Math Background

Lesson: Adjusting the Quotient
Developing the Concept

Your students have now reviewed finding compatible numbers for estimating answers to division problems with two-digit divisors. Now have them learn to adjust their estimates to find actual quotients.

Materials: model multiplication facts table from Introducing the Concept

Preparation: Post your model multiplication fact table where students can see it.

  • Say: You are going to learn how to find out whether your estimate is too large or too small. You'll find out how to adjust your estimate so that it is just right!

    Write 556 ÷ 29 on the chalkboard, as shown below.

  • Say: Suppose you use the estimate 600 ÷ 30 = 20 to find the solution.
    Write 600 ÷ 30 = 20, as shown below. Point to the problem.
  • Ask: Will 2 be “just right” as the first digit in the quotient of 556 ÷ 29?
    As a class, complete the first step in the division algorithm, as shown below.
    division problem
  • Say: Notice that 58 is greater than 55. You cannot subtract 58 from 55. This means that your estimate needs to be a lesser number. In other words, 2 is too great, so try using 1. Point to 55 − 58.

    Rewrite the problem and solve using 1 as the first digit in the divisor.

    division problem
  • Write 1,312 ÷ 26 on the chalkboard, as shown below.
  • Say: Suppose you use the estimate 1,200 ÷ 30 = 40 to find the solution.
    Write 1,200 ÷ 30 = 40 on the chalkboard. Point to the problem.
  • Ask: Will 4 be "just right" for the first digit in the quotient of 1,312 ÷ 26? As a class, complete the first step in the division algorithm as shown below.
    division problem
  • Say: Notice that 27 is greater than 26. When this situation happens, it means that your estimate needs to be greater. In other words, 4 is too small, so try using 5. Point to 131 − 104 = 27.

    Rewrite the problem and solve using 5 as the first digit in the divisor.

    division problem
  • Point out that, after bringing down the 2, 12 < 26.
  • Ask: Are there enough ones to divide?
    Students will realize that there are not enough ones.
  • Say: So, write 0 in the ones place. Then write the remainder.
    Emphasize how students will know when their estimates are too great or too small.
  • Say: 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.

    Remind your students that there is more than one way to begin solving a division problem. Tell your students to "just begin" and then to adjust as needed.

    Give your students these problems: 531 ÷ 16; 364 ÷ 22; 1,506 ÷ 24; 3,186 ÷ 45. Ask them to first estimate the quotients and then to find the actual quotients. Tell students to refer to their multiplication fact table when necessary.
    (531 ÷ 16 = 33 R3; 364 ÷ 22 = 16 R12; 1506 ÷ 24 = 62 R18; 3,186 ÷ 45 = 70 R36)

Wrap-Up and Assessment Hints
Create several division problems and the problem used to estimate the quotient (see the example, below). These problems should include a mix of estimates that are too large and too small. For each problem, ask students to first tell if the estimate is too large or too small. Then have them write a sentence telling how they know. Finally, have students find the actual quotient.

Problem:

division problem

Houghton Mifflin Math Grade 4