by Dave Mitchell  143 White Pine Cr.,  Waterloo, ON  N2V 1B3,  519-885-5700
email:  mitchell@arithmecode.com


THE FRACTION RAP  ( Generate rap noise in the background.)

A Fraction is a division, so you don't have to make a decision.
You just take the numerator and divide by the denominator,
And then sooner or later, you get a repeater or terminator.
'Cause a fraction is a division, so you don't have to make a decision.
That's a rap  - 2 - 3 - 4 .  That's a rap.

MATHEMATICAL PI     ( To the tune of American Pie )

A long, long time ago, I can still remember when those numbers used to make me cry.
But I knew if I had my chance, I could make those figures dance and in math class,
I'd be happy for a while.
But studying just made me quiver, and each math fact would send a shiver __
Bad news on my math test, it sent me on a new quest.
I still recall with greatest pride, the day my mind was opened wide.
Math's importance got inside, the day I really tried.
And now I'm singing, "Why, why don't you learn about Pi ? " 
Get ecstatic 'bout quadratics and let calculus fly.
Take a swig of trig and you will be on a high, singing,
"This'll be the day that I try ! "

BEDMAS  ( To the tune of the Hokey - Pokey  )

You do the brackets first, exponents then take flight,
Next you divide and multiply in order left to right.
You add 'em and subtract 'em as you go and then you shout,
That's what BEDMAS is all about.
You work in BEDMAS order, you work in BEDMAS order,
You work in B - E - D - M - A - S order, work in BEDMAS order.
You work in B - E - D - M - A - S order, that's what BEDMAS is all about.


Here is a circle, it knows how to get around.
It has a radius from centre to rim.
And its diameter's a line that goes from side to side
while passing through the centre.
Now isn't that sim - ple?
Pi R squared sounds like area to me, when I need a circumference I'll just use Pi D.
Pi R squared sounds like area to me, when I need a circumference I'll just use Pi D.

ZERO SONG  ( Author unknown. To the tune of Rudolph )

Zero, that funny number, has a shape that looks like oh.
And if you want to use it, there are things you need to know.
Never divide by zero. If you do you will be sad,
Getting a crazy answer, making your report look bad.
But treat zero as your friend, use it carefully.
Safe to multiply or add, that's the rule for zero lads and lasses.
Zero, that funny number, wants to be a comrade true,
But never divide by zero, or you'll be getting zero too.


See, I have a rhyme assisting my feeble brain its tasks oft'times resisting.
(Count the number of letters in each word in the sentence above to get Pi to
12 decimal places)

113355 - (first three odd natural numbers, each one duplicated)
Turn it into a division:  113 into 355, in other words, 355 ÷ 113.  Try it.

E-X-C-E-L-L-E-N-T  Words
1 2 3 4 5 6 7 8 9 10 11 12 13

14 15 16 17 18 19 20 21 22 23 24 25 26

The value of a word is the sum of the values of its letters. A word is excellent if its
value is exactly 100. Find the value of each of the following words: EXCELLENT,
your students find?

MATH and MOZART (The Mozart Effect)

Researchers at the University of Southern California have found that students do
better on arithmetic calculations if Mozart is being played in the background while
they work. You can test this theory by having students do arithmetic without Mozart
and then doing questions of comparable difficulty while Mozart is being played. A
Mozart CD should be easy to find. You might extend this by testing other types of
music. (Students might search for the Mozart Effect on the internet.)


Using all four and only the four digits 1,9,9,7 each time, form expressions for
all the numbers from 1 to 100. You may use + , - , x , ÷ , ( ) , ! , radicals and
exponents as appropriate if they involve only 1,9,9,7.
Examples:  99 - 17 = 82,  9 ÷ 9 + 7 - 1 = 7, 97 - 91 = 89, etc.
Post a chart at the back of the room. Make it an ongoing class project to get
expressions for all the numbers from 1 to 100.  Each time a student posts a correct
expression, he/she initials it.


- one cup of cereal on a paper towel on each desk
- students count and tally the number of each letter ( - discount the broken pieces )
- calculate the percent of their own pieces which are As, Bs, Cs, ... Zs
- add up all student totals to get the total number of As, Bs, Cs, ... Zs  in the box
- suppose there are 2000 pieces in the box and Betty found that 5 % of her pieces
  are As. Based on this, she might predict that 5 % of the 2000 pieces or 100 As
  should be found in the box.  How does this compare with the actual total number
  of As in the box ?
- repeat for the other letters
- would the students in Betty's row make better predictions if they used the totals
  for their row converted to percents rather than just their own individual totals ?
- students eat the cereal and will then ask to be excused to get a drink of water
- have one of the students call the 1 - 800 number on the box to ask why there are
  not approximately the same number of each letter in a box  (report back)


In the 1980s, Canada trust had a contest. For each $50 you deposited, you got a
ballot.  All ballots from branches across Ontario were collected when the contest
ended.  From the big drum of ballots, one was drawn and the person whose name
was selected was asked to report to test his skill in order to win the condominium
prize which, at the time, was worth $80 000 U.S.  He was given a pencil, paper and
fifteen minutes. No calculator was allowed. Here is the question he was given :

1. multiply 228 by 21
2. add 10 824
3. divide by 12
4. subtract 1 121
He got the wrong answer and was, of course, terribly
disappointed. Another name was drawn and this time
the contestant was successful in obtaining the correct
answer, along with the condo in the sunny south !

Have your students try the question to see if they can "win" the condo.


Students work in groups to create a map of a fictitious town called Mathton. They
make up a set of questions based on a unit of study and create a rally directed by
answers to these questions.
Example: Begin at the corner of Triangle Street and Equation Boulevard and head
north. If the answer to question #1 is x = 5, then turn right at the first opportunity.
If not, turn left at the first street after Polygon Pizza. etc.
Groups exchange and try the rallies. They also grade the rallies based on criteria
supplied by you. You collect the rallies and evaluation sheets and add your own
component to the mark by looking at the quality of the presentations submitted.


Take any number and add 14. Multiply the result by 2. Subtract 8. Divide by 4.
Subtract half the original number. Think of the number-letter association: 1 = A,
2 = B, 3 = C etc. Now think of an animal that starts with the letter which
corresponds to your answer above. Back up one position in the alphabet and think
of a country which starts with that letter. ... Are you thinking of elephants in
If you do the arithmetic properly, no matter what number you start out with, you
should end up with 5. This will correspond to E and most people think of Elephant.
The D usually leads people to think of Denmark. (You might get a few Eagles or
Emus in the Dominican Republic)
Why does it work?  It's simple algebra.
Let the number be n. Then you get n + 14. Next you get 2n + 28 followed by
2n + 28 - 8 or 2n + 20. When you divide by 4 you get 2n/4 + 5 which reduces to
n/2 + 5. Now when you take away half of n (i.e. n/2) you get 5.


What is the answer to this question?  (8)(3) ÷ (2)(4)  Do you say it's 48 or do you
say it's 3? ...  BEDMAS would suggest working left to right on this one because it's
all multiplication and division. However, it is suggested that implied mutliplication
should be done before multiplication and division notated by ×, ÷.  Therefore, you
get 24 ÷ 8 or 3.

GREAT DECODING PROBLEM :  ( Found on an old ditto.)


1. Copy the code on your paper leaving two blank lines between each line
    of letters. (Check)
2. Count the number of As, Bs, Cs, ... , Zs. ( Make a chart. )
3. The most frequently used letter in English is the letter E.  Which letter
    in the code represents the letter E?  Print an E underneath each place this
    letter appears in the code.
4.  The second most frequently used letter is T.  Print a T underneath each place 
    this letter appears.
5.  A very common word is THE. If you can figure out where the word THE
    appears, you now know which letter represents H.
6.  This code contains the words UNITED STATES. If you can find these words
    you now know several more letters.
7.  Besides E and T, the most frequently used letters in English are A, O and N.
    Look at your chart and you should be able to figure out which letter
    represents O.
8.  Solving for the remaining letters is up to you. What is the message?  Try it
    on your own first to get the idea and then give it to your class.


Find the call numbers for your area at www.cbc.ca  and turn your students
on to some of these great shows.
- Quirks and Quarks - science show - Saturdays at noon after the news
- This Morning Tonight - evenings at 8:00 p.m.
- As It Happens - Evenings at 6:30 p.m.

- Are you looking at 30 sets of glazed eyes? Try handing
back the tests and posting solutions at a few stations around the room. Allow a few
minutes for students to check their work, correct their solutions and negotiate
any mark changes. The ball is now in their court.

SIMPLE MIND   (Like the game MasterMind but easier. A good way to work
into MasterMind which is available at toy and game stores.)

- the secret code is three digits made up using 1,2,3,4 with repetition allowed
- score each guess:  R means right digit in right place,  W  means right digit
  in wrong place
- example : if the code is 324 and guess is 332, clue is  RW because the 3 is in the
  right place and the 2 is in the wrong place. The middle 3 does not get a score because
  there is only one 3 in the code and it has been accounted for.
NOTE : the order in which the Rs and Ws are given is of no significance.
Sample Puzzles :
i ii iii
121 W
411 W
132 WW
(answer is 243)
433 R
232 W
131 RWW
(answer is 113)
434 RR
(answer is 444)

- graduate to MasterMind

ArithmeCode  © 1997  by Dave Mitchell

For sample puzzles and information on booklets, see the main menu of this web site.
Note: Songs and music which accompany many of the ideas I have presented are available on my CDs, Math, Music and Mayhem, Go Forth and Multiply and Class Actions. Information on these items can be found by going to the CD Information Page .

You're a teacher!  Your attitude and enthusiasm for math
will make a difference for your students.

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