The two sides of a coin

Sundaram S

One of the purposes of learning mathematics in school is to apply the concepts and procedures of mathematics to “quantify and understand” real life situations. These situations are usually presented to students in simplified versions as “word problems” at two levels – the learning level and the application level.

In primary and lower middle schools, students need to master the basic concepts and procedures of mathematics. For this they are given “raw problems” like 3+4=? as well as word problems reflecting simple life situations where these concepts and procedures can be used. We can call this the learning level, as the main objective here is to master computations and procedures. A problem such as, “If a shop has 8 baskets with 10 apples in each basket, what is the total number of apples in the shop?” is an example.

By upper middle school, students are expected to have mastered all the basic concepts and procedures of basic mathematics. They are now exposed to a variety of life situations which can be understood and explored using all the mathematics that has been learnt until then. We can call this the application level which may also need and nurture higher order thinking skills. A problem like, “If Rs. 10,000 is deposited in a bank account, what interest would it earn if the interest rate is 8%?” is an example.

These application level situations can be broadly classified as

• Situations needing measurement of physical aspects of objects and shapes like length, volume, and surface area with appropriate units of measurement.
• Physical situations like speed, time and distance as well as work, wages, and time taken.
• Commercial situations involving income, expenditure, sale, purchase, savings, banking, investment, etc.

This gradation in the use of word problems (learning vs application) has not been brought out very well in the current syllabus. Word problems related to both “learning and application levels” occur at random stages in many chapters without a unifying theme. For example in class 8, many commercial applications of mathematics come in a chapter called “Comparing Quantities” (Ch 8). Lack of a unifying theme could also be a factor in making the problems difficult for students to understand. A unifying theme like “Application of Mathematics to Life” at the middle school stage could enable better focus and understanding.

Apart from grouping them as “Application of Mathematics to Life”, many of the word problems based on commercial situations can also be used as a starting point for thinking about several basic ideas in economics, which is introduced as a formal subject from class 9. A few questions for thinking and discussion at the end of some of the word problems would achieve the purpose. It is important to note that this however will only be in the nature of informal introduction. This should be done only at the class 8 level where students have sufficient life experiences to understand basic ideas in economics as well as mastery of the mathematical procedures.

These examples could ideally be done in class 9 as an introduction to ideas in economics, taking off from word problems of “applications of mathematics” which they have done in earlier classes. These could be taught jointly by the math and economics teachers. Students could also be encouraged to discuss these problems with their parents to get better insights regarding the economics of managing a home.

Let us consider a few examples.

Informal introduction to some ideas from the field of economics

Problem 1

Ramu goes to the vegetable market with Rs 150. In one shop tomatoes are available at Rs 25/kg and in another they are available at Rs 30/kg. What quantity of tomatoes can he buy at each of the shops?

Questions for discussion

1. If the amount of money Ramu has is fixed, how would you describe the relation between the price and quantity of tomatoes that could be purchased with that amount?
2. Assuming that Ramu has sufficient money, how would you describe the relation between the quantity purchased and the amount required for the purchase?
3. Can you think of other areas of life where you can see such relations?

Problem 2

At the end of November 2015, Ramu’s parents made a summary of the income and expenses for the month. The total take-home-salary of both his father and mother was Rs 28,455. They spent Rs 11,235 on food, Rs 5,000 on house rent, Rs 1,000 on Ramu’s school fees and Rs 10,345 on other expenses. How much did they save in that month?

Questions for discussion

1. What do you think Ramu’s parents could do with the money they save?
2. What would have happened if the food expenses had come to Rs 13,500 instead of Rs 11,234?
3. What could the family do if they wanted to buy a refrigerator costing Rs 12,000?
4. If they want to save more money in February 2015, what expenses would be relatively easier to reduce?
5. Why are the salaries called take-home-salaries?

Problem 3

Ramu purchased 5 kgs of tomatoes for Rs 200. After a week his mother purchased 4 kgs of tomatoes for Rs 140. Who purchased tomatoes at a cheaper price?

Questions for discussion

1. Why does the price of tomatoes vary from season to season (summer, winter, monsoon)? Would the same thing happen to tomato soup packets?
2. What do you think would be the differences in producing and selling tomatoes vs tomato soup packets?
3. Why does the price of tomatoes vary from day to day? Would the same thing happen to tomato soup packets?
4. Why does the price of tomatoes sometimes vary from shop to shop on the same day?
5. If you were a seller of tomatoes, how would you decide the price at which you would sell them on any particular day?

Problem 4

Ramu’s family wanted to buy a 150 lt refrigerator and they visited several showrooms and checked the prices. Big Bazaar was offering a 150 lt refrigerator for Rs 12,000 and offering an end of season discount of 15%. Reliance Mart was offering the same refrigerator for Rs 11,500 with a discount of 10%. In which shop should the family buy the refrigerator?

Questions for discussion

1. Why do shops offer discounts?
2. Why do discounts in the same shop change from month to month?
3. What could be the meaning of the term “end of season”?
4. Reliance Mart was offering to sell a refrigerator on EMIs. What could an EMI mean?
5. Would Ramu’s family buy a refrigerator just by comparing prices?

Problem 5

A few days before Diwali, Ramu’s father announced that he had received a bonus of Rs 15,000. His mother suggested that they deposit the entire amount in the bank and use it in the future for some major purchase. His parents had accounts with State Bank of India which offered the following savings schemes.

Questions for discussion

1. Why would a bank offer interest on saving schemes?
2. What would be the interest earned per year in both the schemes?
3. Why does the bank offer a higher rate of interest in Plan B?
4. What factors must Ramu’s parents consider before making the decision to invest in Plan A or B?
5. Why do you think Ramu’s father’s company decide to give him a bonus?

Remarks

These are just a few illustrative examples. Mathematics and economics teachers could write many such problems for exploring other basic ideas in economics. I am not giving any suggestions for discussions (in the classroom) in this article and am leaving it to the teachers to work them out as appropriate.

I feel that looking at the word problems done in earlier classes, from the perspective of commerce and economics would enable a smooth “from known to unknown” transition to an entirely new topic like economics.

Such an approach can also involve the family in discussing with their children such issues in economics which impact their day-to-day lives.

The author started his school career, lasting 24 years, with the Rishi Valley School. He has worked as a Principal, Teacher Trainer and Educational Consultant in several schools in India. His areas of interest are primary mathematics, school leadership, and quality in education. He conducts workshops for teachers of primary and middle School on the theme of “Understanding Primary Math for Effective Teaching”. He can be reached at sundaram48@yahoo.com.