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
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.