AC9M8A03 · YEAR 8 · ALGEBRA

Mathematical Modelling

ACARA v9 CONTENT DESCRIPTION “use mathematical modelling to solve applied problems involving linear relations, including financial contexts; formulate problems with linear functions, choosing a representation; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model”

Modelling a situation with a straight line

Many everyday situations change by a constant amount at every step, and these are exactly the situations a straight line describes well. A phone plan, a taxi fare or a club membership often has a fixed amount to begin with and then a steady charge added for each extra unit. Mathematical modelling means taking such a situation and writing it as a linear relation, a rule of the form y = mx + c, so that we can reason about it with algebra. Take a plan that costs $20 to join and then $5 for each item bought. The total cost depends on the number of items n, and because every item adds the same $5, the cost climbs in equal steps. Writing this as cost = 20 + 5n captures the whole situation in one line, and plotting it shows a straight line that starts at 20 and rises steadily. Recognising that the change is constant is the signal that a linear model is the right tool.

From a real plan to a straight line
A fixed start plus a steady amount per item is a linear relation.
A situation that changes by the same amount each step is linear and can be modelled as cost = 20 + 5n.

Rate of change and starting value

Every linear model carries two key numbers, and each has a plain meaning in the situation. In cost = 20 + 5n the number 20 is the value of cost when n = 0, that is the cost before any items are bought, so it is the starting value c. The number 5 is how much the cost rises for each extra item, so it is the rate of change m. On the graph these two numbers are easy to see: c = 20 is the height where the line meets the cost axis, the intercept, and m = 5 is the steepness, the rise of $5 for every one step to the right. This is the same intercept and steepness idea met when graphing lines, now read in the language of the situation. Naming m as a rate and c as a start makes a model easy to interpret, because the moment you see the rule you know both where the cost begins and how fast it grows.

Reading m and c from the graph
The start value c is where the line meets the cost axis; m is the rise per step.
c = 20 is the starting value and m = 5 is the rate of change, the extra cost for each additional item.

Three ways to show the same relation

The same linear relation can be presented in three connected ways, and a good modeller moves between them freely. The first is a rule, the equation cost = 20 + 5n, which is compact and works for any number of items. The second is a table of values, pairing n = 0, 1, 2, 3 with cost = 20, 25, 30, 35, which makes the constant step of 5 plain to see. The third is a graph, the straight line through those points, which shows the trend at a glance. Choosing a representation is part of formulating the problem: a table is handy for a few exact values, a graph is best for spotting trends, and a rule is best for calculating or for very large numbers. Each representation can be turned into the others, and because they all describe the one relation they must always agree.

One relation, three representations
The rule, the table and the graph all describe cost = 20 + 5n.
A linear model can be shown three ways - a rule, a table of values, and a graph - and each agrees with the others.

A worked financial model

A worked example shows the whole modelling cycle on one financial question. First formulate the model: the plan gives cost = 20 + 5n. To predict a value, substitute the number you know; for 6 items the cost is 20 + 5(6) = 20 + 30 = 50, so 6 items cost $50. To find when a target is reached, set the model equal to that target and solve; to reach a cost of $45 we write 20 + 5n = 45, subtract 20 to get 5n = 25, then divide by 5 to get n = 5, so $45 buys 5 items. The first question runs the model forwards from a known n to a cost, and the second runs it backwards from a known cost to find n. Both use the same single rule, which is the power of having a model.

Using the model to answer a question
Substitute to predict a cost, or solve to find when a target is reached.
Substitute into the model to predict a cost, or solve to find when a target is reached: 6 items cost $50.

Interpreting and reviewing the model

A number on its own is not the end of modelling; the answer must be stated in the words of the situation and then judged. For the prediction above we say the cost of 6 items is $50, not merely 50, so the meaning is clear. Next check that the answer is reasonable: $50 is 6 steps of $5 added to a $20 start, which fits, and 5 items for $45 also makes sense. Finally, review the model itself, because a linear relation is only appropriate while the rate of change truly stays constant. This model assumes the per-item cost stays $5 every time. If the plan offered a bulk discount once you bought ten items, the rate would change and the single straight line would no longer fit, so the model would need adjusting. Deciding whether a constant rate really describes the situation is a genuine part of the work, not an afterthought.

Interpreting and reviewing the answer
State the answer in context, check it, and review the assumptions.
State the answer in context, check it is reasonable, and review whether a constant rate really fits.

Why this matters

Linear models describe a remarkable range of everyday situations, from the cost of a plan to wages earned by the hour, fuel used over a distance, or savings that grow by a fixed amount each week, all with one simple straight-line idea. Learning to formulate a situation as a rule, to choose between a rule, a table and a graph, to solve the model and then to review whether it is appropriate turns algebra into a practical tool for the real world. This builds directly on solving linear equations and reading their graphs, and it leads on to experimenting more freely with linear functions in the next unit. For now we stay within linear relations with one independent variable that change at a constant rate; questions about percentages of an amount or about sharing in a ratio belong to other units and are not the focus here.

Teaching tip: keep returning to the two questions that unlock every linear model. What is the starting value, the amount before anything happens? And what is the constant rate, the amount added at each step? Once a student can name these two numbers in a story, writing cost = 20 + 5n becomes natural rather than a guess, and reading m and c back out of a rule becomes just as easy.

A powerful habit is to ask whether the rate is really constant. Encourage children to imagine a bulk discount, a joining offer or a price rise, and to notice that any of these would bend the line. Spotting when a straight line stops being a fair description is the heart of reviewing a model, and it is a skill that carries far beyond this one example.

Builds on: Linear Equations and Graphs (AC9M8A02). That unit solved linear equations and drew their graphs; here we use those same lines and equations to model real situations and to interpret what the answers mean.
Quick self-check
1. A plan costs $20 to join plus $5 per item. Which rule models the total cost for n items?
2. In cost = 20 + 5n, what does the 20 represent?
3. Using cost = 20 + 5n, what is the cost of 6 items?
4. Using cost = 20 + 5n, how many items give a total cost of $45?
5. Before trusting a linear model, what should you review?