Turning a pattern into a formula that predicts

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What to assume they know

A letter-number stands for any number, not one number — letter-numbers, and writing a relationship as an expression
The dropped multiplication sign, and what it silently means — the shortened notation, so 3n reads as 3 × n
Collecting like terms, and what "simplest form" is for — simplifying, which is what settles whether two formulas are the same formula
Multiples of a number; division with quotient and remainder
Reading a two-row table of values

What they should be able to do

Describe a growing pattern in words before writing any symbols
Write an expression for the general term of a pattern, naming the letter yourself and saying what it counts
Test a candidate formula against every case that was given, and say why passing those tests is not yet a proof
Use a formula to answer a case far outside the ones drawn
Write the positions at which a repeating design occurs, using multiples
Go the other way: given a position, decide which item of a repeating pattern sits there, using the remainder on division
Show that two differently-built formulas for the same pattern are equal by simplifying

Where it usually goes wrong

"If a rule fits the cases I was shown, it is the rule." The second row of machines on p.96 is a live case: several rules fit four pairs. Fitting is how you find a candidate; it is not how you finish. The chapter itself refuses to settle the matchstick question by more testing and simplifies instead.
"The formula comes first and the words are an explanation of it." §4.5 opens by saying the opposite in its own way, and every worked pattern in the section describes the relation in ordinary language before a letter appears.
"n means the answer." In 3n the letter counts which appearance is wanted, and the expression gives the position. Mixing the two is the commonest error in repeating-pattern questions: at 3n – 1, n = 41 gives position 122, not appearance 122.
"Two different-looking formulas mean one of us is wrong." 3 + 2 × (y – 1) and 2y + 1 are the same formula. Simplification is the test, and it is available precisely because of §4.4.
"Step 33 means drawing 33 triangles." The point of the whole section is that it does not. Ask for Step 108 early, let the class feel the drawing get silly, and only then build the formula.
"The pattern grows by 2, so the formula is 2y." It is the commonest wrong guess and worth showing: 2y gives 2 at Step 1, where the picture plainly has 3. Getting the growth right is only part of it — the starting point has to be right too.

Questions to check understanding

Given a table of values for a growing pattern, write a formula for Step y
Given a drawing of the first few steps of a matchstick pattern, say how many sticks Step 10 needs and write a general formula — the chapter asks this again on p.104 with squares instead of triangles
Given a repeating pattern, write an expression for the positions of one of its items
Given a position number, say which item of a repeating pattern sits there, and justify it by a remainder
Show that two given formulas for the same pattern agree, by simplifying
Find the rule of a number machine from its inputs and outputs

Examples worth working on the board

The solved number machine (Part I, §4.5, p.95). Five Y-shaped machines in a row, each with two numbers above the fork and a pink output box below. Input pairs, left to right: (5, 2), (8, 1), (9, 11), (10, 10), (6, 4). Outputs printed for the first four: 8, 15, 7, 10; the fifth box is empty. The page gives the rule in words — twice the first number, less the second — and writes it 2a – b, then shows the first case worked. Checked against p.95; the machines are drawn art and the pairing of inputs to outputs is lost in extraction.
The two unsolved rows of machines (Part I, §4.5, p.96). Same layout, ten more machines in two rows. Upper row: (5, 2) → 5, (8, 1) → 7, (9, 11) → 18, (10, 10) → 18, then a machine whose inputs are the letters a and b. Lower row: (4, 1) → 5, (6, 0) → 1, (3, 2) → 7, (10, 3) → empty, then the letter machine again. Checked from p.96. The page prints no rules for these; it then asks the student to invent a machine of their own.
The saree border (Example 12, Part I, §4.5, p.96). Six floral motifs printed in a row, labelled with the first six letters of the alphabet, in which the fourth repeats the first, the fifth the second and the sixth the third. Checked against p.96 — the motifs are artwork, and only their labels extract, so the repetition is invisible in the text.
The design formulas (Part I, §4.5, p.97). Design C first appears at position 3, then 6, so its nth appearance is at 3n. Design B appears at positions 2, 5, 8, 11, 14, and its nth appearance is at 3n – 1. Design A's is 3n – 2.
Going backwards (Part I, §4.5, p.97). A three-row table: position 99 gives quotient 33 and remainder 0; position 122 gives 40 and 2; position 148 gives 49 and 1. The page asks which design sits at each and does not answer. Checked from p.97.
The matchstick triangles (Part I, §4.5, p.100). A strip of triangles that share sides, drawn for Steps 1 to 4. Step 1 is one triangle. The page states that Step 5 uses 11 matchsticks, and prints the counts for the first six steps: 3, 5, 7, 9, 11, 13. A second table underneath re-writes those counts as 3, then 3 + 2, then 3 + 2 + 2, and so on. Checked from p.100 — the matchstick strip is a photograph-style drawing and extracts as nothing.
The two formulas (Part I, §4.5, p.101). From the growth pattern: at Step y there are one fewer 2s than y, added to a starting 3, giving 3 + 2 × (y – 1). Noticing that the 3 of Step 1 is itself 1 + 2 gives 2y + 1. The page simplifies the first into the second rather than testing it on more steps.
The second way of counting (Part I, §4.5, p.101). The same strip is redrawn with the horizontal matchsticks distinguished from the diagonal ones. The page states the Step 2 counts — 2 horizontal, 3 diagonal — and then asks for Steps 3 and 4, for an expression for each orientation at Step y, and whether the two expressions add to 2y + 1. It answers none of these. Checked from p.101.
The far cases (Part I, §4.5, p.100). Step 33, Step 84 and Step 108 are the ones the page asks for, and it asks twice — both times on p.100, and both times before any formula exists. Checked against pp.100–101. The first asking comes straight after Step 5 and adds "is there a quicker way"; the second comes at the foot of the page, once the repeated-addition table has supplied a method but still no closed expression. The formula arrives on p.101, as 3 + 2 × (y − 1) and then as 2y + 1 — and the chapter never goes back to 33, 84 and 108 to spend it. That unreturned-to question is the argument of this topic.

Figures to have open

The Y-shaped number machine, drawn once and reused. It needs two input slots, a sealed middle, and one output slot; the seal is the point.
The row of five solved machines with their printed inputs and outputs, and the two unsolved rows. Redraw.
A repeating border of three distinguishable motifs, six positions long, with positions numbered. The motifs need only be distinguishable — they do not need to be the textbook's floral art, which should not be reproduced.
The matchstick triangle strip for Steps 1 to 5, then the same strip with horizontal and diagonal sticks coloured differently. Standard schematic, but it must be drawn accurately: shared sides are shared, not doubled.
The two-row step/count table. Redraw.
No photograph 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", §4.5 Pick Patterns and Reveal Relationships, pp.95–105. This topic uses the unnumbered blocks Formula Detective (pp.95–96), Algebraic Expressions to Describe Patterns (pp.96–97) and Matchstick Patterns (pp.100–101), together with the pattern items of the closing Figure it Out (pp.102–105).
The Patterns in a Calendar block (pp.97–99), which is also inside §4.5, is carried by Why a formula says in one line what words take a paragraph to say.
Backward pointer: Part I, §4.1, p.82, where the Ls made of matchsticks gave the chapter its first formula, and p.84, where the word formula is introduced.