Problem Solving or the ART of THINKING ON IT “mathematical thinking can be improved by tackling questions conscientiously; reflecting on this experience; linking feelings with action; studying the process of resolving problems; and

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10 Οκτ 2013 (πριν από 4 χρόνια και 1 μήνα)

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Problem Solving or the ART o
f THINKING ON IT


“mathematical thinking can be improved by



tackling questions conscientiously;


reflecting on this experience;


linking feelings with action;


studying the process of resolving problems;


and


noticing how wh
at you learn fits in with your own experience.”
1



The subject of Geometry is best understood by those who are willing to learn and understand definitions,
theorems and postulates and who can apply this knowledge to problem solving. Problem solving may be

in the
form of a formal two
-
column proof or a paragraph proof or in the form of developing an equation and solving.


There are some strategies

to problem solving:
(1) being stuck is good as it generates thinking skills, (2) focused
“thinking on it” is a
start to becoming unstuck, (3) specializing or trying some specific cases leads to possible
solutions and/or ideas, (4) try to generalize after specializing and noting a pattern, (5) jot down ideas as they
flow from the brain, (6) write down everything tha
t you know about the problem, (7) ask yourself WHY
,

(8) draw pictures and/or re
-
arrange the information, (9) reduce the problem to a smaller problem, (10) walk
away from the problem for a period of time to clear the mind.


1.

How many squares are there on a s
tandard chessboard?


2.

Draw a square on your paper. Draw a line across it. Draw several more lines through the square so that

several regions are formed. The task is to color regions in such a way that adjacent regions are never
colored the same. How few d
ifferent colors are needed to color any arrangement?


3.

A goat is tethered by a 6 meter rope to the outside corner of a shed measuring 4 meters by 5 meters in
the middle of a pasture. What is the area of grass on which the goat can graze?


4.

Five women have l
unch together seated around a circular table. Ms. Osborne is sitting between Ms.
Lewis and Ms. Martin. Ellen is sitting between Cathy and Ms. Norris. Ms Lewis is between Ellen and
Alice. Cathy and Doris are sisters. Betty is seated with Ms. Parkes on
her left and Ms. Martin on her
right. Match the surnames with the first names.


5.

Ross collects lizards, beetles and worms. He has more worms than lizards and beetles together. All
together there are twelve heads and twenty
-
six legs. How many lizards are

there in the collection?


6.

Small booklets can be made by folding a single sheet of paper several times and then cutting and
stapling. I would like to number the pages before making the folds. How can this be done? (consider
booklets of various number of

pages)








7.

A secret number is assigned to each vertex of a triangle. On each side of the triangle is written the sum
of the secret numbers at the endpoints. Give a rule for finding the secret numbers if you know the sums.







8.

Twenty
-
f
ive coins are placed in a 5 by 5 array. A fly lands on one coin and tries to hop onto every coin

exactly once by only moving to an adjacent coin in the same row or column. Is it possible?




9.

Take a strip of paper and fold it in half several times. Unfol
d it and notice how many folds are “in” and

how many folds are “out”. What sequence would arise from 8 folds?, 9 folds?, 10 folds?




10.

Some numbers can be written as the sum of consecutive numbers. For example: 9 = 4 + 5

11 = 5 + 6


21 = 6 + 7 + 8


Ar
e there any numbers that cannot be written as the sum of consecutive numbers?


What are the properties of numbers for which this can be done?
















1
John Mason, L. Burton, K. Stacey, THINKING MATHEMATICALLY, Addison
-
Wesley Pub. Co.

1982


Problems taken from

THINKING MATHEMATICALLY