Why every rule of arithmetic carries over to algebra

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

A letter-number stands for any number, not one number — what a letter-number is, and what an algebraic expression is
Every expression can be rewritten as a sum of terms and the rest of Part I, Chapter 2 — reading an arithmetic expression as a sum of its terms
Negative numbers: adding a negative is the same as subtracting
Multiplication before addition when both appear without brackets

What they should be able to do

Rewrite a subtraction as the addition of a negative term, and say why that makes the expression easier to read
Evaluate an arithmetic expression by choosing a convenient order of terms
Remove a bracket that has a minus sign in front of it, and check the answer by the other route
State the distributive property in your own words
Explain why a rule proved for all numbers may be used on a letter-number
Say what it means for an algebraic expression to take a value
Predict which manipulations will still be legal once letters replace numbers, and why

Where it usually goes wrong

"Algebra has its own rules, which is why it is hard." It has none of its own. Everything used in this chapter is a fact about numbers that the student already met in Chapter 2.
"You can only rearrange an expression once you know what the numbers are." Exactly backwards. A rule that holds for every number holds without knowing which number, which is precisely why it survives the arrival of letters.
"Minus is a thing you do, not a thing a term carries." The sum-of-terms reading moves the minus onto the term. That single move is what makes swapping safe: 83 + 28 – 13 + 32 can be reordered only because –13 travels with its sign.
"A bracket with a minus in front just loses the bracket." The p.86 pair of columns exists to head this off: every term inside changes sign, not only the first. Show the wrong version once, get 68 – 18 + 13 = 63, and put it next to the two correct routes.
"Grouping is a trick for getting the answer faster." It is that, but the reason it is allowed is what matters here.
"Swapping works for subtraction too." It does not: the sum-of-terms reading is what makes the reordering legal, and it works because the expression has been turned into an addition first.

Questions to check understanding

Evaluate an arithmetic expression that mixes multiplication with addition and subtraction
Evaluate an expression containing a bracket with a minus sign in front, and show the working by both routes
Rewrite a given expression so that every one of its terms is added
Say whether a proposed rearrangement of an expression is legal, and why
State the distributive property and use it on a numerical example
Given an algebraic expression and a value for its letter-number, find the value the expression takes — the shape §4.3's diagnostic block uses on p.87

Examples worth working on the board

The seven expressions the section sets (Part I, §4.2, p.85). In printed order: 23 – 10 × 2; 83 + 28 – 13 + 32; 34 – 14 + 20; 42 + 15 – (8 – 7); 68 – (18 + 13); 7 × 4 + 9 × 6; 20 + 8 × (16 – 6). The section works the first, the second and the fifth and hands the other four to the student.
The first one worked (Part I, §4.2, p.85). 23 – 10 × 2 is rewritten so that both terms are added: the second term is –10 × 2, which becomes –20, and the total is 3. The step.
The second one worked (Part I, §4.2, p.86). 83 + 28 – 13 + 32 is drawn as four boxed terms — 83, 28, –13, 32 — with arcs pairing the first with the third and the second with the fourth, giving 70 and 60, and then 130. Checked against p.86; the boxes and arcs are drawn artwork and extract as bare numbers with the pairing lost.
The fifth one worked twice (Part I, §4.2, p.86). 68 – (18 + 13) is set out in two parallel columns joined by the word OR. The left column adds inside the bracket first: 68 + (–31), then 37. The right column removes the bracket first: 68 + (–18) + (–13), pairs 68 with –18 to make 50, and then 50 + (–13) = 37. Checked from p.86. Both columns must be shown — the point is that the two routes are forced to agree, not that one of them is faster.
The bridge back to the ages (Part I, §4.2, p.86). The section closes by recalling Example 1: with 23 put in place of a, the expression a + 3 takes the value 26. This is the sentence that ties the arithmetic revision to the letters, and it should be the last beat.
What §4.2 says it is recalling (Part I, §4.2, p.85). Writing expressions as sums of terms; swapping; grouping; brackets, including a bracket with a minus in front; and the distributive property. The three of these that the page sets in bold are swapping, grouping and the distributive property.

Figures to have open

The four boxed terms with pairing arcs, redrawn. This is the clearest single picture of grouping in the chapter and cannot be replaced by a line of text.
The two-column bracket comparison, redrawn with the OR between the columns.
A distributive-property strip: a multiplier outside a bracket, then the same multiplier written against each term inside. Standard schematic.
No photograph or dataset is needed for this topic.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 4 "Expressions using Letter-Numbers", §4.2 Revisiting Arithmetic Expressions, pp.85–86.
Backward pointer: Part I, Chapter 2 Arithmetic Expressions, which is where swapping, grouping, brackets and the distributive property were introduced; §4.2 names them as recalled material rather than teaching them again.
Forward pointer inside the same chapter: Part I, §4.4, p.88, where the licence is first used on letters — the rectangle perimeter is rearranged without either side being known.