Linear growth and linear decay

What to assume they know

• A constant difference is the signature of a linear pattern — linear patterns, constant differences, and reading a rule off a step and a starting value
• A polynomial as an input–output machine — substituting into a rule, and the idea of a function
• Turning a situation into an expression: terms, variables, coefficients — the constant term as the part that does not vary
• Arithmetic with decimals, for a step of 0.5
• Negative coefficients, and subtracting a multiple from a starting amount
• Reading a two-row table of paired values

What they should be able to do

• Build a table of values for a given linear rule over a stated range of inputs
• Decide from a table or a rule whether a quantity is growing or decaying linearly
• Identify the coefficient as the amount gained or lost per interval, and the constant as the value before the process begins
• Write a rule for a described situation from its starting value and its per-interval change, using the correct sign
• Name the interval a rate is quoted over, and explain why the rate is meaningless without it
• Find when a linearly decaying quantity reaches zero, by solving the rule set equal to 0
• Distinguish linear decay from a decline that is proportional rather than fixed
• Read and use the chapter's bracket notation for a function of one variable

Where it usually goes wrong

• "Decay means shrinking by a percentage." In this chapter it means losing the same absolute amount each interval. Show the ₹800-a-year phone beside a tenth-a-year phone; the gaps differ from the second step onwards.
• "The rate alone tells you the change." "Rises by 60" is not a claim until the interval is named. Per kilometre, per month, per year, per day — the chapter uses all four across §2.4, deliberately.
• "The constant term is part of the growth." It is the amount already there before anything happens. Setting the input to zero isolates it, and in Example 9's table it is the first entry.
• "Growth needs a positive constant and decay a negative one." The sign that decides growth from decay sits on the coefficient, not on the constant. The tank has constant 3 and coefficient –0.5.
• "A decaying model keeps working forever." 3 – 0.5t returns a negative height from seven months on. The rule is fine; the situation stops matching it. Compute the zero and mark it.
• "The tank example's step is 0.5 because the coefficient is 0.5." They agree here only because the input steps by one month. Look back at the square in §2.2, where a coefficient of 4 and a step of 0.5 produced a step of 2.
• "A table starting at 1 shows the initial value." Item 4's table starts at day 1 and shows ₹585, not the ₹600 recharge. The starting value has to be read out of the rule.
• "C(d) is C multiplied by d." It is the value of the cost rule at the distance d. This is the first place in the chapter that brackets are used this way, and it is worth naming rather than assuming.
• "Population grows linearly." Item 3 counts migration only, and only under a stated assumption. Say so.

Questions to check understanding

• Build a table of values for a stated linear rule over a given range
• Write the rule for a described situation and state whether it is growth or decay
• Give the value of the quantity after a stated number of intervals
• Find when a decaying quantity reaches zero
• Justify, in words, why a given situation counts as linear growth or linear decay — all three parts (iii) of Exercise Set 2.4 ask for the reason, not just the expression
• Read the rate and the initial value off a rule, and say what each means in the situation

Examples worth working on the board

Inputs below. §2.4 prints no figure — it runs on two worked examples, two tables and an exercise set. Values marked verified are worked out here; the chapter prints no answers and this book has no appended key.

• Example 9, the journey (p. 24). The chapter gives the cost of a journey as the linear function C(d) = 100 + 60d, with C the total cost in rupees and d the distance in kilometres. The printed table gives: d = 0, 1, 2, 3, 4, 5 and C = 100, 160, 220, 280, 340, 400. The chapter states that each extra kilometre lifts the cost by a fixed ₹60, and names this linear growth.
• A mismatch to hand the teacher. The prose introducing the table says it will run d from 0 to 10 km; the table as printed stops at d = 5.
• Reading the rule back. Verified — the 100 is charged before any distance is covered, and the 60 is the price of one kilometre. Here the table does show d = 0, so the constant term is visible in the data, unlike the chess club table on p. 19.
• Think and Reflect, p. 24. What does 15 km cost, and how far can be covered for ₹700? Hand over 15 and 700 as inputs. Verified — ₹1,000, and 10 km. Note that the second is the reverse question again, and the answer happens to be the distance the prose promised the table would reach.
• Example 10, the water tank (pp. 24–25). At summer's start a cylindrical tank stands with water 3 m deep, and its height in metres after t months is given as h(t) = 3 – 0.5t. The printed table gives t = 0, 1, 2, 3, 4 and h = 3, 2.5, 2, 1.5, 1. The chapter states that as t rises by one month the height falls by a fixed 0.5, and names this linear decay.
• Think and Reflect, p. 25. The height at the end of 5 months. Hand over the 5. Verified — 0.5 m.
• The two examples side by side. Verified — both rules are a constant plus or minus a coefficient times the input. What differs is the constant they start from (100 against 3), the sign and size of the coefficient, and the units — of which only the sign changes the behaviour from climbing to falling. Showing them as a matched pair is the topic's central move; the chapter presents them consecutively but does not align them.
• Where the tank runs dry. Verified, and an added extension — 3 – 0.5t = 0 at t = 6, so the model reaches zero height at six months. At t = 8 it returns –1 m, which is not a height. The chapter never asks this about the tank. But it does ask the equivalent question in Exercise Set 2.4 item 4(ii) about a prepaid balance, so the question is one the chapter invites, not one the explanation is importing.
• The definitions (p. 25). Linear growth is defined as a linear pattern whose quantity gains the same amount across each equal interval, and linear decay the same with a loss. The phrase about equal intervals is what makes the definitions work; without it, "gains a constant amount" says nothing.
• The contrast the chapter does not draw. Verified, and not in the book — set a starting value of 10,000 losing ₹800 a year against one losing a tenth of its value a year. Linear: 10,000; 9,200; 8,400; 7,600. Proportional: 10,000; 9,000; 8,100; 7,290. Both fall; only the first has equal gaps, and only the first reaches zero. The chapter names linear decay without ever showing a decay that is not linear, so the definition's force is easy to miss. This is an added contrast — flag it as such.
• Exercise Set 2.4 (pp. 25–26), four multi-part items, all inputs to hand over:
  • A plant 1.75 feet tall gaining 0.5 feet a month. (i) Its height after 7 months. (ii) A table for t from 0 to 10 months. (iii) An expression relating h to t, with a reason why it is growth.
  • A phone bought for ₹10,000 losing ₹800 of value a year. (i) Its value after 3 years. (ii) A table for t from 0 to 8 years. (iii) An expression relating v to t, with a reason why it is decay.
  • A village of initial population 750, joined by 50 people a year from a nearby city. (i) The population after 6 years. (ii) A table for t from 0 to 10 years. (iii) An expression relating P to t, with a reason why it is growth.
  • A ₹600 recharge whose balance drops ₹15 a day. (i) An equation for the remaining balance b(x) after x days, with a reason why it is decay. (ii) The day the balance runs out. (iii) A table for x from 1 to 10 days.
• The exercise answers, as checks. Verified — item 1: h = 1.75 + 0.5t, and 5.25 feet after 7 months. Item 2: v = 10000 – 800t, and ₹7,600 after 3 years; note this rule hits zero at 12.5 years, past the table's range. Item 3: P = 750 + 50t, and 1,050 after 6 years. Item 4: b(x) = 600 – 15x, running out at x = 40 days.
• One detail in item 4 worth noticing. Its table is asked for from x = 1, while items 1, 2 and 3 all begin at 0. So the one item that explicitly asks where the quantity runs out is also the one whose table hides the starting value.
• A modelling caveat on item 3. Verified — the item counts only arrivals from the city. A real population also has births, deaths and departures, so the linear rule is a description of the migration alone. Worth one line, because it is the clearest case in the chapter of a linear model being an approximation rather than a fact.

Figures to have open

• A drawn tank with a movable water level tied to a t slider, and a floor line at zero that the model visibly crosses. Standard schematic, and it carries sections 6, 10 and 11.
• A paired-rule display that can hold 100 + 60d and 3 – 0.5t side by side with matching parts aligned — constant above constant, coefficient above coefficient. Standard schematic; the chapter prints them a page apart and never aligns them.
• Two four-entry decline tables, fixed-amount and fixed-fraction, with the gaps measured between entries. Standard schematic; this comparison is not in the book.
• No artwork from the textbook is required; §2.4 prints none.

Where this sits in the book

• NCERT Ganita Manjari, Class 9, printed Chapter 2, "Introduction to Linear Polynomials", §2.4 "Linear growth and linear decay", pp. 24–26. Example 9 and its table are on p. 24; Example 10 begins at the foot of p. 24 with its table at the top of p. 25; the two definitions are on p. 25; Exercise Set 2.4 runs from the middle of p. 25 to the middle of p. 26.
• Two Think and Reflect boxes: p. 24 (the 15 km cost and the ₹700 distance) and p. 25 (the height at 5 months).
• Backward links inside the chapter: linear patterns and constant differences are established in §2.3, pp. 21–23, covered in A constant difference is the signature of a linear pattern; the distinction between an expression and an equation is on p. 20.
• Forward link inside the chapter: on p. 31 the chapter states that linear growth shows up as a line of positive slope and linear decay as a line of negative slope. That is where this topic's two signs become two directions, and it lands in What a and b do to the line: slope, y-intercept, and parallel families.
• End-of-Chapter Exercises, item 5 on p. 37, is a savings problem of the same shape.
• Chapter summary, pp. 39–40, restates both definitions and the slope-sign link.