 Lesson: Adjusting the Quotient Introducing the Concept

Your students will have divided by multiples of 10 and by one-digit divisors before. Draw on this background to help them divide by two-digit divisors. Students MUST know the basic multiplication and division facts. Knowing the division algorithm but using incorrect facts leads to incorrect answers and frustration for your students. Have your students use a multiplication fact table, as necessary. Knowing how to use compatible numbers to make an estimate of a quotient will help to relieve the frustration of knowing how or where to start.

Materials: a large model of the multiplication fact table, prepared 11-by-11 grids for students to make their own fact tables (see below). Preparation: Construct a large multiplication fact table and post it where students can see it. Prepare grids, like the one shown above, for students' fact tables.

Prerequisite Skills and Concepts: Students should be familiar with the basic addition, subtraction, multiplication, and division facts. They should be able to divide by multiples of 10 and by one-digit divisors.

Write the following problem on the board. • Say: Point to 38. Notice that the divisor is close to 40. Write 40 on the chalkboard and cover the 0 with your hand. Think about all the numbers 4 divides into evenly. Point to the row of multiples of 4 on your multiplication table.
• Ask: Which of these numbers could you multiply by 100 to make a number that is close to 1,399?
Someone will probably respond with either 12 x 100 or 16 x 100 to make 1,200 or 1,600. Encourage both responses if only one is mentioned. Under the problem, write both 1,200 ÷ 40 and 1,600 ÷ 40.
• Ask: Why can we use either 1,200 ÷ 40 or 1,600 ÷ 40 to find our estimate?
Direct the response to include the fact that an estimate is not exact and that there is always more than one way to approach a problem.
• Ask: Could I use 1,400 ÷ 40 to help estimate the quotient of 1,399 ÷ 38?
Whether the response is “yes” or “no,” explain that using 1400 would not make finding an estimate very easy.
• Ask: Why is 1,400 ÷ 40 not easy to compute?
Point out that 1,200 and 1,600 are compatible with 40 for division. Forty divides evenly and easily into 1,200 and 1,600, but this is not so for 1,400, since 14 is not in the row of multiples of 4. Tell students that it is all right for them to choose numbers different from those chosen by the student sitting next to them. What is important is that they choose numbers that work well together.
• Ask: What is 1,200 ÷ 40? 1,600 ÷ 40?
(1,200 ÷ 40 = 30; 1,600 ÷ 40 = 40) Review division by multiples of 10 if necessary.

On the chalkboard, write and estimate with several more division problems. As a class, brainstorm division problems that could be used to estimate the quotients. Then work as a class to find the estimates.

• Say: Now that you have a good idea of how to estimate the quotient of a division problem using basic multiplication and division facts, each of you will make a basic multiplication fact table using the form I give you.
Tell them to write neatly and take care to write the correct answer to each multiplication fact on the table.