039. Algebraic Modelling with Expressions

Learning Intentions

• To know that algebra can model a variety of situations
• apply an expression in a modelling situation
• construct 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 $3x+5$
• 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 $g$ 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 $k$.
a) Write an expression for the fare.
b) Find the fare when $k=7$.

Worked Example 3

A school buys $n$ 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 $x$ cm and width $x+3$ 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 $w$ weeks.
a) Write an expression for the total amount saved.
b) Find the amount saved after $6$ weeks.

Worked Example 6

A bag contains $r$ red marbles and $5$ 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 $m$.
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 $h$.
a) Write an expression for the total fee.
b) Find the fee when $h=5$.

Problem 3

A shop sells $p$ 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 $y$ cm and width $y+4$ 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 $t$ weeks.
a) Write an expression for the total amount of money.
b) Find the amount after $4$ weeks.

Problem 6

A container has $b$ blue counters and $3$ 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

1. Write an expression for each situation:
   a) $6 per ticket for $t$ tickets
   b) $9 fixed charge plus $2 per item $i$
   c) $5$ more than a number $x$

2. Write an expression for each situation:
   a) $12 already saved and then $4 each week for $w$ weeks
   b) a number $n$ multiplied by $7$
   c) $3$ less than a number $p$

3. Apply each expression by substituting the given value:
   a) $4x+3$ when $x=5$
   b) $2n+8$ when $n=7$
   c) $6p$ when $p=4$

4. Apply each expression in context:
   a) A fare is $3k+5$. Find the fare when $k=6$.
   b) A cost is $4m+2$. Find the cost when $m=8$.
   c) A total is $7w+10$. Find the total when $w=3$.

5. Construct an expression from each description:
   a) A gym charges $25 joining fee and $12 per week for $w$ weeks
   b) A school hires $c$ buses at $90 each
   c) A person has $40 and spends $x$ dollars

6. Construct an expression from each description:
   a) The total number of legs on $n$ chairs
   b) The perimeter of a square with side length $s$
   c) The total cost of $q$ notebooks at $3 each and one pen for $2

Reasoning

7. Explain why the expression $5x+12$ could model a situation with a fixed amount and a changing amount.

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

9. Explain what the variable represents in the expression $8n+15$ for a real-life model.

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

Problem-solving

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 $t$ tickets, then find the cost for $5$ tickets.

12. A plumber charges a $60 call-out fee and $25 per hour for $h$ hours. Write an expression for the total charge, then find the charge for $3$ hours.

13. A rectangle has length $x$ cm and width $x+2$ cm. Write an expression for its perimeter.

14. A school fundraiser starts with $150 and then raises $35 each day for $d$ days. Write an expression for the total money raised, then find the total after $4$ days.

15. A shop sells muffins for $3 each and cakes for $8 each. Write an expression for the total cost of buying $m$ muffins and $2$ cakes.

16. A bag contains $r$ 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 “$5$ more than $x$” as $5-x$
• 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