Solution Using Verbal Argument and ModelDate: 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 |
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