Seeking Depth in Algebra II

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Seeking Depth in Algebra II
Naoko Akiyama nakiyama_at_urbanschool.org Scott
Nelson snelson_at_urbanschool.org Henri
Picciotto hpicciotto_at_urbanschool.org www.picciot
to.org/math-ed

• The Urban School of San Francisco
• 1563 Page Street
• San Francisco, CA 94117
• [phone]
• www.urbanschool.org

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The ProblemTeaching Algebra II

• Too much material
• Too many topics
• Superficial understanding
• Poor retention
• Loss of interest

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A Partial Solution

• Choose Depth over Breadth

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Our Hopes

• Access for everyone
• No ceiling for anyone
• Authentic engagement
• Real retention
• Depth of understanding

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Math 3 Course Overview

• Themes
• Functions
• Trigonometry
• Real World Applications

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Math 3A

• Linear Programming
• Variation Functions
• Quadratics
• Exponential Functions, Logarithms
• Unit Circle Trigonometry

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Math 3B

• Iterating Linear Functions
• Sequences and Series
• Functions Composition and Inverses
• Laws of Sines and Cosines
• Polar Coordinates, Vectors
• Complex Numbers

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The course evolves

• Collaboration makes it possible

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Our Colleagues

• Richard Lautze
• Liz Caffrey
• Jee Park
• Kim Seashore

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Workshop Outline

• Iterating Linear Functions
• Quadratics
• Selected Labs
• Complex Numbers

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Iterating Linear Functions

• Introduction to Sequences and Series

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The Problem Opaque Formulas
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The Birthday Experiment

• Select the number of the day of the month you
  were born
• Divide by 2
• Add 4
• Repeat!

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Time Series Tablefor the Birthday Experiment
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Iterating a linear function
input
y mx b
output
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Time Series Graph for the Birthday Experiment
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Modeling MedicationFluRidder

• 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.

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FluRidder Problem
What equation did we iterate to model this?
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Recursive Notation
for the FluRidder Model
for n 1
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Special Case m 1Iterating y x b
Example b 4
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Time Series Graph for y x 4
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Special Caseb 0Iterating y mxfor 0 lt m lt1
Example m 0.5

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Time Series Graph for y .5x
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Special Casesb 0Iterating y mxfor m gt 1
Example m 1.5

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Time Series Graph for y1.5x
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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!

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Introducing Arithmetic and Geometric Series
Algorithms vs. Formulas
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Arithmetic Series

• Staircase Sums

3579
3 5 7 9 9 7 5 3 (12 12 12 12)/2
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Geometric Seriesmultiply, subtract, solve
a1 3, r .2, n 4

• S 3 .6 .12 .024
• .2S .6 .12 .024 .0048 ? multiply
• .8S 3 .0048 ?
  subtract
• S 2.9952/.8 3.744 ? solve

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Generalize
S a1 a2 a3 an r S r (a1
a2 a3 an) ? multiply
a2 a3 an an1 (1-r)S a1 an1 ?
subtract ? solve
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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

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Quadratics

• Completing the Square

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The Problem
What does this mean?
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We use a geometric interpretationto help
students understand this.
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The Lab Gear
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Make a rectangleusing 2x2 and 4x
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x (2x 4) 2x2 4x
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2x (x 2) 2x2 4x
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Lab Gear
The Box
Algebra
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Making Rectangles
Make as many rectangles as you can with an x2, 8
xs and any number of ones. Sketch them.
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Solving Quadratics Equal Squares
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Making Equal Squares


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Completing the Square
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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

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Selected Labs

• Inverse Variation
• Exponential Decay
• Logarithms

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Perspective

• Collect data apparent size of a classmate as a
  function of distance

• Look for a numerical pattern
• Notice the (nearly) constant product
• Find a formula

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Review Similar Triangles
? Constant product ? Inverse variation
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Application
If the front pillar is 15 meters away,how far is
the back pillar on the left?
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Dice Experiment

• Start with 40 dice
• Shake the box, remove dice that show 0
• Record the number of dice left
• Repeat!

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Outcomes

• Hands-on experiments motivate the concepts
• They are good for the long period
• They give students something to think, talk, and
  write about

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Scientific Notation1200 1.2 (103)

• Super-Scientific Notation 1200 10?

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1200 10?
Figure it out graphically,by looking for the
intersectionof two functions
( MODE FUNC )
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1200 10?
103lt1200 lt 104 x must be between 3 and 4y is
between 1000 and 1400
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Graph
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2nd CALC
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Back on the home screen
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Super-Scientific Notation

• Do 5-9, as a student might.

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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

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Complex Numbers

• A Graphical Approach

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The Problem
What does this mean?
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The Leap of Faith

• Go Graphical

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The Complex Number Plane
The Real Number Line
x
0
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Multiplication of Complex Numbers
An Example
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Multiplication of complex numbersworks for real
numbers!
Multiply
(2, 0) (5, 0) (2, 0) (5, 180) (2,
180) (5, 180)
(10, 0) (10, 180) (10, 360) (10, 0)
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One (1,0), remains the identity multiplier.
Reciprocals are well-defined.
So division works.
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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)
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(1,90)2 ?
(1,90)2 (1,180) (1, 90)2 -1
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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

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Summary

• Depth and breadth balance
• Access and challenge low threshold, high ceiling
  
• Keeping students in math past the required
  courses
• Preparation for Calculus