Extending Indian place value to the right of the point

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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 tenth part and A hundredth part, and continuing the split — tenths and hundredths, and the ten-to-one trade between them (Part I, §3.2–§3.3)
The Land of Tens: each place is ten of the one before — each place in the Indian system holds ten of the one to its right (Part I, Chapter 1)
Writing a whole number in expanded form and reading its place values back
Writing and saying large numbers with Indian digit grouping, such as 1,00,000
The idea of a fraction with denominator 10, 100 or 1000

What they should be able to do

State that a unit may be split into any number of equal parts, and give the chapter's reason for always choosing ten
Extend the ten-to-one chain of places rightwards past the units and name the places it produces
Explain why the chain has no last place in either direction
Say why the system is called decimal, and give the word's ancestry as the chapter gives it
Show, with a worked case, that writing digits without a separator leaves the quantity ambiguous
Use the decimal point to write quantities that would otherwise be confused, and read the point as the marker of where the units place ends
Fill in a place value table for a given decimal number and read off what each digit contributes
Read a decimal number aloud digit by digit after the point, and say why the digits after the point are not read as one whole number
Convert a count of one size of part into decimal form, as with 234 tenths

Where it usually goes wrong

"You are not allowed to split a unit into four." The chapter says the opposite in as many words and prints the quarter and sixteenth scales to prove it. An explanation that turns the ten-split into a law has inverted the section: the point is that ten is chosen, for a reason, and the reason is stated.
"The decimal point is a new operation." It is a mark of position. Nothing is multiplied, divided or joined by it.
"The point sits in the middle of the number." It sits immediately right of the units place, and the number is not symmetric about it. There is no place between the units and the tenths, and no place answering to the units on the right-hand side.
"7.05 is seven and five." It is seven units and five hundredths, with an empty tenths place. The chapter's three readings of 705 exist for exactly this student.
"After the point, read the digits as one whole number." The page rules that out for 0.274 and explains why: those three digits name three different sizes of part, so each is said on its own.
"234 tenths cannot be written with a point because 234 is bigger than 10." It becomes 23.4. Counts larger than the base are normalised, exactly as 23 hundreds becomes 2300.
"The places stop at thousandths." The chapter's last chain runs to ten-thousandths and says the extension continues at both ends.
"Decimal is a European idea because the word is Latin." The chapter's own account is more careful than that: it gives the Latin word for ten, links it to Sanskrit, and notes the related words in eight Indian languages. The history of the notation itself is taken up at Part I, §3.8, p.78.

Questions to check understanding

Fill in a place value table for a given decimal number
Write a quantity described in words in decimal form, and the reverse
Say what a named digit of a given decimal number is worth
Convert a count such as 234 tenths or 105 tenths into decimal form
Answer counting questions of the "How Big?" kind, and invent one
Read a decimal number aloud correctly, and write down one read aloud
Explain why a separator is needed, using a number that would otherwise be ambiguous
State how many equal parts a unit is split into in this system, and give the reason

Examples worth working on the board

The pencil on three scales (Part I, §3.4, p.59). The same pencil is drawn three times. First above a scale marked 0 to 3 whose last unit alone carries fine tick marks. Second above a scale marked 0 to 3 with quarter marks labelled 2¼, 2 2/4 and 2¾. Third above the quarter scale with a further cluster of fine ticks beyond the last quarter. The accompanying text says each quarter may itself be cut into four, giving parts of which sixteen fill a unit. The whole figure is artwork; its labels extract as scattered fragments.
The whole-number place values (Part I, §3.4, p.59). 281, with 2 in the hundreds place, 8 in the tens place and 1 in the ones place. Then the three statements: ten ones for a ten, ten tens for a hundred, ten hundreds for a thousand.
The first chain diagram (Part I, §3.4, p.59). Five boxes in a row — 10,000, 1000, 100, 10, 1 — with a ×10 arrow curving leftwards over each gap and a ÷10 arrow curving rightwards under each.
The second chain diagram (Part I, §3.4, p.60). The same row continued rightwards by two boxes, one-tenth and one-hundredth, arrows intact.
The magnification cascade (Part I, §3.4, p.60). Checked, and it extracts as nothing at all. Three stacked number lines: 0 to 1 marked in tenths; the stretch from 6 tenths to 7 tenths blown up and marked in hundredths from 60 hundredths to 70 hundredths; the stretch from 66 hundredths to 67 hundredths blown up again and marked in thousandths from 660 to 670. Blue arrows join each level to the next. This is the best single figure in the chapter for showing that the splitting never has to stop.
The third chain diagram (Part I, §3.4, p.61). Ten boxes running from 1,00,000 down to one ten-thousandth, arrows in both directions across every gap. Note the Indian digit grouping in the leftmost box.
The name (Part I, §3.4, p.61). The system is named for ten; the chapter gives the Latin word for ten, links it to Sanskrit daśha, and names eight Indian languages carrying related words — alphabetised here as Assamese, Bodo, Gujarati, Hindi, Kashmiri, Konkani, Marathi and Odia, though the page sets them in a different order.
How Big? (Part I, §3.4, p.61). Five counting questions: how many thousandths fill a unit, a tenth, a hundredth; how many tenths fill a ten; how many hundredths fill a ten. Students are asked to invent more.
The ambiguity (Part I, §3.4, p.61). Written bare, 42 could be four tens and two units or four units and two tenths. Then 705, offered with three readings: seven hundreds with five ones; seven tens with five tenths; seven units with five hundredths.
The two tables (Part I, §3.4, p.62). The first pairs each of the three quantities with its decimal form: 705, 70.5, 7.05. The second is a place value table with columns headed Hundreds, Tens, Units, Tenths, Hundredths and a dot printed in the header between Units and Tenths; each cell holds a digit times its place, not a bare digit. Below the tables sits a strip of coloured cells with a point between the fifth and sixth, labelled with what each side counts. All of this is artwork.
Reading aloud (Part I, §3.4, p.63). 705, 70.5, 7.05 and 0.274 with their spoken forms; the page states outright that the digits after the point in 0.274 are not read as a single whole number, because they name tenths, hundredths and thousandths separately.
Eight quantities to write in decimal form (Part I, §3.4, p.63). Given in words or as fractions: 2 ones with 3 tenths and 5 hundredths; 1 ten and 5 tenths; 4 ones and 6 hundredths; 1 hundred, 1 one and 1 hundredth; 8 hundredths together with 9 tenths; 5 hundredths; 1 tenth; and finally a set of three mixed quantities — 2 and 1 hundredth, 4 and 1 tenth, and 7 and 7 thousandths, giving 2.01, 4.1 and 7.007. Item (h) is the only one of the eight set that way: each of its three entries prints a full-size whole number in front of a stacked fraction, where (e), (f) and (g) print bare fractions. Read as bare fractions it would come out as 0.02, 0.4 and 0.007 — a different answer to the chapter's own question, and the one an explanation would show. Verified on the printed page of p.63 and against the appended key.
Counts of one kind of part (Part I, §3.4, pp.63–64). 23 hundreds is 2300; 23 tens is 230; 234 tenths is 23.4. Each is set in a small place value table in which the whole count is first written in one cell and then normalised across the row. Two more are left to the student: 234 hundredths and 105 tenths.

Figures to have open

The pencil on a quarter-marked scale beside the same pencil on a tenth-marked scale. This carries sections 2 to 4 and must be drawn accurately enough that both readings are believable. Standard schematic; the printed version on p.59 is artwork.
The chain of places with ×10 and ÷10 arrows, built once and extended in stages. Standard schematic. It should end with boxes visibly continuing off both edges of the frame.
The three-level zoom cascade. Redraw from p.60. Nothing else in the chapter makes the endlessness of the splitting visible, and the printed version extracts as nothing, so it must be rebuilt from the printed page.
The place value table with the point in the header. Redraw from p.62; the printed cells hold products, not digits, and that detail is the whole reason the table teaches anything.
The coloured place strip from the foot of p.62, with the four labels. Redraw.
No photograph is required.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 3 "A Peek Beyond the Point", §3.4 Decimal Place Value, pp.58–64. The quarter and sixteenth scales are on p.59 with the first chain diagram; the second chain and the zoom cascade are on p.60; the third chain, the naming of the system and the How Big? questions are on p.61; the unnumbered subheading Notation, Writing and Reading of Numbers opens on p.61 and runs to p.64, carrying the ambiguity argument, the two tables on p.62, the reading conventions on p.63 and the counts of tenths on pp.63–64.
Backward pointer: Part I, Chapter 1, for Indian place value itself, and for writing counts such as 23 hundreds.
Forward pointers inside the same chapter: Part I, §3.5, pp.64–69, where the metric units turn out to be these places renamed; Part I, §3.8, p.78, for the history of the separator itself.