As we explore new mathematical tools, we're balancing understanding with practice through Worksheets and Workouts.
Exponents, logarithms, and trigonometry are typically found in the curriculum after algebra and geometry, but before calculus. This sometimes gets them lumped into a class with the name "precalculus." And certainly their place before calculus gives us a rich variety of non-polynomial functions to explore with calculus tools. But these topics are a beautiful area of mathematics in their own right, and it is a pleasure to have the opportunity to explore them with my students this block.
There is a delicate balance that needs to be struck in this class. On the one hand, it takes practice to learn to work with trigonometric and logarithmic expressions quickly and accurately, and this is definitely a skill worth developing. On the other hand, it's easy to get lost in the algebra and lose sight of the mathematical reality that our equations are trying to describe. To maintain this balance, our in-class assignments come in two varieties: Worksheets and Workouts.
On Worksheets, students are expected to explore and learn about new mathematical ideas, and expand their mathematical worldview. On Workouts, students build their mathematical muscles by working through exercises of gradually increasing difficulty. Usually, we'll do at least one Workout before diving into the major Worksheet for the day. This way, new concepts get revisited and become familiar over time. And even three weeks in, we've already seen a lot of new concepts!
We began the block by recalling facts about how exponents behave. Starting with our intuitive definition of a^n as "n copies of a multiplied together," we were able to develop a consistent definition for negative and fractional exponents. We puzzled over what an irrational exponent would look like, and saw our first example of a limit.
We then introduced the logarithm into our conversation. We know that the equation 2^x=8 has a solution: x=3. The equation 2^x=9 should have a solution as well. It's a number we can't solve for using any methods from their earlier algebra classes, but we recognize that it's a real number, it's between 3 and 4, and it's much closer to 3 than it is to 4. Since exponential equations are so important in mathematics, we need to have a name for this number. So we call it log_2(9), the power we need to raise 2 to in order to get the number 9.
This is a consistent theme in this class. We have a specific number that we would like to find, but our existing algebraic tools don't give us an exact value for the number, so we give our number a name. Then we have a new tool for studying numbers of this kind. A lot of the work of this class is developing an intuitive understanding of these new tools. Students are encouraged to give exact values when they solve problems, and reminded that sometimes exact answers may not be pretty answers. At the same time, we are working to develop enough number sense to sanity-check our answers. If we start with 1000g of radioactive material, and it decreases by 10% each year, we should be suspicious if we find that half of it is gone after 2.06712485 years—surely it should take at least five years to get down to half.
This week we began our exploration of trigonometry through the unit circle. Working with ten-inch embroidery hoops and special measuring tapes, we were able to find radian angle measures by stretching the measuring tapes around the outside of the circles and marking the end points of these arcs. We need tools for relating the arc lengths around the outside of the circle with the x and y values of the corresponding points. Since we can't find an exact value for these numbers (except for a few very special cases that we'll be talking about next week) we invent new functions called cosine and sine to describe the x and y values, respectively.
Using our embroidery hoops, we were able to find surprisingly accurate decimal approximations for the sine and cosine of various values. After that, we had a Workout on sine and cosine number sense. In one part of this worksheet, students were supposed to look at three trigonometric expressions, and figure out which two of them represented the same number. For example, we don't know exact values for sin(2), sin(-2), or -sin(2), but if we think carefully about their placement on the unit circle, we can figure out which two of them are equal to each other.
We have a lot more ideas to see in the remainder of the block. We'll be talking about applications of trigonometric functions to geometry. We'll introduce the tangent function, and explore some trigonometric identities. We'll talk about the importance of trigonometric functions in modeling periodic behavior. This block is packed with some really beautiful mathematics, and I am looking forward to watching our students continue to learn, explore, and grow.
— Susan Durst