When a sum of consecutive numbers is a multiple of something

What to assume they know

• Adding and subtracting whole numbers and integers, negatives included
• Standing a letter in for an unknown whole number, and collecting like terms — the chapter calls such a letter a letter-number
• Even and odd numbers, and reading "even" as "has 2 as a factor"
• Multiples and factors: reading "multiple of 4" as "4 times some whole number" (What makes a number a perfect square sets up factor pairs)
• Division leaving a remainder

What they should be able to do

• Write a chosen number as a sum of two or more consecutive numbers, and say when this can be done in more than one way
• Show that a sum of two consecutive numbers is always odd, and use that to explain why no even number is such a sum
• Express a run of k consecutive numbers starting at n using letter-numbers, and simplify its total
• Deduce from that expression that an odd-length run totals its length times its middle term
• Deduce that an even-length run totals half its length times an odd number, and hence that a four-term run is even but never a multiple of 4
• Recover the terms of a run from its total, and describe the neighbours of a named term in terms of that term
• Distinguish a claim tested on examples from a claim settled by an argument about the general form

Where it usually goes wrong

• "Consecutive means any increasing list." Consecutive means step exactly one. The moment the step is 2 — as in the five consecutive even numbers of chapter-end item 9 — every formula in this topic changes, and students who learned the four-term result as "add 6" will misapply it.
• "You test it on a few runs and then you know." Trying three runs of four shows the pattern; it does not establish it. The chapter is explicit that the number of runs is unlimited, so the general form is the only thing that can settle the question. Make the switch from checking to arguing visible.
• "Odd-length runs are multiples of their length, so even-length runs are too." They are not, and the reason is structural rather than accidental: an even-length run has no middle term for the total to be a multiple of. Show the missing middle.
• "An even number is a sum of consecutive numbers, so it is a sum of two." A sum of two neighbours is always odd. Even numbers need three or more terms — or none, in some cases.
• "The sum of four consecutive numbers is a multiple of 4 because there are four of them." The count of terms is not automatically a factor of the total. This is the specific error the next topic and "Pairs to Make Fours" are built to correct.
• "Negative numbers cannot be part of a run." Anshu's fourth question opens the door deliberately. Whether 0 is a sum of consecutive numbers has a different answer depending on whether negatives are admitted, and noticing that the answer depends on the rules is the mathematical move.

Questions to check understanding

• Given a total, find the run of a stated length that produces it
• Given one term of a run — greatest, least or middle — write the others in terms of that term, for runs of consecutive numbers and of consecutive even numbers
• Decide whether a named number is a sum of two consecutive numbers, and justify it by parity rather than by search
• Show that the total of any three consecutive numbers is a multiple of 3, using letter-numbers
• Show that the total of any four consecutive numbers is even, and give a reason it cannot be a multiple of 4
• Find every way of writing a given number as a sum of two or more consecutive natural numbers, and say how you know the list is complete
• Products, not sums: decide whether a product of consecutive integers must be a multiple of 2, of 6, and so on, and say why

Examples worth working on the board

Inputs. Where a value is the chapter's own printed working I say so; the rest are the chapter's questions, left open on the page.

• Anshu's six equations (Part I, §5.1, p.112, in a pale yellow tinted box). Printed exactly: 7 as 3 + 4; 10 as 1 + 2 + 3 + 4; 12 as 3 + 4 + 5; and 15 three ways — as 7 + 8, as 4 + 5 + 6, and as 1 + 2 + 3 + 4 + 5. Note the shape of the list: 15 is the one number given more than one decomposition, which is the whole point of Anshu's second question.
• Anshu's four questions (same page, four bullets). Can every natural number be written this way; which numbers admit more than one way; whether every even number can be written this way given that every odd one can be written as two neighbours; and whether 0 can be written this way if negative numbers are allowed. The chapter answers none of them — the page sends them to class discussion under a "Math Talk" flag.
• The run of four the chapter picks up next: 3, 4, 5 and 6 (Part I, §5.1, Part I p.112). Their plain total is the number.
• The general run, for the algebra section.: k × n plus 1 + 2 + ⋯ + (k − 1), and then k × (2n + k − 1) ÷ 2. Everything in sections 8–10 is read off the second form. This general form is added here; the chapter asks for algebra here (Part I, §5.1, p.113, hint line) without setting it out.
• Three lengths to instantiate it on: k = 3, k = 4, k = 5. For odd k the second factor is even, so the total is k times a whole number. For k = 4 the second factor 2n + 3 is odd, so the total is 2 × odd.
• Exercise inputs, §5.1 "Figure it Out" (Part I p.122). No. 1 gives a four-term run with total 34 and asks for the terms. No. 2 names the greatest of five consecutive numbers p and asks for the other four in terms of p.
• Exercise inputs, chapter-end "Figure it Out" (Part I pp.133). No. 7 wants three consecutive numbers whose first is a multiple of 2, second a multiple of 3, third a multiple of 4, then asks how often such triples occur. No. 9 names the middle of five consecutive even numbers as 5p and asks for the other four in terms of p — note the step between consecutive even numbers is 2, not 1, which is the trap. No. 14 turns to products of consecutive integers: two in a row, then (as printed) "these" consecutive integers against a multiple of 6, then four in a row, then five.
• The chapter prints no answers to any of the above.

Figures to have open

• A number-line strip carrying a movable window of k consecutive counters, reusable at k = 2, 3, 4, 5. Standard schematic; it is the spine of sections 2–6 and nothing in the chapter substitutes for it.
• A levelling diagram: a stepped bar chart of a run, redrawn as equal bars of the middle height, with the outer bars' surplus and deficit shown cancelling. Standard schematic. This is the picture the chapter's algebra hint implies but does not draw.
• Anshu's six-equation panel, redrawn rather than reproduced from Part I, §5.1, p.112.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 5, "Number Play", §5.1 "Is This a Multiple Of?", opening subheading "Sum of Consecutive Numbers", Part I pp.112–113.
• §5.1 "Figure it Out", Part I p.122, items 1 and 2.
• Chapter-end "Figure it Out", Part I p.133, items 7, 9 and 14.
• The chapter's SUMMARY (Part I p.134) lists four divisibility properties and the three shortcut divisors; it does not mention consecutive numbers, so this topic has no line in the summary to point back to.