We're always on the lookout for more interesting or clearer ways to explain murky concepts in mathematics. So I was really excited to recently come across recently what I think is a nice way to make better sense of domain and range. I have to imagine that many readers already know about this. If so, let me know how your approach is the same as or different from the one I present below.
To illustrate this 'way', I’ve written a few incomplete sentences. Take a moment to think about different ways you could complete these sentences. To get the most out of it, do this with a partner or group or think about two or three examples yourself for each sentence.
______ learned how to _____.
_____ is taller than _____.
_____ erupted on _____.
For the first sentence, I wrote:
"Mike learned how to swim."
"The dog learned how to fetch."
"My class learned how to factor an expression."
Now, let's switch the order of our slots to see if the sentences make any sense. For example: "Fetch learned how to the dog." Nope.
But when we play with "is taller than," we should notice that many of the pairs are freely reversible: "Marissa is taller than no one" makes as much sense as "No one is taller than Marissa," and in that case, the meaning is reversed by transposing the slots. Interesting. (You have to know Marissa.)
What we see is that the phrase "learned how to" is a relation that connects certain subjects with certain objects. Oh, the cross-curricular connections!
But what we should ultimately think about are the kinds of things that are typically written in the blanks to the left and right of "learned how to."
Mike, the dog, and my class are all sentient individuals or collections of sentient individuals. And swim, fetch, and factor an expression are all actions. So, we could say that the domain of the relation "learned how to" is the set of all sentient individuals. That is the set we can draw from to fill the first blank. And the range of "learned how to" is the set of all actions.
These aren't the correct answers by any means. There isn't a correct answer. We are simply making an analogy to highlight how domain and range operate.
What we can notice by playing around with different relations is that not only are some relations "reversible" in the sense that their domains and ranges can be flipped around and still make sense, but some relations have more restricted domains or ranges than others. For the relation "_____ erupted on _____," we may decide that the domain is the set of all active volcanoes and the range is the set of all dates. So, Mt. Kilauea is in the domain, but Mt. Everest is not.
How have you made domain and range relevant and interesting to your students? How about subject and object in English?
Josh is an author and instructional designer with Carnegie Learning's Instructional Design team. He's been an instructional designer for more than 20 years and has worked with Carnegie Learning for 7 years, most recently helping to author the Middle School Math Solution series and upcoming High School Math Solution series. Josh and his wife, Vicki, both write for Carnegie Learning and have 4 children, which keep them pretty busy outside of work.
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