Seeking Depth in Algebra II

Naoko Akiyama nakiyama@urbanschool.org

Scott Nelson snelson@urbanschool.org

Henri Picciotto hpicciotto@MathEducationPage.orgwww.MathEducationPage.org

The ProblemTeaching Algebra II

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

A Partial Solution

Choose Depth over Breadth

Our Hopes

• Access for everyone

• No ceiling for anyone

• Authentic engagement

• Real retention

• Depth of understanding

Math 3 Course Overview

Themes:

• Functions

• Trigonometry

• “Real World” Applications

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 course evolves

Collaboration makes it possible

Our Colleagues

• Richard Lautze

• Liz Caffrey

• Jee Park

• Kim Seashore

Workshop Outline

• Iterating Linear Functions

• Quadratics

• Selected Labs

• Complex Numbers

Iterating Linear Functions

Introduction to Sequences and Series

The Problem: Opaque Formulas

The Birthday Experiment

Select the number of the day of the month you were born

– Divide by 2

– Add 4

– Repeat!

Time Series Tablefor the Birthday Experiment

Iterating a linear function

input

output

y = mx + b

Time Series Graph for the Birthday Experiment

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.

FluRidder Problem

What equation did we iterate to model this?

Recursive Notation for the FluRidder Model

for n ≥ 1

Special Case: m = 1Iterating y = x + b

Example: b = 4

Time Series Graph for y = x + 4

Special Case:b = 0Iterating y = mx

for 0 < m <1

Example: m = 0.5

Time Series Graph for y = .5x

Special Cases:b = 0Iterating y = mx

for m > 1

Example: m = 1.5

Time Series Graph for y=1.5x

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!

Introducing Arithmetic and Geometric Series:

Algorithms vs. Formulas

Staircase Sums

Arithmetic Series

3 + 5 + 7 + 9

9 + 7 + 5 + 3

(12 +12 +12 +12)/2

3+5+7+9

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

Quadratics

Completing the Square

The Problem

What does this mean?

We use a geometric interpretationto help students understand this.

The Lab Gear

Make a rectangleusing 2x2 and 4x

x (2x + 4) = 2x2 + 4x

2x (x + 2) = 2x2 + 4x

Lab Gear

“The Box”

Algebra

x +5

x x2 5x

+5 5x 25

Making Rectangles

Make as many rectangles as you can with an x2, 8 x’s and any number of ones.

Sketch them.

Solving Quadratics:

Equal Squares

Making Equal Squares

=

=

Completing the Square

x

x x2 x

x

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”

Selected Labs

Inverse Variation

Exponential Decay

Logarithms

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

Review Similar Triangles

Constant product

Inverse variation

Application

If the front pillar is 15 meters away,how far is the back pillar on the left?

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

Super-Scientific Notation 1200 = 10?

Scientific Notation1200 = 1.2 (103)

Figure it out graphically,by looking for the intersectionof two functions:

1200 = 10?

( MODE: FUNC )

103<1200 < 104

x must be between 3 and 4y is between 1000 and 1400

1200 = 10?

Graph

2nd CALC

Back on the home screen:

Super-Scientific Notation

Do #5-9, as a student might.

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

Complex Numbers

A Graphical Approach

The Problem

What does this mean?

The Leap of Faith

Go Graphical

The Complex Number Plane

The Real Number Line

real axis

imaginary axisz

r b

a

0 x

Multiplication of Complex Numbers

An Example:

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°)

• One (1,0°), remains the identity multiplier.

• Reciprocals are well-defined.

• So division works.

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°)

(1,90°)2 = (1,180°)(1, 90°)2 = -1

(1,90°)2 = ?

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

Summary

• Depth and breadth: balance• Access and challenge: low threshold, high

ceiling • Keeping students in math past the required

courses• Preparation for Calculus