Evaluating expressions that mix integers and brackets

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What to assume they know

Why a negative times a negative must be positive and The sign rule for division, and why it follows from multiplication: multiplying and dividing integers, and the sign of the result
Commutative, associative, and distributive over the integers: reordering, regrouping and distributing, which is what licenses rewriting an expression at all
Writing a situation as an expression before computing it and Comparing two expressions by reasoning, not by evaluating: writing a situation as an expression, and comparing two expressions without evaluating them — this chapter leans on both and restates neither
The convention that a multiplication inside a longer expression is carried out before the additions and subtractions around it
Reading a scale that runs above and below a zero, such as a thermometer

What they should be able to do

Turn a described situation into a single expression before evaluating it
Evaluate an expression that mixes products, sums, differences and brackets, in the right order
Model a rate that acts downward as a negative number, and multiply it by an elapsed time
Compare two solution methods for the same situation and say which one survives a change to the starting conditions
Recover the rule a pattern machine is applying, from its inputs and outputs
Predict the output of a machine for a new set of inputs once the rule is known
Compare the values of several expressions built from the same two numbers and order them, without computing any of them fully
Say how a stated value changes when brackets are moved or a sign is flipped throughout

Where it usually goes wrong

"Work left to right." Example 1 does not: the two products are formed first and then added. The chapter never states the convention here — it is carried over from Part I's chapter on arithmetic expressions.
"Negative marks mean you subtract, so write a minus in the expression." Either works, but only one scales. The chapter's choice is to make the value itself −2 and then add throughout, so that no step ever has to decide between adding and subtracting. Show the alternative and show it costing more thought.
"Method 1 and Method 2 are the same thing written differently." They are not, and part (b) is where they part company: the subtracting method has to be reasoned about again when the lift starts above ground, while the signed-rate method takes a starting position plus a product and stops.
"Speed cannot be negative." In Example 2 the single number carries both how fast and which way — the same compression the carrom coin used on p.26. It is a modelling choice, and worth naming as one.
"Brackets are just there to make it look tidy." Item 15 on p.44 hands you four numbers and three operations and asks for the largest and the smallest results; the only thing you get to change is where the brackets go. That item is the strongest possible answer to this misconception.
"If I can't see the rule, I should guess a formula and hope." The machines reward a systematic read: look for a row where one input is zero-like or repeated, compare two rows differing in one input, and see what moved. Teach the method, not the answer.
"A pattern that fits five rows must be the rule." It is a rule. The chapter asks the reader to invent machines and challenge classmates precisely because a finite table does not pin down a formula. Say this once — it costs ten seconds and it is honest.

Questions to check understanding

Write an expression for a described situation, then evaluate it (p.43 item 8 asks only for the expression, which is the more revealing half)
Evaluate an expression mixing brackets, products and sums (p.42 item 1)
Given a scoring rule, a total and a count of correct answers, find the count of wrong answers and the length of the test (p.43 item 6)
Given one expression's value, write down a related expression's value without recomputing (p.44 items 13 and 14)
Arrange several expressions in increasing order (p.44 item 12)
Find three consecutive numbers with a stated product (p.43 item 9)
Decide which totals can be paid with two coin denominations of opposite sign, and say why every total can be (pp.43–44 item 10 — the +85 in its stem, seven further targets, and a "Try This" on 1568 pibs; the richest open item in the chapter)
Recover the operations a pattern machine performs and complete its table (p.43 item 7)
Place brackets so that a result is as large, or as small, as possible (p.44 item 15)
Fill blanks in an expression skeleton in several different ways (p.44 item 16)
Competency papers at this level lean heavily on the "model it, then evaluate it" shape — a rate with a direction, a running total, and one question that changes the starting conditions

Examples worth working on the board

Example 1, the test (Part II, §2.2, p.35). Inputs: a paper of 50 multiple choice questions; a right answer earns 5 marks and a wrong one carries a penalty worth 2; Mala answers 30 correctly and 20 wrongly. The chapter's printed route is to fix the two per-question values as 5 and −2, write one expression combining both counts, and only then evaluate.
The open follow-up (Part II, p.36). Immediately after the solution the chapter asks for the highest and the lowest score the paper allows, and prints no answer. It does have a definite answer: the highest score comes from all 50 answered correctly, the lowest from all 50 answered wrongly. Whether the paper allows a question to be left blank changes neither bound, because a blank earns 0 and 0 is better than the −2 a wrong answer costs, so leaving one blank can only raise a total, never lower it.
Example 2, the mine shaft (Part II, pp.36–37). Inputs: above ground counts positive and below ground negative; the lift travels 3 metres a minute; part (a) starts at ground level and descends for one hour; part (b) starts 15 m above ground and descends for 45 minutes.
The shaft illustration (Part II, p.36). Checked against the printed page. A painted cut-away of a hillside with a vertical shaft down its middle and horizontal galleries running off it, each gallery labelled with its level: +180 m, +125 m, +100 m, +40 m, 0 m, −50 m, −80 m, −125 m, −175 m, with the lift car drawn near the −125 m gallery and a tipper truck at the 0 m surface. Every one of those labels is lettering inside artwork and none of it survives text extraction. It is the best picture in the chapter of a signed scale, and worth redrawing faithfully.
The two methods, side by side (Part II, pp.36–37). Method 1 inputs: 60 minutes at 3 metres a minute gives a distance, which is then subtracted from the starting level. Method 2 inputs: the direction is folded into the rate, which becomes −3 metres a minute, and the rate is multiplied by 60. Part (b) is then worked by Method 2 only, as a starting position plus a signed product, and the reader is asked to redo it by Method 1. That asymmetry is the argument of sections 7 and 8.
Machine 1 (Part II, "Pick the Pattern", p.41). Checked against the printed page; the machines are hand-drawn artwork and none of their numbers extract. Machine 1 takes three inputs and returns one output, drawn as a star. Its six rows, in order: (5, 8, 3) → 10; (10, 11, 12) → 9; (5, 8, −3) → 16; (−3, 10, 2) → 5; (−4, −1, −6) → 1; and (−10, −12, −9) → a blank pink star. The chapter prints the rule on p.42 — first input plus second input minus third input — and leaves the last row for the reader.
Machine 2 (Part II, "Pick the Pattern", p.41). Checked against the printed page. Rows, in order: (4, 8, −3) → −29; (6, −11, 12) → 54; (5, 3, 7) → −22; (−3, 9, −8) → 35; (−7, 4, 6) → 22; and (−10, −12, −9) → blank. The chapter prints no rule for this machine and Part II has no answer key. A rule that fits all five given rows is: multiply the first two inputs, add the third, and negate the whole thing. If the explanation states it, it must state it as its own answer to the book's question.
The exercise machine (Part II, "Figure it Out" item 7, p.43). Checked against the printed page. Rows, in order: (4, 8, −3) → 28; (6, 9, 6) → −48; (2, 3, −2) → 8; (−9, 5, −8) → 31; (7, −4, −6) → −17; and (−16, −6, −9) → blank. A rule fitting all five is: subtract the product of the second and third inputs from the first. Again, derived here and not printed. This machine is the better one, because recovering its rule requires knowing that the multiplication happens before the subtraction.
The sign-flip item (Part II, item 4, p.42). Input: a long alternating sum is given with its value, and the same expression with every sign reversed is asked for without recomputing. The point is that reversing every sign multiplies the whole expression by −1 — the distributive property doing a day's work in one line.
The ordering item (Part II, item 12, p.44). Checked against the printed page. Inputs: six expressions to be arranged from smallest to largest — three additive and three products, an even split. The additive three are (−348) + (−1064), (−348) − (−1064) and 348 − (−1064). The products are (−348) × (−1064), 348 × (−1064) and 348 × 964. That last factor is 964, not 1064, and it is not a misprint to be tidied away: with 1064 there, the last product would equal (−348) × (−1064) exactly and the item would have no single answer. The intended method is to settle each expression's sign and rough size rather than to compute six values.
The alien currency (Part II, item 10, pp.43–44). Checked against the printed page. Inputs: a society whose only two coins are worth +13 pibs and −9 pibs, and several of each. The chapter asks whether +85 can be paid, answers itself that ten +13 coins and five −9 coins do it, and then sets seven more targets — +20, +40, −50, +8, +10, −2 and +1 — with a hint to write down some multiples of 13 and of 9, closing with a "Try This" asking about 1568 pibs. It prints no answers. The reason they all work is added here to supply, not the book's: 13 and 9 have no common factor, so a handful of coins can be worth exactly +1 pib — seven of the +13 coins with ten of the −9 coins gives 91 − 90 = +1. That alone does not finish the argument, and the gap is worth being careful about: repeating a +1 bundle only ever builds totals of 0 or more, and you cannot use a bundle a negative number of times. Two of the seven printed targets are negative, so the explanation also needs the mirror bundle, which coprimality equally supplies — two of the +13 coins with three of the −9 coins gives 26 − 27 = −1. With a +1 bundle and a −1 bundle available, every integer total is reachable. Say which part of that is the chapter's and which is added here.
The consecutive-product item (Part II, item 9, p.43). Input: find three consecutive numbers whose product is −6, and then three whose product is 120. Short, and the best small item in the set for making a student reason about how many negative factors a product needs.
The Collatz variant (Part II, item 5, p.42). Inputs: start anywhere; halve an even number; multiply an odd number by −3 and add 1; repeat. Printing caution: the printed example chain runs −7, 22, 11, then a term set as 32, then −16, −8, −4, −2, −1, 4, 2, 1, looping back to −2. Under the stated rule the term after 11 is −32, and −32 is also what is needed for the next term to be −16. The printed term is missing its minus sign. Confirmed against the printed page p.42. Use −32; do not show the chain as printed.

Figures to have open

A vertical signed scale with labelled levels above and below a zero line, and a marker that can travel along it. Redraw from the p.36 shaft illustration; the painted original is the book's artwork and its labels are the only place those numbers appear.
A machine box with three input slots and an output star, with rows that can be filled one at a time and a blank final row. The book's are hand-drawn; a clean redraw is better for showing it and loses nothing.
An expression line that can collapse one layer at a time, with the layer being worked highlighted.
A bracket pair that can be dragged to different positions in a fixed sequence of numbers, with the value recomputing as it moves.
No photograph or dataset from the textbook is needed.

Where this sits in the book

NCERT Class 7 Mathematics, Ganita Prakash Part II, printed Chapter 2 "Operations with Integers", §2.2: Example 1, p.35, and its open follow-up, p.36; Example 2, pp.36–37, with both methods.
Part II, the unnumbered sub-heading "Pick the Pattern", pp.41–42, for the two machines and the printed rule for the first of them.
Part II, the closing "Figure it Out" block, pp.42–44, for items 1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15 and 16. The rest of the block is spoken for elsewhere, and this brief is not the only other claimant: item 2 (the division set) and item 11 belong to The sign rule for division, and why it follows from multiplication, item 3 (the integers whose product with −1 is given) to Why a negative times a negative must be positive, and item 1(c) is also drawn on by Magic grids of integers. Between the four briefs the block is fully covered. Verified against the printed page of p.42.
Part II, p.39, items 2 and 3, for the temperature and the cement-company situations, which are the same modelling shape one page earlier.
Backward pointer: Part I, printed Chapter 2 "Arithmetic Expressions", for the order-of-operations convention this chapter uses without restating.