Order is the point: a sequence carries position as well as value

What to assume they know

Reading and continuing simple number patterns from Classes 6–8
Whole numbers, integers and their arithmetic, including negative differences
Unit fractions and their order (What "rational" means, and why the denominator cannot be zero)
Reading an ordered pair as position-then-value (Two axes, an origin, and why the order of the pair matters)
Square and triangular numbers as figurate counts, met in Class 6

What they should be able to do

State what makes a list a sequence, and identify the first, second and fifth term of a given sequence
Use subscript notation to name a term by its position, and read a statement like t₄ = 7 as a claim about position 4
Compute the successive differences of a given sequence and describe the pattern in words
Rewrite each triangular number as a running total of counting numbers, and each square number as a running total of odd numbers
Distinguish a finite sequence from an infinite one, and say what the three dots assert
Produce examples of sequences that decrease, that carry negative terms, and that carry fractions
Explain why a position must be a counting number while a term need not be

Where it usually goes wrong

"A sequence is just a set of numbers." Then 1, 4, 9, 16 and 16, 4, 9, 1 would be the same object. They are not: the first has gaps 3, 5, 7 and a rule, the second has nothing. Show the reordering and watch the pattern die.
"The three dots mean a few more numbers follow." On this page they assert that the list never ends. A sequence with a last term is written out and finished, like the five-term example on p. 174.
**"t₄ means t multiplied by 4."** The subscript is an address. Reading it as a product is the single most common notational slip at this stage; say out loud "the term in position 4".
"Different letters mean different kinds of sequence." t, s and u are just three names, introduced so two sequences can be compared without their subscripts colliding.
"Sequences go up." The unit fractions go down and the list starting at −7 starts below zero. Growth is not part of the definition.
"The position can be 0, because the chapter says a position is a non-negative integer." The stated range does include zero, but every sequence subscript in this chapter starts at 1. Stage numbering in §8.6.1 does start at 0, and that is a stage label, not a term position — keep the two apart from the beginning.
"A triangular number is a triangle." It is a count. The triangle is one way of laying that many dots out so the running total becomes visible.

Questions to check understanding

Continue a given sequence by four terms and state the rule in words
Write named terms of a sequence in subscript form, and read a subscript statement back into words
Given a sequence, write its row of successive differences
Decide whether a stated list is finite or infinite, and justify it
Rewrite a triangular number as a sum of counting numbers, and a square number as a sum of odd numbers
Build the running-total sequence of a given sequence, as the exercise on p. 176 does for 1, 4, 7, 10, 13, …
Produce a sequence meeting a stated condition — decreasing, or with negative terms, or with fractional terms

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated data; the chapter leaves them as questions.

The four opening sequences (§8.1, p. 174), printed in a boxed panel with each list labelled: 1, 2, 3, 4, 5, 6, …; 1, 3, 5, 7, 9, 11, …; 1, 3, 6, 10, 15, 21, …; 1, 4, 9, 16, 25, 36, …. On the page the first list is labelled Natural Numbers; the second, Odd Numbers; then come the triangular and the square ones, each named the same way. Note the first label in particular: this brief says counting numbers for it throughout, which is the friendlier phrase for an explanation and means exactly the same set — but it is this brief's substitution and not what the panel says, so text reproducing the panel should carry the printed word instead. The three dots are glossed on the same page.
The gap patterns (§8.1, pp. 174–175). Counting numbers: every gap is 1. Odd numbers: every gap is 2. Triangular numbers, across the first six terms: gaps 2, 3, 4, 5, 6. Square numbers, across the first six terms: gaps 3, 5, 7, 9, 11. Note: the last two lists have gaps that change, and the gaps are themselves recognisable sequences — that is the hinge of the section.
Triangular numbers as running totals (§8.1, p. 175): 1; 1 + 2 = 3; 1 + 2 + 3 = 6; 1 + 2 + 3 + 4 = 10; and 15 = 1 + 2 + 3 + 4 + 5, which the page states explicitly for the fifth term.
Fig. 8.1 (p. 175). Five dot triangles on a single brushed teal panel, labelled 1, 3, 6, 10, 15 beneath them; the dots are drawn as small yellow discs. The page asks the student to draw the next two arrays. Verified: the next two counts are 21 and 28, needing rows of 6 and 7 dots added.
Square numbers as running totals of odds (§8.1, p. 175): 1; 1 + 3 = 4; 1 + 3 + 5 = 9; 1 + 3 + 5 + 7 = 16.
Fig. 8.2 (p. 175). A red square panel carrying an 8 × 8 array of round counters — 64 in all — with the counters coloured in alternating L-shaped shells, yellow and blue, each shell traced by an olive line. Read off the printed page: the innermost shell is a single counter, and the shells then hold 3, 5, 7, 9, 11, 13 and 15 counters. Verified: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8 × 8, so the figure is a picture of the previous bullet carried out to eight terms. The caption asks the student to explain the relationship.
A finite sequence (§8.1, p. 174): 6, 12, 24, 48, 96 — stated on the page as having five terms, with no dots after it.
Subscript notation (§8.1, p. 176). For the odd numbers, t₁ = 1, t₂ = 3, t₃ = 5, t₄ = 7. The page also sets up s₁, s₂, s₃, … and u₁, u₂, u₃, … so that more than one sequence can be discussed at once.
Two more kinds of sequence (§8.1, p. 176): the decreasing unit fractions 1, ½, ⅓, ¼, … with first term 1; and −7, −3, 1, 5, 9, … written with s₁ = −7 and a gap of 4 throughout. The second one crosses zero, which is the point of printing it.
The running-total exercise (§8.1, p. 176). Start from 1, 4, 7, 10, 13, …; the page asks for the next four terms and then for the first ten terms of the sequence of running totals, with the hint that the totals begin 1, then 1 + 4 = 5, then 1 + 4 + 7 = 12. Verified: the original continues 16, 19, 22, 25; the running totals are 1, 5, 12, 22, 35, 51, 70, 92, 117, 145.
The triangular-number exercise (§8.1, p. 176): write t₅, t₆, t₇ and t₈ for the triangular numbers. Verified: 15, 21, 28, 36.

Figures to have open

Fig. 8.1 redrawn as a schematic: five dot triangles with their counts, and the sixth and seventh outlined faintly for the student to complete. Standard schematic; the printed art is a decorative brushed panel and does not need reproducing.
Fig. 8.2 redrawn as an 8 × 8 grid of counters with the eight L-shaped shells shown separately and then nested. This one carries the argument of section 6 and must keep the shell structure exactly: shells of 1, 3, 5, 7, 9, 11, 13, 15.
A position-and-value strip: a row of boxes labelled 1, 2, 3, 4, … above and the terms below, so that "term" and "position" are visibly two different rows. Standard schematic.
No photograph is needed anywhere in this topic.

Where this sits in the book

Chapter 8, "Predicting What Comes Next: Exploring Sequences and Progressions", §8.1 Introduction to Sequences, pp. 174–176. Figures 8.1 and 8.2 both sit on p. 175.
Three Think and Reflect boxes belong to this topic: describing and continuing the four opening sequences (p. 174), listing further kinds of sequence (p. 176), and the two inline Exercise prompts on p. 176.
Forward pointers, deliberate cross-references out of this topic: the two rule types are §8.2 (p. 176) and §8.3 (p. 178); the summary restates the definition of a sequence and of a term on p. 196.
The chapter names Class 6, Chapter 1 as the earlier home of the odds-and-squares relationship (p. 175).