Geometry, simply described as the study of shapes can be found everywhere you look. Literally. Geometry is vital in architecture, construction, city planning, furniture design, you want to build it - you'll probably need some geometry at some point.
That being said geometry is also extremely polarizing. Students begin learning geometric concepts as soon as they hit the classroom. Kindergartners learn the basic shapes (circle, triangle, etc.). By third grade students begin their introduction to angles, learning about right, acute, and obtuse. (We love angles! Check out our Angle Personality t-shirt collection when you have a sec.) Any good elementary school curriculum will begin introducing transformations to students. These are the images that occur when you move a shape on a plane. For instance, a reflection flips the shape over a line mapping each point to a new one along the perpendicular bisector, while a rotation spins the shape about a point. Before they leave elementary school they are familiar with 3D solids like cubes and rectangular prisms.
Geometry doesn't stop when kids hit middle school. Common core requires students in grades 6-8 to master real-world area and volume problems, understand angle relationships (don't get us started on this! Check out our flat-Earth post to come!) and know about perpendicular and parallel lines in algebraic forms. But high school, that is where the divide generally comes.
High school geometry is a catch all. It generally covers everything students should have already learned, and stacks onto it basically the entirety of Euclid's Elements Book 1. The issue is, geometry is what we call an axiomatic system. What does that mean? In this post we explain what geometry really is, shock you with the fact they teach every one flat-Earth math, test your spacial skills, give you tips on how to survive geometry class, and go over one of our most popular social media puzzles - The Neighborhood.
Axioms and Proof
An axiomatic system is best described as a house of cards. You start with a solid foundation (the fewer cards the more impressive) and proceed to very carefully and delicately build the structure atop.
The most common geometry, the geometry of high school, is based on Euclid. He started with five basic axioms, good for a foundation. For the most part, he didn't assume too much so everyone would believe him when he arrived at the results. His first four axioms were:
I can draw a straight line between two points
My line can go on and on forever if I want it to
Give me a point and a length and I can give you a circle
All right angles have the same measure
Most mathematicians of the time agreed that these, what we call axioms or basic truths, were true. Euclid then very deftly began to prove every belief about geometry (and number theory) using only his axioms. Once his axioms led to a theorem, he could use that too in his next proofs. Do you see the house of cards we are building?
If every single (or almost every single) conclusion is built on a set of predetermined beliefs, what happens if one, just one, of those is wrong? Catastrophe! The whole thing collapses. It is important to note that Euclid wasn't some great sage. Most of what he developed had already been done before. What was spectacular about Euclid's work, was that he codified and proved the many beliefs and ideas that were floating around in his time.
Now, there are a few things we should tell you.
Euclid's 5th postulate, the one we didn't mention here was very controversial.
Eventually someone (actually multiple someones) came along and said "hey! you can't do that" and they were right, sort of.
We teach students Euclid's geometry without explaining to them the very important story behind it. Very important indeed. As Euclid's geometry is the geometry of a flat Earth. That's right. It is 2D geometry. Since we live in a 3D world, that seems kind of... important?
Well, we can't give it all away. You'll have to pick up your copy of Marco the Great and the History of Numberville to learn more. But we will tell you this, although Marco is introduced to the idea of flat Earth math, you won't believe the journey he has in store. The third book in the series will focus totally on geometry and boy we can't wait for you to explore that world!
Why Students Struggle with Geometry
Alright, we have a basic understanding of what geometry is: it an axiomatic house of cards for which we study shapes. But why do some students flourish in geometry while others simply hate it?
There are two main reasons: proof and spacial thinking. Let's start with space. Spatial thinking is the ability to imagine or visualize objects, how they move or transform, their shapes, and their relationships with everything else. Let's test your thinking.
The Stop Sign Challenge
You have a stop sign. Easy enough. If most of us remember our geometry we know it is an octagon, but not only that, a regular one (a regular polygon is one such that all its sides have equal length. A square is a regular polygon. A rectangle - that is not a square - is not.)
Imagine there is a point in the center of the octagon. We are going to spin the sign around this point. (This should be easy to imagine. Just think about the guy standing on the corner - we'll add a video in case you need help).
How many degrees would I need to spin the sign, assuming it started right-side up, for the letters to be vertical with P on top? (See image)
Some of us are very good at this type of thinking. What did you pick? If you selected 90° then nice work. You see, since a circle has 360° (and we could draw a circle around this octagon so that each of its corners, its vertices, were on the circle), if we divide that by the eight angles of an octagon we know - remembering this octagon is regular as that is important - each angle has a measure of 45°. We moved each vertex two down the line, so that is 45+45=90.
But that was an easy one. Can you tell me what we did to the stop sign to produce this image?
If you can, you are probably pretty good at spacial thinking. If you can't, you are like the rest of us. In fact, high marks in spacial thinking is often an indicator of exceptional talent (also known as "giftedness"). But believe it or not, the founders of MathBait aren't super strong spacial thinkers! Turns out that is not a prerequisite to be good at math - or geometry, geometry is one of our favorites.
However, many curriculum treat it that way. If a student has difficulty thinking spatially, geometry (because of how it is taught not the subject itself) can be really tricky and downright frustrating for students. For students who are skilled spacial thinkers, the subject is a lot easier and even intuitive!
What can you do if you are not a spatial thinker?
If you are not superbly gifted at thinking spatially, that doesn't mean geometry is something you have to simply suffer through. There are a few things that might help you to get the most out of the subject.
Doodle. Yes, we said it. Fill your math binder with doodles. Drawing things out allows you to take the imagining and guessing out of it! It is much easier to understand when you can see it right in front of you.
Find something tactile. Shapes are literally everywhere. While we are not recommending taking out your tools and grabbing that sign at the end of the street to complete your homework, if you do happen to have something lying around that matches what you are working with, use it. Check these pattern blocks out, they make for all sorts of fun plus can be super helpful.
Use technology! It is actually crazy that we make students do so much with pen and paper. We live in a technological world and we should certainly use the tools at our disposal. In fact, that is a Standard of Mathematical Practice! At MathBait we love GeoGebra. You can translate, transform, rotate, and reflect to your heart's desire.
Okay, we've tackled almost everything we set out to do. There is one more key concept hanging over our heads - proof. This is the second major hold up in geometry class. It is sad really, proofs can be the most beautiful things. They force students to explain their thinking, to consider not just memorizing something (we hate that here) but strategically and logically thinking through why we know it must be true.
Unfortunately, we don't have a nice clean list of how to help with this one. A lot has to do with the teacher and the pedagogy. And a lot has to do with the approach. The good news is - this is exactly what we created MathBait for! We focus on a conceptual understanding: having fun, exploring, and actually "getting it". Instead of the more traditional memorize, study, cram, and forget. When you understand the ideas, proving them is nothing more than making a list. A journal of your travels. So, if proofs are what's getting you down - join MathBait! You'll be glad you did.
My Neighborhood
One of our most popular Social Puzzles was the My Neighborhood Challenge. We told viewers it took only one minute to solve, they could do it by the end of the video. Across platforms we received lots of guesses. Before we dive into the attempts and the solution, check out the puzzle for yourself. Can you solve before time runs out?
Did you figure it out? Guesses from our accounts include:
27?
22
26.8 (and we are guessing the more exact 26.83)
"Sqrt of 720 if my mental maths correct"
21.63
First, we have to say, can anyone solve for something like 26.83 or √(720) in under a minute in their head? That might have been a clue you are just a bit off.
The beauty of this problem is how simple and yet deceiving it is. For those spatial thinkers among us, it may have been more obvious, the answer was right there in the question. You see, what makes this difficult is our ability to piece together clues to make sense of what is lying directly in front of us!
Let's explore the clues again.
The bank is exactly halfway between the church and downtown.
The church to downtown is the same distance as the church to the school.
My house is on the same street as the school.
It is 12 miles from the bank to downtown.
Our goal is to find the distance from the school to the bank. Let's make a simple diagram.
Is it the triangle that throws our brain off? We are so trained to remember a²+b²=c² that our mind automatically starts calculating the value without considering (1) is that even a right triangle? and (2) is that even what I need to find?
Good news. No Pythagorean Theorem necessary, but if you want to actually understand the theorem make sure to check out our awesome Pythagorean visuals (two ways!) that make this memorized skill seem so much more natural.
Why did they tell us it is 12 miles from the bank to downtown? That seems important. Since we also know the church to the bank is the same distance, that must be 12 too, making the distance from the church to downtown 12+12=24. Wait! Is that some sort of curved superhighway? The whole neighborhood is the sector of a circle. And you know what Euclid said about circles, give me a point (the church) and a length (24 miles) and I can draw one!
The beauty of a circle is it is the set of all points equidistant from the center. That means My House is also 24 miles from church as is the store. And what if I ignore that triangle completely? I choose to see the rectangle instead, and the diagonals of a rectangle are congruent (the same in size and shape). That means... if from the church to My House is 24 miles, then from the school to the bank is 24 miles too.
With some basic geometry, and a minute, you can navigate even the strangest of neighborhoods. 😉
We Want to Know Your Thoughts!
Thanks for reading friends! Don't forget to snag your copy of Marco the Great and the History of Numberville and join in all the fun at The Kryptografima. If you made it down here because you are wondering about the stop sign (hey! they never told us the answer!) we applaud your tenacity. Here it is.
To obtain the image shown, we picked two antipodal (opposite essentially) points and reflected the STOP sign about them. Here is our line of reflection. And while you're down here, vote in our poll! What are your feelings about Geometry? We can't wait to see if the answers are as polarizing as we think!
What's your take on Geometry?
Love it! Absolutely essential.
Hate it! Throw it away.
I could go either way.
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