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Chapter 5 · Period 1 of 12 · 45 min

Can Every Number Be a Sum of Consecutive Numbers?

Is This a Multiple Of? · Inductive

Students often assume the parity observation breaks for larger numbers; encourage a second pair to test a set starting above 20 to widen the evidence base. Watch for the common error of listing fewer than eight sign combinations — a quick class count before evaluation saves time. The conjecture that all eight expressions are even (rather than merely sharing parity with one another) is a productive over-generalisation worth noting without resolving yet.

0–8Math Talk Q1, section 5.1 p.112 ("Explore whether every natural number, every even number, and 0....") — students try small cases (1, 2, 3, …, 10) to decide which can and cannot be expressed as sums of consecutive numbers, recording findings in pairs.
8–20Math Talk Q2, section 5.1 p.112 ("Take four consecutive numbers; place '+' and '−' signs....") — groups list all eight sign combinations for a chosen set of four consecutive numbers, then share and compare lists to confirm all eight are accounted for.
20–35Math Talk Q3, section 5.1 p.113 ("Evaluate each of the eight expressions formed from 3, 4, 5, 6....") — students evaluate all eight expressions for {3,4,5,6} and a second self-chosen set, recording each result; the class pools results and notices all outcomes are even.
35–45Math Talk Q4, section 5.1 p.113 ("Take any four consecutive numbers; place '+'/'-' signs in all eight ways....") — pairs check parity for a third set of four consecutive numbers, then a brief whole-class share surfaces the conjecture: all eight expressions share parity.
Assessment

Math Talk Q1, section 5.1 p.112 — Explore whether every natural number, every even number, and 0 (using negatives) can be written as a sum of consecutive numbers; discuss with class.

Which of the following correctly explains why all eight expressions formed by placing '+' or '−' between four consecutive numbers always share the same parity?

  • A Adding or subtracting consecutive numbers always produces a result divisible by 4.
  • B Consecutive numbers alternate between odd and even, so their total is always odd.
  • C Switching one sign from '+' to '−' changes the expression's value by twice the affected number, which is even; so parity cannot change.
  • D The four numbers cancel each other in pairs, making every expression equal to zero.

C. Switching one sign from '+' to '−' changes the expression's value by twice the affected number, which is even; so parity cannot change.

What each wrong option reveals

  • A The student conflates 'same parity' with 'divisible by 4', not recognising that parity is about divisibility by 2 only.
  • B The student incorrectly extends the alternating odd–even pattern of the numbers themselves to the result, ignoring the interaction of the signs.
  • D The student believes consecutive numbers cancel to zero, confusing a balanced sum with a zero sum.
Chapter 5 · 12 periods of 45 min

Also saved as 7 periods or 10 periods.

  • 2Why Parity Never Changes: Three Algebraic Proofs
    Is This a Multiple Of?
  • 3Breaking Even and Pairs to Make Fours
    Is This a Multiple Of?
  • 4Always, Sometimes, or Never: Five Divisibility Statements
    Is This a Multiple Of?
  • 5What Remains? Remainder Arithmetic and Figure it Out
    Is This a Multiple Of?
  • 6Completing Section 5.1 and Place-Value Roots of Divisibility
    Is This a Multiple Of?
  • 7Why the Digit Sum Rules Work: Proofs for 9 and 3
    Checking Divisibility Quickly
  • 8The Alternating Digit Rule: Proving Divisibility by 11
    Checking Divisibility Quickly
  • 9Digital Roots, Composite Shortcuts, and the Divisibility Table
    Checking Divisibility Quickly
  • 10Digital Root Puzzles and the First Cryptarithms
    Checking Divisibility Quickly
  • 11Multiplication Cryptarithms: Reasoning Digit by Digit
    Digits in Disguise
  • 12Number Play: Four Problems, Every Method
    Synthesis
Ganita Prakash · Class 8 · 14 chapters

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