In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Donna House. Her topic, from Pre-Algebra: adding a mixture of positive and negative numbers.

**How can this topic be used in your students’ future courses in mathematics or science?**

Version 1

It would be difficult to do well in any future course in mathematics or science without understanding the concept of adding and subtracting positive and negative numbers. This concept is used for temperature, altitude, growth and decay, magnitude, distance, size, profit and loss, and many other topics.

An example from physics would be the students solving a problem involving force. They need to discover how much drag force is needed to stop a drag racing car at the end of the track. The forward force (speed) of the car is positive and needs to be “cancelled out” or reduced to zero in order to stop the car. The students will need to determine if the regular brakes (negative) on the car are sufficient to stop the racer in time, or if additional drag forces (negative) need to be added.

Version 2 (Written as an Engage)

How much force is needed to stop a drag racing car? If you do not stop the car in time, it will crash into the wall, or the fence, or maybe even the water tower and then where would you declare your undying love to Betty Sue? If you stop the car too quickly, you will lose the race and Betty Sue won’t love you anymore. And all this happened because you did not know about adding and subtracting positive and negative numbers!

To stop that racing car you will need to know how much drag force is needed to cancel out the forward force (speed) of the car. Since the forward force is positive, the drag force is negative. But the regular brakes may not give enough drag force to stop the car in time. You may need to add some negative numbers! Remember, your entire future with Betty Sue depends on adding positive and negative numbers!

**C1. How has this topic appeared in pop culture?**

What if you thought you won the lottery, but found out you were wrong? In November of 2007 a scratch-off lottery card game in England was pulled from the shelves because customers did not understand the concept of negative numbers. Many people tried to claim their prizes only to be told they did not win. Why was there so much confusion? The cards involved negative numbers!

The “Cool Cash” scratch-offs had a cute picture of a penguin on the front. One scratched off windows trying to reveal a temperature that was lower than the temperature revealed on the card. All of the temperatures revealed were below zero and had negative signs. The problem was that many people could not understand whether -6 was larger or smaller than -8. What do you think?

http://www.manchestereveningnews.co.uk/news/greater-manchester-news/cool-cash-card-confusion-1009701

**D3. How did people’s conception of this topic change over time? **

Negative numbers have struggled for recognition since ancient times. Negative numbers were ignored by mathematicians for centuries, and considered to be false, non-existent, or simply absurd. Whenever a solution was found to be negative, it was discarded as nonsense. They got no respect. Eventually, the concept of adding and subtracting negative and positive numbers was used to indicate debt and payment, but not much else. Mathematicians just did not quite understand what negative numbers are, even though the negatives cried out to be seen as real numbers.

As time passed, negative numbers began to be recognized as useful, but were generally considered imaginary. They were not accepted as real numbers until the middle of the 18^{th} century, but were still commonly ignored as solutions. The negatives protested peacefully. In the 19^{th} century, negative numbers were finally accepted, but still not widely liked. However, their usefulness caused them to be recognized and they happily indicated the weather, the distance below sea level, and whether or not a golfer’s score was below par.

Today negative numbers have a very good relationship with positive numbers, and are loved by many people. The usefulness of adding and subtracting positive numbers cannot be denied. (Just try to break a world record in racing without this concept!)

http://webspace.ship.edu/kgmcgi/m400/Presentations/Chapter 5 Something Less Than Nothing.ppt

http://en.wikipedia.org/wiki/Negative_number#History