133r. Review Evaluating and Comparing Algebraic Expressions

Learning Intentions

• Substitute values to Evaluate algebraic expressions
• To understand what it means for two expressions to be equivalent
• To understand how the commutative and associative laws for arithmetic can be used to Determine equivalence
• show that two expressions are not equivalent Use substitution

Pre-requisite Summary

• Know that a pronumeral or variable stands for a number
• Be able to replace a pronumeral with a given value
• Understand that multiplication may be written without the sign, for example
• Know the difference between an expression and an equation
• Recall the commutative law for addition and multiplication
• Recall the associative law for addition and multiplication
• Be able to evaluate numerical expressions accurately
• Understand that equivalent expressions always have the same value for the same substitution

Worked Examples

Worked Example 1

Substitute the given value and evaluate:

a) Solve when

b) Find when

c) Find when and

Worked Example 2

Substitute the given value and evaluate:

a) Find when

b) Find when

c) Find when

Worked Example 3

Decide whether each pair of expressions is equivalent:

a) and

b) and

c) and

Worked Example 4

Use the commutative or associative law to show that the expressions are equivalent:

a) and

b) and

c) and

Worked Example 5

Use substitution to show that two expressions are not equivalent:

a) and

b) and

c) and

Worked Example 6

For each pair, decide whether the expressions are equivalent. Use a law if possible, or substitution if needed:

a) and

b) and

c) and

Problems

Problem 1

Substitute the given value and evaluate:

a) Find when

b) Find when

c) Find when and

Problem 2

Substitute the given value and evaluate:

a) Find when

b) Find when

c) Find when

Problem 3

Decide whether each pair of expressions is equivalent:

a) and

b) and

c) and

Problem 4

Use the commutative or associative law to show that the expressions are equivalent:

a) and

b) and

c) and

Problem 5

Use substitution to show that two expressions are not equivalent:

a) and

b) and

c) and

Problem 6

For each pair, decide whether the expressions are equivalent. Use a law if possible, or substitution if needed:

a) and

b) and

c) and

Problems

Understanding and Fluency

Exercise 1.

Substitute the given value and evaluate:

a) when

b) when

c) when and

Exercise 2.

Substitute the given value and evaluate:

a) when

b) when

c) when

Exercise 3.

Decide whether each pair of expressions is equivalent:

a) and

b) and

c) and

Exercise 4.

Decide whether each pair of expressions is equivalent:

a) and

b) and

c) and

Exercise 5.

Use the commutative or associative law to justify equivalence:

a) and

b) and

c) and

Exercise 6.

Use substitution to show that the expressions are not equivalent:

a) and

b) and

c) and

Exercise 7.

For each pair, decide whether the expressions are equivalent. Use a law or substitution:

a) and

b) and

c) and

Exercise 8.

Evaluate both expressions for the same substitution:

a) and when

b) and when

c) and when

Reasoning

Exercise 9.

Explain what it means for two expressions to be equivalent.

Exercise 10.

A student says that and are equivalent because they both use and . Explain the mistake.

Exercise 11.

Noah says that if two expressions give the same value for one substitution, then they must be equivalent. Is he correct? Explain.

Exercise 12.

Explain how the commutative law can be used to show that and are equivalent.

Exercise 13.

A student says that and are not equivalent because the brackets are different. Describe the error.

Problem-solving

Exercise 14.

A taxi fare can be written as , where is the number of kilometres travelled. Find the fare when .

Exercise 15.

Two students write the total number of apples as and . Explain whether these expressions are equivalent.

Exercise 16.

A pattern rule is written as . Another student writes . Use substitution to decide whether the two rules are equivalent.

Exercise 17.

A builder writes the total length as . Another builder writes . Explain whether the expressions are equivalent.

Exercise 18.

A shop writes one total cost as and another as . Use substitution to show whether the two expressions are equivalent.

Exercise 19.

A student claims that and are equivalent. Test the claim using substitution.

Potential Misunderstandings

• Students may think two expressions are equivalent if they look similar
• Students may think two expressions are equivalent if they use the same variables and numbers
• Students may confuse evaluating an expression with solving an equation
• Students may substitute a value into one expression correctly but not the other
• Students may forget that multiplication is implied in expressions such as
• Students may think the commutative law applies to subtraction in the same way as addition
• Students may think the associative law changes the order of terms, rather than the grouping
• Students may believe that matching one substitution is enough to prove equivalence
• Students may not Recognise that one counterexample is enough to show two expressions are not equivalent

Next: 134r. Review Like Terms and Simplifying Expressions