You should be able to see how many of the things the young student would normally learn in grade school apply to a high school lesson. Sadly most students don't make the connection or have forgotten the previous lessons and don't see how they apply. So for the younger students we start off with a lesson that's easy and fun and we can make a bit of a game of it...once this is digested and mastered we add a little more covering the same ground and a bit more and then the same ground and a bit more and and so on...until the unit circle is filled in. But we start with just a few points and add more each time.
All we are doing is decoding a point on a plane. ( 6 , π/2 ) for example means get a six block and point it to the π/2. Notice the little boy has to be taught where to start from (the origin) and what the symbols mean. On the second lesson he will "get it" more and the third more than that.
Counting by 1/12, 1/6, 1/4, 1/3 and 1/2 are handy skills to have...so when we first start out we can count out a few of the easier ones and get more and more complex each time we come back to visit the concept. And we can see 3/6 = 1/2 from a very different perspective than fooling with pieces of a pie even though we are fooling with pieces of pi. Fractions teach multiplication too.
Many teachers don't know what is going to be taught in high school or have forgotten so they fail to prepare the young students for what comes and never make any attempt to show the younger students what they can expect when they get older. Teaching math has become so compartmentalized that students get lost going from what compartment to another. I often see this in the text books themselves chapter to chapter these is no thread or common theme except that it's all math. I often as how do they teach addition and subtraction and integers as three separate and distinct lessons? How do you teach multiplication without teaching factors and division...?
Why not use the so called higher math to teach the basic operations instead of thinking you have to know the basic operations first?
We can do many lessons on "same" or the concept of same...π/2 = 90º and π/6 + π/6 + π/6 = 3π/6 which is same as π/2 which is the same as 90º...it's not hard and can be fun. Rather than give them a circle like that to memorize why not have them build the circle themselves? They can fill in more and more information with each lesson. Dividing by two is important to be able to do here...and you'd be surprised at how many "ah-ha" moments can be created whe they find out that 1/2 of π/2 is π/4 and 1/2 of 90º is 45º...then 1/2 of π/6 is π/12 and all kinds of wheels turn when you mention division is just multiplication by the inverse... π ÷ 2 = π x 1/2 = π/2 and if you look closely that first set of symbols should make you go, "of course." It's right there in that division symbol: ÷ put one over the other.
Counting in degrees is also EASY and converting back and forth and how to do so should be discovered by the students so they make up their own rules (which will become the formula most are given to memorize and then forget). Once they have made up a formula for going from radians to degrees a lesson on converting from polar coordinates to Cartesian coordinates or rectangular coordinates would be in order. Most of the time they get all of this at once in one chapter. Confusion ensues.
Also I note a lesson on polar coordinates is often not introduced until high school, then they see it ONCE and they may not see it again until college. One exposure is insufficient to learn ANYTHING. Then they will become confused when they begin teaching vectors and spherical geometry because the text books assume they know this much because it should have been covered in high school. It was, once, and then it was forgotten. Anybody besides me see the problems here?
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