Enhancing Algebra Instruction Through the Use of Graphing Technology Bill Gillam 10/18/02

Carla’s Lemonade Stand # glasses$ Profit Input: number of glasses Output: Profit Initial costs are $9.50. Each glass costs $0.75

Carla’s Lemonade Stand # glasses$ Profit Input: number of glasses Output: Profit Initial costs are $9.50. Each glass costs $0.75

Initial costs are $9.50. Each glass costs $0.75 Carla’s Lemonade Stand Input: number of glasses Output: Profit

Initial costs are $9.50. Each glass costs $0.75 Carla’s Lemonade Stand Input: number of glasses Output: Profit

P = g Carla’s Lemonade Stand Input: number of glasses Output: Profit

Initial costs are $9.50. Each glass costs $0.75 Carla’s Lemonade Stand Input: number of glasses Output: Profit Rule: P(n) =0.75n n = 0, 1,2,3….. \Y1= 0.75X – 9.50 \Y2= \Y3= \Y4= \Y5= \Y6=

Initial costs are $9.50. Each glass costs $0.75 Carla’s Lemonade Stand Input: number of glasses Output: Profit Rule: P(n) =0.75n n = 0, 1,2,3…

Initial costs are $9.50. Each glass costs $0.75 Carla’s Lemonade Stand Input: number of glasses Output: Profit Rule: P(n) =0.75n n = 0, 1,2,3…..

Variables Related by Rules # Lawns$ Profit Rule:

Variables Related by Rules # Lawns$ Profit Rule: x = profit

Variables Related by Rules Summary 1. Tables, Graphs & Rules 2. Income, Expense & Profit Power Point by Bill Gillam Adapted from Concepts in Algebra: A Technological Approach

Variables Related by Data in Graphs: Part II - Interpreting Graphs X Y AGE

Variables Related by Data in Graphs: “The Hiker”

Variables Related by Data in Graphs: Hike This! Time

Variables Related by Data in Graphs: Hike This!

Time Variables Related by Data in Graphs: Hike This!

Time Variables Related by Data in Graphs: Hike This!

Time Variables Related by Data in Graphs: Hike This!

Time Variables Related by Data in Graphs: Hike This!

Time Variables Related by Data in Graphs: Hike This!

Time Variables Related by Data in Graphs: Hike This!

Time Variables Related by Data in Graphs: Hike This!

Time Variables Related by Data in Graphs: Hike This!

Time Variables Related by Data in Graphs: Hike This!

Time Variables Related by Data in Graphs: Hike This!

Variables Related by Data in Graphs: Interpreting Graphs 1. CBL Stations 2. Graph Stories

Variables Related by Data in Graphs: Interpreting Graphs Summary 1. Speed 2. Rate of Change

Variables Related by Data in Graphs: Interpreting Graphs Power point by Bill Gillam

Rule: Y = *x + f(0) Variables Related Linearly # Lawns$ Profit

Rule: Y = 10x Variables Related Linearly # Lawns$ Profit

Variables Related Linearly # Lawns$ Profit Rule: Y = + f(0)

Variables Related Linearly # Lawns$ Profit Rule: Y = 12x

Variables Related Linearly Rule: Y = 12x ProfitProfit lawns =195/x x =16.25

Variables Related by Rules Fundraiser Activity

Variables Related Linearly Summary 1. Ratio of change between any two points is constant. 2. Rule: y = Mx + b 3. Graphs a straight line.

Variables Related by Rules Summary Powerpoint by Bill Gillam

Investigating Linear Patterns in Data 7/22/2002

“If he was all on the same scale as his foot, he must certainly have been a giant.” - Sherlock Holmes The Adventure of Wisteria Lodge. Investigating Linear Patterns in Data

Footlength and Height Question: Is the length of a person’s foot a useful predictor of his/her height?

Footlength and Height 1. Measure and record each student’s footlength and height in a table. 2. Graph the ordered pairs: (footlength, height). 3. Discuss

Strategies: 1. Studying numerical patterns (table) 2. Matching a graphical pattern (graph) 3. Using function fitting tools (symbolic) Investigating Linear Patterns in Data

Studying numerical patterns (table) FootHeight

Matching a graphical pattern (graph)

Using function fitting tools (regression) y = mx + b

Symbolic Representation of Line of Best Fit y = mx + b

Strategies: 1. Studying numerical patterns (table) 2. Matching a graphical pattern (graph) 3. Using function fitting tools to obtain a modeling function (symbolic) Investigating Linear Patterns in Data

Power point by Bill Gillam Investigating Linear Patterns in Data

Exponential Growth According to legend, chess was invented by Grand Vizier Sissa Ben Dahir, and given to King Shirham of India. The king offered him a reward, and he requested the following: "Just one grain of wheat on the first square of the chessboard. Then put two on the second square, four on the next, then eight, and continue, doubling the number of grains on each successive square, until every square on the chessboard is reached."

According to legend, chess was invented by Grand Vizier Sissa Ben Dahir. He presented the game to King Shirham of India. The king offered him a reward, and he requested the following: Exponential Growth eight, and continue, doubling the number of grains on each successive square, until the last square is reached." ”Place one grain of wheat on the first square. Put two on the second square, four on the third, then

Exponential Growth You may give me the wheat or its equal value on the 64th day. This is all I require for my services. The king agreed, but he lost his entire kingdom to Sissa Ben Dahir. Why?

Exponential Growth square/dayriceSum __________ How much wheat did the King owe for 64th day? How much wheat in all?

Exponential Growth In all, the king owed about 18,000,000,000,000,000,000 grains of wheat. This was more than the worth of his entire kingdom!

Exponential Growth There is a function related to this story: f(x)=2^x day rice sum rice dayrice2^(day-1) 2^day-1 112^0 = ____ 2^1 - 1 = ____ 222^1 = ____ 2^2 - 1 = ____ 342^2 = ____ 2^3 - 1 = ____ 482^3 = ____ 2^4 - 1 = ____... 64____2^63 = ____2^64-1 = ____ Copy and fill out this chart.

Exponential Growth

Moore's Law (from the intel website):

Exponential Growth Gordon Moore (co-founded Intel in 1968) made his famous observation in 1965, just four years after the first planar integrated circuit was discovered. The press called it "Moore's Law" and the name has stuck. In his original paper, Moore predicted that the number of transistors per integrated circuit would double every 18 months. He forecast that this trend would continue through Through Intel's technology, Moore's Law has been maintained for far longer, and still holds true as we enter the new century. The mission of Intel's technology development team is to continue to break down barriers to Moore's Law.

Exponential Growth chip Year Transistors , , , , , processor , DX processor1989 1,180,000 Pentium® processor ,100,000 Pentium II processor ,500,000 Pentium III processor ,000,000 Pentium 4 processor ,000,000 Produce a plot of year vs. transistors

Exponential Growth- Moore’s Law chip Year Transistors , , , , , processor , DX processor1989 1,180,000 Pentium® processor ,100,000 Pentium II processor ,500,000 Pentium III processor ,000,000 Pentium 4 processor ,000,000 Produce a plot of year vs. transistors (from the intel website):

Exponential Growth Review of how to do a point plot: "STAT" "Edit" enter year in L1 and transistors in L2. "2nd" "Y=" "Plotsoff" "Enter" “Enter" "2nd" "Y=" Choose Plot1 {On, Scatterplot, L1, L2, mark} "Zoom" 9

Exponential Growth Moore’s Law indicates that the growth should be modeled by: Y=2250*2^((12/18)x) or Y=2250*2^(x/1.5) Is this accurate? Use exponential regression to find a curve that fits.

Exponential Growth 1. Describe the graph: 2. How does this relate to the rice problem? 3. Can you think of other things that “grow” this way (ie. Doubling over a constant period of time?)

Exponential Graphs *************************************

Exponential Graphs 1. Which graph is most likely 4^x? 2. Which is most likely 0.25^x? 3. Which is most likely 1^x? ************************************* Y= 4^xY = 0.25^x Y = 1^x

Exponential Graphs Graph the following pairs of functions on your calculator: A) y = 0.5^x and y = 2^x B) y = 0.25^x and y = 4^x C) y = (3/4)^x and y = (4/3)^x ************************************* Make a conjecture about the relationship between these graphs.

Exponential Graphs ************************************* Make a statement about the relationship between these graphs.

Exponential Graphs *************************************

Exponential Graphs ************************************* 1. Sketch a guess for the graph of y = 4*3^x.

Exponential Graphs ************************************* Sketch a guess for the following graphs, then check them on your calculator to see how close you came. Y = 5^x; Y = 2*5^x; Y =0.2^x; Y = 6*(0.2^x); Y = -3^x and Y = -0.2^x. Do exercises 1 - 4