110r. Operations with Fractions and Mixed Numerals

Learning Intentions

• add and subtract fractions by first finding a lowest common multiple of the denominators
• multiply and divide fractions
• To understand that an integer can be written as a fraction with as the denominator
• perform the four operations on mixed numerals, converting to improper fractions as required

Pre-requisite Summary

• Know that equivalent fractions have the same numerical value
• Be able to Solve the lowest common multiple of two numbers
• Be able to Simplify fractions fully
• Understand the difference between proper fractions, improper fractions and mixed numerals
• Know how to convert between mixed numerals and improper fractions
• Recall that multiplication and division facts can help simplify fraction calculations
• Understand that division by a fraction is related to multiplying by its reciprocal
• Know that any integer can be written as a fraction over

Worked Examples

Worked Example 1

Add or subtract by first finding a lowest common multiple of the denominators:

a)

b)

Worked Example 2

Add or subtract by first finding a lowest common multiple of the denominators:

a)

b)

Worked Example 3

Multiply or divide fractions:

a)

b)

c)

Worked Example 4

Write each integer as a fraction and Evaluate:

a)

b)

c)

Worked Example 5

Convert to improper fractions, then evaluate:

a)

b)

Worked Example 6

Convert to improper fractions, then evaluate:

a)

b)

Problems

Problem 1

Add or subtract by first finding a lowest common multiple of the denominators:

a)

b)

Problem 2

Add or subtract by first finding a lowest common multiple of the denominators:

a)

b)

Problem 3

Multiply or divide fractions:

a)

b)

c)

Problem 4

Write each integer as a fraction and evaluate:

a)

b)

c)

Problem 5

Convert to improper fractions, then evaluate:

a)

b)

Problem 6

Convert to improper fractions, then evaluate:

a)

b)

Exercises

Understanding and Fluency

Exercise 1.

Add or subtract the fractions:

a)

b)

c)

Exercise 2.

Add or subtract the fractions:

a)

b)

c)

Exercise 3.

Multiply the fractions:

a)

b)

c)

Exercise 4.

Divide the fractions:

a)

b)

c)

Exercise 5.

Write each integer as a fraction with denominator , then evaluate:

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 cannot be found by adding the denominators.

Exercise 10.

A student says that . Explain the mistake.

Exercise 11.

Explain why the integer can also be written as .

Exercise 12.

A student converts to . Describe the error.

Exercise 13.

Noah says that to divide by a fraction, you divide both numerators and both denominators. Is he correct? Explain.

Problem-solving

Exercise 14.

A recipe uses cup of milk in one step and cup in another step. How much milk is used altogether?

Exercise 15.

A ribbon of length m is cut and m is used. How much ribbon is left?

Exercise 16.

A container holds L of juice. It is shared equally among people. How much juice does each person get?

Exercise 17.

A student walks km in the morning and km in the afternoon. How far does the student walk altogether?

Exercise 18.

A baker makes trays of muffins, and each tray holds of a dozen muffins. How many dozens of muffins are made altogether?

Exercise 19.

A rope m long is cut into pieces of length m. How many pieces can be cut?

Potential Misunderstandings

• Students may add or subtract denominators when working with fractions
• Students may forget to find a common denominator before adding or subtracting unlike fractions
• Students may Use a common denominator that is not a common multiple
• Students may forget to simplify the final answer
• Students may multiply fractions correctly but not simplify before or after multiplying
• Students may confuse dividing by a fraction with dividing the numerators and denominators separately
• Students may forget that dividing by a fraction means multiplying by its reciprocal
• Students may not Recognise that an integer such as can be written as
• Students may convert mixed numerals to improper fractions incorrectly
• Students may forget to convert mixed numerals before multiplying or dividing
• Students may convert correctly but then make errors with the fraction operation
• Students may leave answers as improper fractions when a mixed numeral is expected, or vice versa

Next: 111. Operations with Negative Fractions