Multiply as whole numbers, then count the decimal digits

What to assume they know

Extending Indian place value to the right of the point — reading a decimal as place-value parts, so that 27.53 is 2 Tens, 7 Ones, 5 Tenths and 3 Hundredths
the chapter's opening recap of decimal place value — the chapter's own opening recap: every decimal is a sum of tenths, hundredths and thousandths, and writing it as a fraction over 10, 100 or 1000 is always possible
A fraction of a fraction, and why the numerators and denominators multiply — a fraction times a fraction: numerators together, denominators together
Multiplying two whole numbers of three or four digits by hand
Knowing that 10 × 10 = 100 and 10 × 100 = 1000, so a product of powers of ten is again 1 followed by zeros

What they should be able to do

Convert a decimal into a whole number over a power of ten, and back
Compute a decimal product by the fraction route, showing every step
Read off, from a worked product, how many digits sit after the point in each factor and in the answer
State the placement rule and justify it from the denominators rather than asserting it
Given a whole-number product, write down the decimal products built from the same digits without multiplying again
Explain why writing ₹56.50 instead of ₹56.5 changes the digit count but not the answer
Predict what multiplying a decimal by 10, 100 or 1000 does to its point

Where it usually goes wrong

"Line the decimal points up, the way you do when adding." Addition needs the places aligned; multiplication does not, and the chapter's method deliberately strips both points off before a single digit is multiplied. Showing 9.5 × 5 written in an aligned column and then crossed out is worth ten seconds of video.
"The product has as many decimal places as the longer factor." The table on Part II pp.70–71 is built to kill this: 5.7 × 13.35 has factors with one and two places and a product with three. They add; they do not compete.
"Adding a trailing zero changes the number." ₹56.50 and ₹56.5 are the same amount. Q6 on Part II p.73 is asking exactly this. The digit count changes, the count in the product changes with it, and the value does not — the extra zeros end up at the tail of the answer where they can be written or dropped.
"The rule is arbitrary, so it has to be memorised." It falls out of the denominators in three lines. A student who can write 5.96 as a fraction can re-derive it in the exam hall.
"596 × 248 has six digits, so 5.96 × 24.8 must too." The digits are fixed; where the point lands is not, and for 0.018 × 0.012 the point lands so far left that zeros have to be written in front. Digit count and place count are different counts.
"You must convert to fractions every time, or it is not proper working." The chapter offers the fraction route as the justification and the digit count as the working method. Both are legitimate; the fraction route is what you fall back on when you doubt the count.

Questions to check understanding

Multiply two given decimals and show the fraction working
Given a whole-number product, write down four decimal products built from the same digits (this is the shape of Q8 on Part II p.73 and of Q3 on Part II p.87)
State how many digits will stand after the point in a stated product, before computing it
Word problems that end in a unit change — millimetres to centimetres, rupees and paise, metres to kilometres
Cost and profit problems: buy at one decimal price, sell at another, scale by a whole number
Explain why two differently written decimals give the same product
The SUMMARY bullet on Part II p.95 states the placement rule in one sentence and is the most likely source of a one-mark recall question

Examples worth working on the board

Example 1, the five pens (Part II, §4.2, p.69). One pen costs ₹9.5; five pens are bought. The book shows the answer twice: once as 9.5 added five times, and once as the fractions 5 over 1 and 95 over 10 multiplied together. Both routes matter — the first says what multiplication means, the second says how to do it when neither factor is whole.
Example 2, the car (Part II, §4.2, p.69). 12.5 km per litre of petrol, 7.5 litres of petrol. Fraction form: 125 over 10 times 75 over 10.
Example 3, Ajay's week (Part II, §4.2, p.70). School is 827 m from home; he walks there and back, six days a week; the answer is wanted in kilometres. The book converts 827 m to 0.827 km first, then multiplies by 2, then by 6 — two multiplications, not one, so the explanation can show the point holding still while the digits grow.
Example 4, the rectangle (Part II, §4.2, p.70). Checked against the printed page. A plain unshaded rectangle sits in the right margin with 5.7 cm printed up its left side and 13.3 cm printed under its base; no grid, no shading, nothing else in the picture. The two lengths are the inputs; the area is the thing to find.
The digit-count table (Part II, §4.2, pp.70–71). Checked against the printed page; it straddles the page turn. Four rows, each carrying a fully worked fraction computation on the left and three counts on the right — digits after the point in the multiplier, in the multiplicand, and in the product. The four multiplications are 9.5 × 5 (counts 1, 0, 1), 12.5 × 7.5 (1, 1, 2), 1.64 × 6 (2, 0, 2) and 5.7 × 13.35 (1, 2, 3). Read the third column as the sum of the first two; that is the whole point of printing four rows.
The brace diagram (Part II, §4.2, p.71). Checked against the printed page. Under the line 596 × 248 = 147808 the book sets 5.96 × 24.8 = 147.808 with a small brace under each of the three numbers, labelled 2 decimal places, 1 decimal place and 3 decimal places, and a fourth brace joining the first two under the caption 2 + 1 = 3. This single figure is the argument of the topic; redraw it.
Example 5, the bare product (Part II, §4.2, p.72). Find 5.8 × 1.24. Checked against the printed page: the example carries no story at all — no purchase, no journey, no measurement, just the two numbers and two green callouts counting the digits after each point. The book supplies the whole-number product 58 × 124 = 7192 and asks the reader to place the point, then to check the placement by redoing the sum in fractions.
"Figure it Out" Q8 (Part II, §4.2, p.73). Given 18 × 12 = 216, six products are asked for: 18 × 1.2, 18 × 0.12, 1.8 × 1.2, 0.18 × 0.12, 0.018 × 0.012 and 1.8 × 12. Every one uses the digits 216. The follow-up question — which of them fall below 1 — hands over to Why multiplying by a decimal below 1 shrinks a number.
"Figure it Out" Q6, the oranges (Part II, §4.2, p.73). Oranges cost ₹56.50 a kilogram and 2.250 kg are bought. The book then asks whether rewriting these as 56.5 and 2.25 changes the product, and why. Inputs only — the resolution is section 10's work.
"Figure it Out" Q10, the times-ten table (Part II, §4.2, pp.73–74). Checked against the printed page. A five-row table with column heads × 10, × 100 and × 1000, and the left column filled with 5.7, 23.02, 0.92, 0.306 and 24.67. Every other cell is blank on the printed page.
Two other exercise inputs worth using (Part II, §4.2, p.73). A rupee coin is 1.45 mm thick and 36 of them are stacked, answer wanted in centimetres (Q5); notebooks are bought at ₹23.6 each and sold at ₹30 each, 50 in a week (Q7). Both make the point that the unit, not the arithmetic, is where marks are lost.

Figures to have open

The 5.7 cm by 13.3 cm rectangle, redrawn plain with the two lengths labelled. Standard schematic — do not lift the book's own margin figure.
A four-row comparison table carrying the multiplication, and the three counts. It must be re-drawable row by row; the reveal is the teaching.
The brace diagram for 5.96 × 24.8. This one is load-bearing and must keep the book's structure: braces under the two factors and under the product, with the 2 + 1 = 3 brace joining the factor counts.
A five-row × three-column grid for the × 10, × 100, × 1000 table, fillable live.
A number line fine enough to resolve 127.125 for the trailing-zero comparison.
No photograph and no data set from the textbook is needed for this topic.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 4 "Another Peek Beyond the Point", §4.2 "Decimal Multiplication", pp.69–72 — Examples 1 and 2 (p.69), Examples 3 and 4 and the first half of the digit-count table (p.70), the rest of the table, the stated rule and the brace diagram (p.71), and Example 5 (p.72)
Same part, same chapter, §4.2, "Figure it Out", pp.73–74 — questions 5, 6, 7, 8 and 10
Same part, same chapter, §4.1 "A Quick Recap of Decimals", pp.67–68, for the place-value reading of a decimal and the division-by-powers-of-ten rule the explanation leans on in section 11
Same part, same chapter, SUMMARY, p.95, bullet 2
Backward pointers: Extending Indian place value to the right of the point and A fraction of a fraction, and why the numerators and denominators multiply
Forward pointer: Why multiplying by a decimal below 1 shrinks a number, which takes up the question printed immediately after Example 5