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Long multiplication and short division

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給老師： 為 Long multiplication and short division（Primary Maths、Grade 5）準備好可直接使用的課程投影片, 複習筆記——用於你的課程，或把這個主題當成互動班級活動，讓學生以即時遊戲的方式進行。

課程筆記

What Are Long Multiplication and Short Division?

  • Long multiplication is a step-by-step way to multiply big numbers. You multiply each digit of one number by each digit of the other, then add the results.
  • Short division is a quick way to divide a number by a one-digit number. You work from left to right, carrying remainders.
  • You will learn to multiply numbers with up to 4 digits by a 2-digit number, and to divide numbers with up to 4 digits by a 1-digit number.
  • Always estimate your answer first. This helps you check if your final answer is sensible.
  • After you finish, check your work. You can use the inverse operation (multiplication for division, division for multiplication) to verify.

Long Multiplication: Step by Step

  • Write the numbers one under the other, lining up the ones, tens, hundreds, etc.
  • Multiply the multiplicand (the top number) by each digit of the multiplier (the bottom number), starting with the ones digit.
  • When you multiply by the tens digit, put a zero in the ones place of that row. This is because you are really multiplying by tens.
  • Add all the rows together to get the product (the answer).
  • Example: 23 × 45. First, 23 × 5 = 115. Then, 23 × 40 = 920 (write 920, with the 0 in the ones place). Add: 115 + 920 = 1035.

Short Division: Step by Step

  • Short division is used when dividing by a one-digit number. Write the dividend (the number being divided) inside the division bracket, and the divisor outside.
  • Work from left to right. Divide each digit, write the answer above, and carry any remainder to the next digit.
  • Example: 847 ÷ 7. 7 goes into 8 once, remainder 1. Carry the 1 to make 14. 7 goes into 14 twice, remainder 0. 7 goes into 7 once. So 847 ÷ 7 = 121.
  • If there is a remainder at the end, you can write it as a remainder (e.g., 3 R 2), as a fraction (e.g., 3 2/5), or as a rounded decimal (e.g., 3.4).
  • When dividing by a two-digit number, you may use long division (a similar method with more steps) or break the problem into easier parts.

Estimating and Checking

  • Before you calculate, estimate by rounding the numbers to the nearest ten or hundred. For example, 23 × 45 is about 20 × 50 = 1000.
  • After you calculate, check if your answer is close to your estimate. If not, you may have made a mistake.
  • Use the inverse operation to check: for multiplication, divide your product by one of the factors; for division, multiply your quotient by the divisor.
  • Example: 23 × 45 = 1035. Check: 1035 ÷ 45 = 23. ✓

Order of Operations and Brackets

  • Sometimes a problem has brackets ( ) to show which part to do first.
  • Always do the part inside the brackets first.
  • If there are no brackets, do multiplication and division before addition and subtraction. This is called order of operations.
  • Example: (2 + 3) × 4 = 5 × 4 = 20. Without brackets, 2 + 3 × 4 = 2 + 12 = 14.

Solving Word Problems

  • Read the problem carefully and find the numbers and the question.
  • Decide whether to multiply or divide. Look for clue words: 'each', 'per', 'total' often mean multiply; 'share', 'split', 'how many in each' often mean divide.
  • Write a number sentence with the operation, then solve it.
  • Check that your answer makes sense in the story. For example, if you are sharing 20 sweets among 4 friends, each gets 5, not 80.

Practice Tips

  • Memorize your multiplication tables up to 12 × 12. This makes long multiplication much faster.
  • Practice with small numbers first, then move to bigger ones.
  • Always show your working out. This helps you find mistakes.
  • Use estimation to see if your answer is reasonable.

投影片

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練習題

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  1. 1.What is the name of the method that multiplies every digit in the first number by every digit in the second number and then adds the results?

    Easy
    • ALong multiplication
    • BShort division
    • CAddition
    • DSubtraction

  2. 2.In long multiplication, you need to memorize the multiplication table for single digits.

    Easy

    True or false?

  3. 3.In the long multiplication example, the multiplicand is 23,958,233 and the multiplier is 5,830. What is the first partial product (when multiplying by the ones digit of the multiplier)?

    Medium
    • A0
    • B71874699
    • C191665864
    • D119791165

  4. 4.Which of the following are true about long multiplication? (Select all that apply)

    Medium
    • AIt is also called grade-school multiplication.
    • BIt is the usual algorithm for multiplying larger numbers by hand in base 10.
    • CIt uses only addition, no multiplication.
    • DIt requires memorizing the multiplication table.

  5. 5.Match each term with its meaning.

    Medium
    • Multiplicand
    • Multiplier
    • Product
    • The number being multiplied
    • The number you multiply by
    • The result of multiplication

  6. 6.Put these steps of long multiplication in the correct order.

    Medium
    • Multiply the multiplicand by each digit of the multiplier.
    • Write down the partial products, shifted properly.
    • Add up all the partial products.
    • Start with the ones digit of the multiplier.

  7. 7.The product of any number and zero is zero.

    Easy

    True or false?

  8. 8.Match each part of a multiplication problem to its correct name.

    Easy
    • 23,958,233
    • 5,830
    • 139,676,498,390
    • Multiplier
    • Product
    • Multiplicand

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