Common multiples: why the first "idli-vada" is 15

What to assume they know

Skip-counting in threes, fives and tens from primary school
The word multiple: the numbers you reach by counting in steps of a fixed size
Reading a two-circle overlap diagram — which region means "both"
Every number sequence is a rule, not a list: a sequence is a rule, not a list, so it can be continued without being written out
Multiplication tables up to 10, well enough to recognise a multiple on sight

What they should be able to do

Decide, for any number, which of two counting rhythms land on it
Define a common multiple as a number both of the given step sizes land on
Find the first common multiple of a small pair by listing, and continue the list of later common multiples without listing
Explain why the common multiples of a pair are themselves evenly spaced
Place numbers correctly in the two regions and the overlap of a Venn-style diagram, and read such a diagram backwards when numbers have been erased
Count how many multiples of a number fall below a given limit without writing them all out
Predict, for a new pair, whether the first common multiple will be as large as the product or smaller, and say what the answer depends on

Where it usually goes wrong

"15 is first because 15 is a special number." It is first because it is the first place a run of threes and a run of fives can meet, and they cannot meet earlier because neither run can be shortcut — 3 and 5 have nothing in common to shorten it with. Swap in 4 and 6 and the meeting happens at 12, not 24.
"Common multiples run out." They do not. Once you have one, add the same gap again and you have another, for ever. The two-circle picture cannot show this, because a drawn circle holds finitely many numerals — say so out loud when the figure appears.
"The overlap region means 'either'." It means "both". A child new to these diagrams often reads the lap as a merge rather than an intersection. Section 5 should place a number into the lap by checking it against both lists, twice.
"To count the double calls up to 90 I have to write the game out." Counting by division is the point of question 2. Writing ninety turns is the thing the question is designed to make unnecessary.
"Every multiple of the second number should give a lone second call." The pp.108–109 puzzle exists to break this. When the smaller number divides the larger, the second word never gets said by itself — every one of its turns is already a double.
"The first common multiple is always the two numbers multiplied." True sometimes and false often, and which one it is depends on the pair, not on the size. This is the misconception.

Questions to check understanding

Given two step sizes, write the first four common multiples
Say which of a list of numbers would receive the double call, and why
Count the multiples of a given number below a stated limit, without listing
Complete a two-circle diagram when the two headings are given
Recover the headings when only the overlap is filled in
Given that one number is fixed, find every second number for which the larger number's turns are always double calls
Find the least number that every member of a short list divides
The appendix bound with this chapter answers the §5.1 questions on footer pages 1 to 3, which is the closest thing this section has to a marked scheme

Examples worth working on the board

The counting game (§5.1, p.107). Children in a ring count 1, 2, 3, … in turn. On a multiple of 3 the player calls out a food word instead of the number; on a multiple of 5, a different food word; on a number that is both, a hyphenated word combining the two. A mistake puts the player out, and the game runs in rounds until one player is left. The chapter's own text answers "which is the first double call?" with 15 and then leaves a blank for the student to name that class of numbers.
How the opening page is drawn (p.107). Checked against the printed page. The two columns of text sit inside a ring of children's faces, and each child has a speech bubble: a green bubble carrying a numeral for an ordinary turn, and a bubble carrying a picture of the food on a plate for a special turn. The turns visible run 1, 2, then a plate, then 4 and on round to 16, 17, a plate, 19. Redraw it; do not lift it.
Fig. 5.1 (p.108). Two overlapping circles headed "Multiples of 3" and "Multiples of 5", with an arrow from the lap down to a caption naming the overlap. Checked against the printed page. Left region as printed: 3, 9, 12, 18, 21, 24, 27. Overlap as printed: 15, 30. Right region as printed: 5, 10, 20, 25. Note: 6 is a multiple of 3 below 30 and it is not in the printed figure. Either include it and say the figure is being completed, or reproduce the omission deliberately and ask the class what is missing. Do not silently show a figure that differs from the student's book.
The counting question (p.108). Inputs: the game runs over 1 to 90; "idli" turns are the multiples of 3, "vada" turns the multiples of 5, and double calls the multiples of 15; the counts asked for include the double calls. A second question rescales the same setup to 900. A third asks for the number at which the tenth double call happens.
Three more pairs (p.108). The game is replayed with 2 and 5, with 3 and 7, and with 4 and 6, the smaller number taking the first food word. Two of these three pairs have their first collision at the product; one does not.
The pair that never produces a lone second call (pp.108–109). A speech bubble reports a game in which only the first word and the double word were ever heard. A second bubble supplies that one of the two numbers was 4, and the reader is offered 2, 3, 5, 8 and 10 as candidates for the other. The condition is that every multiple of the larger number is already a multiple of the smaller, i.e. the smaller divides the larger. Both 8 (with 4 as the smaller) and 2 (with 4 as the larger) meet that condition; the appendix on footer page 2 records only 8. Treat this as a "find all that work" question, not a single-answer one.
The erased diagram (p.111, question 8). Checked against the printed page. Two overlapping circles with both headings left blank; only the overlap is filled, with 24, 48 and 72. The reader supplies the two headings and the numbers in the outer regions. Many pairs fit. The reasoning to show is that the overlap must be the common multiples of whatever pair is chosen.
Two closing challenges (p.111, questions 9 and 10). The least number divisible by every one of 1 to 10 with 7 struck out, and then the same with 7 restored.

Figures to have open

A ring of children with numbered turns. Redraw; the printed opening art on p.107 is the book's own illustration and must not be reproduced.
A single number track from 1 to at least 60 that can carry two rhythms of ticks at once. Standard schematic. This is the workhorse figure of the topic and everything from section 2 to section 8 can be staged on it.
A two-circle overlap diagram with headed regions, appearing three times: as Fig. 5.1 (p.108), as the same diagram with 6 restored, and as the erased version from p.111. Standard schematic, but the contents must match the printed contents exactly where the printed contents are being shown.
No photograph or data table from the textbook is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 5 "Prime Time", §5.1 "Common Multiples and Common Factors", pp.107–111 — the opening counting game (p.107), Fig. 5.1 and the first Figure it Out (p.108), the replayed pairs and the speech-bubble puzzle (pp.108–109), and the closing Figure it Out (p.111)
Forward pointer inside the same chapter: §5.3, p.116, where the first-collision question is answered in terms of co-primality
Chapter Summary, p.128, for the printed wording of factor
Solutions appendix bound with this chapter file, footer pages 1–3, for §5.1