We have been picturing subtraction problems using green
squares and gray squares. You can also use
pink squares for an amount to subtract. For example, there are
$5$ green squares and $3$ pink
squares on the grid to the left, representing $5-3$. If we combine $3$
green squares and $3$ pink
squares to form $3$ gray squares, only $2$
green squares would remain. So this is a picture of the
subtraction problem $5-3=2$.
For each row in the table below, how many green
squares remain after combining with the pink squares?
Click the Collapse button to move the
pink squares on top of the green
Click to display the subtraction problem $5-6$. If we combine the
green and pink squares,
we are now left with one pink square rather than any number of
green squares. This means the answer is the
negative number $-1$.
By thinking of pink squares as negative numbers, you can do
addition problems involving negative and positive numbers.
Solve the addition problem in each row of the table below, counting
pink squares as $-1$ each and green
squares as $+1$ each.
Solve the addition problems in the table below.
We call two numbers opposites if they’re the same size, but one is positive and the
other is negative.
Notice that opposites can be drawn on the grid in exactly the same way, except that the
positive number uses green squares and the negative number uses
Solve each subtraction and addition problem in the table below, as
shown on the grid.
This tells you another way to do subtraction. You already know how to
do addition; to subtract a number, just add its opposite!
Let’s now see why the “add its opposite” rule still works when the number you’re subtracting
is negative. We’ll look at another way of thinking about subtracting a negative number, by
closely examining the subtraction problem $3-(-2)$.
We want to subtract $-2$ from $3$. Think of this as removing some squares on the grid with
value $-2$: that is, removing $2$ pink squares. Click
to see a way of writing $3$ that uses $2$
In each row of this table, click the
Expand button to split the number up into some
green squares and some pink
squares. Then click the Remove button to remove some of those
pink squares. What is the total value of the squares that remain
on the grid after removing these pink squares?
Notice that your answers in Questions subNegQ and
addPosQ are the same. This is because
subtracting a negative number means removing pink
squares, which has the same effect as adding green
squares. So this is another way to see that subtracting a number means the same thing as
adding its opposite.
For each subtraction problem shown, turn it into an addition problem and then
Solve each subtraction problem.
One common reason to add and subtract negative numbers is to solve problems related to
debt. Owing somebody money is the same thing as having a negative amount of money. Some
specific ways this works include:
Translate each word problem into an arithmetic problem, and solve it.