Using a Number Line for Mental Math

I am sure there are multiple definitions out there. Growing up, I thought mental math was doing the exact same work I would have done on paper but doing it in the air above my head so I mentally “saw” the work. My brother, on the other hand, actually decomposed numbers and saw the patterns to make his math go twice as fast as mine.

Here is an example to try.

- On paper, multiply out 364 by 3. Time yourself.

- Now try to do the same process in your mind. Stare at the ceiling or a blank wall and imagine the work your just did. Repeat every step. Time yourself.

- Now on paper or mentally add up 300(3) + 60(3) + 4(3). Time yourself.

Look at the times. While, yes, we did the same exact problem all three times, the processes were different. Which was faster? If you look back at the third example, what pattern do you notice? Why were the 300, 60, and 4 broken up? How did I know to break them up that way?

Practice it that pattern – Pick paper or mental math, and multiply 423 by 5.

Did you feel more confident? Did you break down the 400, 20, and 3 by their place values when multiplying by 5? THAT is the mental math my brother did. Recognizing the small patterns is not necessary in math, BUT it sure saves time and helps you avoid human error in clicking the wrong button on a calculator (we have ALL done it – even math teachers).

Let’s combine those small patterns that help us mentally with a number line.

What number lines do you see around you? What shape do you mentally bring up when you think of a number line? Most textbooks, and also when I teach number lines, we think left to right and horizontal. I do like this for notes since we read from left to right, so it makes the most sense. Or does it?

The number line I hold mentally is like a THERMOMETER and is vertical up and down. Negative values are lower; positive values are higher. Subtraction or a negative means I move down. Addition or a positive means it is getting warmer, so I move higher up.

Bank balances rise or lower. To owe someone means you have worked yourself “into a hole”. Holes are below ground, so the vertical number lines make more sense to my brain when I do math.

Try this with your child:

You have $14 and spend $5. How much do you have?

Yes, this is subtraction, but what number line did you think of? Were you starting and 14 and moving LEFT 5 or DOWN 5?

Try thinking LEFT on this one:

You have $3 and paid $5 to your friend for their new gel pen. How much do you owe them tomorrow?

Did thinking left help or confuse you? Read it again and try thinking DOWN.

Come up with your own scenarios as a family. When do you see numbers? Money? Temperature? Adding cups or taking teaspoons out of a recipe? Create your own list of at least 5 questions and practice visualizing your new number line. The more you practice, the sooner this mental tool will become a reflex!