Using Technology to Uncover Mathematics August 3-6, 2015 Dave Brownslides available at Professor, Ithaca Collegehttp://faculty.ithaca.edu/dabrown/geneva/

Standards for Mathematical Practice Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. 2

Goals for the Week Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Goals for the Week NCTM Position (on role of technology in teaching & learning) It is essential that teachers and students have regular access to technologies that support and advance mathematical sense making, reasoning, problem solving, and communication. Effective teachers optimize the potential of technology to develop students’ understanding, stimulate their interest, and increase their proficiency in mathematics. When teachers use technology strategically, they can provide greater access to mathematics for all students.

Goals for the Week eMathInstruction (philosophy, excerpt) eMathInstruction was founded on the simple premise that 21st century technology can and will change the entire landscape of mathematics education. What was once a field dominated by hardbound textbooks, rote practice, and predictable tests has been replaced by standards based and ever changing assessments. Graphing calculator technology and dynamic geometry software allow us to explore, conjecture, and solve more types of problems. But too often states “raise the bar” by packing more and more content into the school year, sacrificing depth for breadth. Never has the saying a mile wide but only an inch deep been more appropriate.

Goals for the Week Use technology for – Computation – Modeling – Exploration (conjecture and reasoning) Technologies – Graphing Calculator – Desmos (laptops or tablets) – GeoGebra (laptops or tablets) Making Connections Design your own

For Day 4 - Reminder Second half of day – bring own idea to develop into a technology exploration Work on creating your own activity in one session Sharing out in subsequent session

Goals for Day 1 Morning Session 1 – Modeling with data – Graphing calculator Morning Session 2 – Modeling project – stopping distance – Graphing calculator Afternoon Session 1 – Recursion – Graphing calculator Afternoon Session 2 – Implicit Curves with parameters – Desmos

Definition of a Function A function is a rule that takes certain numbers as inputs and assigns to each a definite output number. The set of all input numbers is called the domain of the function and the set of resulting output numbers is called the range of the function. The input is called the independent variable and the output is called the dependent variable.

Table Daily snowfall in Syracuse, December 5–16, 2010 Date (Dec. 2010) Snowfall in inches The domain is the set of December dates, and the range is the set of daily snowfalls. The daily snowfall in Syracuse satisfies the definition of a function: Each date, t, has a unique snowfall, S, associated with it.

Functions can be represented by tables, graphs, formulas, and descriptions in words. For example, the function giving the daily snowfall in Syracuse can be represented by the graph in this Figure, as well as by the Table. The Rule of Four Tables, Graphs, Formulas, and Words Figure: Syracuse snowfall, December, 2010

Example with a Formula As another example of a function, consider the snow tree cricket. Surprisingly enough, all such crickets chirp at essentially the same rate if they are at the same temperature. That means that the chirp rate is a function of temperature. We write C = f(T ) to express the fact that we think of C as a function of T and that we have named this function f. The graph of this function is in this Figure.

Day 1, Session 1 Consider the following scatterplots. Describe the data and possible functions to model each set of data. (A)(B) (C) (D) (E) (F)

Day 1, Session 1 Key features of data to consider in model building – Direction: increasing/decreasing – Shape: linear or concave up/down – End-behavior: Increase/decrease without bound or asymptotes

Day 1, Session 1 (A) xy Increasing No concavity End-behavior: Increases without bound Linear – confirmed via First Differences y=ax+b a=? b=? y=3x-1

Day 1, Session 1 (B) xy Increasing Concave up End-behavior: Increases without bound Could be quadratic – confirm? y=ax 2 +bx+c a=? b=? c=? c=1 a + b = 2 4a+2b=5 a=1/2 b=3/2

Day 1, Session 1 (C) xy Increasing, decreasing, increasing Concave down then up (Inflection) End-behavior: Increases without bound Could be cubic – confirm? y=ax 3 +bx 2 +cx+d a=? b=? c=? d=?

Day 1, Session 1 (C) xy Linear Algebra & Graphing Calculator y=ax 3 +bx 2 +cx+d 0a + 0b + 0c + d = 0 a + b + c + d = -1 8a + 4b + 2c + d = 4 27a + 9b + 3c + d = 21

Day 1, Session 1 (C) xy y=ax 3 +bx 2 +cx+d 0a + 0b + 0c + d = 0 a + b + c + d = -1 8a + 4b + 2c + d = 4 27a + 9b + 3c + d = 21 Solve for this!

Day 1, Session 1 (C) xy y=ax 3 +bx 2 +cx+d 0a + 0b + 0c + d = 0 a + b + c + d = -1 8a + 4b + 2c + d = 4 27a + 9b + 3c + d = 21 Use MATRX key to solve:a=1, b=0, c=-2, d=0

Day 1, Session 1 (C) xy y=ax 3 +bx 2 +cx+d 0a + 0b + 0c + d = 0 a + b + c + d = -1 8a + 4b + 2c + d = 4 27a + 9b + 3c + d = 21 Use MATRX key to solve:a=1, b=0, c=-2, d=0 BUT, there must be an easier way!!

Day 1, Session 1 (C) xy y=ax 3 +bx 2 +cx+d BUT, there must be an easier way!! Appeal to a bit of data modeling – regression! Using the STATbutton on calculator

Day 1, Session 1 (D) xy Increasing Concave up End-behavior: Increases without bound Could be quadratic – confirm? No constant differences! Even if we continue.

Day 1, Session 1 (D) xy Increasing Concave up End-behavior: Increases without bound Could be quadratic – NO!! 1.25/1 = /1.25 = / = / = / = / = / = 1.25 Try something else! Exponential with base 1.25 Repeated multiplication! f(x) = 1.25 x For comparison, do a regression for both Quadratic and Exponential

Day 1, Session 1 (E) xy Increasing Concave down End-behavior: Increases without bound? Slow growth Let’s use regression to confirm a choice Definitely logarithmic!

Day 1, Session 1 (F) xy Increasing Concave up then down (I.P.) End-behavior: Increases with a bound Limiting value - asymptote Use regression to confirm a choice Logistic confirmed! Any ideas of function type?

Day 1, Session 1 Time# shares 9:00am1516 9: : : : : : : :00pm1525 1: : : During the month of November 2008, the NASDAQ was experiencing roller coaster days with the number of shares traded rising and falling dramatically each days, usually ending down. Find a model for the accompanying data, justifying your choice.

Day 1, Session 2 Distance travelled by car from when a “hazard” is noticed and car stops. What affects stopping distance? Stopping Distance=Reaction distance plus braking distance Physical law underlying this We are going to discover this! Stopping distance for an automobile

Day 1, Session 2 Make recommendation on where to place warning lights on new super highway Account for weather conditions and reaction time Modeling Work in teams of three Report out the final results

Day 1, Session 2 Stopping distance for an automobile ndex.html

Day 1, Session 2 REPORTING – Where will you place the warning system? – Explain your reasoning – What is your model? Stopping Distance Models

Day 1, Session 2 Which of the MP Standards did we experience? 1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning.

Day 1, Session 3 Recursion What is it? – Way to define values of a function in terms of other values of the function. – Eg: n! = n*(n-1)! Why do we care? – Discrete vs continuous modeling – Lots of applications! Investment strategies Spread of disease Washing effects on my fading jeans Reinforces exponential models as repeated multiplication

Session 3 - Recursion Most commonly associated with sequences Example: – A n =A n-1 +2 with A 1 =1 – Use Calculator, MODE  SEQ

Session 3 - Recursion Example: – A n =A n-1 +2 with A 1 =1 – Choose Y= – Enter as below

Session 3 - Recursion Example: – A n =A n-1 +2 with A 1 =1 – Choose TBLSET – Enter as below

Session 3 - Recursion Example: – A n =A n-1 +2 with A 1 =1 – Choose TABLE – Enter the n-values yourself What do we see as outputs? What happens if we change A 1 to 2?

Session 3 - Recursion Example: – A n =A n-1 +2 with A 1 =1 – We can plot too! – Choose WINDOW

Session 3 - Recursion Example: – A n =A n-1 +2 with A 1 =1 – We can plot too! – Choose GRAPH What kind of function is this? Can you find an explicit formula, f(n)?

Session 3 - Recursion On to Activities!

Session 3 - Recursion Modeling Population – Fish and Wildlife Management monitors trout population in a stream, with its research showing that predation along with pollution and fishing causes the trout population to decrease at a rate of 20% per month. The Management team proposes to add trout each month to restock the stream. The current population is 300 trout. 1.If there is no restocking, what will happen to the trout population over the next 10 months? 2.What is the long-term of effect of adding 100 trout per month? 3.Investigate the result of changing the number of trout introduced each month. What is the long-term effect on trout population? 4.Investigate the impact of changing the initial population on the long- term trout population. 5.Investigate the impact of changing the rate of population decrease on the long-term trout population.

Day 1, Session 4 Implicit Curves – Technology as exploration – Intro to Desmos – Desmos.com Desmos.com – Play a little

Day 1, Session 4 Implicit Curves – Technology as exploration – Is this the graph of a function? Why or why not? This is the curve y 2 =x 3 +x 2 -3x+2 What does this mean?

Day 1, Session 4 Implicit Curves Plotting and exploration using parameters On to Activities!