To introduce teaching reciprocals of numbers and terms I begin the lesson explaining that everything has an opposite. The opposite of shutting a door is to open it. The opposite of saying hello is to say goodbye. Numbers have opposites too we call them reciprocals.
To start the lesson I say to the class,
“I want you to discuss what you think might be the opposite of 2 and you must explain why you think this.”
In your explanation try to use diagrams such as the place value table or a number line to back up your argument. I asked the class to present their reasoning on mini-whiteboards for me and other students to read.
The most common response at this point is to say negative two. We discuss how this could work on a number line.
Negative two is the opposite of positive two because it is the same magnitude of distance away from zero on the number line. This makes sense. Or does it?
If the reciprocal of a number is the same distance of that number from zero what is the reciprocal of zero? If zero represents ‘nothing’ or no place value how can it be positive? If zero cannot be positive, then how can the opposite of zero be negative?
I encourage students to think some more.
After a short time, we discuss how the number two can also be written as a fraction . The almost immediate response now is the opposite of two is one half because you can flip the fraction. This makes sense. I remind the class we thought we had it last time but got stuck on the opposite of zero.
If we take the reciprocal of a number to be the flipped fraction when the number is written over 1. Then the opposite of zero must be infinity. When you write zero as and you flip it to make the question now becomes how many zeros go into 1? Infinite zeros do. The reciprocal of zero is therefore infinity. More simply – the opposite of nothing (zero) is everything (infinity).
Now we understand the reciprocal of a number to be defined as one divided by that number we can look at more complex values such as finding the reciprocal of ordinary and top-heavy fractions, mixed numbers, decimals, powers and even algebraic expressions.
When teaching reciprocals of numbers and terms I know it would have been much easier for me, and possibly for the students too, to say the reciprocal of a number is 1 divided by that number. I could even have stated n × 1/ n = 1. Sometimes I will do this depending on the students I’m teaching. However, I have found though that when there is an opportunity to explore mathematics in this way students become engaged much quicker and are more likely to maintain their engagement for longer.
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