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.
Im aginary Num bers
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
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
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
Continuous Random Variables in Medical Devices
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
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.
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.