Subtraction Metaphors 2

Concrete to Abstract

Both "take sway" and "less than" situations are relatively "concrete" situations which can be modelled and understood easily.

Compared to these, the following 2 situations are more abstract and need more life experience and abstract thinking. Therefor they are best taught in later grades.

Comparison/ Difference/ Distance situations

This life situation has many variations in real life.

Comparison"

Here we compare one collection with another to find out the bigger and the smaller ones. Let us see the following example.

Ram has a collection of 530 stamps whereas Shyam's collection is 235.

First question can be "whose stamp collection is bigger?" The answer to the question automatically has the answer to the related question "whose stamp collection is smaller?".

Difference

The "comparison" situation can be analysed further with another question "By how much is Ram's stamp collection bigger or smaller that Shyam's?".

This can be called "Difference situation" which is an extension of the "comparison situation".

Distance

When applied to linear measurements, the idea of "difference" can be modified into a "distance situation." Take the following problems.

The km stone at Town A reads 540 and at Town B, which is further down the road, the km stone reads 670. What is the distance between Towns A & B?

While measuring the length of a table Ram used a meter scale. The reading at one end of the table was 12 cm & the reading at the other end was 72 cm. What was the length of the table? (Students could offer reasonable guesses as to why the starting reading was 12 instead of 0!)

Distance measured with a ruler

Students need to realise that the distance between two points is actually the "total of the number of spaces between the readings". Keeping the 0 mark of the ruler at the starting point of the distance just makes the "difference/ distance" easy to calculate because one of the values is 0.

This concept is tested with the idea of measuring with a "broken ruler".

Distance on the road

Students also need to realise that this can cause 2 variations in problems involving travel along routes with km stones.

The distance covered is always the difference between the "starting reading & the "ending reading".

But if the problem asks for the number of km stones crossed from the starting point & the ending point, they need to add 1 to the distance answer. A distance of 5 km will have 6 km stones including the one at the starting point.

Subtractions involving Negative number

The "subtraction as distance" concept makes it easy to perform subtractions involving negative number, by marking the numbers on a number line which extends to both sides of zero.

Additive Comparison Vs Multiplicative Comparison

In daily life we do two kinds of comparisons. This is called "Additive Comparison" which uses the idea of the difference.

The other kind of comparison is "Multiplicative Comparison". In this we would say that Ram got twice as many marks as Shyam or Shyam got half the marks which Ram got. We will deal with this idea in chapters 10.5 & 10.6 on multiplication metaphors.

How much/ How many more? OR Adding to Subtract OR Complementary Addition situations

Let us consider the following problem.

At the end of the year, class 35 Class 5A students will move to Class 6A. Class 6A has only 32 student chairs. How many more student chairs are required to seat all the students?

In this situation we have 2 (concrete or abstract) collections. The task is to find out how much should be added to the smaller collection to make it equal to the bigger collection.

Adding to Subtract

This is the concept behind using addition (as Counting On) to perform subtraction. A problem like 18 - 15 can be thought of as the distance between 15 & 18, which can be found by adding from 15 till 18 is reached. We would have added 3 while reaching from 15 to 18.

Difficulty with this situation

The term "more" is usually associated with addition. In this subtraction situation, the term "more" is used but the operation required in subtraction. This is obviously a difficult concept for children to understand.

It would help if examples can be discussed from the life experiences of children themselves; the quantity of rice or milk that needs to be poured into a partly-filled container in order to fill it, finding the distance yet to be covered in the midst of a journey, the number of days still left from the school vacations etc.

Perils of mechanical rules

This also shows the perils of teaching a "rule" that if a problem has the word "more" then it is an addition problem. Understanding this situation is very important.

Number Sense Strategies for Subtraction

All the above life situations also provide us various strategies to perform subtraction using number sense. The chapter on Number Sense details several of them.

Use of the term "Take Away"

Like in the case of Put Together, we have avoided the term "Take Away" as it is both a process and concept also. Use of the term "Separate" makes it easier to internalise the other metaphors of subtraction.