Solution Using Verbal Argument and Model

Date: 02/06/2002 at 19:53:12
From: Tansy Woo
Subject: Whole numbers

Dear Dr. Math,

I am a teacher of primary school in Singapore. I have a challenging 
question and I am trying to explore all the possible heuristic 
approaches so that I can explain better to my students. They are using 
systematic list (table) and model drawing.

Mr. Li has fewer than 200 sweets. He wants to give each pupil in his 
class an equal number of sweets. If he gives each pupil 2 sweets, he 
will have 70 sweets left over; if 4, he will need 10 more sweets. How 
many pupils are there in the class? How many sweets does Mr. Li have?

Date: 02/06/2002 at 23:37:26
From: Doctor Peterson
Subject: Re: Whole numbers

Hi, Tansy.

I'll first use algebra, which often can be turned into a more concrete 
approach with little effort. Say there are N sweets and P pupils. Then

    2P = N-70
    4P = N+10

We can eliminate N by subtracting the first equation from the second:

    2P = 80

Therefore, P = 40 pupils, and N = 2P+70 = 150.

Now as I said, we can do the same with a verbal argument or with a 
model. Verbally, we see that if we first gave every pupil 2 sweets, we 
would have 70 left; giving each ANOTHER 2 sweets (making a total of 4 
each), we would need 10 more, meaning that we need 70+10 = 80 to give 
them 2 each. There must be 40 pupils.

Using a model, we can make a box representing one sweet per pupil; 2 
of these boxes are 70 less than the number of sweets we have:

    +-------+-------+------------+
    | class | class |     70     |
    +-------+-------+------------+
    \____________________________/
                sweets

but 4 of the boxes are 10 more than the number of sweets:

    +-------+-------+-------+-------+
    | class | class | class | class |
    +-------+-------+-------+-------+
    \____________________________/\_/
                 sweets            10

Comparing these, twice the class size plus 80 will be the same as 4 
times the class size:

    +-------+-------+---------------+
    | class | class |       80      |
    +-------+-------+---------------+
    +-------+-------+-------+-------+
    | class | class | class | class |
    +-------+-------+-------+-------+

Clearly, twice the class size is 80, and the class size is 40.

Both of these methods are equivalent to the algebraic method, and just 
require a little more imagination. Since there are several ways to 
solve the equation, there are probably also several equivalent ways to 
do it without equations.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/06/2002 at 23:44:56
From: Tansy Woo
Subject: Whole numbers

Dear Dr. Math,

I really appreciate your great help! Your workings are clear and I 
think the children will be able to understand it.

Is it possible to solve this question by using drawing pictures/
illustrations or by simplifying the question? If yes, how should I go 
about it?

Regards,
Wendy

Date: 02/07/2002 at 08:48:39
From: Doctor Peterson
Subject: Re: Whole numbers

Hello again!

I don't think you can get much closer to actual pictures than my 
visual solution; since you don't know how many of anything there is, 
you can't very well draw individual sweets or children, but have to 
draw something to represent groups of them, as I did. You might try 
using smaller numbers to simplify the question, but that would only 
make it simpler if you are using a table or try-and-adjust method.

One thing that might help is to choose some small numbers, say 5 
pupils and 18 sweets, and see that with 2 each there are 8 extra, and 
with 4 each you need 2 more. Then putting those figures into my 
drawing, children might get a feel for what is happening before trying 
the same method with unknown numbers. This is a good way to start 
problem solving; I call it "playing with the situation" in order to 
understand the problem before actually trying to solve it. Although 
this is not part of the actual problem solving method, it can help 
tremendously in seeing how to solve the problem.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/07/2002 at 20:34:29
From: Tansy Woo
Subject: Whole numbers

Dear Dr. Math,

I would like to express my sincere thanks for your prompt reply, and 
to say that your suggested solutions are clear and interesting.

Best regards,
Tansy