We have looked at how to multiply whole numbers when they are positive. Now let's look at what happens when negative integers are included in questions. Remember the product is the answer when two numbers are multiplied together.
The applet below allows you to select two integers to multiply by using the sliders to change the value and the check boxes to change the sign.
Guiding questions
What is the sign of the product of one positive and one negative integer?
Is the product of $3\times\left(-4\right)$3×(−4) the same as $-4\times3$−4×3? Check this for more products.
What is the sign of the product of two negative integers?
To get comfortable with multiplying integers, we can use tools like the applet above to help us identify the patterns. Unlike adding and subtracting integers where we can use the number line or counters, multiplication comes down to looking at the sign of factors.
Preciously, we have seen that the product of two positive integers is a positive integer
The product or quotient of a positive integer and a negative integer is a negative integer.
The product of two negative integers is a positive integer.
If we are working with integers, we just need to determine what the sign will be and then we can multiply the absolute values of the integers as we already know how to do.
Evaluate: $-4\times5$−4×5
Think: We know that $4\times5=20$4×5=20, so we just need to determine what the sign will be. We have the product of negative integer and a positive integer, so the product will be negative.
Do:
$-4\times5$−4×5 | $=$= | $-20$−20 |
Evaluate: $-7\times\left(-5\right)$−7×(−5)
Think: We know that $7\times5=35$7×5=35, so we just need to determine what the sign will be. We have the product of two negative integers, so the product will be positive.
Do:
$-7\times\left(-5\right)$−7×(−5) | $=$= | $35$35 |
Evaluate: $8\times\left(-12\right)$8×(−12)
Evaluate: $-9\times\left(-11\right)$−9×(−11)
The same principals that help us to multiply integers also apply to divide. Check that the rules for multiplication also work for division.
We have seen that the quotient of two positive integers is a positive integer
The quotient of a positive integer and a negative integer is a negative integer.
The quotient of two negative integers is a positive integer.
Evaluate: $48\div\left(-6\right)$48÷(−6)
Think: We know that $48\div6=8$48÷6=8, so we just need to determine what the sign will be. We have the quotient of one positive and one negative integers, so the quotient will be negative.
Do:
$48\div\left(-6\right)$48÷(−6) | $=$= | $-8$−8 |
Evaluate: $21\div\left(-7\right)$21÷(−7)
Evaluate: $\frac{-60}{10}$−6010
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide integers and other rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.