Hands on Teaching and Curriculum Development M ISTI Highlights for High Schools Classes January 2012 Review of Interactive M athem atics Activities Instructor Adwoa B Boakye I traveled to Mantua Italy to teach mathematics courses after being selected as a teacher for the MIT MISTI Highlights for High Schools Program I structured all of my courses as physical workshops where we developed intuition for the math we were learning through derivations and projects My previous instructor Dr Elizabeth Cavicchi of Reconstructing Galileo and Professor Peter Dourmashkin were involved in brainstorming activities and assessing classroom dynamics before I went abroad The snapshots below are overviews of a few of the interactive math activities I designed and taught over my three week stay Courses Im aginary Num bers Statistics Probability Com position of Functions Continuous Random Variables Imaginary Numbers from the need for i to Mathematical Modeling The Derivation of the Imaginary Number The Unit Circle Sinusoids and Circular Motion Modeling with Sinusoids The Derivation of i On the first day we started with a question Squareland has a debt of 250 000 and a population of 6 400 How much debt does each lane have and what will they have to contribute in order to pay it off To check your answer is debt lane payment 2

Checking this problem geometrically is debt lane payment 2 leads to a contradiction You cannot have debt 250 000 lane payment 2 or take the square root of a negative number without introducing a new construct i Getting the students to discover this allowed me to justify the exploration of i while making the students excited to learn about something they discovered The Unit Circle Sinusoids and Circular M otion Before I set off to Italy Dr Cavicchi and I built a circular wheel with a nail to represent a point revolving around the unit circle During my imaginary numbers class I showed the students the footage I captured of the shadow created by the projection of the nail onto the horizontal plane With my coaching the students analyzed the footage noted how the shadow moved faster in the center than at the sides and postulated why this was happening A Physical Introduction to the Unit Circle Sines and Cosines as Horizontal and Vertical Projections Design Project Crude Solar Pane Using a blank sheet of paper a marker and a friend I asked the students to replicate the motion that they described while their friend slowly pulled the paper away from them The images they produced were sinusoids Deriving this understanding physically allowed me to talk about horizontal and vertical projections very easily NOTE Having the students build the unit circle would have been an excellent activity if we had the materials M odeling with Sinusoids The final assignment of the class was for the students to develop an equation that modeled the sun exposure that a solar panel receives over the span of a week Without suggesting how the students should model the sun exposure I allowed the students to brainstorm individually what the output of their equation would be Using the concepts we learned i when combined with e can be used to construct a sinusoid and concepts they already knew the absolute value Design Project Crude Solar Panel Sun Exposure Model

makes all values positive I coached the students by walking around the tables brainstorming with them asking them their thought process and encouraging students with complimentary ideas to work with each other while asking them to refine their models noting peak intensity and possible failures of the model Continuous Random Variables from Discrete to Continuous Deriving Distributions Continuous Random Variables in Medical Devices Deriving Distributions The University of Colorado s Plinko Applet shows what would occur if a stream of balls were dropped into a system of pegs By setting up the problem on the chalk board and asking the students to work together to figure out the percentage of balls that would end up in each collection bin I was able to get the students to work together to postulate the result CRVs in M edical Devices Random walks occur in many environments electrical wires the air we breathe and in pools of water By noting the connection between continuous and discrete random variables in previous lectures and thought experiments I built the students knowledge so that they could accomplish the final project determining the likelihood that a highly concentrated bolus of medication would be delivered within a span of time t Conclusion Mathematics is often about data representation By engaging the students in the activities featured above and more I worked to show how math is fun philosophically and in application especially when you can discover how it works through your own questions observations and understanding