For the time being we're going to talk about the problem 5 x 23.
The most basic definition of multiplication is counting equal sized groups, so 5 x 23 really means I have 5 groups of 23 or 23+23+23+23+23. Initially students tend to show this with beans counting each group by ones to decide the total number in all - accurate but not the most time effective.
Eventually students discover that they can use base ten blocks to represent the same numbers and piles of beans are replaced by blocks allowing students to county by tens - a little more efficient.
With some questioning and redirecting, we move from a group mentality to showing the problem in arrays - square patterns of blocks. The same blocks are used, but instead of five groups of blocks I have five rows with 23 blocks in each row - I just have to figure out how many blocks my box is made of. This doesn't do much initially, students are still counting individually blocks instead of really multiplying.
Then they discover that they can split their array into two - it doesn't change the amount of blocks they used just to move them around. Now we're getting somewhere! Since we have previously discovered the zeros trick in multiplication (when I am multiplying by a number that ends in zero then I can ignore the zeros and tack them back on at the end: 20 x 100 is the same as 2 x 1 with three zeros on the end) this creates two very easy multiplication problems. All they have to do is figure out how many blocks are inside of each array and add the two together.
When they understand the idea of moving the blocks around we use grid paper instead.
The next step on our progression is called an Open Area Model and it is described in the video below.
It doesn't take long after they get the hang of an open area model for students to be ready to move on without a picture. Their first picture-less solutions usually look something like this distributive model.
Not all students use this exact distributive method, they have the freedom and flexibility to structure it however it makes most sense to them. The important part is that they're transferring their understanding of the more concrete models into numbers without a picture.
The fancy mathematical name for the final method we teach is called the "partial products" products method because we're once again finding parts of the final product and then adding those parts together.
Since we’re focusing so much on understanding what is going on instead of just manipulating numbers, this method is the closest we teach in fourth grade to the traditional method of multiplying. In reality the traditional method just combines the multiplying step with the addition step so you do them both at the same time. Some students will still choose to use the traditional method and that’s great as long as they understand what they’re doing.
This last video shows multiple methods of solving the largest size of single-digit multiplication problem students will see this year.
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