Linear Relationships in Action

Overview

Students model and solve practical problems involving linear relationships, using tables, equations and graphs. They interpret the rate of change and initial value, solve for an unknown, and evaluate whether their answer makes sense in context.

Learning intentions

Students will:

• identify the initial value and constant rate of change in a situation
• represent a linear relationship using a table, equation and graph
• solve practical problems involving linear relationships
• explain and check solutions, including units and reasonable limits

Success criteria

• I can describe what the starting value and rate of change mean in a real situation.
• I can write a rule such as (y = mx + b) from a table or written problem.
• I can use an equation or graph to find an unknown value.
• I can check whether my answer is realistic and communicate it with correct units.

Curriculum links

• Algebra — mathematical modelling of applied problems involving linear relations and financial contexts.
• Algebra — graphing linear relations, solving linear equations and verifying solutions.
• Algebra — experimenting with linear functions using digital tools and testing conjectures.
• Number — choosing efficient calculation strategies and interpreting financial solutions.

Lesson structure (110 minutes)

1. 0–8 min · Hook and diagnostic. Display a practical comparison using the opening hook and comparison slide: a rideshare service charges a $7 starting fee plus $3 per kilometre, while a second service charges a $5 starting fee plus $4 per kilometre. Ask, “Which is cheaper for 3 km? What about 9 km?” Students estimate individually, then discuss with the group of five. Collect quick responses to identify prior understanding of tables, coordinates and substitution.

2. 8–22 min · Build the model. Use the modelling and vocabulary slides to introduce the terms initial value, rate of change, independent variable and dependent variable. Teacher models the rideshare example with a table for kilometres (x=0,1,2,3,4), identifies the constant difference, writes (C=3x+7), and plots the points. Students annotate their own copy of the linear relationships modelling worksheet and explain what each number in the equation means.

3. 22–40 min · Guided representation practice. Teacher presents three situations on the worked examples slides: a museum membership with an enrolment fee, wages paid at a constant hourly rate, and a streaming plan with a monthly fee plus a viewing charge. For each, students help complete a table, equation and graph on the worksheet. Pause after each representation and ask, “What does the gradient represent?” and “What does the intercept represent?” Students verify one table value by substitution and identify appropriate units.

4. 40–58 min · Matching and reasoning activity. Give students the Linear Graph Matching Cards. Students work collaboratively to match descriptions, tables, equations and graphs, explaining each match rather than relying on appearance. Teacher listens for misconceptions, particularly confusing the initial value with the rate of change, and asks students to justify matches using several pieces of evidence. Groups record a match and its explanation on the worksheet.

5. 58–68 min · Break and reset. Students take a short break. On return, display the mid-lesson check slide with (y=6x+11). Students independently answer: “What is the value when (x=4)? What does 11 represent? What is the rate of change?” Review answers together and address errors before continuing. The value is 35, 11 is the initial value, and the rate of change is 6.

6. 68–94 min · Applied problem-solving. Students complete the worksheet’s scaffolded problems, first independently and then checking with a partner. Problems include comparing two delivery costs, calculating pay for different hours, and deciding when one membership becomes cheaper than another. Require students to show a table or graph, write an equation, solve the question and state a contextual answer. Teacher conferences with each student, prompting: “What is changing?”, “What is fixed?”, “What values are sensible?” Students may use a graphing tool to check their graph and solution.

7. 94–110 min · Plenary and assessment. Use the discussion and exit-ticket slides to revisit the opening rideshare comparison. Students explain which option is better for different distances and why the answer changes. They complete the worksheet exit ticket: “A kayak rental company charges $9 to collect a kayak and $5 per hour. Write the rule, find the cost for 7 hours, and explain whether 18 hours is a sensible value for the model.” The rule is (C=5h+9), and the cost for 7 hours is $44. Invite several students to share different representations and collect responses.

Resources

• the complete linear relationships slide deck
• the linear relationships modelling worksheet
• the Linear Graph Matching Cards
• Mini-whiteboards and markers
• Graph paper, rulers and pencils
• Calculators
• Devices with a graphing tool, if available
• Projector or interactive whiteboard

Assessment

• Listen to explanations during the hook, guided examples and card matching; note whether students distinguish initial value from rate of change.
• Check tables, equations, graphs, substitution and units during the applied problems, conferencing with each student.
• Use the exit ticket to assess independent modelling, calculation, interpretation and judgement about the model’s reasonable domain.

Differentiation

• Support students with a partially completed table, labelled axes, a formula prompt (y=mx+b), colour-coding for the starting value and rate, and sentence starters such as “The initial value means…” and “The answer is reasonable because…”.
• Read problems aloud, clarify financial vocabulary, and allow students to explain reasoning orally before recording it. Provide enlarged graph grids and calculator access where required.
• For students ready for extension, ask them to find the break-even point between two pricing plans, represent it algebraically and graphically, and explain which plan is preferable for different ranges of use.
• Advanced learners can design a new pricing problem using equations such as (A=4x+13) and (B=7x+1), determine the intersection at (x=4), identify a realistic domain, and swap it with a peer to solve and critique.

Extension

• Investigate how changing the initial value or rate of change affects the graph using a digital graphing tool. Students make and test a conjecture, such as “Increasing the initial value moves the line vertically without changing its gradient.”
• Compare the new plans (A=4x+13) and (B=7x+1). Students create a table, graph both rules, solve (4x+13=7x+1), and explain which plan is preferable before and after the break-even input.
• Create a short written recommendation for a new real-world plan, supported by a table, graph, equation and realistic domain.