076. Equations and Expressions

Learning Intentions

• To understand the difference between an equation and an expression
• Determine if an equation is true or false, substituting for pronumerals if required
• write an equation given an English description

Pre-requisite Summary

• Understand that a pronumeral or variable can represent a number
• Be able to Substitute a number for a pronumeral in an algebraic statement
• Recall the meaning of the operation symbols , , , and
• Understand that an expression does not include an equals sign
• Understand that an equation states that two quantities are equal
• Be able to Evaluate simple expressions Use order of operations
• Be able to Interpret words such as sum, difference, product, quotient, and equals

Worked Examples

Worked Example 1

a) State whether is an expression or an equation.

b) State whether is an expression or an equation.

c) Explain the difference between an expression and an equation.

Worked Example 2

Determine whether each equation is true or false when :

a)

b)

c)

Worked Example 3

Determine whether each equation is true or false when and :

a)

b)

c)

Worked Example 4

Write an equation for each description:

a) a number plus equals

b) three times equals

c) less than is

Worked Example 5

Write an equation for each description and test whether it is true for the given value:

a) twice plus equals , when

b) , when

c) , when

Worked Example 6

For each statement:

a) write it as an equation

b) substitute the given value

c) decide whether the equation is true or false

“ more than is ”, when

Problems

Problem 1

a) State whether is an expression or an equation.

b) State whether is an expression or an equation.

c) Explain the difference between an expression and an equation.

Problem 2

Determine whether each equation is true or false when :

a)

b)

c)

Problem 3

Determine whether each equation is true or false when and :

a)

b)

c)

Problem 4

Write an equation for each description:

a) a number plus equals

b) four times equals

c) less than is

Problem 5

Write an equation for each description and test whether it is true for the given value:

a) twice plus equals , when

b) , when

c) , when

Problem 6

For each statement:

a) write it as an equation

b) substitute the given value

c) decide whether the equation is true or false

“ more than is ”, when

Exercises

Understanding and Fluency

Exercise 1.

State whether each is an expression or an equation:

a)

b)

c)

Exercise 2.

State whether each is an expression or an equation:

a)

b)

c)

Exercise 3.

Determine whether each equation is true or false for the given value:

a) , when

b) , when

c) , when

Exercise 4.

Determine whether each equation is true or false for the given value:

a) , when

b) , when

c) , when

Exercise 5.

Write an equation for each description:

a) a number plus equals

b) two times equals

c) less than is

Exercise 6.

Write an equation for each description:

a) more than is

b) three times equals

c) minus equals

Exercise 7.

Write an equation, then test whether it is true for the given value:

a) more than is , when

b) twice is , when

c) less than is , when

Exercise 8.

Write an equation, then test whether it is true for the given value:

a) , when

b) , when

c) , when

Reasoning

Exercise 9.

Explain why is an expression but is an equation.

Exercise 10.

A student says that is an equation because it has a pronumeral. Explain the mistake.

Exercise 11.

Explain why substituting a value into both sides of an equation helps decide whether it is true or false.

Exercise 12.

A student substitutes into and writes . Explain the error.

Problem-solving

Exercise 13.

A taxi fare is described by “$6 plus $2 for each kilometre equals the total fare $14”. Write this as an equation.

Exercise 14.

A student says, “Three times my number is .” Write an equation for this statement.

Exercise 15.

A phone plan cost is described by “$10 plus $3 for each gigabyte used equals $19”. Write an equation.

Exercise 16.

A rectangle has width . “The width plus equals .” Write this as an equation and test whether it is true when .

Exercise 17.

A game score is described by “Twice the score minus equals .” Write the equation and test it for .

Exercise 18.

A savings problem says, “$8 more than the amount saved equals $23.” Write the equation and test it when .

Potential Misunderstandings

• Students may think any algebraic statement is an equation, even when there is no equals sign
• Students may think an expression must include a pronumeral, even though numerical expressions also exist
• Students may confuse the equals sign with a signal to Calculate rather than a statement that two quantities are equal
• Students may substitute into only one side of an equation when deciding whether it is true or false
• Students may treat as or as the two-digit number formed by and
• Students may reverse word descriptions such as “ less than ” and write instead of
• Students may think an equation is false just because it is not yet solved

Next: 077. Solving Equations by Inspection