Introduction
This is a question involving whole numbers, for primary 5 to 6 (» grades 5 or 6 for 11 to 12 year olds). Besides showing the full thinking process of solving this mathematics question with metacognition and heuristics, I use a Before-After Comparison Bar Model Diagram with big and small units (quite a common useful technique used in solving Singapore school word problems). Once again, the planning and analysis stage (with the help of the bar model) is crucial, but once this is done, the calculations come out rather naturally.
Stage 1: Understanding the Problem
Can you explain the problem in your own words?
{ Read the questions two or three times. Make sure you can paraphrase it, not just parrot it. Make sure you understand all the words and make sure you know what the problem is talking about }
What topic is this on? What concepts are involved?
Whole numbers, ratio, before-vs-after comparison.
What are you supposed to find?
The total number of scouts and guides at the beginning of the camp (i.e. the “before” period).
Stage 2: Planning the approach
What are some heuristic(s) you can use?
· Bar-model diagrams
· before vs after comparison
· solve part of the problem (first)
· restating the problem in another way
· others as and when necessary
Let’s read the word problem sentence by sentence and try to translate the information into the bar diagram. “At the beginning of a camp, the number of boy scouts was five times the number of girl guides.” How do you draw that?
Well, I divide the working space into two. On the top half, is the “before” stage. On the bottom half is the “after” stage. For each of the two stages, reserve two lines: one for boy scouts, the other for girl guides, and I label “boys” and “girls” accordingly. For the “before” stage, draw five bars for the boys and one bar for the girls.
 |
| Figure 1. Interpreting the given sentences into the bar diagram. |
Next sentence … “After 81 boy scouts left the camp, the number of girl guides was twice the number of boy scouts.” How do you represent that on the diagram?
Er … in the “after” stage, there are 81 boy scouts less, so I need to draw a shorter bar … but how short? Anyway, only a number of boys went away but all the girls stayed put. So I draw one bar for the girls in the “after” stage, exactly the same length as for the “before” stage. But how to draw the bar for the boys in the “after” stage … hmmm …
Can you rephrase the problem?
In the after stage, the number of girl guides is twice of the number of boys, so … er … the number of boys must be half of the number of girls. So I draw a short bar for the boys that is half of the length of the girls’ bar. Since the difference between the “before” and “after” numbers for the boys is 81, I draw a horizontal arrow spanner to indicate that. I draw a vertical dotted line to help me compare these quantities more clearly.
Last sentence … “What was the total number of scouts and guides at the beginning of the camp? ” How do you draw that?
Total number … to represent that, I draw a vertical brace (or curly bracket) combining the boys and the girls in the “before” stage. Then I put a question mark (‘?’) to remind me that this is actually what the question is asking for.
Now do you observe anything?
I now have a short bar (a small unit) for the “after” boys. The bar for the “after” girls is one big (original size) unit, which is two small units. OK, let’s divide each big unit into two small units (using short dotted vertical dividers) and see what we get.
 |
| Figure 2. Further refinement of the bar diagram. |
Now do you observe anything?
Ah-hah! (*epiphany* light-bulbs flashing in the brain) The 81 actually corresponds to 9 small units!
Can you solve the problem now?
Yes! Yes! Yes!
Stage 3: Execution
 |
| Figure 3. Doing the calculations |
Stage 4: Evaluation
Is your answer correct? Does it fit the conditions of the posed problem?
Using 1 small unit « 9 people, and looking back at the diagram, we calculate as shown below (a rough pencil working will do, and this should not take very long). We systematically check against all the given clauses in each sentence of the given problem.
 |
| Figure 4. Checking the solution |
Final Presentation
The final presentation is short and sweet. The hard part was actually the thinking and the observation, which was explained in painful detail above.
 |
| Figure 5. Final Presentation |
Stage 5: Reflection
What did you learn by solving this problem?
I practised more and got better at the problem solving process.
I learned to use metacognition to monitor myself and ask myself questions as I am mulling over the maths problem. I also learned to be careful and to check my own work systematically.
I learned how to use the following heuristics:-
· Bar-model diagrams
· before vs after comparison
· solve part of the problem (first)
· restating the problem in another way (especially when I get stuck)
· Using big units and small units
What else did you learn that you can use in future problems of this sort?
I learned to cut big units into small units. I learned that if I can find the link between the big unit and the small unit and if I can figure out the small unit, then I can figure out everything else in the problem! In future if I encounter this sort of problem, I can try out this powerful tactic.
[ Remark: Actually, the drawing of the bar diagram is already part of the execution i.e. in practice, stages 2 and 3 merge, or shall we say go back and forth in tiny loops of interpretation followed by part drawing, followed by having greater understanding and insights, making observations, which leads to further planning and refinement of the diagram and so on, until the solution becomes obvious. In problems involving big units and small units, the solution becomes obvious once we can figure out what the small unit represents. ]
Is there another way to solve the problem?
Can you make up a similar problem?
Please write your comments below.
