A hundredth part, and continuing the split

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What to assume they know

A tenth part — tenths, and trading ten tenths for a unit (Part I, §3.2)
Why whole units are not enough to measure with — why a length between two marks cannot be pinned down (Part I, §3.1)
Adding and subtracting whole numbers in expanded form, reordering and regrouping the terms of the sum freely, as in Commutative and associative: why order and grouping are free
Halving a length, and halving a number written with a fractional part
Comparing fractions with denominators 10 and 100

What they should be able to do

Explain why splitting each tenth into ten equal parts produces a part of which a hundred make the unit
Write one length as units, tenths and hundredths; as units and hundredths; and as a plain count of hundredths, and move freely between the three
State the two trade rates the chapter relies on: ten hundredths for one tenth, a hundred hundredths for one unit
Compare lengths in which tenths and hundredths are mixed, including the pairs designed to be confused
Add and subtract such lengths by at least two different routes and check that they agree
Point out the step-for-step match between a whole-number sum in expanded form and the same sum done in tenths and hundredths
Break a unit or a tenth apart mid-subtraction and say what was traded for what

Where it usually goes wrong

"3 tenths and 3 hundredths are close." They differ by a factor of ten, and group (a) on p.55 exists to make a student say so out loud. An explanation that treats the two as near-neighbours has lost the section.
"Adding a hundredths part is a new procedure." It is the same procedure with one more column. The chapter's own evidence is the pair of whole-number sums it sets alongside; use them, and use them at full size.
"You must convert everything to hundredths before adding." One of the four printed routes does; three do not. Insisting on conversion turns a choice into a rule.
"Twelve hundredths in the middle of a sum is a mistake." It is a legitimate intermediate count that gets traded up. The same was true of thirteen tenths in §3.2.
"Once the parts are small enough you can stop." The folded sheet needed a second split; the chapter goes on to a third in §3.4. There is no last split.
"A length between two marks has no length." It has one; the scale merely cannot report it. This is the misreading the folded sheet is designed to catch, and it recurs whenever a student writes "cannot be measured".
"Adding zeros or dropping them changes nothing anywhere." In this section nothing is written with a decimal point yet, so the question does not arise — but the moment it does, at Part I, §3.6, p.70, it needs a careful answer. Do not pre-empt it here.

Questions to check understanding

Write one quantity in all three available forms and say which is most useful for what
Identify the longest and shortest in a group whose entries mix tenths and hundredths
Mark several such lengths on a supplied scale, having first worked out what one division of that scale is worth
Add or subtract two quantities carrying both tenths and hundredths, showing the trades
Repeat the same computation by a second route as a check
Given a whole-number computation in expanded form, produce the matching computation for a quantity with fractional parts
Say how many hundredths make a tenth, and how many make a unit

Examples worth working on the board

The folded sheet (Part I, §3.3, pp.52–53). A sheet 8 and 9 tenths units long, doubled over the long way so that its length halves. The page first narrows the answer to somewhere between 4 and 4 tenths and 4 and 5 tenths, then refines it once the tenths have been split. Checked pp.52–53: the sheet is photographed lying on a ruler running 0 to 10, and the folded sheet appears twice more, the second time above a magnified strip of the ruler between 4 and 5.
The magnified strip (Part I, §3.3, p.53). An oval blow-up of the stretch from 4 to 5 with every tenth mark labelled — 4 and 1 tenth through 4 and 9 tenths — and each of those tenths visibly subdivided again. All lettering is inside the artwork and extracts as fragments.
The counting argument (Part I, §3.3, p.53). Ten small parts in each tenth, ten tenths in a unit, therefore a hundred small parts in a unit.
The reading of the fold (Part I, §3.3, p.53). Four units, four tenths and five hundredths, with the page then asking whether it may equally be called four units and forty-five hundredths.
The hundredth-marked line (Part I, §3.3, p.54). A finely divided line whose selected ticks are labelled in tenths above and in hundredths below; the values printed below are 1 hundredth, 20 hundredths, 99 hundredths and 130 hundredths, and the values printed above include 1 tenth, 2 tenths, 1, 1 and 3 tenths, and 2. Five boxes are left empty for the student — one between the 20 hundredths and 99 hundredths labels, and four to the right of 130 hundredths.
The wire in three notations (Part I, §3.3, p.54). One length, printed three ways: as 1 unit with 1 tenth and 4 hundredths; as 1 unit with 14 hundredths; and as a bare count of 114 hundredths. Each is set beside its spoken form in bold. Checked: the wire lies on a ruler labelled in tenths from 0 past 1 to 1 and 2 tenths, with the wire's ends marked in red.
Two more wires to read (Part I, §3.3, p.54). One against a scale running from 4 and 9 tenths to 6 and 1 tenth, one against a scale running from 14 and 4 tenths to 15 and 6 tenths. Checked; the answers are blank on the page.
Longest and shortest, seven groups (Part I, §3.3, pp.55–56). The groups, in order: (a) 3 tenths, 3 hundredths, 33 hundredths; (b) 3 and 1 tenth, 30 tenths, 1 and 3 tenths; (c) 45 hundredths, 54 hundredths, 5 tenths, 4 tenths; (d) 3 and 6 tenths, 3 and 6 hundredths, 3 and 6 tenths and 6 hundredths; (e) 8 tenths and 2 hundredths, 9 hundredths, 1 and 8 hundredths; (f) 7 and 3 tenths 5 hundredths, 7 and 5 tenths, 7 and 41 hundredths; (g) 65 tenths 15 hundredths, 5 and 87 hundredths, 5 and 7 hundredths. Each group has its own blank scale to mark up, and the scales are not all labelled the same way — group (d)'s runs from 30 tenths to 40 tenths.
The four-route sum (Part I, §3.3, pp.56–57). 15 and 3 tenths 4 hundredths plus 2 and 6 tenths 8 hundredths. Method 1 groups units, tenths and hundredths separately; Method 2 is the same thing stacked in a column; Method 3 converts the fractional parts to hundredths; Method 4 converts both numbers entirely to hundredths. Two speech balloons carry the trade rates: ten hundredths for one tenth, a hundred hundredths for one unit.
The whole-number sum set alongside (Part I, §3.3, p.57). 483 + 268, written as (400 + 80 + 3) + (200 + 60 + 8) and normalised line by line down to 751.
The first subtraction (Part I, §3.3, p.57). 25 and 9 tenths, minus 6 and 4 tenths 7 hundredths, shown in three stacked stages.
The second subtraction (Part I, §3.3, p.58). 15 and 3 tenths 4 hundredths, minus 2 and 6 tenths 8 hundredths, again in three stages, with red and green loops in the printed artwork showing which place was broken open at each stage. Checked p.58.
The whole-number subtraction set alongside (Part I, §3.3, p.58). 653 − 268, written out in the same expanded style and normalised to 385.
The closing practice set (Part I, §3.3, p.58). Six items mixing sums and differences. One of them, item (a), has a hundredths part on one side only; the other five carry hundredths on both sides. See Notes on item (f).

Figures to have open

A unit bar cut into ten, with one tenth pulled out and itself cut into ten. This single graphic carries sections 3 and 4 and should be built once and reused. Standard schematic.
A magnified stretch of ruler between two whole numbers, with the tenths labelled and each tenth subdivided. Redraw from the p.53 blow-up; the printed version is a photographic oval and its labels will not survive scaling.
One shared number line carrying the three lengths of group (a). Standard schematic.
The four-route sum laid out as four parallel columns. Redraw; the printed version runs down two pages and cannot be seen at once, which is exactly what the explanation can fix.
The expanded whole-number sum and the expanded sum in parts, aligned line for line. Redraw. This is the argument of the topic.
No photograph from the textbook is required.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 3 "A Peek Beyond the Point", §3.3 A Hundredth Part, pp.52–58. The folded sheet begins on p.52; the split into hundredths and the magnified strip are on p.53; the hundredth line and the three-notation wire on p.54; the seven grouping items on pp.55–56; the four-route sum on pp.56–57; 483 + 268 on p.57; the two subtractions on pp.57–58 with 653 − 268 and the practice set on p.58.
Backward pointers: Part I, §3.2, for the tenth and its trade rate; Part I, Chapter 2, §2.2, for the commutative and associative properties of addition, which are what license the reordering and regrouping these expanded sums do.
Forward pointers inside the same chapter: Part I, §3.4, pp.58–61, which turns the repeated split into a place value system; Part I, §3.7, pp.74–75, where these same additions reappear in decimal notation.