mathematics and technology(2)

Technology & Mathematics: Synergistic Motivators for High

School Algebra

Michael HerzogQiyam Tung

Introductions

• Michael Herzog

• Qiyam Tung

• G-TEAMS

• Computers in the classroom

You harvest 100 lbs of tomatoes from your garden. Tomatoes are 1% solid and 99% water. You dry the tomatoes in the sun for 1 week until they are 98% water. How much do they weigh?

If you have a computer, please go to my website and enter your answer:

• http://tinyurl.com/AAIS-Conference

• Google Doc spreadsheet of results

http://tinyurl.com/AAIS-Conference

Google docs for Collaborative Review

• Post and shared a Google document with all the material and practice problems that students need to know for an upcoming assessment.

• Students collaborate online with solutions and comments on this document.

• Review in next class focused on areas where students needed help as seen by reviewing the document in class.

Letting go!

• Create a real problem that requires– Research– Data Collection– Analysis

• Have students use a computer to do this and create a report or presentation to share their results

Project Example

• Growth of the Internet

Technology in Our Classroom

• Smart Board and projector• Airliner wireless slate• Remote control lighting• Document camera• Bamboo pad• Software

– Google docs, GSP, Wolfram-Alpha, Join.Me, etc.

● Technology is all around us.● Laptops, Kindles, cars, watches,

phones.

● Mathematics is the "universal" language that all computers follow.

● By studying mathematics, we can understand how our technology works.

● However, we can also improve our understanding of mathematics by studying it with the technology we have.● WolframAlpha● Programming

Technology and Mathematics

Video Games and Mathematics• Gaming: not just for the hardcore player anymore

• iPhone and Facebook apps and Flash games• Games are illusions• Glitches break the illusions

• Reveal the underlying mathematics that govern the game’s universe

Video Games and Mathematics

http://www.youtube.com/watch?v=qcc3kSG2nms




• Glitch: walking (not swimming) underwater

• Why did this happen?


Pictorial ExplanationBanjo “switches gears” once he touches the surface of the water.

● Simplify it to 2D.● Water Surface =

● y=2x+3 , if 0<=x<=1.5● y=6, if 1.5<x<=2.2

● Assume the character is a single point (x1, y1) on the graph. ● Explain, in mathematical terms, how the game knows when the

character is on the surface of the water.


(x1,y1)

● Banjo “switches gears” when he is touches the water.● Banjo is a point● Water is composed of two lines.

● Mathematically, this means the point (Banjo) is one either one of the lines (water).

● In other words, one of these two must be true● y1=2x1+3● y1 = 6


(x1,y1)

● 6=2(0.5)+3 = 4● False!

● 5=2(1)+3 = 5● True!

● 1=2(2)+3 =7● False!


(1,5)

(0.5,6)

(2,1)

y1=2x1+3, y1 = 6


(1,5)

(0.5,6)

(2,1)

y1=2x1+3

y1 = 6

We want this to be true as well!


(1,5)

(0.5,6)

(2,1)

y1=2x1+3

y1 = 6

We want this to be true as well!

y1 <=2x1+3

y1 <= 6

Why Study <Math Concept>?

Why Study <Math Concept>?

• What to do?o Give an application after it is asked.o Give a problem that forces them to use this concept so

that this question doesn't come up in the first place!

• Ask them the question!• Is there a useful way to describe this phenomena?

Inquiry: Algorithm Analysis

• Motivate the need for logarithms in algorithm analysis.

• Algorithm analysis is the study of the amount of resources (such as time and storage) necessary to execute an algorithm.

• Example: phone book search.

• Find a name x in a phone book with n names.

• How would you search for the name?

Binary SearchSearch for: Zack0 seconds passed

Binary SearchSearch for: Zack1 second passed

Binary Search•Binary search is faster than brute force (linear) search.

•Not very informative.

< <

What does fast mean here?

Binary Search

More precise question: how long will it take in the worst case scenario?

•1 second to read a page•Book has n pages.

● Linear search: n seconds● Binary search:

● Worst case: we keep dividing by two until there is only 1 answer left.

● The number of times we have to divide by two to get the answer.

● We have a name for that: logarithm. Specifically, log2(n).

● What does this mean?

Binary Search

Programming and Mathematics● Teach math through programming

● Why should we do it?● It forces students to be precise.

● Question: define multiplication of exponential functions with the same base.

● ax x ay = ax+y

● Now do it in English:

● A sample student definition: “add the exponents.”● Add whose exponents? The answer is not a number.

Is it an operation?

Programming and Mathematics● Ambiguous answer

● 25 x 23 = 8● 25 x 23 = “add the exponents”● 25 x 23 = 28

● Which one is the right answer?

● Better answer: the multiplication of two powers of the same base is the base to the power of the sum of the exponentials.

Programming and Mathematics

● Computers are unforgiving: they do not do what you meant to say, but exactly what you said.

● Computers can simulate. ● Free Fall of objects (quadratics and gravity)● Monty Hall problem (probability).

Physics in Video Games.

Development of the original Super Mario Bros

● Can we simulate jumping with linear equations?● Unrealistic

● Quadratics are a far better approximation.

Simulating Jumping with Quadratics


Less talking, more doing!

Simulate a bouncing ball.

height = acceleration*t2 + initial_velocity*t + initial_height


height = acceleration*t2 + initial_velocity*t + initial_height

• The values for initial velocity and height are straightforward (just positive)

• What about acceleration? Is it positive or negative?

• We can test this with our program!

Programming and Mathematics

• They got to see the code• They could learn more if they wrote the code.

Technology and Mathematics

o Technology helps us manage the classroom in general.

o Understanding the technology around us is a great motivator for learning mathematics.

o Technology itself is very helpful is helping understanding mathematics.

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