Adding and subtracting decimals by aligning place value

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What to assume they know

A tenth part and A hundredth part, and continuing the split — adding and subtracting in tenths and hundredths, with the trades written out (Part I, §3.2–§3.3)
Extending Indian place value to the right of the point — the point, and the place value table (Part I, §3.4)
Column addition and subtraction of whole numbers, with carrying and borrowing
When an approximate answer is the better answer — using an estimate as a check on a computed answer (Part I, Chapter 1)
Continuing a sequence by identifying a constant step

What they should be able to do

Add and subtract decimal numbers set out in columns, aligning by place rather than by the ends of the digits
Handle a sum in which the two numbers have different numbers of digits after the point, including the case where one has none
Say what is being traded whenever a carry or a borrow is made, in the language of tenths, hundredths and thousandths
Produce the fully expanded place value working for a given sum, and match it line for line against the compact column form
Continue a decimal sequence, whether it rises or falls, and state the step
State a bound on the size of a sum before computing it, and test whether a proposed bound always holds

Where it usually goes wrong

"Line the numbers up at their right-hand ends." This is the error the practice set is built to expose. In 18 + 8.8 and in 17 − 16.198 the right-hand ends are in different places entirely, and a student who aligns them gets an answer wrong by a factor of ten or worse.
"You cannot add a whole number to a decimal number." Four items in the practice set do exactly that. A whole number simply has nothing in the places after the point.
"The point moves during the working." The points of both numbers and of the answer sit in one vertical line and never move. Draw that line and leave it in view.
"Carrying is a trick you learn." It is the trade the last two sections spent six pages on. The expanded working on p.75 shows ten tenths going up as one unit and ten hundredths going up as one tenth, in the same figure.
"9.9 − 9.09 is 0.9 because 9 minus 0 is 9." A worked counter-case belongs in the explanation; the item is in the printed set.
"A falling sequence is not a sequence." Three of the eight printed sequences fall, and one alternates its step size in the closing practice set at Part I, §3.8, p.79.
"Sonu's statement is a rule the chapter has taught." It is not. It is attributed to a character, followed immediately by a boxed question asking whether it holds generally, and left open. An explanation that reports it as a rule reverses the point of the passage, which is that a plausible generalisation has to be tested before it is believed.

Questions to check understanding

Add or subtract two decimal numbers, at least one pair having different numbers of digits after the point
Add or subtract where one of the two is a whole number
Write the expanded place value working for a given sum and its compact form
Continue a decimal sequence, rising or falling, and state the step
Estimate a sum before computing it, and say which whole numbers it must lie between
Given a worked solution containing a misalignment, find the error and say what it cost
Decide whether a stated generalisation about decimal sums always holds, and produce a case that settles it

Examples worth working on the board

The cloth (Part I, §3.7, p.74). One length of 2.7 m, another of 3.5 m; the question asks for the total, and then for how much the second exceeds the first. The page states the total as 6.2 m.
The paired layouts (Part I, §3.7, p.74). For the sum: on the left, 2 and 7 tenths above 3 and 5 tenths, totalling 5 and 12 tenths and then 6 and 2 tenths; on the right, the same sum with the point, with a small carried 1 set above the units column. For the difference: on the left, 3 and 5 tenths rewritten as 2 and 15 tenths before 2 and 7 tenths is taken away; on the right, 3.5 with a small 2 and a small 1 written above it before 2.7 is taken away. All four layouts are artwork and the carry and borrow digits extract as loose numerals.
The chapter's conclusion (Part I, §3.7, p.74). The method used for whole numbers serves for decimals as well.
The expanded sum (Part I, §3.7, p.75). Checked; this is the most information-dense figure in the section and extracts as scattered fragments. 75.345 and 86.691 are written out place by place — 7 tens with 8 tens, 5 units with 6 units, 3 tenths with 6 tenths, 4 hundredths with 9 hundredths, 5 thousandths with 1 thousandth. The row beneath gives each column total after the carry from the column to its right has been added: 16 tens, 12 units, 10 tenths, 13 hundredths and 6 thousandths, together with the hundred carried out of the 16 tens. Do not read these as the bare column sums — 7 tens and 8 tens are 15 tens, and the sixteenth ten is the one arriving from the units. Red arrows curve leftwards from each overflowing column to the carry it produces, and the leading 1 of 16, 12, 10 and 13 is ringed in red. The compact column form is set beside it, ending in 162.036.
The matching subtraction (Part I, §3.7, p.75). A Try This asks for the same treatment of 84.691 − 77.345, expanded and compact.
The practice set (Part I, §3.7, p.75). Sums: 5.3 + 2.6; 18 + 8.8; 2.15 + 5.26; 9.01 + 9.10; 29.19 + 9.91; 0.934 + 0.6; 0.75 + 0.03; 6.236 + 0.487. Differences: 5.6 − 2.3; 18 − 8.8; 10.4 − 4.5; 17 − 16.198; 17 − 0.05; 34.505 − 18.1; 9.9 − 9.09; 6.236 − 0.487. Verified against the printed page of p.75. Four of the sixteen have a whole number on one side (18 + 8.8, 18 − 8.8, 17 − 16.198, 17 − 0.05). Of the rest, the unequal pairs are three-place against one-place (0.934 + 0.6; 34.505 − 18.1) and one-place against two-place (9.9 − 9.09) — not one item pairs a two-place number with a three-place one, so an explanation that wants that case has to invent it.
The opening sequence (Part I, §3.7, p.75). 4.4, 4.8, 5.2, 5.6, 6.0, with the step named as 0.4 and three more terms asked for.
Eight more sequences (Part I, §3.7, p.76). Their opening terms: 4.4, 4.45, 4.5; 25.75, 26.25, 26.75; 10.56, 10.67, 10.78; 13.5, 16, 18.5; 8.5, 9.4, 10.3; 5, 4.95, 4.90; 12.45, 11.95, 11.45; 36.5, 33, 29.5. Three of the eight fall rather than rise, and two have steps larger than one. The page asks for these to be done mentally, and then for students to invent their own.
Sonu's bound (Part I, §3.7, p.76). The claim, as the page reports it: add two decimal numbers and the total always exceeds what you would get by adding only the parts before their points, while always falling short of that figure plus two. Worked against 25.936 and 8.202, the claim says the total lies above 33 and below 35. The page asks whether it holds for any two decimal numbers, and then asks the same for 25.93603259 and 8.202. It gives no answer.
The teacher's note (Part I, §3.7, p.76). Estimating first can reveal that a computed answer has gone wrong.
The boundary case, to supply. The upper limit always holds, because two fractional parts each below one cannot together reach two. The lower limit fails in exactly one situation: when both numbers have nothing after the point, so that the sum equals the sum of the whole parts rather than exceeding it. This analysis is added here, not the chapter's — see Notes.

Figures to have open

The paired layouts from p.74: the same computation in tenths and in decimal notation, shown in step. Redraw; the printed pair sits across the width of the page and the pairing is the argument.
The expanded place value working for 75.345 + 86.691, with the carry arrows. Redraw from p.75. This is the figure the topic is built on and it must be legible at full width.
A single vertical alignment line through two decimal points and the answer's. Standard schematic, reusable throughout.
One misaligned sum shown deliberately wrong, then corrected. Standard schematic; not in the textbook, and it should be labelled as an added counter-example.
A bound bar for the estimation claim. Standard schematic.
No photograph or textbook table is required.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 3 "A Peek Beyond the Point", §3.7 Addition and Subtraction of Decimals, pp.74–76. The cloth problem and the paired layouts are on p.74; the expanded sum, the Try This subtraction and the practice set on p.75; the two unnumbered subheadings Decimal Sequences, opening on p.75, and Estimating Sums and Differences, on p.76, close the section.
Backward pointers: Part I, §3.2, p.50, and §3.3, pp.56–58, for the same computations before the point existed; Part I, Chapter 1, §1.4, for estimation as a check.
Forward pointers: Part I, §3.8, pp.76–77, for what an unchecked answer can cost; Part II, printed Chapter 4, §4.4, where estimating before computing returns as a section of its own.