Turning a situation into an expression: terms, variables, coefficients

What to assume they know

• Arithmetic with whole numbers and decimals, including multiplying a decimal by a whole number
• Perimeter and area of a rectangle, and that a rectangle has two sides of each measurement
• Area measured in square units, and that a per-square-metre cost is multiplied by an area
• Substituting a number for a letter in a simple expression, from earlier classes
• Multiplying a bracket by a term, as in x times (10 – x)

What they should be able to do

• Build an expression from a described situation by naming the quantities that vary and writing the arithmetic that combines them
• Split a given expression into its terms, and for each term name the variable and the coefficient
• Identify the constant term of an expression and say what feature of the situation it corresponds to
• Explain why a coefficient in a modelled expression is a rate — pens per box, rupees per metre, rupees per square metre
• Count how many distinct letters an expression uses, and distinguish that count from the highest power appearing in it
• Derive that a rectangle cut from a 20 cm wire with one side x must have the other side 10 – x, and state the range of x for which a rectangle exists
• Expand x(10 – x) and explain why the result is no longer first-degree

Where it usually goes wrong

• "x is a specific unknown number I am supposed to find." Nothing in Example 1 can be solved. Raju could buy any number of boxes; the expression is a standing instruction that works for all of them at once. Finding a value becomes possible only when a total is fixed, which is two topics away.
• "4x means 4 and then x." It means 4 multiplied by x. Read it aloud as "four pens for every red box, and there are x red boxes".
• "The constant 3 can be absorbed into the other terms." It cannot — it is the one part of the total that survives when x and y are both zero. Free pens do not scale with purchases.
• "Two letters means degree two." Example 1 has two letters and no multiplication between them; Example 3 has one letter and does multiply it by itself. Counting letters and reading powers are two different measurements, and the chapter separates them deliberately.
• "50lw is just another term like the others." It is the only term in the garden expression that changes when either measurement changes. Doubling the length doubles the wire cost and the seed cost but leaves the wooden fencing alone — worth showing numerically.
• "The wire example gives one rectangle." It gives a whole family. Every choice of x strictly between 0 and 10 gives a different rectangle of the same perimeter and a different area.
• "Width is 20 – x." A frequent slip. The 20 is the whole way round, not half of it; the width pairs with x to make ten, not twenty.
• "Letter-numbers and variables are two different things." They are the same thing under two names. The chapter introduces the first word and then tells you it is switching to the second.

Questions to check understanding

• Given a described situation, write the expression and state its terms, variables, coefficients and constant term
• Name the coefficient of a stated letter in a given expression, including cases where the sign is negative
• Say how many variables a given expression uses
• Given a fixed perimeter and one side as a letter, write the other side and then the area
• Explain in words what a particular coefficient or constant means in the situation being modelled — the competency-based form the chapter's Think and Reflect boxes on p. 17 are already asking for
• Compare two expressions and say what structural difference makes them different kinds of object

Examples worth working on the board

Anything computed here is marked verified; the chapter prints the totals but leaves most of the checking to the student.

• Example 1, the sealed boxes (p. 16, with Fig. 2.1). A shop sells sealed boxes in different colours. Every red box holds 4 pens; every blue box holds 5 pencils. Raju buys x red boxes and y blue boxes, and is given 3 extra pens free. The chapter's expression for the combined count of pens and pencils is 4x + 5y + 3. Its terms are 4x, 5y and 3; the variables are x and y; the coefficients of x and y are 4 and 5; the constant is 3.
• How to read that expression back. Verified — the 4 is pens per red box, the 5 is pencils per blue box, and the 3 is the only part of the total that does not depend on how much Raju buys. Setting x = 0 and y = 0 leaves 3, which is the free pens and nothing else. A good check: x = 2, y = 3 gives 8 + 15 + 3 = 26.
• A caution about Fig. 2.1. The prose says the boxes are sealed, but the drawn figure shows two open containers with the pens and pencils standing up in plain view — a red one lettered PENS and a blue one lettered Pencils. The premise of the example is that you cannot see inside.
• Example 2, the garden (pp. 16–17, with Fig. 2.2). A rectangular garden is l metres long and w metres wide. Wire fencing runs along the length at ₹100 per metre; wooden fencing runs along the width at ₹80 per metre; seed is sown over the whole garden at ₹50 per square metre. The chapter's costs are 2l × 100 = 200l, 2w × 80 = 160w, and 50 × l × w = 50lw, giving a total of ₹(200l + 160w + 50lw).
• Why the twos are there (Fig. 2.2, p. 17). The painted figure labels four edges of the garden, not two: the left and right edges are both labelled l metres, the top and bottom edges both w metres. Describe them by position, not by length — the artwork is drawn in perspective, and there the top and bottom edges are the longer pair on the page, while nothing in Example 2 fixes which of l and w is bigger. That is the whole justification for 2l and 2w — the fence has to go along both of each pair.
• The third term is the odd one (p. 17). Verified — 50lw is the only term that answers to both letters: 200l moves when l changes, 160w moves when w changes, and 50lw moves when either does, so doubling both quadruples it. Watch the arithmetic here rather than reasoning from the shape of the term. Doubling l alone does double 50lw, exactly as it doubles 200l; the term that sits still is 160w. What fails to double is the total, because one of its three parts is untouched. Numerical check: at l = 10, w = 4 the three costs are ₹2,000, ₹640 and ₹2,000, total ₹4,640; doubling only l to 20 gives ₹4,000, ₹640 and ₹4,000, total ₹8,640 — not double the first total. The chapter's Think and Reflect box on p. 17 asks the student to say how this expression differs from Example 1's, and this is the difference.
• Example 3, the bent wire (p. 17). A 20 cm wire is bent into a rectangle. The chapter offers 7 cm by 3 cm and 5.5 cm by 4.5 cm as two of the many possibilities and invites more. Taking the length as x cm, it states the width as (10 – x) cm, and the area as x(10 – x), which it also writes as 10x – x².
• The missing step in 10 – x. Verified — the perimeter is 2(x + width) = 20, so x + width = 10 and the width is 10 – x. The chapter prints the conclusion without the halving. Check against the chapter's own pairs: 7 + 3 = 10 and 5.5 + 4.5 = 10.
• The range nobody states. Verified — a genuine rectangle needs x > 0 and 10 – x > 0, so x runs strictly between 0 and 10. At x = 5 the shape is a square of area 25 cm², which is the largest area available. The chapter sets no such bounds anywhere in §2.1.
• Expanding the product. Verified — x(10 – x) = 10x – x². The x² arrives from x multiplying itself, which is exactly what makes this expression differ in kind from 4x + 5y + 3.
• The three shapes side by side. Verified — Example 1 uses two letters and no letter is multiplied by another; Example 2 uses two letters and one term multiplies them together; Example 3 uses one letter but multiplies it by itself. The chapter states on p. 18 that Examples 1 and 2 involve two variables while Example 3 involves only one, and then restricts the rest of the discussion to one variable.

Figures to have open

• A schematic rectangle for the garden with all four edges labelled and the two per-metre rates attached to the correct pair of edges, plus the interior shaded to carry the per-square-metre rate. This replaces the chapter's painted landscape (Fig. 2.2, p. 17); redraw it rather than reproducing the artwork, but keep the four labels, because the factor of two depends on them.
• Two closed boxes, red and blue, with count labels on the outside. Standard schematic; do not copy Fig. 2.1's open containers, which contradict the word "sealed".
• A 20 cm loop of wire changing into several rectangles of the same perimeter — 7 by 3, 5.5 by 4.5, and the 5 by 5 square — with a running x + width = 10 strip beneath. Standard schematic; the chapter names the first two pairs and leaves the rest open.
• An annotation layer for 4x + 5y + 3 that can bracket terms, ring coefficients and flag the constant independently. Standard schematic.

Where this sits in the book

• NCERT Ganita Manjari, Class 9, printed Chapter 2, "Introduction to Linear Polynomials", §2.1 "Introduction", pp. 16–18. Examples 1, 2 and 3 sit on pp. 16–17; Fig. 2.1 is on p. 16 and Fig. 2.2 on p. 17.
• Two Think and Reflect boxes on p. 17 ask the reader to name the parts of the garden and wire expressions and to compare each with Example 1. They set the agenda for sections 6 to 9 here.
• The sentence that closes the topic — that Examples 1 and 2 use two variables while Example 3 uses one — opens p. 18 and leads directly into the next topic, Univariate polynomials and what degree names.
• Chapter summary, pp. 39–40, restates the anatomy of an expression using a two-variable example of its own.