In this guide
A learner points to the smaller starting fee: “This one is cheaper.” Another points to the smaller hourly charge: “No, this one is.” Both have noticed something useful. Neither has yet answered the question that matters: cheaper for how long?
Try it in Class 9: allow 45 minutes after learners have met coordinates and linear equations. Pairs compare two fictional cycle-hire plans using a table, a graph and an equation. Prepare squared paper, rulers and the plan cards below. The learning goal is a decision supported by a model, including what the model assumes.
Give the class two plans and a clear boundary
Imagine a school activity using a cycle for a whole number of hours. These are invented classroom prices, not offers from a business. Plan A charges a fixed ₹20 plus ₹4 for each hour. Plan B charges a fixed ₹50 plus ₹2 for each hour. For this task, the booking can last from 0 to 20 whole hours; there are no extra charges or discounts.
| Plan | Fixed charge | Charge per hour | Total cost for x hours |
|---|---|---|---|
| A | ₹20 | ₹4 | A = 20 + 4 x |
| B | ₹50 | ₹2 | B = 50 + 2 x |
Use the zero-hour case to interpret the fixed charge mathematically. It is the starting value of the model, not a suggestion to rent a cycle without using it. Ask pairs to predict a winner for a short booking and a long booking before calculating. Keep those first predictions visible so they can revise them with evidence.
Build a table that can challenge a first impression
Give learners the five hour values below, but initially leave both cost columns blank. Each partner checks one plan, then exchanges calculations. Require units in the headings. A result of “60” becomes more useful when it is clearly a cost in rupees for a particular number of hours.
| Hours x | Plan A cost (₹) | Plan B cost (₹) |
|---|---|---|
| 0 | 20 | 50 |
| 5 | 40 | 60 |
| 10 | 60 | 70 |
| 15 | 80 | 80 |
| 20 | 100 | 90 |
Cover the completed columns when sharing the pupil task; the table here is also the teacher check. Ask: “Which rows support your prediction? Which row makes you reconsider it?” A pair that checks only 0 and 20 hours has found a change in the cheaper plan but has not yet located exactly where the change happens.
Let the graph tell the same story
Draw hours on the horizontal axis from 0 to 20 and cost on the vertical axis from ₹0 to ₹110. One convenient scale is one large square for five hours and one large square for ₹10. Label both axes and plot the five points for each plan. Use a solid line and a dashed line as well as different colours, so the traces remain distinguishable in a photocopy.
Ask learners to mark the point where the traces meet and write a sentence beginning “At this point…”. For this whole-hour booking rule, the permitted bookings are the integer hour values. The drawn lines help reveal the linear relationship between them. They do not silently change the billing rule to allow fractional bookings.
Invite a pair to explain the shapes without recalculating every point. Plan A starts lower, at ₹20, but rises by ₹4 per hour. Plan B starts at ₹50 and rises by ₹2 per hour. The steeper rise eventually removes A’s starting advantage. A graph with swapped axes or an inconsistent scale may conceal that relationship; correct the representation before judging the decision.
Find the meeting point, then test either side
Teacher check: the crossing and the decision
At equal cost, 20 + 4 x = 50 + 2 x. Subtract 2 x and 20 from both sides: 2 x = 30, so x = 15. Substituting into either plan gives ₹80. The graph should meet at (15,80), with hours first and rupees second.
At 14 hours, A costs₹76 and B costs₹78. At 16 hours, A costs₹84 and B costs₹82. Therefore A is cheaper for the permitted bookings below 15 hours, the plans tie at 15 hours, and B is cheaper above 15 hours, up to the 20-hour limit. “B is always cheaper because 2 is less than 4” ignores its larger fixed charge.
The difference A − B = 2 x −30 offers another explanation: it is negative before 15, zero at 15 and positive after 15. This extension should explain the decision, rather than replace the table and graph with an unexplained rule.
Run it as a 45-minute investigation
- 0–5 minutes: read the cards and make a justified prediction.
- 5–15 minutes: complete and cross-check the table.
- 15–27 minutes: draw and compare labelled graphs.
- 27–37 minutes: solve for equal cost and test neighbouring hours.
- 37–42 minutes: change one rule and predict its effect.
- 42–45 minutes: write the exit explanation.
For the changed-rule round, let Plan B’s fixed charge fall to₹40 while its hourly charge remains₹2. Ask pupils to predict whether the crossing moves earlier or later before calculating. It moves earlier:20+4 x=40+2 x gives x=10 and cost₹60. Pairs should connect the changed starting value to their revised graph.
Notice the reasoning, not just the final number
Offer a pre-drawn axis and partially completed table to learners who need help with graph organisation. Let a partner dictate a calculation or explain a point orally. Stretch others by asking whether either plan is preferable at every permitted hour after a different fixed charge is chosen. Keep the original model and the extension clearly separated.
The exit prompt is: “For a 12-hour booking I choose___ because___. The plans cost the same at___ hours. My conclusion assumes___.” A strong response chooses A at₹68 rather than B at₹74, identifies 15 hours and mentions the stated charges or whole-hour rule. A correct plan choice with no reason is a useful starting point for the next lesson.
The modelling focus connects with CBSE’s Class IX Mathematics curriculum for 2026–27. Extend discussion with mathematical thinking and context, or use the self-assessment routine to help learners identify which representation clarified their decision.
AI assistance was used to finalize this contribution. The views expressed are personal views of the author; official statements are identified through linked official sources.
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