If you look at all of the area formulae that we foist onto kids, we generally start with letters that represent the dimensions we tell them to search for, length (l), width (w), perpendicular height (h), base length (b), radius (r):
Rectangle: A = l x w
Parallelogram: A = l x h
Triangle: A = bh/2
Trapezium: A = (a + b)h/2
Circle: A = Pi r^2
It's well intended, and might be simpler at first, but as you can see above the result is a disconnected mess of isolated ideas that - experience will tell you - are very difficult to remember.
When you start introducing formulae for surface area and volume it only gets worse.
Instead I focus on the idea that area always results from the multiplication of two perpendicular dimensions. For the kids that's just 'area is always length times height.' I wrote about it for the first time back in 2013 here:
Thinking about it in those terms, it doesn't matter how we label the lengths we're multiplying, so long as kids can identify them, and we're consistent.
If we do that then those formulae look like this instead:
Rectangle: A = ab
Parallelogram: A = ab
Triangle: A = ab/2
Trapezium: A = (a1 + a2)b/2
Circle: A = Pi ab
And now that core truth runs through all of the formulae. Length times height (ab.) With a couple of easy mnemonics it makes them much easier to remember and (I think) reveals more mathematical meaning.
We'll be sure to bring in ideas like 'what we've been calling the length and height of a circle are always equal to its radius,' which will eventually get us back to the Pi r^2 simplification - necessary because it's such a strong convention - but for the others there isn't much need to alter them.
So in sum I used b instead of h because I was thinking ahead to what all these formulae were going to look like once they've seen a dozen of them. I'd share the slides here that I have that I think make that even more apparent, but I can't paste images into the comments box.
What do you think? I know some teachers love this idea when they see it, and I've found others are very uncomfortable with it at first.
That is really helpful thank you, I had misread the b as representing base.
What do I think, I think it makes perfect sense. I’m not sure if I’d be tempted to stick with b and h, I like the connection between the height is perpendicular for the base. I also think height gives us an intuitive connection. Eg the height of someone.
But if I’m reading this correctly that could then be applied to say the rectangle and the circle, the trapezium would then be b1 + b2.
It might be that not using b for base would remove the confusion around the base being the bottom length which is parallel to say the bottom of the page and that moving to ab could then focus more on the shape.
i started with b and h, I think (or a and h.) Maybe I even tried l and h... 🤔 Can't quite recall now.
Either way, I ran into a problem when I got to volume.
I had something like this:
Rectangle: A = bh
Parallelogram: A = bh
Triangle: A = bh/2
Trapezium: A = (a + b)h/2
Circle: A = Pi bh
I started with this so that formulae for triangle and trapezium would remain in their most common form.
But then, I usually communicate volume as 'area times height.'
And that then gives you things like:
V = (bh)h
And the problem is that 'h' there is representing two different numbers, two different 'heights.'
This problem arises because of course none of these lengths is a 'height' in the strictest sense, they're all just lengths, perpendicular to each other. So in the end I opted for the completely arbitrary a, b, and c, so that volume becomes:
V = abc
I would even be content to move away from the language of 'height' altogether and just focus on 'lengths,' and identifying the correct lengths, it's just that anything I've tried always seems more confusing rather than helpful in the beginning.
Also yes - I've never been a fan of 'base length.' Why does a triangle have a base length but not the rectangle or parallelogram? And to your point, 'base' implies a natural bottom side, when you can pick any side you like, so long as you then multiply it be a perpendicular.
Great article around checking the individual part. But, why did you choose to use b rather h in the formula for trapezium?
Thanks!
If you look at all of the area formulae that we foist onto kids, we generally start with letters that represent the dimensions we tell them to search for, length (l), width (w), perpendicular height (h), base length (b), radius (r):
Rectangle: A = l x w
Parallelogram: A = l x h
Triangle: A = bh/2
Trapezium: A = (a + b)h/2
Circle: A = Pi r^2
It's well intended, and might be simpler at first, but as you can see above the result is a disconnected mess of isolated ideas that - experience will tell you - are very difficult to remember.
When you start introducing formulae for surface area and volume it only gets worse.
Instead I focus on the idea that area always results from the multiplication of two perpendicular dimensions. For the kids that's just 'area is always length times height.' I wrote about it for the first time back in 2013 here:
https://tothereal.wordpress.com/2013/06/23/what-does-simplicity-look-like/
Thinking about it in those terms, it doesn't matter how we label the lengths we're multiplying, so long as kids can identify them, and we're consistent.
If we do that then those formulae look like this instead:
Rectangle: A = ab
Parallelogram: A = ab
Triangle: A = ab/2
Trapezium: A = (a1 + a2)b/2
Circle: A = Pi ab
And now that core truth runs through all of the formulae. Length times height (ab.) With a couple of easy mnemonics it makes them much easier to remember and (I think) reveals more mathematical meaning.
We'll be sure to bring in ideas like 'what we've been calling the length and height of a circle are always equal to its radius,' which will eventually get us back to the Pi r^2 simplification - necessary because it's such a strong convention - but for the others there isn't much need to alter them.
So in sum I used b instead of h because I was thinking ahead to what all these formulae were going to look like once they've seen a dozen of them. I'd share the slides here that I have that I think make that even more apparent, but I can't paste images into the comments box.
What do you think? I know some teachers love this idea when they see it, and I've found others are very uncomfortable with it at first.
That is really helpful thank you, I had misread the b as representing base.
What do I think, I think it makes perfect sense. I’m not sure if I’d be tempted to stick with b and h, I like the connection between the height is perpendicular for the base. I also think height gives us an intuitive connection. Eg the height of someone.
But if I’m reading this correctly that could then be applied to say the rectangle and the circle, the trapezium would then be b1 + b2.
It might be that not using b for base would remove the confusion around the base being the bottom length which is parallel to say the bottom of the page and that moving to ab could then focus more on the shape.
I don’t know if I am over thinking this?
i started with b and h, I think (or a and h.) Maybe I even tried l and h... 🤔 Can't quite recall now.
Either way, I ran into a problem when I got to volume.
I had something like this:
Rectangle: A = bh
Parallelogram: A = bh
Triangle: A = bh/2
Trapezium: A = (a + b)h/2
Circle: A = Pi bh
I started with this so that formulae for triangle and trapezium would remain in their most common form.
But then, I usually communicate volume as 'area times height.'
And that then gives you things like:
V = (bh)h
And the problem is that 'h' there is representing two different numbers, two different 'heights.'
This problem arises because of course none of these lengths is a 'height' in the strictest sense, they're all just lengths, perpendicular to each other. So in the end I opted for the completely arbitrary a, b, and c, so that volume becomes:
V = abc
I would even be content to move away from the language of 'height' altogether and just focus on 'lengths,' and identifying the correct lengths, it's just that anything I've tried always seems more confusing rather than helpful in the beginning.
Also yes - I've never been a fan of 'base length.' Why does a triangle have a base length but not the rectangle or parallelogram? And to your point, 'base' implies a natural bottom side, when you can pick any side you like, so long as you then multiply it be a perpendicular.