111. Operations with Negative Fractions

Learning Intentions

• To understand that the techniques for adding, subtracting, multiplying and dividing positive fractions also Apply to negative fractions
• To understand that the rules for positive and negative integers also apply to fractions
• add, subtract, multiply and divide negative fractions and mixed numerals

Pre-requisite Summary

• Know how to add and subtract positive fractions Use a common denominator
• Know how to multiply and divide positive fractions
• Understand that dividing by a fraction means multiplying by its reciprocal
• Know that integers can be positive or negative
• Know the sign rules for multiplication and division of integers
• Be able to compare fractions on a number line
• Be able to convert between mixed numerals and improper fractions
• Be able to Simplify fractions fully

Worked Examples

Worked Example 1

Evaluate each expression:

a)

b)

c)

Worked Example 2

Evaluate each expression:

a)

b)

c)

Worked Example 3

Multiply or divide:

a)

b)

c)

Worked Example 4

Multiply or divide:

a)

b)

c)

Worked Example 5

Convert to improper fractions, then evaluate:

a)

b)

c)

Worked Example 6

Convert to improper fractions, then evaluate:

a)

b)

c)

Problems

Problem 1

Evaluate each expression:

a)

b)

c)

Problem 2

Evaluate each expression:

a)

b)

c)

Problem 3

Multiply or divide:

a)

b)

c)

Problem 4

Multiply or divide:

a)

b)

c)

Problem 5

Convert to improper fractions, then evaluate:

a)

b)

c)

Problem 6

Convert to improper fractions, then evaluate:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

Evaluate each sum:

a)

b)

c)

Exercise 2.

Evaluate each difference:

a)

b)

c)

Exercise 3.

Multiply the fractions:

a)

b)

c)

Exercise 4.

Divide the fractions:

a)

b)

c)

Exercise 5.

Determine whether each answer will be positive or negative, then Calculate:

a)

b)

c)

Exercise 6.

Convert each mixed numeral to an improper fraction:

a)

b)

c)

Exercise 7.

Perform the operation on mixed numerals:

a)

b)

c)

Exercise 8.

Perform the operation on mixed numerals:

a)

b)

c)

Reasoning

Exercise 9.

Explain why is found using the same common denominator method as .

Exercise 10.

A student says that must be negative because both fractions are negative. Explain the mistake.

Exercise 11.

Noah says that because he subtracted both numerators and denominators. Explain why this is incorrect.

Exercise 12.

Explain why dividing by gives a negative answer when the dividend is positive.

Exercise 13.

A student converts to . Describe the error.

Problem-solving

Exercise 14.

A scuba diver is at m relative to sea level and rises by m. What is the diver’s new position?

Exercise 15.

A bank balance changes by of a dollar each day for days. What is the total change?

Exercise 16.

A temperature of drops by . What is the new temperature?

Exercise 17.

A length of rope is cut into pieces of length m. If the total change in measured length is m, how many pieces does this represent?

Exercise 18.

A machine multiplies an input by . What is the output when the input is ?

Exercise 19.

A hiker is km relative to a checkpoint and then moves forward km. What is the new position?

Potential Misunderstandings

• Students may think the sign rules for integers do not apply to fractions
• Students may forget to Solve a common denominator when adding or subtracting negative fractions
• Students may add or subtract denominators instead of using equivalent fractions
• Students may ignore the negative sign when simplifying
• Students may think two negative fractions multiplied together give a negative result
• Students may forget that dividing by a fraction means multiplying by its reciprocal
• Students may reverse the wrong fraction when dividing
• Students may convert a negative mixed numeral to an improper fraction incorrectly
• Students may place the negative sign on only one part of a mixed numeral instead of on the whole number
• Students may simplify arithmetic correctly but leave the final sign incorrect
• Students may think subtraction with negative fractions follows different fraction rules from subtraction with positive fractions
• Students may not Recognise that the same fraction methods are used first, and then the sign rules are applied

Next: 112r. Review Decimals, Place Value and Conversions