In my last post, I described an example of using an in-store coupon to explain addition and subtraction using a negative number. Let’s consider multiplications and divisions of negative numbers this time. I think we have many teaching materials showing how, the mechanics of working with negative numbers in math such as flipping signs based on certain rules. Here the goal is to demonstrate why. How would we show a new student why a negative and a negative make a positive? What real life situations can we use to describe ( – 200 ) x ( – 7 ) = +1400?

For this, I want to use a cell phone data story. I subscribe to a family data plan. To avoid excessive use of data, I turned off “Data” in all of our phone settings so we only use the internet at home via WiFi. However, one day, I noticed that we are still losing data faster than I expected. I’ve looked into the usage log and found that at around 2 am the night before, about 200 MB data was consumed. Apple phones have a setting called “WiFi Assist” which uses the cellular data to accelerate the internet speed when WiFi is slow. I did not know this and thus had not turn it off. Our phones have been using the cellular data to update their apps at night.

I lost (-) 200 MB data one night. Let’s say I wanted to forecast how much data I will lose in 7 days if I didn’t turn off “WiFi Assist”. I am losing (-) 200 MB a day. Advancing forward (+) 7 more days, I will lose (-) 1400 MB, or ( -200 MB ) x ( +7 days ) = ( -1400 ). See that a negative number multiplied by a positive number results in a negative number. 

Now let’s say I wanted to know how much data I had had before “WiFi Assist” consumed it. My monthly plan refreshed just 7 days ago. If about the same amount of data was consumed every day, I need to backtrack (-) 7 days to figure out how much in total I had. So ( -200 MB per day ) x ( -7 days ) gives me +1400 MB of data I had. A negative number multiplied by another negative number resulted in a positive number.

Now try a division this time. My goal is to figure out how many days I have been losing data because of this “WiFi Assist.” Assume I already know that I used to have (+) 1400 MB of data in total. I also know that I lost (-) 200 MB last night. Assuming that the same amount is being consumed every day, I would calculate: ( +1400 MB ) ÷ ( -200 MB per day) = ( -7 days). Or it was 7 days ago that I had full 1400 MB of data. See that a positive divided by a negative results in a negative number.

Now let’s forecast how many more days it will take to lose (-) 1000 MB more if I continue to have “WiFi Assist” turned on. I would calculate it as follows: ( -1000 MB ) ÷ ( -200 MB per day ) = ( +5 days ). 5 more days need to advance forward before I lose (-) 1000 MB more. Note this time that a negative number divided by another negative results in a positive number.

Working with negative numbers increases the awareness of your starting position whereas such awareness may not have existed before. Knowing where you are now is an essential piece of information to solving many problems. That’s the Theory of Negativity introduced by A. Neinstein. (ha, ha, ha)

I hope this example is helpful. You may also have good examples to share. If you do, please share it in the comment section.