039. Algebraic Modelling with Expressions

Learning Intentions

• To know that algebra can model a variety of situations
• Apply an expression in a modelling situation
• Draw an expression from a problem description

Pre-requisite Summary

• Understand that a variable can represent a changing number in a real situation
• Be able to Interpret simple algebraic expressions such as
• Know the meaning of terms, coefficients and constant terms
• Be able to Substitute values into an expression
• Understand common operation words such as sum, difference, product, total, per, each and fixed cost
• Be able to translate short verbal descriptions into arithmetic operations

Worked Examples

Worked Example 1

A phone plan costs a fixed $12 plus $3 for each gigabyte used.

a) Write an expression for the total cost.

b) Explain what each part of the expression represents.

Worked Example 2

A taxi fare has a $5 flagfall and costs $2 per kilometre .

a) Write an expression for the fare.

b) Solve the fare when .

Worked Example 3

A school buys notebooks at $4 each and one folder for $6.

a) Write an expression for the total cost.

b) State what the variable represents.

Worked Example 4

A rectangle has length cm and width cm.

a) Write an expression for the perimeter.

b) Explain how the expression models the situation.

Worked Example 5

A person has $20 already saved and then saves $8 each week for weeks.

a) Write an expression for the total amount saved.

b) Find the amount saved after weeks.

Worked Example 6

A bag contains red marbles and more blue marbles than red marbles.

a) Write an expression for the number of blue marbles.

b) Write an expression for the total number of marbles.

Problems

Problem 1

A movie streaming service costs a fixed $10 plus $4 for each month .

a) Write an expression for the total cost.

b) Explain what each part of the expression represents.

Problem 2

A parking fee has a $3 entry charge and costs $2 per hour .

a) Write an expression for the total fee.

b) Find the fee when .

Problem 3

A shop sells pencils at $2 each and one ruler for $5.

a) Write an expression for the total cost.

b) State what the variable represents.

Problem 4

A rectangle has length cm and width cm.

a) Write an expression for the perimeter.

b) Explain how the expression models the situation.

Problem 5

A student has $15 already and then earns $7 each week for weeks.

a) Write an expression for the total amount of money.

b) Find the amount after weeks.

Problem 6

A container has blue counters and more green counters than blue counters.

a) Write an expression for the number of green counters.

b) Write an expression for the total number of counters.

Exercises

Understanding and Fluency

Exercise 1.

Write an expression for each situation:

a) $6 per ticket for tickets

b) $9 fixed charge plus $2 per item

c) more than a number

Exercise 2.

Write an expression for each situation:

a) $12 already saved and then $4 each week for weeks

b) a number multiplied by

c) less than a number

Exercise 3.

Apply each expression by substituting the given value:

a) when

b) when

c) when

Exercise 4.

Apply each expression in context:

a) A fare is . Find the fare when .

b) A cost is . Find the cost when .

c) A total is . Find the total when .

Exercise 5.

Construct an expression from each description:

a) A gym charges $25 joining fee and $12 per week for weeks

b) A school hires buses at $90 each

c) A person has $40 and spends $x$ dollars

Exercise 6.

Construct an expression from each description:

a) The total number of legs on chairs

b) The perimeter of a square with side length

c) The total cost of notebooks at $3 each and one pen for $2

Reasoning

Exercise 7.

Explain why the expression could model a situation with a fixed amount and a changing amount.

Exercise 8.

A student says that “$4 per ticket and a $3 booking fee” should be written as . Explain the mistake.

Exercise 9.

Explain what the variable represents in the expression for a real-life model.

Exercise 10.

A student writes “$20 already saved and then $6 each week for weeks” as . Explain why this is incorrect.

Problem-solving

Exercise 11.

A concert ticket costs $18 and there is a one-time booking fee of $4. Write an expression for the total cost of buying tickets, then find the cost for tickets.

Exercise 12.

A plumber charges a $60 call-out fee and $25 per hour for hours. Write an expression for the total charge, then find the charge for hours.

Exercise 13.

A rectangle has length cm and width cm. Write an expression for its perimeter.

Exercise 14.

A school fundraiser starts with $150 and then raises $35 each day for days. Write an expression for the total money raised, then find the total after days.

Exercise 15.

A shop sells muffins for $3 each and cakes for $8 each. Write an expression for the total cost of buying muffins and cakes.

Exercise 16.

A bag contains red counters and twice as many yellow counters. Write an expression for the number of yellow counters and for the total number of counters.

Potential Misunderstandings

• Students may think algebra only applies to number puzzles and not to real situations
• Students may confuse a fixed amount with a changing amount in a model
• Students may reverse operation words, for example writing “ more than ” as
• Students may use the variable inconsistently within the same situation
• Students may forget that “per” usually indicates multiplication
• Students may omit a constant term such as a joining fee or starting amount
• Students may substitute into an expression incorrectly when applying a model
• Students may write an arithmetic answer instead of a general algebraic expression when asked to model a situation