Seeking Depth in Algebra II Naoko Akiyama [email protected] Scott Nelson...
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Transcript of Seeking Depth in Algebra II Naoko Akiyama [email protected] Scott Nelson...
Seeking Depth in Algebra II
Naoko Akiyama [email protected]
Scott Nelson [email protected]
Henri Picciotto [email protected]
The ProblemTeaching Algebra II
• Too much material• Too many topics• Superficial understanding• Poor retention• Loss of interest
Our Hopes
• Access for everyone
• No ceiling for anyone
• Authentic engagement
• Real retention
• Depth of understanding
Math 3A
A. Linear Programming
B. Variation Functions
C. Quadratics
D. Exponential Functions, Logarithms
E. Unit Circle Trigonometry
Math 3B
A. Iterating Linear Functions
B. Sequences and Series
C. Functions: Composition and Inverses
D. Laws of Sines and Cosines
E. Polar Coordinates, Vectors
F. Complex Numbers
The Birthday Experiment
Select the number of the day of the month you were born
– Divide by 2
– Add 4
– Repeat!
Modeling Medication:FluRidder
• FluRidder is an imaginary medication• Your body eliminates 32% of the FluRidder in your
system every hour• You take 100 units of FluRidder initially• You take an additional hourly dose of 40 units
beginning one hour after you took the initial dose
Make a time-series table and graph.
Outcomes
• Grounds work on sequences and series
• Makes notation more meaningful
• Enhances calculator fluency
• Introduces convergence, divergence, limits
• Makes arithmetic and geometric sequences look easier!
Geometric Series:multiply, subtract, solve
S = 3 + .6 + .12 + .024
.2S = .6 + .12 + .024 + .0048 multiply
.8S = 3 – .0048 subtract
S = 2.9952/.8 = 3.744 solve
a1 = 3, r = .2, n = 4
Generalize:
S = a1 + a2 + a3 + … + an
r ·S = r (a1 + a2 + a3 +… + an) multiply
= a2 + a3 + …+ an + an+1
(1-r)S = a1 – an+1 subtract
solve
Outcomes
• A way to understand — the algorithms are more meaningful than the formulas for most students
• A way to remember — the formulas are easy to forget, the algorithms are easy to remember
• A foundation for proof of the formulas
Making Rectangles
Make as many rectangles as you can with an x2, 8 x’s and any number of ones.
Sketch them.
Outcomes• Concrete understanding of completing the
square and the quadratic equation
• Connecting algebraic and geometric multiplication and factoring
• Connecting factors, zeroes and intercepts
• Preview of moving parabolas around and transformations
• Better understanding of “no solution”
Perspective
• Collect data: apparent size of a classmate as a function of distance
Distance (sidewalk squares)
Apparent height (cm)
3 41
6 20
9 13
12 10
15 8
18 7
• Look for a numerical pattern
• Notice the (nearly) constant product
• Find a formula
Dice Experiment
• Start with 40 dice
• Shake the box, remove dice that show “0”• Record the number of dice left
• Repeat!
Outcomes
• Hands-on experiments motivate the concepts
• They are good for the long period• They give students something to think, talk,
and write about
Figure it out graphically,by looking for the intersectionof two functions:
1200 = 10?
( MODE: FUNC )
Outcomes
• Postponing the terminology and notation allows us to build on what the students understand
• The approach gives meaning to logarithms, emphasizing that logs are exponents
• It helps justify the log rules
• When terminology and notation are introduced, some students forget this foundation, but reminding them of it remains powerful
Multiplication of complex numbersworks for real numbers!
(2, 0°) · (5, 0°) =
(2, 0°) · (5, 180°) =
(2, 180°) · (5, 180°) =
Multiply:
(10, 0°)
(10, 180°)
(10, 360°) = (10, 0°)
Powering
(1,45°)1 = (1,45°)(1,45°)2 = (1,90°)(1,45°)3 = (1,135°)(1,45°)4 = (1,180°)(1,45°)5 = (1,225°)(1,45°)6 = (1,270°)(1,45°)7 = (1,315°)(1,45°)8 = (1,360°)
Outcomes• Depth in understanding i and complex numbers
• Review/preview polar coordinates• Trigonometry review, including special right
triangles• Review/preview vectors• Understanding basic operations• Binomial multiplication• Completing a quest that started in kindergarten