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Flashcards in this deck (38)

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  • multiplicação

    multiplicação

    noun math core

  • vezes

    vezes

    preposition math core

  • produto

    produto

    noun math core

  • fator

    fator

    noun math

  • tabela

    tabela

    noun reference core

  • número

    número

    noun math common

  • par

    par

    adjective math pattern

  • ímpar

    ímpar

    adjective math pattern

  • identidade

    identidade

    noun math concept

  • quadrado

    quadrado

    noun math square_numbers

  • comutatividade

    comutatividade

    noun math property

  • simetria

    simetria

    noun math pattern

  • padrão

    padrão

    noun math core

  • dobrar

    dobrar

    verb strategy mental_shortcut

  • metade

    metade

    noun math strategy

  • atalho mental

    atalho mental

    noun study core

  • memorizar

    memorizar

    verb study core

  • prática

    prática

    noun study core

  • dobro

    dobro

    noun math strategy

  • dez menos

    dez menos

    phrase mental_shortcut core

  • tabuada

    tabuada

    noun math core

  • linha

    linha

    noun layout reference

  • coluna

    coluna

    noun layout reference

  • resultado

    resultado

    noun math common

  • multiplicar

    multiplicar

    verb action

  • múltiplo

    múltiplo

    noun math

  • atalho

    atalho

    noun study_notes:mental_shortcuts_(doubling,_10-minus)

  • memorização

    memorização

    noun study_notes:memorization_&_practice_strategies

  • estratégia

    estratégia

    noun study_notes:memorization_&_practice_strategies

  • número quadrado

    número quadrado

    noun math squares

  • 10 menos

    10 menos

    phrase strategy

  • praticar

    praticar

    verb study

  • multiplicando

    multiplicando

    substantivo matemática papel

  • multiplicador

    multiplicador

    substantivo matemática papel

  • identidade (elemento identidade)

    identidade (elemento identidade)

    substantivo matemática propriedade

  • quadrado (número quadrado)

    quadrado (número quadrado)

    substantivo matemática padrão

  • par (número par)

    par (número par)

    adjetivo matemática número

  • tabela de multiplicação

    tabela de multiplicação

    substantivo recurso memorização
学习笔记

Overview

  • The input is the multiplication table for 1 through 20: each line shows \(a \times b = c\) for \(1\le a,b\le20\).
  • Multiplication gives repeated addition and has predictable patterns useful for fast calculation and memorization.

Fundamental properties to remember

  • Commutative property: \(a \times b = b \times a\), so the table is symmetric across the diagonal.
  • Identity element: \(a \times 1 = a\) for any integer \(a\).
  • Squares: numbers on the diagonal satisfy \(n \times n = n^2\) (e.g., \(12 \times 12 = 144\)).
  • Zero property (not in this table): \(a \times 0 = 0\).

Useful patterns in the 1–20 table

  • Evens and multiples of 2: multiply by 2 is the same as doubling: \(a\times2 = 2a\).
  • Multiples of 5 and 10: end in 0 or 5: \(a\times10\) ends with 0, \(a\times5\) ends with 0 or 5.
  • Multiples of 3: digit-sum rule helps check (sum of digits divisible by 3).
  • Multiples of 9 (single-digit times): for \(1\le n\le9\), digits of \(9\times n\) sum to 9 (e.g., \(9\times7=63\), \(6+3=9\)).
  • Doubling/halving pairs: \(a\times b = (2a)\times(b/2)\) when \(b\) is even — useful to simplify mental steps.

Mental-math shortcuts and tricks

  • Use commutativity: pick the factor pair that is easiest (e.g., prefer \(4\times25\) as \(25\times4\)).
  • Multiply by 10 and adjust: \(a\times9 = a\times(10-1) = 10a - a\).
  • Multiply by 5: \(a\times5 = (a\times10)/2\).
  • Double-and-double: to multiply by 4, double twice: \(a\times4 = 2(2a)\).
  • Split numbers: \(a\times(b+c) = a\times b + a\times c\) to break hard problems into known facts.
  • 11 trick (two-digit): for \(11\times (10a+b)\) when \(a+b<10\), result is \(a\,(a+b)\,b\) (e.g., \(11\times23=253\)). If \(a+b\ge10\), carry the tens.

Memorization strategies

  • Learn key anchors first: memorize 1, 2, 5, 10, squares (1,4,9,16,25,...), and 11–12 rows.
  • Use symmetry: once you know row 3, column 3 gives same values for 3×1 and 1×3 etc.
  • Chunking: learn in small groups (e.g., 2–4, 5, 6–8, 9, 11–12, then 13–20).
  • Skip counting: practice counting by 2s, 3s, 4s, 5s, etc., to internalize multiples.
  • Active recall: timed drills, flashcards, and teaching someone else speed up retention.

Common patterns to practice (examples)

  • Squares to remember: \(7\times7=49\), \(8\times8=64\), \(9\times9=81\), \(10\times10=100\), \(12\times12=144\).
  • Handy conversions:
  • \(a\times10 = 10a\) (just append a 0)
  • \(a\times5 = \frac{10a}{2}\)
  • \(a\times4 = 2(2a)\)
  • \(a\times9 = 10a - a\)

Short practice set (do mentally)

  1. Compute \(6\times7\).
  2. Compute \(9\times8\) using the 10-minus trick.
  3. Compute \(15\times4\) using doubling.
  4. Compute \(11\times14\) (use 11 trick with carry).

(Answers: 1. \(42\); 2. \(72\); 3. \(60\); 4. \(154\).)

Quick error checks and tips

  • Check with reverse multiplication by commutativity.
  • Use digit-sum or divisibility rules for plausibility (3, 9 tests).
  • For two-digit results, verify approximate size: \(a\times b \approx\) (round) helps catch large mistakes.

How to use the full 1–20 table effectively

  • Use it as a reference while practicing mental shortcuts until you can recall the common rows.
  • Focus memorization on small building blocks (1–12) first; larger rows often follow by adding known values (e.g., \(13\times n = 10n + 3n\)).
  • Practice real-world problems (money, measurements) to reinforce retention.

Summary tips for fast mastery

  • Master 1, 2, 5, 10 rows and squares first.
  • Use commutativity to reduce memorization load.
  • Apply doubling/halving and splitting strategies for hard products.
  • Drill regularly with short, timed sessions.