136e. Algebraic Fractions and Common Denominators
Learning Intentions
- To understand what an algebraic fraction is
- Solve the lowest common denominator of two algebraic fractions
- find equivalent algebraic fractions with different denominators
- add and subtract algebraic fractions and Simplify the result
Pre-requisite Summary
- Know that a fraction has a numerator and a denominator
- Understand that a pronumeral can stand for a number
- Be able to simplify numerical fractions
- Know how to find the lowest common multiple of two numbers
- Understand that equivalent fractions have the same value
- Be able to add and subtract numerical fractions with a common denominator
- Be able to simplify algebraic expressions by combining like terms
Worked Examples
Worked Example 1
State what makes each one an algebraic fraction:
a)
b)
c)
Worked Example 2
Find the lowest common denominator of each pair:
a)
b)
c)
Worked Example 3
Write an equivalent algebraic fraction with the given denominator:
a)
b)
c)
Worked Example 4
Add the algebraic fractions and simplify:
a)
b)
c)
Worked Example 5
Add or subtract the algebraic fractions and simplify:
a)
b)
c)
Worked Example 6
Add or subtract the algebraic fractions and simplify the result:
a)
b)
c)
Problems
Problem 1
State what makes each one an algebraic fraction:
a)
b)
c)
Problem 2
Find the lowest common denominator of each pair:
a)
b)
c)
Problem 3
Write an equivalent algebraic fraction with the given denominator:
a)
b)
c)
Problem 4
Add the algebraic fractions and simplify:
a)
b)
c)
Problem 5
Add or subtract the algebraic fractions and simplify:
a)
b)
c)
Problem 6
Add or subtract the algebraic fractions and simplify the result:
a)
b)
c)
Exercises
Understanding and Fluency
Exercise 1.
Identify which of the following are algebraic fractions:
a)
b)
c)
d)
Exercise 2.
Find the lowest common denominator of each pair:
a)
b)
c)
Exercise 3.
Find the lowest common denominator of each pair:
a)
b)
c)
Exercise 4.
Write an equivalent algebraic fraction with the stated denominator:
a)
b)
c)
Exercise 5.
Write an equivalent algebraic fraction with the stated denominator:
a)
b)
c)
Exercise 6.
Add or subtract and simplify:
a)
b)
c)
d)
Exercise 7.
Add or subtract and simplify:
a)
b)
c)
d)
Exercise 8.
Add or subtract and simplify:
a)
b)
c)
d)
Reasoning
Exercise 9.
Explain why
Exercise 10.
A student says that the lowest common denominator of
Exercise 11.
Noah says that to make an equivalent algebraic fraction, you only need to multiply the denominator. Is he correct? Explain.
Exercise 12.
Explain why
Exercise 13.
A student adds
Problem-solving
Exercise 14.
A formula includes the expression
Exercise 15.
A student writes two parts of a calculation as
Exercise 16.
In a pattern rule, the total change is given by
Exercise 17.
A science formula is written as
Exercise 18.
A quantity changes by
Exercise 19.
A designer combines two lengths written as
Potential Misunderstandings
- Students may think an algebraic fraction must have pronumerals in both the numerator and denominator
- Students may confuse the lowest common denominator with simply adding the denominators
- Students may forget that equivalent fractions are made by multiplying both numerator and denominator by the same expression
- Students may multiply only the denominator when making an equivalent algebraic fraction
- Students may try to add numerators and denominators separately
- Students may not Recognise when denominators already share a common factor
- Students may simplify too early and lose the common denominator structure
- Students may make arithmetic errors when combining the numerators after rewriting the fractions
- Students may forget to simplify the final result when possible