What makes a polynomial linear, and the equation you get by fixing its value

What to assume they know

• Univariate polynomials and what degree names — degree, and that degree 1 is called linear
• Turning a situation into an expression: terms, variables, coefficients — terms, coefficients and constants, and reading a coefficient as a rate
• Perimeter of a square as four times its side
• Arithmetic with decimals, for a side that grows in steps of 0.5 cm
• Collecting like terms, so that x + (x + 10) becomes 2x + 10
• Undoing addition and multiplication to isolate a letter, from earlier classes
• Reading a two-row table in which the last column is written with a letter in it

What they should be able to do

• Recognise a linear polynomial from its degree and give examples in several letters
• Tabulate the values of a linear polynomial at evenly spaced inputs and show that the successive differences are equal
• State the size of the output step given the input step and the coefficient, and explain why the two are proportional
• Distinguish a linear polynomial from a linear equation, and say what act turns one into the other
• Translate a worded situation into a linear equation by naming the unknown
• Solve a linear equation in one variable and check the solution against the wording of the problem
• Explain why substituting into a quadratic is the same procedure as substituting into a linear polynomial, even though the differences no longer come out equal

Where it usually goes wrong

• "An expression can be solved." 4x and 200 + 50m have no answers. They have values, one for each input. The word "solve" only becomes available once a total is fixed, and the chapter marks that moment on p. 20.
• "2x + 10 = 64 and 2x = 54 are different problems." They are different equations with the same solution. Every legal step produces a new equation and preserves the answer — that is what makes the steps legal.
• "The step in the output equals the coefficient." Only when the input steps by 1. The chapter's own square steps by 0.5 and moves by 2 while the coefficient is 4. Put the two examples side by side.
• "The joining fee shows up in the table." It does not — the printed table begins at one match. The constant term of a linear model is the value before the process starts, and it is often the one number the data never displays.
• "If a pattern rises by a fixed amount, the polynomial is just that amount times the input." The club's amounts rise by 50 each time, but the totals are 250, 300, 350 — not 50, 100, 150. The constant shifts the whole run.
• "The letter must be x." The chapter uses x for the square, m for the matches, and s in the exercise set. The letter is a label chosen to suit the situation.
• "Substitution is only for linear polynomials." Item 2 of the exercise set is quadratic and substitutes identically. What changes with degree is the pattern in the answers, not the procedure.
• "7s² – 4s + 6 at s = –3 needs a minus sign somewhere." Both 7s² and –4s come out positive at a negative input. Work it slowly.
• "Any two numbers adding to 64 will do for Example 6." The second condition is doing real work. Check both, always.

Questions to check understanding

• Evaluate a given polynomial, linear or quadratic, at a stated value including a negative one
• Tabulate a linear polynomial at evenly spaced inputs and state the common difference
• Form and solve a linear equation from a worded situation — ages, ratios, coin counts, a cut length, a rectangle's dimensions, all of which appear in Exercise Set 2.2
• Given a total, work backwards to the input that produces it
• State whether a given statement is an expression or an equation, and justify it
• Check a claimed solution against every condition in the original wording

Examples worth working on the board

Inputs below. There is no figure in §2.2 except the input–output machine, which belongs to the next topic. Values marked verified are worked out here; the chapter prints answers only where noted.

• Example 4, the square (p. 19). A square of side x has perimeter 4x, a linear polynomial in x. The Think and Reflect box on the same page asks for the perimeters of squares whose sides measure 1, 1.5, 2, 2.5 and 3 cm, and what becomes of those perimeters when each side gains 0.5 cm.
• The square's numbers. Verified — the perimeters are 4, 6, 8, 10 and 12 cm. Each step of 0.5 cm in the side lifts the perimeter by 2 cm. The chapter states the 2 cm step itself, lower on p. 19.
• Why 2 and not 0.5 (the argument to make). Verified — the coefficient is 4, the input step is 0.5, and 4 × 0.5 = 2. The output step is the coefficient multiplied by the input step, which is why the step size is fixed but is not the same number as the coefficient unless you happen to be stepping by one. The chapter does not spell this out; it is the reason its own two examples show steps of 2 and of 50.
• Example 5, the chess club (p. 19). Joining costs ₹200 once; every match played costs a further ₹50. The printed table has two rows. Number of matches played: 1, 2, 3, 4, 5, …, m. Amount paid in rupees: 250, 300, 350, 400, 450, …, 200 + 50m. The chapter states that the total is ₹(200 + 50m), that this is a linear polynomial in m, and that the amount rises by ₹50 for each extra match.
• Reading the club's polynomial back. Verified — the 200 is paid whether or not a single match is played, and the 50 is the price of one match. Setting m = 0 gives 200, which is the joining fee alone; the table starts at m = 1 and so never shows it. That gap is worth a beat, because the constant term is invisible in the printed data.
• Think and Reflect, p. 19. A player's total comes to ₹750; the box asks for the number of matches behind it. Hand over the 750 as an input. Verified — 200 + 50m = 750 gives m = 11. This is the first place in the chapter where a value gets fixed and the input is asked for, one page before the word for it appears.
• What "at integers" is doing (p. 19). The chapter's statement of the characteristic feature says the differences between successive values at integers are constant. Verified — the qualifier matters, because the square example is stepping by 0.5, not by 1, and gets a step of 2 rather than 4. The honest formulation: equal input steps give equal output steps, and the output step depends on which input step you chose.
• Example 6, the two numbers (p. 20). Two numbers add to 64; one exceeds the other by 10. The chapter takes the smaller as x, writes the larger as x + 10, forms x + (x + 10) = 64, simplifies to 2x + 10 = 64, notes that 2x + 10 is a linear polynomial, and reports 2x = 54, x = 27, so the numbers are 27 and 37. All of those values are printed.
• The step the chapter skips. Verified — between 2x + 10 = 64 and 2x = 54 sits the subtraction of 10 from both sides. The chapter compresses it into one line.
• Checking against the words, not the algebra. Verified — 27 + 37 = 64 and 37 – 27 = 10. Both original conditions have to be tested, and testing only the sum would pass numbers such as 30 and 34.
• Where the definition lands (p. 20). The chapter's rule is that equating a one-variable linear polynomial to a constant produces a linear equation. Note what this makes of 2x + 10 = 64 and 2x = 54: two different equations that the same value of x satisfies.
• Exercise Set 2.2 (p. 21), seven items. Hand over all the data:
  • The value of 5x – 3 at x = 0, at x = –1, and at x = 2.
  • The value of 7s² – 4s + 6 at s = 0, at s = –3, and at s = 4.
  • Salil's mother's present age is three times Salil's; in 5 years their ages will total 70 years. Find both present ages.
  • Two positive integers, differing by 63, in the ratio 2 : 5. Find them.
  • Ruby's two-rupee coins outnumber her five-rupee coins threefold, and the two piles together come to ₹88. Find how many she holds of each kind.
  • A 300 feet fence is cut into two unequal pieces, the longer being four times the shorter. Find both lengths.
  • A rectangle whose length exceeds twice its width by three has a perimeter of 24 cm. Find its two dimensions.
• What these items are for. Verified — items 1 and 2 are pure substitution, and item 2 is deliberately quadratic so that the procedure is seen to be indifferent to degree. Items 3 to 7 all require naming an unknown first: the choice of which quantity to call x is the real work, and in item 5 the natural choice is the number of five-rupee coins, since the other count is described in terms of it. Item 2's middle case, s = –3, is the sign trap — 7 × 9 and –4 × –3 both come out positive.
• A note for item 1. Verified — 5x – 3 gives –3, –8 and 7 at the three inputs. Successive differences here are not equal because the inputs 0, –1, 2 are not evenly spaced; a nice quiet reminder that the constant-step property is a statement about evenly spaced inputs.

Figures to have open

• A square whose side is a draggable length, with the four sides all marked and the perimeter reported below. Standard schematic, and it carries sections 2 to 4.
• A price-list table for the chess club that can grow a leftmost m = 0 column, so the invisible constant term becomes visible. Standard schematic; the chapter's own table on p. 19 starts at one match.
• A two-line balance or step diagram for solving 2x + 10 = 64, showing the same operation applied to both sides. Standard schematic.
• No textbook artwork is needed for this topic; §2.2's only figure is the input–output machine on p. 20, which belongs to A polynomial as an input–output machine.

Where this sits in the book

• NCERT Ganita Manjari, Class 9, printed Chapter 2, "Introduction to Linear Polynomials", §2.2 "Linear Polynomials", pp. 19–21. Example 4 and Example 5 are on p. 19; the definition of a linear equation and Example 6 are on p. 20; Exercise Set 2.2 occupies the upper half of p. 21.
• The closing lines of §2.1 at the top of p. 19 hand the topic its starting point by restating that degree 1 means linear.
• Two Think and Reflect boxes bear on this topic, both on p. 19: the square's perimeters and the ₹750 player. The statement of the constant-difference feature is not a third box but running prose, beginning at the foot of p. 19 and finishing at the top of p. 20.
• Forward pointers inside the chapter: the constant-difference idea is developed in §2.3 (pp. 21–23, see A constant difference is the signature of a linear pattern), and the same coefficient reappears as the slope of a line on p. 31.
• End-of-Chapter Exercises, items 3, 4 and 6, all three printed on p. 37, are further worded linear equations, item 6 being starred.