On December 19, 2012, in her Confession from a Homeschool Mom: Singapore Math Stumped Me Today, Monise L. Seward, blogged that her 6th grader woke her up to ask her for help on a word problem in her Singapore Math book. The nonroutine question she shared was the following:
Mrs. Pappas had some apples. She sold 1/3 of the apples plus 5 more on the first day. She sold 1/3 of the remaining apples plus 5 more on the second day. She had 125 apples left in the end. How many apples did Mrs. Pappas have in the beginning?
If you use algebra to solve this problem, it’s unlikely that the working will arouse any excitement; in fact, you may find this method of solution to be somewhat uninteresting or boring. Yes, algebra does religiously solve the problem, but the solution is anything but elegant. Moreover, most average grade five or six students wouldn’t have acquired the maturity to solve it algebraically.
An algebraic check
Besides working backwards to check the answer, after solving the question, using the Singapore model method, I also checked it out by algebra.Looking at the symbolic clutter, I guessed then that even our Singapore grades 7 and 8 average students would likely be challenged to solve this problem by algebraic means.If we stick to working with only one variable, then we may end up with an unappetizing equation such as the following to solve.
A mum’s solution
Without a good working knowledge with the Singapore model method, we can expect most teachers and parents (who still remember their school math) to solve the question in a way similar to the one sketched and tweeted by my Pinterest and Twitter friend at PragmaticMom.com, as shown below:
2/3x – 5 = 125
2/3x = 125 + 5
2/3x = 130
x = (130 x 3)/2
x = 195
You add the 5 because you have to subtract the 5 apples from the 2/3 calculation because they were added in additionally. Then you do it again …
2/3x – 5 = 195
2/3 x = 195 + 5
2/3 x = 200
x = (200 x 3)/2
x = 300
No algebra, please!
Pretend for a while that algebra is an alien language to you! And you can only use a non-algebraic explanation to communicate your solution to a ten-year-old child! How would you go about doing it? Can you think of some intuitive methods?
After coming across this grade six question via @pragmaticmom, I tweeted a quick-and-dirty solution to the above problem, using the model, or bar, method. Give it a try first, before comparing yours with mine.
I also remarked that we could also solve this question by the Sakamoto method, which I assumed most of you in the United States might not be familiar with; so I’ll skip presenting the Japanese method of solution here.
Working backwards
Had the above grade five or six question involved the fraction 1/2 instead of 1/3, then using the “work backwards” strategy, via a flow chart, would have yielded an equally decent method of solution as the model approach.
Mrs. Pappas had some apples. She sold 1/2 of the apples plus 5 more on the first day. She sold 1/2 of the remaining apples plus 5 more on the second day. She had 125 apples left in the end. How many apples did Mrs. Pappas have in the beginning?
See a sketch of a common method of solution below.
Two bonus questions
Let me leave you with two grade 3/4 problems that lend themselves easily to the model method.1. A shop owner sold 2 more iPads than half the number of iPads in his stock. He then sold 2 fewer iPads than half of the remaining iPads. If he was left with 28 iPads, how many iPads did he have in his stock in the beginning?