Tag: numberline

Posted in Elementary School Math

How should students understand the subtraction operation?

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Studies show that students whose understanding of subtraction is only to take away will have difficulty learning other mathematical concepts.

There are three ways by which subtraction can be understood: (1) Taking Away, (2) Difference, and (3) Inverse relation to addition operation. Pupils’ first experience with subtraction involves taking away. The dash sign means minus and minus is taken to mean ‘take away’.

Subtraction as ‘taking away’

Here are formats of subtraction tasks that involve taking away:

  1. Marco has 12 twelve marbles. He gave 5 to his friend Precy. How many does he have left? (This problem is represented by the equation, 12 – 5 = ____.)

    subtraction as taking away
  2. Marco has 12 marbles. He gave some to his friend, Precy. If he had 7 marbles left, how many did he give to Precy? (This problem is represented by the equation, 12 – ____ = 7.)
  3. Marco gave his friend 5 marbles. If has 7 marbles left, how many did he have at the start? (This problem is represented by the equation,  ____ – 5 = 7.)

Problem situation number 3  require subtraction representation but is actually an addition problem because the solution involve adding 5 and 7 instead of doing subtraction.

For most students this is all they understand about subtraction — to take away. This is probably because the use of subtraction in many daily life situations use this meaning. To compound this situation, many of the subtraction tasks in textbooks are also of this type. Very few, if there is any, will include problems that supports the development of the other meaning of subtraction.  And when for a long time all one know about subtraction is to take away, it would be very hard to accept other meanings. Studies show that students whose conception of subtraction is only to take away will have difficulty learning other mathematical concepts.

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Posted in Elementary School Math

What is an integer?

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Here are some ideas pupils need to learn about integers:

•A number represents a quantity. An integer is a type of number. An integer represents a quantity.

•Integers are useful in representing quantities and includes opposite sense. For example, going up 5 floors and going down 5 floors can be represented by +5 and -5 respectively. The sign ‘+’ represents up and ‘-’ represents  down. The ‘5’ represents the number of floors.

•The integer +5 is read as “positive five” and NOT “plus five”. The integer -5 is read as “negative five” and NOT “minus 5”.

•The words positive and negative are descriptions of the whole number 5 while the words plus and minus describe operation to be done with the numbers. That’s why it doesn’t make sense to read the integer -5 as “minus 5”. From what number are you subtracting it?

•The number 0 is an integer which is neither positive nor negative.

•Integers can be represented in a number line. An integer and its opposite are of the same distance from 0. For example, -4 is 4 units to the left of zero so its opposite must be 4 units to the right of 0. This integer is +4.

integers

Problem: The distance between two integers in the number line is 4 units. If one of the integer is 3 units from zero, what could be these two integers?

•The distance of an integer from zero is called the absolute value of the integer. So the absolute value of -4 is 4 and the absolute value of +4 is also 4.In symbol, /+4/ = 4 and /-4/ = 4.

Of course, merely explaining to students these ideas and giving them lots of exercises will not work. It will never work for many of them. Teachers have to design tasks or activities pupils can work on so that students can construct their own understanding of these ideas. Teachers can help scaffold their learning through problem solving tasks and through the questions and feedback they will provide the students.

Next post on this topic will be about absolute value and operations with integers.