Question Video: Writing Addition and Subtraction Expressions for Rational Number Calculations on a Number Line Mathematics • First Year of Preparatory School

Video Transcript

Observe the calculation shown on the number line. Write a subtraction expression matching the calculation. Convert the subtraction expression to an addition expression.

In this question, we are given a calculation between rational numbers using a number line. The first part of the question wants us to write a subtraction expression representing this calculation. The second part of this question wants us to convert the subtraction expression into an addition expression.

To do this, let’s start by analyzing the calculation that is shown on the number line. The first thing that we can recall is that we can add numbers on a number line by moving right on the number line when they are positive and left when they are negative. We can see that we start at zero and move to 0.75 on the number line. This is the same as adding 0.75 to zero.

We can then see that we move to the left from 0.75 to 0.25. The length of this line is 0.5, and we move to the left. So we can think of this as adding negative 0.5 to 0.75. This gives us 0.75 plus negative 0.5. This is an addition expression. So it is the answer to the second part of the question.

We can see the result of this operation on the number line. If we start at zero then add 0.75 and add negative 0.5, this is the same as starting at zero and just adding 0.25. We can analyze this calculation in a different way to consider it as a subtraction. We could do this by using the properties of addition and subtraction of rational numbers. However, it is useful to see this conversion done on a number line.

To do this, we recall that a subtraction represents a displacement. In general, 𝑎 minus 𝑏 is the displacement when traveling from 𝑏 to 𝑎 on the number line. We can note that moving to 0.75 and then removing the displacement from zero to 0.5 is the same as the displacement as just traveling from 0.5 to 0.75. The displacement when traveling from 0.5 to 0.75 is given by 0.75 minus 0.5. The reason that it is useful to see this conversion on a number line is that it helps justify one of our rules of the addition of rational numbers.

This method gives us a reason why we say that 𝑎 plus negative 𝑏 is equal to 𝑎 minus 𝑏. However, it is not a proof; it is a justification. We have shown that the calculations show that 0.75 minus 0.5 is the same as 0.75 plus negative 0.5.