Using mental math vs. paper and pencil

The example of decimal division I found recently made me think some more about how we teach our students the use of mental math.

For example, how would you want your (current or future) high schooler to find the answer to 5 × 24 or maybe 6 × 71 and such like?

a) I wish he'll use pencil and paper, put 5 under 24 and use the algorithm.
b) I wish he will go 5 × 20 and 5 × 4 in his head and add those.
c) I wish he grabs the calculator.
d) I wish he asks me.

Also think WHICH of the four options is the most efficient 'tool' for finding the answer? After all, just like a carpenter, a good problem solver knows which tools are available and which one is the best to use for a particular task.

Now let's back up in time when the student is in third grade. How are we encouraging third graders to do the same problem? As you may realize, kids usually spend lots of time mastering the multiplication algorithm (the pencil & paper method) and go thru tons of practice problems similar to my examples.

But, what if we FIRST let the third-graders do such problems mentally, and only afterwards taught the algorithm?

Wouldn't that make them more apt in using their head for such problems instead of the long algorithm?

(And they'd also understand the logic behind the algorithm better, I think.)

I tried to check and I didn't find any online activities for this exact thing. I even tried searching for 'distributive property quiz' (since the mental calculation is based on distributive property) but that only brought up algebra type things.

Obviously you can just practice by calling out numbers to your students, but if you want to have a worksheet, you can make such worksheets here: choose multiplication, horizontal, value 1 to be from 2 to 9, and value 2 to be from 11 to 99 - and hit SUBMIT.

You can make easier ones, too, if you choose value 1 to be from 2 to 5, say. Or, choose value 2 to be from 10 to 100 with step 5.

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