The alternating-sum test for 11

What to assume they know

• The digit-sum test for nine and the run-of-nines split of place values (The digit-sum test for 9, and the algebra underneath it)
• Place value, and writing a number as a sum of its place values
• Adding and subtracting integers, and reading a negative total as a shortfall
• Writing "leaves remainder r" in the form of a multiple plus an offset (Writing "leaves remainder r" as an algebraic expression)
• Multiples, factors and remainders

What they should be able to do

• Express each place value as a multiple of eleven with an offset of one, and identify the sign of that offset
• Explain why the sign alternates from one place value to the next, and why the alternation continues indefinitely
• Sort a number's digits into the places whose offset is positive and the places whose offset is negative
• Compute the excess total, the shortfall total, and the difference between them
• Interpret a difference of zero, of eleven, of a multiple of eleven, and of a negative value
• Convert a negative alternating total into the actual remainder on division by eleven
• Carry out the same test in its compact form, signing the digits from the units end, and explain why the two procedures are the same procedure
• Use the test to find missing digits in a number known to be a multiple of eleven or of a composite containing eleven

Where it usually goes wrong

• "A negative answer means I made a mistake." It means the number falls short of the next multiple of eleven rather than running past the previous one. The chapter's own worked example lands on a negative value, and its careful sentence reports the position two ways: so far short, or so far past.
• "The alternating total is the remainder." The compact procedure on Part I p.129 says the result gives the remainder, and taken literally that cannot be right — a remainder is never negative. Add eleven to a negative total to get the remainder. The prose on Part I p.128 phrases it correctly.
• "Start the signs at the left-hand digit." The signing is anchored at the units digit because that is the place value that runs one past. Anchoring at the left flips every sign whenever the digit count is even, and the test then reports the wrong direction. Show the failure once.
• "A difference of zero and a difference of eleven mean different things." Both mean the number is a multiple of eleven. Any multiple of eleven in the difference does. This is the question Part I p.128 asks and leaves open.
• "Eleven cannot have a digit test, because it has two digits." The method never cared how many digits the divisor has. It cares only how each place value sits relative to a multiple of it.
• "The two printed procedures are two different tests." They are the same test, one written as two columns and one written as signed digits. The chapter asks the student to notice this.
• "Zeros can be skipped." A zero digit contributes nothing to its total but it still occupies a place, so it still shifts the signs of everything to its left. Two of the six exercise numbers are built around this.

Questions to check understanding

• Decide whether a given number is a multiple of eleven without dividing
• Give the remainder on division by eleven when the test says no, including from a negative alternating total
• Explain why the sign alternates, using the way each place value sits against a multiple of eleven
• Show that the two-column procedure and the signed-digit procedure agree, on a number of your own choosing
• Find missing digits making a number a multiple of eleven, or of a composite such as 44
• Say what a difference of eleven, or of twenty-two, tells you
• Examine a classmate's conjecture about multiples of eleven under doubling

Examples worth working on the board

Inputs. Values marked "printed" are the chapter's own working.

• The place-value table (Part I, §5.2, p.127, four rows plus a row of ellipses). Printed, row by row: the units place, with 11 × 0 = 0 and 1 written as that product plus one, described as one past a multiple of eleven; the tens place, with 11 × 1 = 11 and 10 written as that product minus one, described as one short; the hundreds place, with 11 × 9 = 99 and 100 written as that product plus one, one past; the thousands place, with 11 × 91 = 1001 and 1000 written as that product minus one, one short. The words for "one past" are set in green and the words for "one short" in red — worth preserving as a colour convention through the whole video. The right-hand column carries pictures: a single loose circle; a chain of eleven circles; a block eleven wide and nine tall with one extra circle beside it; and a block eleven wide and ninety-one tall with one circle missing from it. All of that lettering sits inside artwork.
• The three-digit demonstration, printed (Part I, §5.2, p.127). 400 holds four hundreds, so it runs four past a multiple of eleven, written as 396 plus 4. 60 holds six tens, so it falls six short, written as 66 minus 6. 2 holds two units, so it runs two past, written as zero plus 2. The page then asks, under a "Math Talk" flag, whether 462 is a multiple of eleven, and separately what the general method should be. Both are left open.
• The three-step procedure (Part I, §5.2, p.128, a table with columns for the steps, their purpose, and a worked example). Step one: total the digits sitting in places that run past — the units, hundreds, ten-thousands and so on. Step two: total the digits sitting in places that fall short — the tens, thousands, lakhs and so on. Step three: subtract the second total from the first, and read the result as the remainder position.
• The worked number. The table's heading names the number as 320185. Printed totals: the excess total is 2 + 1 + 5, giving 8; the shortfall total is 3 + 0 + 8, giving 11; the difference is 8 − 11, giving −3, annotated as three short of a multiple of eleven. The paragraph immediately beneath writes the same number as 3,28,105, and the following page uses 328105. See the notes below: this is a real inconsistency in the printed book and it is harmless, because the two spellings put the same digits in the same alternating classes.
• The artwork in step two (Part I, §5.2, p.128). Three place values are drawn: a group of three tall blocks each labelled 9091, a place with nothing drawn, and eight short strips each labelled 11 with one circle missing. The captions above them read as ten-thousands, hundreds and tens. Two of those captions understate the place value — see the notes. The block label 9091 is the giveaway, because one lakh is eleven times 9091 less one.
• The compact procedure (Part I, §5.2, p.129, a three-step table). Printed: place plus and minus signs alternately in front of the digits, starting at the units digit; evaluate; read the result as the remainder position. The worked example signs the digits of 328105 and evaluates the expression to −3, and the page reports the number as three below, or eight above, a multiple of eleven. The page then asks whether this is the same method as the previous one. That question is section 9's whole job.
• The open question about multiples (Part I, §5.2, p.128). If the difference comes out as eleven, or as a multiple of eleven, what does that say about the remainder? Printed as a question and left unanswered. Note that the worked example's own shortfall total is 11, so the number is available.
• Exercise inputs, Part I p.128. Six numbers to test for divisibility by eleven, with the remainder wanted where the test fails: 158, 841, 481, 5529, 90904 and 857076. Note the spread — three digits, four digits, then five and six with zeros inside them, which is where careless signing goes wrong.
• Chapter-end exercise inputs (Part I p.133). No. 6 gives a five-digit number written 3, then a letter p, then 7, then a letter q, then 8, said to be divisible by 44, and asks for every possible pair — 44 being 4 times 11, so this needs two tests. No. 11 is Deepak's claim that some multiples of eleven stay multiples of eleven when doubled while others do not.
• The chapter prints no answers to any exercise item.

Figures to have open

• The place-value ladder for eleven: each power of ten drawn as a block eleven wide, with the leftover circle or the missing circle marked, and the sign attached. This is the chapter's own figure (Part I, §5.2, p.127) and it is what makes the alternation feel inevitable rather than arbitrary. Redraw as a schematic and keep the green-for-past, red-for-short colour coding.
• A two-column sorting board with a number's digits falling into the excess and shortfall columns by place. Standard schematic; the chapter's step table (Part I, §5.2, p.128) implies it without drawing it cleanly.
• A number line around a run of multiples of eleven, with the worked number's position marked and both readings — so far short, so far past — labelled at once. Standard schematic. This is the figure that stops the negative-value confusion.
• A side-by-side alignment of the two printed procedures with matching parts linked. Standard schematic.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 5, "Number Play", §5.2 "Checking Divisibility Quickly", printed subheading "A Shortcut for Divisibility by 11", Part I pp.126–129. The subheading opens at the foot of Part I p.126 with two lines of introduction; the table of place values is on Part I p.127, the three-step procedure on p.128, the compact procedure on p.129.
• Exercise items: Part I p.128, six numbers to test; chapter-end "Figure it Out", Part I p.133, items 6 and 11.
• The fill-in divisibility table on Part I p.129 carries an eleven-column and sits immediately after this material; the table itself is handed to Divisibility by 6 and other numbers, checked through their factors, which owns the factor-based approach it rewards.
• The chapter's SUMMARY, Part I p.134, names 11 alongside 3 and 9.