Math Background

Adjusting the Quotient: When Students Ask

  • What are the different ways to divide?
    You can use many different strategies to find the answer to a division problem. One strategy is to use repeated subtraction. To find 123 ÷ 36, think: How many groups of 36 are there in 123? Start with 123. Subtract 36 repeatedly. Count how many times you subtracted: 123 − 36 = 87 (1); 87 − 36 = 51 (2); 51 − 36 = 15 (3); 15 < 36. There are 3 groups of 36 in 123 with 15 left over. Therefore, 123 ÷ 36 = 3 R15.

    Another strategy is to use what is sometimes called long division. The algorithm for long division is estimate, multiply, subtract, bring down, and repeat as needed. To find 123 ÷ 36, start with the estimate 120 ÷ 40 = 3. Then use this estimate to help you place the first digit in the quotient of the division algorithm.

    long division: 123 divided by 36 = 3 with a remainder of 15

    One other way to divide is to use what is sometimes called short division. This method is very useful when dividing by a one-digit number. The short division method requires you to do the estimation, multiplication, and subtraction steps mentally.

    short division: 1,378 divided by 3. first position 13 divided by 3 = 4 with a remainder of 1

    Think: 13 ÷ 3 = 4; 4 × 3 = 12; 13 – 12 = 1

    Place the 1 next to the 7.

    short division: 1,378 divided by 3. second position 17 divided by 3 = 5 with a remainder of 2

    Think: 17 ÷ 3 = 5; 5 × 3 = 15; 17 – 15 = 2

    Place the 1 next to the 7.

    short division: 1,378 divided by 3. third position 28 divided by 3 = 9 with a remainder of 2

    Think: 28 ÷ 3 = 9; 9 × 3 = 27; 28 – 27 = 1

    Place the 1 next to the 7.

  • How do I know where to place the first digit in the quotient?
    There are a couple of ways to decide where to place the first digit in the quotient. You can use an estimate. To find 389 ÷ 19, first estimate 400 ÷ 20 = 20. Knowing that 389 ÷ 19 is about 20, or 2 tens, helps you know where to place the first digit when finding the actual quotient.

    400 ÷ 20 = 20

    So, 389 ÷ 19 us about 20, or 2 tens.

    short division: 389 divided by 19.

    The first digit in the quotient should be placed above the 8 in 389.

    Another way to decide where to place the first digit is to make some comparisons.

    389 divided by 19 with the 19 and the 3 highlighted in red 19 > 3. There are not enough hundreds.
    389 divided by 19 with the 19 and the 3 and 8 highlighted in red 19 < 38. There are not enough tens.
  • Place the first digit of the quotient, 2, in the tens place.
  • How can I check my answer in division?
    If you multiply the divisor by the quotient and then add the remainder, the result should equal the dividend. You can do this with paper and pencil or with a calculator. For example: 231 ÷ 6 = 38R3; 6 x 38 = 228 and 228 + 3 = 231.

    You can also use a calculator to divide, then you can compare the two answers. However, you must remember that quotients are given only in decimal form on most calculators. Therefore, you must multiply the whole number portion of the quotient by the divisor, take that answer and subtract it from the dividend to get the correct remainder. For example: 231 ÷ 6 = 38.5; 38 x 6 = 228 and 231 − 228 = 3; therefore 231 ÷ 6 = 38.5, is the same as 231 ÷ 6 = 38 R3.

  • How can I decide whether the number I choose for the first digit in the quotient needs to be greater or lesser?
    When the number being subtracted is greater than the number being subtracted from, adjust the quotient to a lesser number. When the answer to the subtraction is greater than the divisor, adjust the quotient to a greater number.

Houghton Mifflin Math Grade 4