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Product Model
Area model is an abstract way of visualising the multiplication process. It can be modified into the Product Model which has many applications in life.
At the primary school we can visualise the product model as any of the following situations.
A car is being driven at a speed of 30 km/hr. What is the distance it would cover in 2 hours & 20 minutes?
A painter charges Rs 100/ m^2 of wall space, all inclusive. The total wall area of a room is 800 m^2. What would be the total cost of painting the room?
In a science test, the average marks scored by 35 students of Class 6A was 45.25. What was the total of all the marks scored by the students?
In a recipe for making kheer, 200g of sugar had to be added to a litre of milk. If 12 litres of kheer had to be prepared, how much sugar would be needed?
A water tap can deliver water at the rate of 10 litres/minute. How long would it take for a drum of 500 litres to be filled?
In the vegetable market the price of tomatoes was Rs 50 /kg. If you order for 2.5 kg of tomatoes, what would be the amount you need to pay the seller?
All the above problems are conceptually a “rate” multiplied by a relevant quantity.
Rate is a general term. Depending on the context it is given different names. Depending on the context, rate is called speed, price, average or some other suitable word.
The typical form of a rate is “some quantity” per “another quantity”.
The last problem is second part of a procedure called “Unitary Method” in commercial applications of arithmetic.
Cartesian Product/ Combinations
Cartesian Product is a way of counting various ways in which 2 events, which are independent of each other, can occur.
Counting in Cartesian Product actually turns out to be multiplication of 2 sets, a topic dealt with in higher classes. In a simplified form it can be understood as the number of different ways or combinations in which certain things can be arranged.
The various combinations can be visually understood by drawing a decision tree.
It leads to ideas of probability, which is studied in the higher classes. "Combinations" also occurs in different forms in different contexts. The following problem gives one such simple form.
For his birthday, Ram gets 3 T Shirts of different designs and 4 trousers in different colours. In how many different combinations can he wear them? Working out all the combinations, we would find that there are 3 X 4 combinations.
It is suggested that the Combinations metaphor may be taught only in Middle School. It leads to an idea of the Cartesian Product used in higher mathematics.
The idea of combinations captures the essence of multiplication in a way "put together" does for addition.
Repeated Addition Procedure
We see that the idea of multiplication arises out of many life situations, other than "repeated addition". Repeated addition is a procedure for getting the result of multiplying any two whole numbers. It is not a fundamental concept observable in the environment around us.
Unfortunately, "repeated addition" has become synonymous with multiplication, obscuring the other important metaphors.
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