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Multiply 2 digits by 2 digits

For the time being we're going to consider the problem 23 x 36.

After gaining familiarity with single by double digit multiplication, most students skip dividing beans and move directly to creating arrays and using grid paper.  The only difference between splitting these arrays and the small ones we started with is that I need to split it in two directions to make really easy problems. 

I can split it vertically like I did before, but it still leaves me with multiplying by 23.

If I split the array in two directions then I have four easy problems.  Now I can multiply the sides of each smaller box to figure out how many blocks are inside those smaller boxes and add them together at the end to decide how many blocks are in my total 23 x 26 box.

I can also use grid paper to show my array, which is nice because moving that many blocks is annoying!
I draw a box that is 23 squares on one side and 36 squares on the other.  I split my box using the place value so I have side lengths of 20 and 3, 30 and 6 - four smaller boxes instead of one big one.  I can multiply to figure out how many squares are inside each small box: the top right box is 20 squares by 30 - so there are 600 squares inside it.  I repeat that step with the other three small boxes, the add the squares inside each one together to find out how many squares are inside the big box.

The same idea works with an open area model, I am just less concerned about the proportions of my representation.  Everything else stays the same.

Moving even farther away from the concrete image of the blocks I can manipulate the just the numbers.  In these distributive examples I split both numbers up and make sure that I multiply each part of the first number by each part of the second number.  Keep in mind this is one strategy shown in two different ways.

This video shows a partial products strategy.  It follows the same rules about splitting up the numbers before we multiply, it simply skips the step of rewriting the problem which makes it more efficient.