally, we add our two partial products together.
The area model of multiplication is based on the idea that finding the area of a rectangle is closely tied to the concept of multiplication.
This rectangle can be split into two rectangles. It's easier to find the area of the two new rectangles, because they use numbers that are easier to calculate mentally.
The diagram below shows how the model is written out by a student. On the top, 20 and 3 are marked because 20 + 3 = 23. Multiplying 20 x 6 is as easy as multiplying 2 x 6 and adding a zero. Multiplying 3 x 6 comes next.
After multiplying to find the area of both rectangles, simply add the two products together to get the answer, 138.
Here is another demonstration that helps students understand this concept. If we are multiplying 23 by 6, that is the same as adding 23 six times. When you add this number repeatedly, you are essentially doing the same multiplication described in the area model above.
How does this model apply to larger numbers? In the following two examples, you can see how the larger rectangle can actually be split into 4 rectangles, all with easy-to-work-with factors and products. This is used for demonstrative purposes. Keep reading to see how a student would write out this work!
When students write out the work for a 2-digit by 2-digit multiplication problem, it typically looks like this. When students are first working with the area model, it helps for them to remember to look "up, then over" to determine which two numbers they multiply together in each box. For example, in the top left box, a student would look "up" to 20 and "over" to 10, multiplying 20 x 10 to get 200. After finding all four products, the student will add the four numbers together to get the final product.
Please see below for more examples of how the area model works with different numbers of digits.
Please click here to view an animation of the area multiplication model.
