The usual problem with this pedagogy is that kids get confused when presented with improper fractions (eg. 11/8), and so on. I guess this is probably still a good intuition to start with, but how did you tackle those extensions later on?
I don't recall that coming up as an issue for him. I don't see why that would be a problem. You can just explain that 8/8 is one whole, leaving 3/8. I don't see why that would be hard to demonstrate.
"These are two pies. Let me cut them into eight pieces each. Let's count to eleven pieces. That's one whole pie plus 3 parts of the second pie, so it's the same as 1 3/8."
Three additional pizza slices automatically appear in my mind when you mention this. I found it more difficult to explain why the multiplication of two negative numbers gives a positive one.
Hm, that's a good one, I'm struggling to explain it to myself now!
My first thought is area, if you imagine four quadrants, draw a rectangle with side lengths positive from the origin, it's top right and multiplying the lengths gives you the area. If you instead take both sides negative then it's bottom left, but the area is the same.
However.. it's also the same if only one side extends negatively, so this is not at all satisfying.
(If that's made it even more confusing, the misleading error there is that side lengths are multiplied to give area, not positions on an axis relative to origin - the rectangle centred at the origin also has the same area.)
First, accept that a negative is the inverse, or opposite of something.
We can intuitively understand that multiplying slices by a negative inverts them - turns them from slices given into slices taken away. (Distribute the multiplication into addition if necessary to drive the point home.)
So it follows that multiplying negative slices by a negative inverts them again, turning them back into slices given.
It’s not a convenient visual, but if a negative slice represents the “taking away” of a slice, then a negative negative takes away the “taking away.”
DoreenMichele|4 years ago
"These are two pies. Let me cut them into eight pieces each. Let's count to eleven pieces. That's one whole pie plus 3 parts of the second pie, so it's the same as 1 3/8."
OJFord|4 years ago
dvfjsdhgfv|4 years ago
OJFord|4 years ago
My first thought is area, if you imagine four quadrants, draw a rectangle with side lengths positive from the origin, it's top right and multiplying the lengths gives you the area. If you instead take both sides negative then it's bottom left, but the area is the same.
However.. it's also the same if only one side extends negatively, so this is not at all satisfying.
(If that's made it even more confusing, the misleading error there is that side lengths are multiplied to give area, not positions on an axis relative to origin - the rectangle centred at the origin also has the same area.)
relaxing|4 years ago
We can intuitively understand that multiplying slices by a negative inverts them - turns them from slices given into slices taken away. (Distribute the multiplication into addition if necessary to drive the point home.)
So it follows that multiplying negative slices by a negative inverts them again, turning them back into slices given.
It’s not a convenient visual, but if a negative slice represents the “taking away” of a slice, then a negative negative takes away the “taking away.”