Math Background

Adjusting the Quotient: Tips and Tricks

  • Have students begin each lesson in this chapter with a different 30-second, 10-problem, basic division fact drill. The goal of this drill is each day to increase the number of correct answers given in 30 seconds.
  • Show a list of dividends like this: 861; 651; 422; 335; 171; 2,376; 7,133. Then have students explain how they would change each number to make it divisible by 8. (800; 640; 400; 320; 160; 2,400; 7,200)
  • Divide students into groups of three or four. Have each student write a three-digit or four-digit dividend on one side of an index card and then to trade cards. Then have students write a one-digit or two-digit divisor on the other side. Have students trade cards again and then write an estimate for the quotient. Have students trade again to perform the division. Finally, have them trade one more time to check the quotient.
  • Have students brainstorm a list of division problems for which they would use the following in order to estimate quotients: 810 ÷ 90; 2,400 ÷ 60; 4,500 ÷ 90. (Problems will vary. 811 ÷ 93; 2,399 ÷ 58; 4,421 ÷ 87)
  • Have students write division algorithms on a large grid, one digit per square in the grid to help them keep their numbers properly aligned.
  • Have groups of students solve similar problems (320 ÷ 8; 3,200 ÷ 80; 32,000 ÷ 800). Then have group members take turns explaining to the class why the quotients are the same (320 ÷ 8 = 40; 3,200 ÷ 80 = 40; 32,000 ÷ 800 = 40). Ask students to adjust the problems by taking away one zero, or adding one zero, so that the quotients will each be different. (320 ÷ 80 = 4; 3,200 ÷ 80 = 40; 32,000 ÷ 80 = 400).
  • When students believe they have correctly found a quotient, have them check their answers by multiplying the quotient by the divisor and then adding the remainder to get the dividend.
  • When introducing the concept of adjusting the quotient, first use several problems all having the number being subtracted greater than the number being subtracted from. Then use several problems all having the answer to the subtraction greater than the divisor. Ask questions that lead students to see the pattern for adjusting the quotient either one lesser or one greater.
  • You may wish to challenge your more advanced students to check their work with a calculator. Students need to know that quotients are given in decimal form on a calculator. Show them how to multiply the whole-number portion of the quotient by the divisor and to 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 = 38R3.)
  • Permit students to keep a multiplication fact table in view for reference. Sometimes the very process of using the division algorithm overwhelms a student so much, that they have trouble recalling basic facts. After students show a degree of mastery, you can put away the fact table.

Houghton Mifflin Math Grade 4