100. Multiplying and Dividing Integers
Learning Intentions
- To understand that the product or quotient of two integers will be positive if the two integers have the same sign
- To understand that the product or quotient of two integers will be negative if the two integers have opposite signs
- Solve the product and quotient of two or more integers
Pre-requisite Summary
- Know that integers can be negative, zero or positive. See 049. Integers on the Number Line
- Be able to Identify the sign of an integer
- Recall basic multiplication facts and related division facts
- Understand that multiplication and division are inverse operations
- Know that a product is the result of multiplication
- Know that a quotient is the result of division. See 007. Division with Remainders and Short Division
- Be able to multiply and divide positive whole numbers accurately
- Understand that sign rules Apply before interpreting the final answer. See 051. Adding and Subtracting Negative Integers or 052e. Multiplying and Dividing Integers
Worked Examples
Worked Example 1
Determine whether each result is positive or negative:
a)
b)
c)
Worked Example 2
Calculate each product:
a)
b)
c)
Worked Example 3
Calculate each quotient:
a)
b)
c)
Worked Example 4
Find each value:
a)
b)
c)
Worked Example 5
Find each value:
a)
b)
c)
Worked Example 6
Evaluate each expression:
a)
b)
c)
Problems
Problem 1
Determine whether each result is positive or negative:
a)
b)
c)
Problem 2
Calculate each product:
a)
b)
c)
Problem 3
Calculate each quotient:
a)
b)
c)
Problem 4
Find each value:
a)
b)
c)
Problem 5
Find each value:
a)
b)
c)
Problem 6
Evaluate each expression:
a)
b)
c)
Exercises
Understanding and Fluency
Exercise 1.
Decide whether each answer will be positive or negative:
a)
b)
c)
d)
Exercise 2.
Calculate each product:
a)
b)
c)
d)
Exercise 3.
Calculate each quotient:
a)
b)
c)
d)
Exercise 4.
Find each product of three integers:
a)
b)
c)
Exercise 5.
Find each quotient of three integers:
a)
b)
c)
Exercise 6.
Evaluate each expression:
a)
b)
c)
Exercise 7.
Write the sign, then calculate:
a)
b)
c)
d)
Exercise 8.
Complete the patterns:
a)
b)
Reasoning
Exercise 9.
Explain why
Exercise 10.
A student says that a negative number times a negative number must be negative because both numbers are negative. Explain the mistake.
Exercise 11.
Explain how you can predict the sign of
Exercise 12.
Noah says that
Exercise 13.
A student works out
Problem-solving
Exercise 14.
A submarine changes depth by multiplying a scale factor of
Exercise 15.
A game score changes by
Exercise 16.
A temperature change of
Exercise 17.
A business makes a loss of $
Exercise 18.
A number machine multiplies by
Exercise 19.
A diver changes depth by multiplying
Potential Misunderstandings
- Students may think two negative integers always give a negative result
- Students may confuse the sign rules for addition and subtraction with the sign rules for multiplication and division
- Students may look only at the first sign and ignore the second sign
- Students may forget that two integers with the same sign give a positive product or quotient
- Students may forget that two integers with opposite signs give a negative product or quotient
- Students may apply the sign rule correctly but make an error with the multiplication or division facts
- Students may not realise that the sign can be determined before calculating the numerical part
- Students may lose track of the sign when working with three or more integers
- Students may think division with negatives follows a different sign rule from multiplication, when the pattern is the same
- Students may not use brackets clearly, causing confusion about which number is negative