For this unit: the book reading is Chapter 3.
Prepare a new short presentation for every class period. As usual, the
handout
contains suggestions for you to work on, but you are welcome to do come up with
your own ideas.
(1) Think about what kind of problems lend themselves well
to various strate-
gies for mental arithmetic , like the ones that are called leading digit ,
compensation
, break-and-bridge, and compatible numbers in the book (or any
others you think of!—be creative). De sign a worksheet that you would use
to help you teach mental arithmetic.
(2) Give an algorithm for how to select coins to give
proper change. For in-
stance, if you need to give 76¢, you can do it with three quarters and a
penny or with seven dimes, a nickel, and a penny. Explain an algorithm
for selecting which coins to use.
Now imagine that there is a huge financial crisis which causes the gov-
ernment to eliminate the nickel and the penny, melt them down, and issue
a new coin called a “pnicky” that’s worth four cents. So now you can make
76¢ with six dimes and four pnickies, for example. Give an algorithm for
making change in the new system .
(3) Explain the “greedy” algorithm for finding an Egyptian
fraction represen -
tation of a standard fraction. Also try writing out a different algorithm ,
like some of the ones we discussed informally in class.
(4) We’ve seen how to do addition and multiplication in
base n by analyzing
the base 10 algorithm and modifying it. Figure out how to do subtraction
and/or long division in base n and make a presentation explaining it.
(5) Examine three alternative subtraction algorithms :
European (p161), In-
dian, and Japanese (we’ll discuss these in class). Make up practice prob-
lems for the three different algorithms. Why does each one work, and when
does it work especially well?
(6) Find patterns in the addition and multiplication
tables in various bases.
Describe and explain.
(7) Read Hyman Bass’s essay on Algorithms and Proficiency
(linked from the
course website). Evaluate some of the algorithms we have encountered
using the attributes discussed by Bass.
