Introduction
This is a challenging question on the topic of ratio, proportion and percentages. I shall again illustrate the process of solving this mathematics question with metacognition and heuristics, which are applicable for all levels, and all topics. The Bar Diagram Model is a very well known heuristic. But there are many other good problem-solving heuristics as well. There is a key to solving this kind of question and in order not to spoil the fun, I shall let my readers have a go at it first. Please think a bit about this problem, and then read on.
Stage 1: Understanding the Problem
Can you explain the problem in your own words?
Siti paid \$ 24 for some towels at a discount of 20%. Now, with the discount, she got three more towels than without the discount.
What concepts is this problem testing you on?
Ratio, proportion, percentages, price, unit price …
Can organise the given information?
We can put the information into a table, like this:-
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| Figure 1. Heuristics: ‘Use a table’ and ‘compare before/after’ |
What are you asked to find?
(a) the number (quantity) of towels Siti bought with the discount
(b) the price of each towel (the unit price) before the discount
Stage 2: Planning the Method of Attack
How are things related?
Total Price = Quantity (how many she bought) x Unit Price (how much each cost)
This relation is similar to
Distance = Speed x Time
Have you solved a similar problem before? How did you solve it the last time?
Yes, a Distance-Speed-Time problem. We used a Triangle Mnemonic.
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| Figure 2. Triangle Mnemonics and Analogous Thinking |
How is this problem different?
Instead of ‘Distance’, we have ‘Total Price’
Instead of ‘Speed’, we have ‘Unit Price’
Instead of ‘Time’, we have ‘Quantity’
Ah! Maybe we can use a similar Triangle Mnemonic for Total Price, Unit Price and Quantity.
Remember: What are you trying to find?
The number of towels - the Quantity. OK … so … if we use a finger to cover ‘Q’ we see ‘T’ over ‘U’. That means
Quantity = Total Price ÷ Unit Price
But this problem looks more difficult, because there are so many unknowns …
Don’t give up. Try some heuristics. (emotional management)
Can you restate the problem in another way? (using a heuristic)
The discount of 20% means she paid 100% – 20% = 80% = 4/5 of the original price. Without the discount, she either would have paid more for the same number of towels, or for the same amount of money, she would have gotten less. …
Can you simplify the problem? (using a heuristic)
Suppose the unit price was halved. Then Siti would get double or 2 times the number of towels. What if … the unit price was 1/3 of the original? Then she would get thrice (3 times) the number of towels. If the price is 1/4 of the original, then she would get 4 times … It looks like the cheaper the things get, the more you can buy … there is a pattern … the numbers seem to go the opposite way … the fraction seems to be inverted (the reciprocal) … why? … why? … why? Oh! It’s because
Quantity = Total Price ÷ Unit Price
The division ‘÷’ causes the fractions to ‘turn over’. I see! J So, since the price is 4/5 of the original, Siti would have bought 5/4 of the number of towels without discount!
Can you draw a bar diagram for this? (using the famous Singapore heuristic)
Yes!
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| Figure 3. Before-After Comparison Bar Model |
Now, can we solve the problem?
Yes! Yes! Yes! It is now very easy.
Stage 3: Execution
(a) 1 part = 3 towels (after discount: 3 more towels)
5 parts = 3 ´ 5 = 15 towels
(b) 4 parts = 3 ´ 4 = 12 towels (before discount)
12 towels ¬¾® \$ 24
1 towel ¬¾® \$ 2
Answer:
(a) She bought 15 towels.
(b) Before the discount, each towel cost \$2.
Stage 4: Evaluation
Let us check the answer:-
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| Figure 4. Checking the answer |
Are we correct?
Yes! The numbers fit nicely.
Final Presentation
The final presentation of the solution can be rather succinct. The foregoing discussion seems long because all the thinking process are explained in full detail. The checking of the answers can be done in pencil as rough work.
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| Figure 5. Final Presentation |
Stage 5: Reflection
What did you learn by solving this problem?
Although all stages are important, the hardest part was stage 2, the planning stage. This involves trying various heuristics to look at the problem from different angles. [This is the real mathematics. Patience, observation and creativity are needed.] The key to solving this problem was to note the inverse ratio relationship between quantity and unit price (given a fixed total price). Once we managed to find the key, we draw the appropriate diagram and the rest of the calculations are pretty straightforward.
In future if you encounter a similar problem, how would you solve it?
If there is a similar problem in future, I can use the 5 stage problem-solving process:-
1. Understanding
2. Planning
3. Execution
4. Evaluation
5. Reflection (learn and transfer for use in future)
Use metacognition (thinking about my own thinking, ask myself questions) to guide myself through the five stages. Manage my own emotions. If I get stuck, do not give up. Look at the problem in different ways.
I can use the following heuristics:-
· Bar model diagram (yes, but there are others …)
· drawing a table
· comparing ‘Before vs After’
· using a Triangle Mnemonic
· thinking of a similar problem
· simplifying the problem
· rephrasing the problem in another way (e.g. convert percentages to fractions)
Dear blogger: why don’t you just present the answer?
Many book authors and teachers already do just that. But do pupils’ actually improve much? Mathematics still remains a mystery to many pupils. They think that certain people are born geniuses and that there is always a way that these geniuses somehow have the knack of finding the right steps straightaway (and they themselves cannot). All they do is try to memorise the algorithm and reproduce it, regardless of whether they understand it. There is research to show that using worked examples alone does not improve pupils’ mathematics much, and that heuristics and metacognition are actually very important in mathematical problem solving.
“Give a person a fish; you have fed him/her for today.
Teach a person to fish; and you have fed him/her for a lifetime”
If you take the time mimic these heuristic and metacognitive processes, your maths will definitely improve by leaps and bounds. You still need to know the basic concepts, which most teachers teach, but you need to learn to put them together. I hope you get empowered in this process.
