Every YouTuber gives the whole ‘Like, Subscribe and Share’ spiel in their videos. Apparently that works in growing their audience and subscribers. So, I am going to use the same tactic here.
Please sign up for the email updates for future posts, like the articles, and share them with those who you think may be interested (and get them to sign up as well).
In Part 1, Fliba and Lagard posed the question: should squares be types of rectangles? With help from Lagard’s sister Glagalbagal, they expanded the scope of the question to cover other shapes like parallelograms, rhombuses and trapeziums. They were also introduced to various ways of representing classificatory systems.
Let us listen in to their conversation a day after they talked to Glagalbagal.
Fliba: So, I drew the diagrams your sister told us about.
The two on the left are the same classification and so are the two on the right.
Lagard: Your handwriting is exactly the same as my sister’s. How come?
Fliba: I think we had the same teacher in preschool. Everybody knows that anybody who learns from the same teacher learns the exact same thing.
Lagard: Good point! Thought, you seem to have missed out rhombuses.
Fliba: Yeah. I got confused about those. I thought we could leave them out for now and get back to them later.
Lagard: Okay. So, what are we supposed to do with these diagrams?
Fliba: Didn’t Glagalbagal tell us to write out definitions. Let’s work on that together.
Lagard: Sure. We already had definitions for squares and rectangles. For squares, I think the definition in both the systems in the same:
A square is a quadrilateral with all sides equal and all angles 90 degrees.
For rectangles, we had two definitions:
A rectangle is a quadrilateral with opposite sides equal and all angles 90 degrees
A rectangle is a quadrilateral with opposite sides equal, all angles 90 degrees, and adjacent sides not equal
Fliba: The first definition is for the system on the right and the second is for the system on the left. However, we can make the definition of square shorter in the system on the right, where a square is a type of rectangle.
A square is a rectangle with adjacent sides equal
Can you do the same thing Fliba did for squares for rectangles as well? And parallelograms and trapeziums? Try to define them in the system on the right.
Once you have done that, you can continue listening in on the conversation:
Lagard: We can do that for a rectangle as well. We can say:
A rectangle is a parallelogram with all angles 90 degrees
We can then go on to say that:
A parallelogram is a trapezium with opposite sides parallel
A trapezium is a quadrilateral with one pair of parallel sides
Fliba: Yeah. That was easy. Let’s try the other system.
Lagard: For parallelograms we can say:
A parallelogram is a quadrilateral with opposite sides parallel
Fliba: That doesn’t work since by this definition, rectangles are types of parallelograms.
Can you fix the definition of parallelogram such that rectangles are not parallelograms?
Once you are done, read on:
Lagard: So, we have to add something in like we did with rectangles.
Fliba: Yes. How about:
A parallelogram is a quadrilateral with opposite sides parallel and all angles not equal to 90 degrees.
Lagard: That seems to work. Now we have to do the same for trapeziums:
A trapezium is a quadrilateral with one pair of opposite sides parallel and the other one not parallel.
Fliba: Exactly. It does seem like the definitions in this system are more complicated. That is why the other one is better.
Lagard: I agree they are more complicated, so maybe for that reason we prefer the other system. However, they are not that complicated. This is not very convincing to me as an argument.
Fliba: Fair enough. Maybe we can try adding in rhombuses and see if that is sufficient to convince you.
Think about how you would add in rhombuses to both the systems. In the left one, we want to separate rhombuses from all the other types of quadrilaterals we are interested in. In the right one, we want to integrate it into the system. This is not straightforward to do. Think about how you would do this and write out definitions of rhombuses in both the systems. We will pick up from here in Part 3.
