A polynomial as an input–output machine

What to assume they know

• What makes a polynomial linear, and the equation you get by fixing its value — linear polynomials, and that fixing a value produces an equation
• Univariate polynomials and what degree names — degree, and the names linear and quadratic
• Substituting a number into an expression and evaluating it
• Multiplying and adding with negative numbers, including 2 × (–6)
• Order of operations — multiply before adding
• Squaring a number, for the quadratic contrast case

What they should be able to do

• Evaluate a linear polynomial at a given input, including a negative input
• Describe substitution as a process with one input and one output rather than as a rewriting of symbols
• Explain what the word function is asserting about a polynomial, at the level this chapter uses it
• Read the labels on the chapter's machine figure and say which part is the input, which the rule, and which the output
• Distinguish a linear function from a quadratic function by the polynomial inside the machine
• Evaluate a quadratic at a given input and observe that the same procedure applies
• Say why the letter naming the input is arbitrary, and evaluate the same rule written with a different letter

Where it usually goes wrong

• "2 × –6 + 3 needs the whole thing negated." It does not. Only the 2x term goes negative; the +3 is untouched. Work it in two visible steps.
• "Substitution means erasing the letter." It means choosing a value for it. The letter is still there in the rule, ready for the next input — which is exactly what the machine picture is for.
• "The machine changes when I change the input." The rule on the display panel never changes. Only the card going in and the tag coming out do. Show several inputs through one unchanged machine.
• "A function is a formula." At this level a function is a rule that commits to exactly one output per input. Two different-looking formulas can be the same function, and a rule that could return either of two answers is not one.
• "Knowing the output tells me the input." For 2x + 3, yes. For 10x – x², no — both 4 and 6 give 24. Show this, because it is the difference between the two machines and it prepares the "which input?" question of the next module.
• "x = 0 is a special case you cannot substitute." It is the easiest one, and it returns the constant term. Feed it in early.
• "5x – 3 and 5s – 3 are different rules." They are the same machine with a different label on the input slot. The exercise set switches letters between consecutive items for exactly this reason.
• "The output has the same units as the input." In the rectangle case a length in centimetres goes in and an area in square centimetres comes out. Label both ends of the machine.

Questions to check understanding

• Find the value of a stated polynomial at one or more given inputs, negative inputs included — the form of Exercise Set 2.2 items 1 and 2, and of End-of-Chapter item 2 on p. 36
• Evaluate a polynomial at an input that is itself a letter, as End-of-Chapter item 2(ii) on p. 36 does with 4t³ – t² + 6 at t = a
• Given a rule and an output, find the input
• State whether a given rule is a linear or a quadratic function
• Explain, for a given rule, whether two different inputs could produce the same output

Examples worth working on the board

Inputs below. Values marked verified are worked out here; the chapter prints the two evaluations of 2x + 3 itself.

• The chapter's machine polynomial (p. 20). The chapter takes 2x + 3 and states that for every x there is a corresponding value. It prints two evaluations: at x = 4, 2 × 4 + 3 = 11; and at x = –6, 2 × –6 + 3 = –9.
• Fig. 2.3 (p. 20, captioned as a linear expression shown as an input–output process). It draws a grey machine on legs. A card lettered x = 4 sits above it with an arrow pointing down into the top. The machine's face carries a lit panel reading y = 2x + 3. A tag hanging off the right-hand side reads y = 11. So the figure supplies the input, the rule and the output as three separate labelled objects.
• What the figure adds that the prose does not. The letter y appears in the artwork — twice — while the prose on p. 20 speaks only of the value of 2x + 3 and never uses y. The lettering is inside the drawing, so it does not read out cleanly; open the image. This matters, because y is exactly the letter §2.5 will adopt on p. 26 for the output of a linear relationship, and Fig. 2.3 is where it quietly arrives.
• The x = 4 run, slowly. Verified — multiply first: 2 × 4 = 8; then add: 8 + 3 = 11. Adding before multiplying gives 14, which is the error the scene exists to block.
• The x = –6 run, slowly. Verified — 2 × (–6) = –12, then –12 + 3 = –9. Two common wrong answers to show and reject: –15, from negating the whole of 12 + 3; and –9 reached by luck from –(12 – 3), which happens to land right and is still wrong reasoning. The chapter prints –9.
• A third input, for the pattern. Verified — at x = 0 the machine returns 3, which is the constant term standing alone. Worth including so the constant term of a linear rule gets a visible meaning: it is the output when nothing is fed in.
• Think and Reflect, p. 20. The box points back at Example 3's rectangle and asks the reader to treat the area 10x – x² as an input–output process, then asks what value it takes at x = 6 cm. Hand over 10x – x² and x = 6 as the inputs.
• The quadratic run. Verified — at x = 6, 10 × 6 = 60 and 6² = 36, so the area is 24 cm². The chapter does not print this. Note the units: x is a length in centimetres and the output is an area in square centimetres, so the machine changes what kind of quantity it is holding.
• Why the quadratic machine repeats itself. Verified and an added observation — 10x – x² also returns 24 at x = 4, since 40 – 16 = 24. So the output 24 does not tell you which input produced it. 2x + 3 can never do this: two different inputs always give two different outputs, because the difference in the outputs is twice the difference in the inputs. The chapter makes no such comparison anywhere in §2.2. This is the strongest single reason to bother drawing the machine.
• The chapter's naming line (the opening line of p. 21, printed above the Exercise Set 2.2 heading rather than after it). It states that 2x + 3 is a linear function while 10x – x² is a quadratic function. That single sentence is the whole justification for section 8 sitting in this topic rather than the previous one.
• Exercise Set 2.2, items 1 and 2 (p. 21) are the drill. Item 1: the value of 5x – 3 at x = 0, x = –1, x = 2. Item 2: the value of 7s² – 4s + 6 at s = 0, s = –3, s = 4. Verified — item 1 returns –3, –8, 7; item 2 returns 6, 81, 102. The letter changes from x to s between the two items, which is the chapter's own way of saying the letter does not matter.
• The chapter's forward pointer (p. 20). It says functions are taken further in later classes. No talk of domain, range, or f(x) notation here — although note that bracket notation does arrive later in this same chapter, as C(d) on p. 24.

Figures to have open

• A redrawn input–output machine with a swappable rule panel, an input card slot and an output tag. This replaces the chapter's Fig. 2.3 (p. 20); redraw rather than reproduce the artwork, but keep all three labelled parts, and keep y on the panel — the figure's y is the letter §2.5 goes on to use.
• A pairing diagram: inputs on the left, outputs on the right, one arrow from each input. Needs a variant where two different inputs arrive at the same output, for the quadratic case. Standard schematic.
• A units badge on the machine's two ends for the rectangle example — cm in, cm² out. Standard schematic.
• No graph is needed yet; the plotting of these pairs is Why two points are enough to draw the line.

Where this sits in the book

• NCERT Ganita Manjari, Class 9, printed Chapter 2, "Introduction to Linear Polynomials", §2.2 "Linear Polynomials", pp. 20–21. The input–output passage and Fig. 2.3 are on p. 20; the Think and Reflect box pointing back at the rectangle is on p. 20; the sentence naming 2x + 3 linear and 10x – x² quadratic opens p. 21.
• Example 3, the 20 cm wire whose area is 10x – x², is on p. 17 and is covered in Turning a situation into an expression: terms, variables, coefficients. This topic reuses its expression.
• Exercise Set 2.2 items 1 and 2, p. 21.
• Forward pointers inside the chapter: bracket notation for a function first appears as C(d) and h(t) on pp. 24–25; p(x) and q(x) are used in End-of-Chapter items 10, 11 and 13 on pp. 38–39, and f(x) in item 14 on p. 39.
• The chapter states on p. 20 that functions are studied further in later classes.