I have a word problem: The price of a pen was $5, The price of a pencil was $2. Miss Lee bought a number of pens and pencils for $26. How many pens and pencils did she buy?
The day before yesterday I taught my pupils this: I made a list for the pens : 1 x $5 = $5 , 2 x $5 = $10 and so on. And also a list for the pencils : 1 x $2 = $2 and so on. Miss Lee bought 4 pens and 3 pencils because (4 x $5) + (3 x $2) = $26. However, I realize that this method cannot be used in big numbers.
Merry
The method you used is called make a list. It is a common problem-solving heuristic. Please continue to use it with your younger students. I wonder if Miss Lee could also buy 2 pens and some pencils. I know 5 pens is not possible because the money left is an odd number $1 and the price of a pencil is $2. Similarly, 1 pen or 3 pens are not possible. Students learn reasoning.
If students know algebra, they can set up equation 5x + 2y =26 where x is the number of pens and y is the number of pencils. As there is only one condition, you still need to use guess-and-check to solve this equation. Unless the problem says something about x + y.
If the value is not 26 but larger then the equation is 5x + 2y = k where k is the large number. A graph can be plotted and possible solutions seen on the graph. (With larger k the number of solutions increases).
For the young students, the method you use is probably the best. When they are older, they will learn to solve the same problem with larger k values.
In my opinion, it is much easier for kids to visualise the mathematical patterns and relationships when the numbers are in the table form such as the guess-and-check method, which I use when coaching kids esp. when they deal with large numbers, when they are in the pre-algebra stage. (:
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