Linear Costs and Relationships

Overview

In lesson 5 of the Design a Sustainable Community unit, students model how sustainable-community costs change over time or usage. They distinguish fixed costs from variable rates, then represent a relationship using a table, graph and linear rule before interpreting predictions in context.

Learning intentions

• WALT model changing costs using tables, graphs and linear rules.
• WALT identify the initial value and rate of change.
• WALT use equations to predict total costs.
• WALT interpret whether a linear model is reasonable in context.

Success criteria

• I can identify the initial value and rate of change in a cost situation.
• I can represent a relationship in a table, graph and equation.
• I can use substitution to check a predicted cost.
• I can explain what the graph means for the community.

Curriculum links

• Linear relations: graph relationships on the Cartesian plane and solve and verify linear equations.
• Mathematical modelling: formulate, represent and interpret linear relationships in practical and financial contexts.
• Digital exploration: use graphing technology to test patterns and compare linear models.
• Linear expressions: simplify and rearrange expressions using inverse and distributive properties.

Lesson structure (90 minutes)

1. 0–8 min · Hook and prior knowledge. Open with the sustainable-community cost hook and display two options: solar lighting with a large installation fee and lower maintenance, or cheaper installation with higher ongoing costs. Students discuss which option may be better and complete two mini-whiteboard prompts: “What stays fixed?” and “What changes?”

2. 8–20 min · Explicit teaching. Use the fixed-and-variable-cost teaching slides to model the rule total cost = initial value + (rate × amount). For example, a community garden costs $120 to install and $15 per month to maintain, so (C=120+15m). Students identify the initial value ($120), rate of change ($15 per month), units and realistic values of (m). Check understanding with mini-whiteboards, including substitution for 4 months.

3. 20–35 min · Guided cost-card investigation. In pairs, distribute sustainable-community cost cards and the linear-cost modelling worksheet. Each pair selects a scenario, such as water tanks, solar lights or garden maintenance, and highlights fixed costs in one colour and variable costs in another. Students complete a table for at least five values, write the linear rule and use the graph template to plot points. Pause twice for students to hold up their tables and compare the first differences.

4. 35–52 min · Graph and interpret. Students plot their relationship by hand, then use the graphing instructions and discussion prompts to enter the rule into graphing technology. They compare the digital graph with their plotted points and answer: What does the vertical intercept mean? What does the slope represent? Which costs are realistic only for whole months or whole items? Students write one context sentence explaining a point on their graph.

5. 52–70 min · Model comparison challenge. Present two competing providers on the model-comparison slides. For example: Provider A, (A=200+10m); Provider B, (B=80+16m). Students use tables, graphs or equations to decide which provider is cheaper after 5, 10 and 20 months. They locate the break-even point by finding when (A=B), verify the result by substitution, and explain which option suits a short- or long-term community project. Confer with each pair and address errors such as confusing the intercept with the rate.

6. 70–82 min · Independent application. Students complete the final modelling task on the independent prediction and explanation section: create a linear cost rule for a sustainable-community service from given fixed and variable costs, predict a specified total, graph the relationship and state one limitation of the model. Students who finish early compare their model with a second option and determine the break-even point.

7. 82–90 min · Plenary and exit check. Return to the plenary and exit-check slide. Students explain to a partner how the initial value and rate affect a graph, then submit a mini-whiteboard or worksheet response: “A bike-share scheme costs $150 to start and $8 per ride. Write the rule, calculate the cost of 12 rides, and explain what $150 represents.” Review responses before the next lesson.

Resources

• the sustainable-community cost slide deck
• the linear-cost modelling worksheet
• Sustainable-community cost cards
• Mini-whiteboards and markers
• Coloured pencils or highlighters
• Graph paper and rulers
• Graphing technology
• Projector or interactive display
• Calculators

Assessment

• Check mini-whiteboard responses for understanding of fixed cost, variable rate, substitution and units.
• During pair work, question students about the meaning of the intercept, slope and realistic domain; inspect tables and graphs for constant first differences.
• Use the exit check to assess whether students can form a rule, predict a value and interpret the initial value in context.

Differentiation

• Support students with colour-coded tables, a partially completed graph template, a word bank and sentence starters such as “The initial value represents…” and “The rate of change means…”.
• Provide guided substitution examples and allow students to use graphing technology alongside hand plotting. Work with one pair at a time while the other pair uses a clearly sequenced task card.
• For EAL learners, pre-teach “installation fee”, “maintenance”, “initial value”, “rate”, “total” and “break-even”; use diagrams, gestures and labelled examples.
• Reduce numerical complexity or provide a completed rule for students requiring additional support. Extension students compare competing linear models, determine break-even points algebraically and graphically, and critique whether the model remains appropriate for all values.

Extension

• Students design two pricing options for a community service and justify which is more sustainable over different time periods.
• Students investigate how changing only the initial value or rate changes the graph, then state and test a conjecture using graphing technology.