(01/10/96)
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MathMagic Cycle 22: Level 4-6 Regular
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BALANCING ACT
A scale can be used to measure the weight of various items, by
comparing the item to a known weight. This cycle's problem will
require you to do some empirical work (empirical is a scienttific
term that means you will experiment and observe and then
report what you see). First you will need to make a scale using a
ruler (the kind with 3 holes), 3 pieces of string (all the same
length) and 2 baskets (like for strawberries, milk , etc that are
the same size). Tie the 3 pieces of string to the ruler through the
3 holes. Attach the two end pieces to the baskets and tie the middle
string to a support. Hopefully, the scale will be balanced--if not, attach weights (paper clips, clay) to one side or the other until the scale is balanced.
Before you begin, explain the differences (if any) between
weight and mass.
Experiment 1) Place 4 or 5 common items that are identical (pencils,
erasers, etc) in basket #1. Decide on another, different item
(marbles are good or paper clips) that can be used to balance
them in basket #2. Then try to balance basket #1 with basket #2.
Did it work right off--If not, add more to basket #1. Explain your
results. Use your results to estimate the weight of 1 basket #1
item in terms of the basket #2 item. Check your results.
Experiment 2) How much does one piece of paper weigh, compared to
a paper clip or a pencil? Explain how you did this experiment? Which
weighs more--500 sheets of paper or 500 paper clips?
Experiment 3) You don't necesarilly need your scale for this. Suppose
6 oranges balance with 4 apples. And suppose 4 apples balance with
8 bananas. How many bananas will be needed to balance 9 oranges?
Experiment 4) Suppose now that 12 pencils balance with 4 erasers,
and that 6 erasers balance with 12 sharpeners. How many pencils would
be needed to balance 6 sharpeners?
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MathMagic Cycle 22: Level 4-6 Advanced
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BEEN AROUND THE BLOCK...
In the diagram, Miguel lives at M and Sonia lives at S. Each
character (either a number or a letter) represents a block in a
neighborhood. This neighborhood is 4 blocks square.
1 S
M 2
Miguel wants to go from his block to Sonia's without passing through
the same block twice. So to get from M to S there are just 2 ways
(cannot go diagonally)--M-1-S or M-2-S.
Now suppose the neighborhood looks like this:
1 2 S
3 4 5
M 6 7
1)In how many ways can we go from M to S, now? List the ways (For
example, one way is M-6-4-2-S)
2)Is there a pattern for square neighborhoods? How about a
neighborhood that is 4 blocks by 4 blocks?
3)Can you find a pattern that could be used to predict the number of
paths from M to S if the neighborhood is 10 blocks by 10
blocks and M and S are in opposite corners of the neighborhood?
4) Suppose Miguel decides to go to Sonia's block by going straight or
turning left only. In how many ways can he get there in a 9
square block neighborhood?
5) Answer questions 1-4 above if Miguel can pass through the same
block more than once.
6)Describe how you might extend this problem for other
neighborhoods, say rectangular, etc.
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Good luck
MrH