Multiplication and Division with Negative Numbers

Multiplication with negative numbers

Remember, one way to think of a multiplication problem like “\$2(3)\$” is as “\$2\$ copies of \$3\$.” When you have a multiplication problem where only the second number is negative, like \$2(-3)\$, this is a good way to think about it. (That is, as finding “\$2\$ copies of \$-3\$.”)

Solve these multiplication problems, as pictured on the grid to the left.

What if the first number in a multiplication problem is negative? You can see what to do in that case by following a pattern.

Solve these multiplication problems. If you’re not sure what to do, use the picture.

Notice that each row of the table has one less copy of \$3\$ than the previous row. That is, each row has either \$3\$ less green squares or \$3\$ more pink squares than the row above it.

Look at the table from negTimesPos. If you take the answer in the first row (\$6\$) and subtract the answer in the second row (\$3\$), you get \$3\$.

 What do you get if you take the answer in the second row and subtract the answer in the third row? \$-\$ \$=\$
 What do you get if you take the answer in the fourth row and subtract the answer in the fifth row? \$-\$ \$=\$
 What do you think you would get if you subtracted the answer in any row from the answer in the row above it?
Solve these multiplication problems.
 Look at the table above. What is \$-2(5)\$?
 What is \$2(5)\$?
 Now, what is the opposite of \$2(5)\$?

So following this pattern gives us a rule for multiplying any positive number by any negative number, in either order:

To multiply a positive number by a negative number, multiply the two numbers as though they were both positive, and then take the opposite.

Solve the following multiplication problems.

What if you want to multiply two negative numbers?

Solve these multiplication problems. Notice that each problem has one fewer copy of \$-2\$ than the one before it. Use the picture if you need help.
 Look at the table above. What is \$-3(-2)\$?
 What is \$3(2)\$?

Following these patterns tells you everything you need to know about multiplying with negative numbers:

• To multiply a positive and a negative number, multiply them as if they were both positive, and then take the opposite.
• To multiply two negative numbers, multiply them as if they were both positive.
Solve these multiplication problems.

Division with negative numbers

When you’re dividing a negative number by a positive number, it’s easiest to think of division in terms of splitting up. That is, when you see something like “\$\${-12}/4\$\$” or “\$-12\$ divided by \$4\$,” think of it as “\$-12\$ split into \$4\$ groups.”

What number is represented by each of the narrow pink rectangles that divides up the wide pink rectangle on the grid? (Remember that each pink square represents \$-1\$.)

When you’re dividing a negative number by another negative number, it’s easiest to think of division in terms of counting out. That is, when you see something like “\$\${-18}/{-3}\$\$” or “\$-18\$ divided by \$-3\$,” think of it as “the number of ‘\$-3\$’s in \$-18\$.”

How many copies of each narrow pink rectangle are there in the wide pink rectangle on the grid?

To divide a positive number by a negative number, you can again use the “pattern” method to figure out what you should do.

Solve these division problems which follow a pattern. Notice that each problem has one fewer copy of \$-2\$ than the one before it.
 Look at the table above. What is \$\$6/{-2}\$\$?
 Now, what is the opposite of \$\$6/2\$\$?

As you can see, the rules for division with negative numbers are exactly the same as the rules for multiplication:

• If one number is positive and one is negative, divide them as if they were both positive, and then take the opposite to get your final answer.
• If both numbers are negative, divide them as if they were both positive.
Solve these division problems.