140. Modelling Situations with Algebra

Learning Intentions

• model simple situations Use algebra
• write expressions from descriptions
• To understand that applying a model requires defining what the variables stand for

Pre-requisite Summary

• Know that a variable or pronumeral stands for a number
• Be able to write simple algebraic expressions such as or
• Understand that multiplication can be written without the sign
• Be able to read word descriptions carefully
• Know that an expression represents a quantity, not a complete equation
• Understand that a model uses algebra to represent a real situation
• Know that variables must be defined clearly before using them

Worked Examples

Worked Example 1

Define a variable and write an expression for each situation:

a) The number of books on a shelf is more than the number of books in a box

b) A taxi fare is a fixed charge of $ plus $ for each kilometre travelled

Worked Example 2

Write an algebraic expression from each description:

a) more than a number

b) times a number

c) less than a number

Worked Example 3

Model each situation using algebra, clearly defining the variable:

a) A student has more stickers than Tom

b) A rectangle has length that is cm more than its width

Worked Example 4

Write an expression from the description, then State what the variable means:

a) The total cost of buying muffins at $ each

b) The number of seats in rows with seats in each row

Worked Example 5

Model each situation with an expression involving two variables:

a) The total number of pencils in red boxes and blue boxes, if each box holds pencils

b) The total cost of adult tickets and child tickets, if adult tickets cost $ and child tickets cost $

Worked Example 6

A model is given. Interpret it and state the variable meaning:

a)

b)

c)

Problems

Problem 1

Define a variable and write an expression for each situation:

a) The number of pens in a drawer is more than the number of pens in a bag

b) A cinema ticket costs $ plus a booking fee of $

Problem 2

Write an algebraic expression from each description:

a) more than a number

b) times a number

c) less than a number

Problem 3

Model each situation using algebra, clearly defining the variable:

a) A child has more marbles than Sam

b) A rectangle has length that is cm more than its width

Problem 4

Write an expression from the description, then state what the variable means:

a) The total cost of buying burgers at $ each

b) The number of chairs in rows with chairs in each row

Problem 5

Model each situation with an expression involving two variables:

a) The total number of apples in green bags and red bags, if each bag holds apples

b) The total cost of adult tickets and child tickets, if adult tickets cost $ and child tickets cost $

Problem 6

A model is given. Interpret it and state the variable meaning:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

State what must be done before using a variable in a model:

a) define what it stands for

b) Choose whether it represents a quantity or a cost

c) make sure the meaning matches the situation

Exercise 2.

Write an algebraic expression for each description:

a) more than a number

b) times a number

c) less than a number

d) the sum of a number and

Exercise 3.

Write an algebraic expression for each description:

a) a number increased by

b) a number decreased by

c) times a number

d) times a number , then add

Exercise 4.

Model each situation using algebra and define the variable:

a) A bag contains more oranges than a basket

b) A movie ticket costs $ for each person

c) A phone plan costs $ plus $ for each extra gigabyte

Exercise 5.

Model each situation using algebra and define the variable:

a) A rectangle has width and length

b) A class has more girls than boys

c) A bike hire costs $ plus $ per hour

Exercise 6.

Write an expression for each real-life situation:

a) The cost of notebooks at $ each

b) The total number of legs on chairs if each chair has legs

c) The perimeter of a square with side length

Exercise 7.

Write an expression for each real-life situation:

a) The total cost of pencils at $ each and erasers

b) The number of pages in books if each book has pages

c) The amount left after spending $ from $

Exercise 8.

For each model, state what the variable could represent in a real situation:

a)

b)

c)

d)

Reasoning

Exercise 9.

Explain why a model such as is incomplete unless the variable is defined.

Exercise 10.

A student says that the expression for “ less than a number ” is . Explain the mistake.

Exercise 11.

Noah says that in every model, the variable must stand for a single object rather than a number of objects. Is he correct? Explain.

Exercise 12.

Explain why the same algebraic expression can model different real situations.

Exercise 13.

A student writes the cost of movie tickets at $ each as . Describe the error.

Problem-solving

Exercise 14.

A fruit shop sells apples for $ each. Write an expression for the cost of buying apples.

Exercise 15.

A school hall has rows with chairs in each row. Write an expression for the total number of chairs.

Exercise 16.

A streaming service charges a fixed fee of $ plus $ for each movie rented. Write an expression for the total cost if movies are rented.

Exercise 17.

A garden is rectangular. Its width is m and its length is m. Write an expression for the perimeter.

Exercise 18.

A student has $ and spends $ on lunch and $ on a drink. Write an expression for the amount left.

Exercise 19.

An event sells adult tickets for $ and child tickets for $ . Write an expression for the total income from adult tickets and child tickets.

Potential Misunderstandings

• Students may think a variable always stands for one fixed value instead of a quantity that can vary
• Students may forget to define what the variable represents
• Students may reverse phrases such as “less than” when writing expressions
• Students may confuse addition and multiplication in models such as
• Students may think a model must always Use only one variable
• Students may not realise that the same expression can represent different situations depending on the variable definition
• Students may write an equation when only an expression is needed
• Students may think the letter chosen for a variable changes the mathematics, rather than just labelling the quantity