Locating a decimal on the number line, and comparing two decimals

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What to assume they know

Extending Indian place value to the right of the point — the decimal point and the place value table (Part I, §3.4)
A hundredth part, and continuing the split — hundredths, and lengths that mix sizes of part (Part I, §3.3)
Marking a fraction on a number line by cutting the unit into equal parts
Comparing whole numbers digit by digit from the largest place
Reading a decimal number aloud, digit by digit after the point

What they should be able to do

Locate a given decimal on a number line by successive splitting, and draw the chain of magnified lines that does it
Work out what one division of an unfamiliar number line is worth before reading anything off it
Decide whether two decimal numbers written with different numbers of digits are equal, and justify the decision from place value
Distinguish a zero written after the last significant digit from a zero written between the point and a digit
Compare two decimals by working leftmost place first, and stop at the first place where they differ
Explain why stopping there is safe
Given a target number and several candidates, decide which candidate is nearest and support it with a distance

Where it usually goes wrong

"Every division on a number line is a tenth." Three printed lines in this section say otherwise — half units between 5 and 10, hundredths between 8 and 8.1, five hundredths between 4.3 and 4.8. Reading the endpoints and counting the gaps is the first step of every one of these items, and the chapter never does it for the student.
"A decimal with more digits is bigger." Asked outright as an exercise at Part I, §3.8, p.79. A worked counter-case belongs in the explanation: 2.5 against 2.05.
"Adding zeros on the right makes the number bigger." This is Zara's worry on p.70, and it is answered by the place value table rather than by assertion. Show the table.
"So zeros never matter." The same table separates 0.2 from 0.02 and 0.002. A zero between the point and a digit pushes that digit into a smaller place; a zero after the last digit does not push anything.
"Compare by counting the digits after the point." 3.81 against 13.800 kills this in one line.
"You have to check every digit before deciding." The chapter's boxed prompt is exactly this question, and the answer is no. Whatever follows the deciding place is smaller than one unit of that place, so it cannot close a gap of at least one such unit.
"Closest means fewest digits different." Nearness is a distance. The page measures it — one hundredth against ten hundredths.
"04.50 is a different number from 4.5." The leading zero adds nothing; the trailing zero adds nothing. Both are in the seven-number list for that reason.

Questions to check understanding

Mark a given decimal on a supplied number line, cutting the line as needed
Read the value of a marked point off a line whose divisions are not tenths
Build a chain of magnified lines that locates a three-place decimal
Decide which members of a list of decimals are equal, and say why
Order a list of decimals, ascending or descending
Compare two decimals and name the place at which the decision was made
Choose the member of a set nearest to a given number, giving the distance
Arrange given digits to make a decimal as near as possible to a target

Examples worth working on the board

Locating 1.4 (Part I, §3.6, pp.69–70). The stretch from 1 to 2 is cut into ten and the fourth mark taken. Checked p.70: a line labelled 1, 1.1 up to 2, with 1.4 set in red and ringed.
The second cut (Part I, §3.6, p.70). A line labelled 1, 1.1, 1.2, 1.3, with the single gap between 1 and 1.1 subdivided into ten and 1.04 arrowed in red. The question printed with it asks for names for all the divisions in that gap.
Four arrows on a line from 5 (Part I, §3.6, p.70). A finely divided line whose labels are 5, 5.1, 5.3 and 5.4 — the 5.2 mark is deliberately left unlabelled — with four red arrows tagged A, B, C and D. The values are blank on the page; do not supply them.
The zero dilemma (Part I, §3.6, p.70). One character says 0.2 may equally be written 0.20 or 0.200; another suspects that adding zeros on the right changes the number. The table that follows has columns for units, tenths, hundredths and thousandths and rows for 0.2, 0.20, 0.200, 0.02 and 0.002. Three of the five are one number; the other two are not.
Seven numbers to sort into equal groups (Part I, §3.6, p.71). 4.5, 4.05, 0.405, 4.050, 4.50, 4.005 and 04.50. Note that one of them carries a leading zero as well as a trailing one.
The magnification cascade for 4.185 (Part I, §3.6, p.71). Checked, and the extraction shows almost none of it. Four stacked lines. The first runs 0 to 10; a rounded box on it marks the stretch that the second line, running 4 to 5, expands. A box on that marks what the third line, running 4.1 to 4.2, expands. A box on that marks what the fourth, running 4.18 to 4.19, expands, and 4.185 is labelled there. Faint grey guide lines join each box to the line that enlarges it. Figure (b) beside it is the same four-level structure with the labels removed and the final mark tagged with a question mark.
Two more cascades to build (Part I, §3.6, p.71). For 9.876 and for 0.407.
The line from 5 to 10 (Part I, §3.6, p.71). The stretch is cut into ten, so each division is worth half a unit, and the page says so and works out that the box marked b is 7.5. Boxes a and c are left to the student. This is the item that breaks the habit of reading every division as a tenth.
Two further lines (Part I, §3.6, p.72). Checked and the ticks counted. The first is labelled 8 and 8.1 at the first and eleventh of its eleven marks, so each division is one hundredth; two boxes, d and e, hang from it. The second is labelled 4.3 and 4.8 at the first and eleventh of its twelve marks, so each division is five hundredths, and the twelfth mark lies beyond 4.8; three boxes, f, g and h, hang from it. The page prints no division size for either line — working it out is the exercise.
6.456 against 6.465 (Part I, §3.6, p.72). A three-row figure, not drawn to scale and saying so. Row one fixes both numbers at 6. Row two narrows both to 6.4. Row three separates them, one falling in the stretch beyond 6.45 and the other beyond 6.46. Each row carries its own sentence on the right.
The stopping rule (Part I, §3.6, pp.72–73). Start from the largest place; while the digits match, move one place right; the first place where they differ decides. The page then asks, in a boxed prompt, whether the digits after that place could ever change the verdict.
Three comparisons (Part I, §3.6, p.73). 1.23 against 1.32; 3.81 against 13.800; 1.009 against 1.090. The middle one differs in the whole part and is there to catch a student counting digits after the point.
Closest to 1 (Part I, §3.6, p.73). From 0.9, 1.1, 1.01 and 1.11. The page sorts them, places 1 among them, and then compares two distances: one hundredth against ten hundredths. Three further targets follow — nearest to 1.09 from the same four; nearest to 4 from 3.56, 3.65 and 3.099; nearest to 1 from 0.8, 0.69 and 1.08.
The digit-arrangement puzzle (Part I, §3.6, p.73). Using 4, 1, 8, 2 and 5 once each, get as near to 25 as possible, in three fixed layouts: two boxes before the point and three after; one before and three after; three before and two after. The layouts are coloured blocks and extract as nothing.

Figures to have open

The four-level zoom cascade for 4.185, with the enlarged interval tinted on each line. Redraw from p.71; this figure carries the thesis and extracts as almost nothing.
A number line whose endpoints are given and whose divisions must be counted, built so the count can be done. Standard schematic; three versions are needed, matching the printed lines from 5 to 10, from 8 to 8.1, and from 4.3 to 4.8.
The five-row place value table for 0.2, 0.20, 0.200, 0.02 and 0.002. Redraw from p.70.
The three-row comparison of 6.456 and 6.465. Redraw from p.72, and keep the book's own warning that it is not to scale.
The digit-arrangement layouts as empty boxes with a point in a fixed position. Standard schematic.
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", §3.6 Locating and Comparing Decimals, pp.69–73. The section carries two unnumbered subheadings: There is Zero Dilemma!, opening on p.70, and Closest Decimals, on p.73. The 1.4 line and the 1-to-1.3 line are on p.70; the seven-number list, the 4.185 cascade and the 5-to-10 line on p.71; the two further lines and the comparison of 6.456 with 6.465 on p.72; the stopping rule, the three comparisons and the nearness items on p.73.
Backward pointers: Part I, §3.4, pp.58–64, for the places and the point; Part I, §3.3, p.54, for an earlier finely divided line.
Forward pointers: Part I, §3.8, p.79, for the ordering and more-digits items in the closing practice set; Part II, printed Chapter 4, for the arithmetic built on top of this comparison.