Irrational and Imaginary Numbers
This unit is currently being taught.
Time: The unit will last approximately 5 weeks. It began on October 4th, 2010, and will conclude around November 5th.
Main resource: Most of the class activities were developed on a weekly (or daily) basis to match the pace of the class as it unraveled. That said, I did find a textbook that would be a good match for the level of curriculum here (**** add this in later).
consider this: http://plus.maths.org/issue45/features/sangwin/index.html
-Why did people invent different kinds of numbers?
-How can you use numbers that you can’t count?
-(something more interesting!)
(continuing numbering from the first unit)
6. Solve equations requiring fractions and decimals
7. Convert between decimals and fractions
8. Solve quadratic equations with no linear term
9. Simplify radical expressions [including fractions, radical in denominator, etc]
future skills (draft stage)
10. Solve radical equations
11. Perform operations on irrational numbers
12. Perform operations on imaginary and complex numbers
13. Determine if a number is a root of polynomial [strike out because this is really just checking standards 11 and 12 in context. this context should be the form of the questions used to assess 11 and 12.]
- SBG #4 — Friday, Oct. 8 — Concepts 3, 5, 6, and 7
- SBG #5 — Friday, Oct. 15 — Concepts 1/4 (combined problem), 6, 7, and 8
- SBG #6 — Friday, Oct. 22
- SBG #7 — Friday, Nov. 5
Read first… Reflection/Teaching Notes
Neither of the projects described below came to full fruition during the current teaching iteration. The “Connections” Mini-Project was launched, but overall class comprehension about how to go about talking about mathematical connections was low. I recommend that if you want to focus on talking about connections, design the conversation around that.
As for the larger planned project, I think we just ran out of steam on this topic. We need something fresh, to be provided by the next unit. Some of the ideas from the planned project were implemented in lesson activities, however (not all is lost!)
“Connections” Mini-Project: Simple prompt idea:
“What is the connection between the number systems unit and the unit on repeating decimals, rational numbers, irrational numbers, etc???”
imagine that your civilization is make the discoveries of new numbers. what would decimals be like? give examples of how they would work. (work this out…)
Develop “decimals” in your number system. Is root 2 still irrational?
this one can be written, blog post, poster, comic, etc…. [??]
Look for the connections between number systems all the way up to complex numbers.
The “Big Project”
Technological Breakthrough : Use “Grapher” (Mac OS X application) to graph complex numbers. It’s marvelous and reasonably intuitive. Plus we have a “Mac cart” available. [or this might just be excellent for the “Prisoners vs. Escapees” activity]
Technology second thought: what is the real objective? What will be the best match for the project? What will be useful as a transferable skill to the final project. The answer likely has to do with XL.
Third thought. Grapher is really cool. Maybe they just need a template to follow. OMG IT CAN CREATE ANIMATION MOVIES!!!
Prompt: Your civilization is developing computer graphics! Or something like that. And they want to design a futuristic, mathematical logo for their government. (make this make sense)
What we need: 1) create a logo in the coordinate plane. 2) sample this logo with points, graphed as complex numbers. 3) consider problem of adding a constant (real or complex) number, multiplying by a real or complex number. what happens to the placement of the logo? 4) respond to your personal “design challenge” (tbd based on level of students. she might have to make it move up, down, scaled, rotated by complex numbers, etc etc)
still working on this.
Two good threads of using complex numbers
1. Fractals, convergence of iterated sequences, using this to practice and justify operations on complex numbers. [this seems to be rich territory for a class activity rather than a full-on project]
2. Multiplication of complex numbers modeling a rotation in the plane. How do you do 180 rotation? 90? how about 60? or weird rotations like 1/7th of 360?
idea: maybe the fractal project is a guided “in class” activity, and the multiplication project is their unit project.
idea 2: I want to make a separate project tackling the “connections” part of the math rubric. connections between number systems and different parts of math we have been learning.
Linear Equation, Integer, Integer Coefficients, Fraction, Egyptian Fraction, Repeating decimal, Terminating decimal, Natural number, Rational Number, Irrational Number, Real Number, …
compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions;
judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities;
learn about decimal representation, recurring or periodic digits, fractional representations [there is a thread here that leads to countability of the rationals] Meanwhile, what if we used a different base (say a base 12 “dozenal” or duodecimal system)
answer, what is a real number? what is a rational number? how do we know irrational numbers are irrational? [we can model and lead to the “proof by contradiction” — showing proof for infinitely many primes, leading to their proof that radical 2 is irrational]
NYS Standards, from the Algebra 2 / Trig document
Starting place: the “Number Sense” standards
|A2.N.1 Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers)|
|A2.N.2 Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form|
|A2.N.3 Perform arithmetic operations with polynomial expressions containing rational coefficients|
|A2.N.4 Perform arithmetic operations on irrational expressions|
|A2.N.5 Rationalize a denominator containing a radical expression|
|A2.N.6 Write square roots of negative numbers in terms of i|
|A2.N.7 Simplify powers of i|
|A2.N.8 Determine the conjugate of a complex number|
|A2.N.9 Perform arithmetic operations on complex numbers and write the answer in the form a + bi
Note: This includes simplifying expressions with complex denominators.
|A2.N.10 Know and apply sigma notation|
Relevant “Algebra” standards
|A2.A.13 Simplify radical expressions|
|A2.A.14 Perform addition, subtraction, multiplication, and division of radical expressions|
|A2.A.15 Rationalize denominators involving algebraic radical expressions|
|A2.A.16 Perform arithmetic operations with rational expressions and rename to lowest terms|
|A2.A.17 Simplify complex fractional expressions|