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The Hispaniola Water Shortage - posted October 16, 2000

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Introduction

Your friend, Rodolfo, lives on the island of Hispaniola. When he cooks he must conserve water, because the family's water supply comes from a rain barrel.

A shop in his town has measuring cups for sale but they come in sets, one set with a 3-oz. cup and an 8-oz. cup, and a second set with a 4-oz. cup and a 10-oz. cup, and Rodolfo has just enough money to buy one set. The store manager understands his problem and agrees that he can test different sets before deciding which ones to buy.

Before going back to the store to make his decision, Rodolfo looks through his recipe file and finds that his recipes call for anywhere from 1 to 12 oz. of water.

Rodolfo hopes to figure out which set of measuring cups will minimize the possibility of spilling water, and yet allow him to measure accurately all the amounts that he needs for his recipes.

Simulation 1

First, Rodolfo tries a 3-oz. cup and an 8-oz. cup. Use the simulation (applet) to determine if he can measure the amounts listed in Table 1.

Click To Show Applet Window

A few answers are filled in to get you started. You might want to copy or print out Table 1.

Table 1
oz. Yes No
1    
2    
3    
4    
5    
6    
7 X  
8    
9    
10 X  
11    
12    

Simulation 2

Next, Rodolfo tries a 4-oz. and a 10-oz. cup. Use the simulation (applet) to determine if he can measure the amounts listed in Table 2.

Click To Show Applet Window

A few answers are filled in to get you started. You might want to copy or print out Table 2.

Table 2
oz. Yes No
1    
2    
3    
4    
5    
6    
7   X
8    
9    
10 X  
11    
12    

Questions:

  1. Do you notice any patterns in Table 1? Explain.

  2. Do you notice any patterns in Table 2? Explain.

  3. Compare the patterns in your two tables. Think about the combinations you used. What do you notice about even and odd numbers?

  4. What are your conclusions about the amounts that can (or cannot) be measured using the two different sets of measuring cups?

  5. Which set of cups would you recommend that Rodolfo buy?

Comments

Teacher Support Page

Most of you made a correct recommendation for the pair of cups Rodolfo should buy: 3 oz. and 8 oz. The other questions we asked were supposed to give you hints as to why that would be a good purchase. Not all of you explained the conclusion we thought was appropriate, that if you only have cups that hold an even number of ounces, you can only get even numbers as a result of combining them in different ways, but having an even and an odd cup would allow you to get both even and odd numbers.

Basically, if you made the correct recommendation and explained why that combination was best, you got credit for this problem.

Some problems you may have had:

  1. Some of you explained that if you had two odd cups, you could only get odd results. Think about what you get if you add 3 and 3. Is that odd?

  2. You can't get 12 oz. from 3-oz. and 8-oz. cups, because they only add up to 11 oz. That was kind of a trick question.

  3. Using 3 oz. and 8 oz. cups, it was a little complicated to get 4 oz. and 10 oz., but it was really possible! If you don't believe me, try again:

    4
    Fill the 3-oz. container. (3 in 3 oz. and 0 in 8 oz.)
    Pour to the 8-oz. container ( 0 in 3 oz. and 3 in 8 oz.)
    Fill the 3-oz. container. (3 in 3 oz. and 3 in 8 oz.)
    Pour to the 8-oz. container again (0 in 3 oz. and 6 in 8 oz.)
    Fill the 3-oz. container. (3 in 3 oz. and 6 in 8 oz.)
    Pour to the 8-oz. container again (1 in 3 oz. and 8 in 8 oz.)
    Empty the 8-oz. container (1 in 3 oz. and 0 in 8 oz.)
    Pour to the 8-oz. container (0 in 3 oz. and 1 in 8 oz.)
    Fill the 3-oz. container. (3 in 3 oz. and 1 in 8 oz.)
    Together you have 4 oz.

    10
    Fill the 8-oz. container. (0 in 3 oz. and 8 in 8 oz.)
    Pour to the 3-oz. container (3 in 3 oz. and 5 in 8 oz.)
    Empty the 3-oz. container (0 in 3 oz. and 5 in 8 oz.)
    Pour again to the 3-oz. container (3 in 3 oz. and 2 in 8 oz.)
    Empty the 3-oz. container (0 in 3 oz. and 2 in 8 oz.)
    Pour again to the 3-oz. container (2 in 3 oz. and 0 in 8 oz.)
    Fill the 8-oz. container. (2 in 3 oz. and 8 in 8 oz.)
    Together you have 10 oz.

Highlighted solutions:

From:  Long N., age 13
Monica M., age 13
School:  Caroline Davis Intermediate School, San Jose, CA
 

Questions:

1. Do you notice any patterns in Table 1? Explain.
I could get all the numbers in the first set by adding or subtracting one cup
by
the other. That is possible, because one cup is an even number and the other
is odd, 3 and 8.

2. Do you notice any patterns in Table 2?  Explain.
In the second set there was a little problem, both cups were even numbered
size cups. I managed to get all the even numbers, 12, 10, 8, 6, 4, and 2, but
I
couldn't get any odd numbers.
3. Compare the patterns in your two tables. Think about the
combinations you used. What do you notice about even and odd numbers?
In the first set when I add the odd with the even I got all the numbers,
because
when you are trying to get a number that is even or odd and you have an even
number and an oddd number to use, it is possible that you could get the
desired number. However, if you only got two numbers that is both evens then
you got a problem. That is because you could only get the even numbers, not
the odds. Even plus even is even. If you got two numbers that is both odd,
there is certain numbers that you can't get.
4. What are your conclusions about the amounts that can (or cannot) be
measured using the two different sets of measuring cups?
in set one I found that it is difficult to get the amount of 1, 4, 7, and10.
Because
I have to do all this pouring back and forth a lot.
The other numbers were easy to get. On set two I think it is hard to get 2,
because of the same reason as above. Plus I couldn't get any odd numbers.
5. Which set of cups would you recommend that Rodolfo buy?
I definitly would recomend that he should the first set, 3 and 8. Because it
is
(versatile?) a set that could be change to the amount he wanted to use to
cook. He could get any amount if he tried hard to figure out. And there is no
need to spill out water.

From:  Allen H., age 12
School:  Woodrow Wilson Middle School, Edison, NJ
 

Questions:

1. Do you notice any patterns in Table 1? Explain.
   Yes, every thing  was yes, becuase if you think about it, you
don't have to put everything into the recipe at the same time. Since
you can get 1 oz. of water with the 3 oz. and 8 oz. measuring cups,
one can just put one cup at a time in to the recipe. (Please accept
this idea, for all my other answers are based on this.)

2. Do you notice any patterns in Table 2?  Explain.
   Yeah,the only amounts of water that the 4 and 10 oz. mesuring cups
can hold, are the even amounts of ounces becuase they can put 2 cups
of water at a time ionto the recipe.

3. Compare the patterns in your two tables. Think about the
combinations you used. What do you notice about even and odd numbers?
   I didn't notice anything about the "odd or even numbers"

4. What are your conclusions about the amounts that can (or cannot)
be measured using the two different sets of measuring cups?
   The 3 and 8 ounce measuring cup can measure all the amounts of
water while the 4 and 10 ounce measuring cups can only measure even
amounts of ounces, therefore, the 3 and 8 measuring cups are better.

5. Which set of cups would you recommend that Rodolfo buy?
   I recomend the 3 oz. and 8 oz. cups because Rodolfo can get all
the certain amounts of water he wants.

From:  Andrei L., age 11
School:  School No. 205, Bucharest, Romania
 

1. From Table 1, all the quantities can be measured using the first
set.
2. From Table 2, only even quantities can be measured using the
second set.
3. Even quantities can be measured using both sets, but odd
quantities can be measured only with the first set (3 and 8 oz.)
4. Both even and odd quantities can be measured with set 1, and only
even quantities can be measured using set 2.
5. I recommend Rodolfo to buy the first set.


1. In the first set, there is a cup for even quantities (8 oz.), and
one for odd quantities (3 oz.).  Using only addition and subtraction
of 8 and 3, all quantities could be obtained, e.g.:

8 - [(8 - 3) + (8 - 3 - 3)] = 1        (1)

then:

8 - 2 x 3 = 2                          (2)
3 = 3                                  (3)
(8 - 3 - 3) + (8 - 3 - 3) = 4          (4)

and so on.

2. Both cups from the second set measure even quantities. By addition
and subtraction of even numbers we cannot obtain odd ones, so that no
odd quantity could be measured.

3. As already said, set one could be used both for even and odd
quantities, while set 2 only for even ones.

4. Any quantity could be measured with set 1, and only even
quantities with set 2.

5. I recommend Rodolfo to buy the first set, in order to be able to
measure all types of quantities.

Note: as my server didn't work properly last week, I was not able to
see the animation, so that I answered only using the text.


15 students received credit this week.

Arda A., age 11 - The Koc Elementary School, Istanbul, Turkey
Chris B., age 11 - School in Rose Valley, Rose Valley, PA
Elizabeth C., age 12 - Wilmington Friends School, Wilmington, DE
Aviva E., age 12 - Wilmington Friends School, Wilmington, DE
Sarah E., age 11 - School in Rose Valley, Rose Valley, PA
Allen H., age 12 - Woodrow Wilson Middle School, Edison, NJ
Justin H., age 12 - Wilmington Friends School, Wilmington, DE
Andrei L., age 11 - School No. 205, Bucharest, Romania
Linda M., age 13 - Issaquah Middle School, Issaquah, WA
Monica M., age 13 - Caroline Davis Intermediate School, San Jose, CA
Henry N., age 13 - Caroline Davis Intermediate School, San Jose, CA
Long N., age 13 - Caroline Davis Intermediate School, San Jose, CA
Chet'la S., age 12 - Wilmington Friends School, Wilmington, DE
Ramjotvir S., age 13 - Caroline Davis Intermediate School, San Jose, CA
Charlene T., age 13 - Wilmington Friends School, Wilmington, DE

View most of the solutions submitted by the students above


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