035. Equivalent Expressions and Algebraic Generalisation

Learning Intentions

• To know what it means for two expressions to be equivalent
• Determine whether two expressions are equivalent Use substitution
• generalise number facts using algebra

Pre-requisite Summary

• Understand that a variable can represent a number
• Be able to Substitute values into algebraic expressions
• Be able to Evaluate expressions using order of operations
• Understand that an expression does not have an equals sign
• Know that two numerical calculations can have the same value even if they look different
• Be able to Recognise simple arithmetic patterns
• Understand that algebra can Describe a rule that works for many numbers
• Be able to Use letters to represent any whole number

Worked Examples

Worked Example 1

a) Explain what it means for the expressions and to be equivalent.

b) Test the expressions when .

c) Test the expressions when .

Worked Example 2

Use substitution to decide whether the expressions are equivalent:

a) and

b) and

Worked Example 3

Use substitution to decide whether the expressions are equivalent:

a) and

b) and

Worked Example 4

Write an algebraic generalisation for each number fact:

a) an even number can be written as

b) the sum of two consecutive numbers

c) the product of and any whole number

Worked Example 5

Generalise the pattern:

a)

b)

c)

Write a rule for the sum of two consecutive whole numbers.

Worked Example 6

Generalise a number fact using algebra:

a) the sum of two even numbers is even

b) the product of an odd number and is even

c) write each statement using algebra

Problems

Problem 1

a) Explain what it means for the expressions and to be equivalent.

b) Test the expressions when .

c) Test the expressions when .

Problem 2

Use substitution to decide whether the expressions are equivalent:

a) and

b) and

Problem 3

Use substitution to decide whether the expressions are equivalent:

a) and

b) and

Problem 4

Write an algebraic generalisation for each number fact:

a) an odd number can be written as

b) the sum of two consecutive numbers

c) the product of and any whole number

Problem 5

Generalise the pattern:

a)

b)

c)

Write a rule for the sum of two consecutive whole numbers.

Problem 6

Generalise a number fact using algebra:

a) the sum of two odd numbers is even

b) the product of any whole number and is a multiple of

c) write each statement using algebra

Exercises

Understanding and Fluency

Exercise 1.

Decide whether each pair of expressions is equivalent by substituting and :

a) and

b) and

c) and

Exercise 2.

Decide whether each pair of expressions is equivalent by substituting suitable values:

a) and

b) and

c) and

Exercise 3.

Use substitution to test equivalence:

a) and

b) and

c) and

Exercise 4.

Use substitution to test equivalence:

a) and

b) and

c) and

Exercise 5.

Write an algebraic expression for each statement:

a) any even number

b) any odd number

c) three times any whole number

Exercise 6.

Write an algebraic generalisation for each fact:

a) the next number after

b) two consecutive numbers

c) three consecutive numbers

Exercise 7.

Generalise each number fact using algebra:

a) an even number plus an even number

b) an odd number plus an odd number

c) an even number plus an odd number

Exercise 8.

Generalise each number fact using algebra:

a) the sum of two consecutive numbers

b) the product of and any whole number

c) the sum of a number and

Reasoning

Exercise 9.

Explain what it means for two expressions to be equivalent.

Exercise 10.

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

Exercise 11.

Explain why substitution can be used to test whether two expressions are equivalent.

Exercise 12.

A student tests and with and gets the same value, then says the expressions must be equivalent. Explain why more care is needed.

Problem-solving

Exercise 13.

A student claims that and are equivalent. Use substitution with two values of to Check the claim.

Exercise 14.

Write an algebraic rule for the perimeter of a square with side length . Explain how this generalises repeated addition.

Exercise 15.

Two consecutive whole numbers are added. Write an algebraic expression for the sum and test it for .

Exercise 16.

Write an algebraic expression for an odd number and the next odd number. Then write an expression for their sum.

Exercise 17.

A teacher says “the sum of any whole number and the next whole number is always odd”. Write this using algebra.

Exercise 18.

A pattern shows:

Write an algebraic generalisation for the sum.

Potential Misunderstandings

• Students may think two expressions are equivalent only if they look the same
• Students may think expressions with the same numbers are automatically equivalent
• Students may use substitution incorrectly by replacing only one occurrence of a variable
• Students may think one successful substitution proves equivalence in every case
• Students may confuse an expression with an equation
• Students may not recognise that algebraic generalisation describes a rule for all suitable numbers
• Students may write examples instead of a general algebraic rule
• Students may confuse consecutive numbers with multiples or with numbers that differ by

Next: 036. Like Terms and Simplifying Expressions