Art of Mathematics How Would You Fit 63 in This Visuvalization
When I expect up "area of a rhomb" on Google images, I find plenty of disappointing images like this ane:
which show the formula, but fail to bear witness why the formula works. That'south why I really appreciate this image instead:
which, with a piddling bit of careful idea, illustrates why the production of the diagonals equals twice the area of the rhombus.
EDIT: Some have mentioned in comments that that second diagram is more complicated than information technology needs to be. Something like this would work as well:
My main objective is to offer students something that encourages them to recall virtually why a formula works, not merely what numbers to plug into an equation to get an answer.
As a side note, the following story is not exactly "visually stunning," but it put an enduring imprint on my listen, and affected the way I teach today. A very gifted Jr. Loftier math teacher was teaching us nearly volume. I suppose just every about school system has a identify in the curriculum where students are required larn how to summate the volume of a pyramid. Sadly, most teachers probably accomplish this by simply writing the formula on the board, and assigning a few plug-and-chug homework bug.
No wonder that, when I ask my higher students if they tin tell me the formula for the volume of a pyramid, fewer than 5% tin.
Instead, building upon lessons from earlier that week, our math teacher began the lesson by saying:
We've learned how to calculate the book of a prism: we simply multiply the area of the base times the height. That's piece of cake. Merely what if we don't have a prism? What if we have a pyramid?
At this indicate, she rummaged through her box of math props, and pulled out a clear plastic cube, and a clear plastic pyramid. She continued by putting the pyramid atop the cube, and and so dropping the pyramid, point-side down inside the cube:
She connected:
These have the aforementioned base, and they are the same height. How many of these pyramids exercise you lot suppose would fit in this cube? 2? Two-and-a-one-half? Three?
Then she picked one educatee from the front row, and instructed him to walk them down the hallway:
Go downwardly to the water fountain, and fill up this pyramid upwardly with water, and tell us how many it takes to fill upwards the cube.
The form sat in silence for about a full minute or so until he walked back in the room. She asked him to requite his report.
"Three," he said.
She pressed him, giving him a hard look. "Exactly three?"
"Exactly three," he affirmed.
Then, she looked effectually the room:
"Who here can tell me the formula I use to become the volume of a pyramid?" she asked.
I daughter raised her hand: "One-3rd the base times the height?"
I've never forgotten that formula, because, instead of having information technology told to the states, we were asked to derive it. Not only have I remembered the formula, but I tin also even tell yous the proper name of the male child who went to the water fountain, and the girl who told u.s.a. all the formula (David and Jill).
Given the upvoted comment, If high schoolhouse math just used a fraction of the resources here, we'd have way more mathematicians, I hope you don't listen me sharing this story here. Powerful visuals can happen even in the imagination. I never got to see that cube filling up with h2o, but everything else in the story I vividly remember.
Incidentally, this same teacher introduced us to the concept of pi by asking us to find something circular in our house ("like a plate or a coffee can"), measuring the circumference and the diameter, and dividing the one number by the other. I can still run into her studying the data on the chalkboard the next 24-hour interval – all 20 or so numbers merely a smidgeon over iii – marveling how, even though we all probably measured differently-sized circles, the answers were coming out remarkably like, "as if maybe that ratio is some kind of constant or something..."
Source: https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain
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