Commutative, associative, and distributive over the integers

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What to assume they know

Why a negative times a negative must be positive: the four sign cases of a product, and the token model behind them
Negative numbers on the number line, and adding and subtracting them: magnitude as distinct from sign, and the additive inverse
That whole-number multiplication is unchanged by reordering and regrouping, and distributes over addition — the properties being tested here
Writing a situation as an expression before computing it: reading an expression, and what brackets do to it
The rectangle-of-objects picture of a product from earlier classes

What they should be able to do

State the commutative, associative and distributive properties for integers in letters, and read each statement aloud correctly
Reproduce the chapter's two-part argument for commutativity: magnitude first, then sign
Evaluate a three-factor product in more than one grouping and confirm the results agree
Explain why associativity plus commutativity lets a long product be taken in any order at all
Determine the sign of a product of several integers by counting how many are negative, and state the condition under which that count decides the answer
Verify the distributive property on a numerical instance with negative terms
Read a rectangle of green and red tokens as a distributive statement
Distinguish, in the chapter's own text, an argument from a checked instance

Where it usually goes wrong

"These laws are obvious, so why prove them?" They are obvious for whole numbers. Multiplication was just extended to numbers it was never defined on, and whether the old behaviour came along is exactly what has to be checked. Open the explanation with that framing or the whole topic looks like bookkeeping.
"Checking three examples proves it." The chapter checks instances for associativity and distributivity and then states the general law. That gap is real. The commutativity argument on pp.34–35 shows what the difference looks like.
"Subtraction and division are commutative too." Nothing on these pages says so, and both fail immediately: 3 − 5 against 5 − 3, or 36 ÷ (−18) against (−18) ÷ 36. The chapter's three laws are stated for multiplication only — the SUMMARY's three bullets each say so.
"An even number of minus signs makes the answer positive." Only if no factor is zero, and only if you are counting negative factors, not minus signs on the page. (−2) × (−1) × (−5) × (−3) has four negative factors and is positive; (−2) × (−1) × (−5) × 0 has three and is zero. Show the zero case explicitly.
"Distributive means you can split any operation over any other." It is stated here for multiplication over addition, in that order.
"a × (b + c) = (a × b) + c." The commonest slip in the whole topic, and the token rectangle is the cure: both blocks are four rows deep, so the multiplier reaches every part of the sum.
"Regrouping is the same thing as reordering." They are two different laws and the chapter treats them separately — brackets moved on p.39, the numbers themselves moved on p.40. Only when you have both may you evaluate a long product in any order you like.

Questions to check understanding

Fill the blanks in a table of swapped products (the chapter's own task, p.34)
Evaluate a three- or four-factor product and state which law let you regroup it
Evaluate an expression such as (−5) × (18 + (−3)) both ways round (p.42)
State the sign of a product of several integers without computing it
Given one product of large numbers, deduce a related one by distributing — the p.44 items that move a factor by one are exactly this
Given a bracketed expression and its value, write the value of a rearranged version without recomputing (p.44, item 14)
Name the property that justifies a stated equality between two expressions
Board-style items usually ask for the property by name and then for one verifying instance; both halves carry marks

Examples worth working on the board

The swap table (Part II, §2.2, p.34). Checked against the printed page. A four-row bordered table, two columns wide, each row holding a product and its swap. Row 1: 3 × (−4) and (−4) × 3, both filled in. Row 2: (−30) × 12 and 12 × (−30), both answers left blank. Row 3: (−15) × (−8) and (−8) × (−15), both filled in. Row 4: 14 × (−5) filled in, and (−5) × ▢ with the factor blank rather than the answer. The blanks are part of the design.
The commutativity argument (Part II, §2.2, pp.34–35). Two inputs, and they must stay separate. (i) Magnitude: the size of a product depends only on the sizes of the two numbers, and whole-number multiplication already survives a swap. (ii) Sign: both-positive and both-negative give a positive answer either way round; one-of-each gives a negative answer either way round. Together they close the case. This is the only one of the three laws the chapter genuinely argues.
Associativity (Part II, "Expressions Using Integers", pp.39–40). Inputs: the expression 5 × (−3) × 4, bracketed first as (5 × (−3)) × 4 and then as 5 × ((−3) × 4); afterwards the same three numbers regrouped as (5 × 4) × (−3). A second expression, 25 × (−6) × 12, is set as an exercise in every order.
The −1 ladder (Part II, p.40). Checked against the printed page. A boxed passage with four lines: two −1s multiplied, then three, then four, then five, followed by the box's own summary that two or four of them give a positive answer and three or five give a negative one, and an invitation to generalise. Printing caution: the fourth line, the one with five −1s, is printed with the answer 1. Five −1s multiply to −1, and the box's very next sentence says so. Confirmed against the printed page p.40.
The sign of a long product (Part II, p.40). The chapter asks, and does not answer, for a simple way to get the sign of a product of many integers. The answer: count the negatives; an even count gives a positive product and an odd count a negative one — provided none of the factors is zero, because a product containing zero is zero and has no sign at all. That proviso is added here, not the chapter's, and it is the difference between a true generalisation and one that merely fits the examples on the page.
Distributivity, numerically (Part II, p.40). Inputs: 5 × (4 + (−2)) set against 5 × 4 + 5 × (−2); then (−2) × (4 + (−3)) set against (−2) × 4 + (−2) × (−3), which the reader is asked to check. Notice that the second instance is the one with a negative multiplier, and the chapter hands it over rather than working it.
The token rectangle (Part II, p.41). Checked against the printed page. A single rounded frame containing two blocks: on the left a four-by-two block of green tokens, on the right a four-by-three block of red. A brace above the whole frame is labelled 4 × (2 + (−3)); braces below label the two blocks 4 × 2 and 4 × (−3). Inputs: the two block sizes and the three labels. This picture is the closest the chapter comes to a reason rather than a check.
The Try This that is left open (Part II, p.41). The reader is asked to draw the same kind of picture for −4 × (2 + (−3)), with a hint pointing back to multiplying by a negative as adding inverses. The chapter prints no such picture and Part II carries no answer key. If the explanation draws it, it must say it is supplying what the book asked for.
The general statements (Part II, pp.35, 40, 41, and SUMMARY p.45). In letters: a × b = b × a; a × (b × c) = (a × b) × c; a × (b + c) = (a × b) + (a × c). Unlike the division identities on p.39, these letters range over all integers.

Figures to have open

Green and red tokens arrangeable into a rectangle with a fixed row count and two adjacent column blocks, with braces above and below. Redraw from p.41 rather than lifting; this is the topic's central picture.
A two-column table of paired products with fillable answer cells, for the swap table.
An expression line on which brackets can slide between positions without the numbers moving — needed for sections 6 and 7 and hard to fake with static frames.
A ladder of repeated −1 factors with a running sign indicator.
A three-branch hierarchy for the closing map.
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, pp.34–35 (commutativity) and the unnumbered sub-heading "Expressions Using Integers", pp.39–41 (associativity, the −1 ladder, and the distributive property with its token picture).
Part II, p.45, SUMMARY, bullets three to five, for the three general statements as the chapter finally words them.
Part II, pp.42 and 44, for the exercise items that spend the three laws.
Sibling topic Magic grids of integers, where these laws do the work in the magic grid.