A classic elementary math problem that folks from a number of professions, from psychologists to professors to priests like to ask is the following:
A bat and a ball cost $1.10 in total.
The bat costs a dollar more than the ball.
How much does the ball cost?
For novice problem solvers, the immediate, intuitive answer is 10 cents. Yet the correct response is 5 cents. Why is that so?
If the ball is 10 cents, then the bar has to cost $1.10, which totals $1.20. Why do most of us jump to the wrong conclusion—that the ball costs 10 cents?
We should expect few students to bother checking whether the intuitive answer of 10 cents could possibly be wrong. Research by Professor Shane Frederick (2005) finds that this is the most popular answer even among bright college students, be they from MIT or Harvard.
A few years ago, I included a similar question for a grade 2 supplementary title, as it was in vogue in some local text papers. See the question below.
Recently, while revising the book, I saw that the model drawing had been somewhat modified by the editor. Although a model drawing would likely help a grade 2 child to better visualize what is happening, however, a better shading, or the use of a dotted line, would have made the model easier to understand. Can you improve the model drawing?
Interestingly, I find out that even after warning students of the danger of simply accepting the obvious answer, or reminding them of the importance of checking their answer, variations of the above question do not seem to help them improve their scores. I recently tickled my Fan page readers with the following mathematical trickie.
Two cousins together are 11.
One is 10 years older than the other.
Find out how old both of them are.
Let me end with this Cognitive Reflection Test (CTR), which is made up of tricky questions whose answers tend to trap the unwary, and which may be suitably given to problem solvers in lower grades.
1. If it takes 5 machines 5 minutes to make 5 bearings, how long would it take 100 machines to make 100 bearings?
2. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?
3. A frog is climbing up a wall which is 12 m high. Every day, it climbs up 3 m but slips down 2 m. How many days will it take the frog to first reach the top of the wall?
4. A cyclist traveled from P to Q at 20 km/h, and went back at 10 km/h. What is his average speed for the entire journey?
5. It costs $5 to cut a log into 6 pieces. How much will it cost to cut the log into 12 pieces?
Expected incorrect answers
1. 100 minutes. 2. 24 days. 3. 12 days. 4. 15 km/h 5. $10
References
Donnelly, R. (2013). The art of thinking clearly. UK: Sceptre.
Yan, K. C. (2012). Mathematical quickies & trickies. Singapore: MathPlus Publishing.
Postscript: What’s your CTR score? Here is something to ponder about: Those with a high CTR score are often atheists; those with low CTR results tend to believe in God or some deity.
© Yan Kow Cheong, May 7, 2013.
