Math Background

Lesson: Division
Developing the Concept

After using manipulatives to introduce the division algorithm for multidigit numbers, the concept should be fully developed. It is important to give students plenty of time to master the division of multidigit numbers. The development of this concept should not be rushed.

Materials: overhead base-ten blocks, overhead projector, base-ten blocks for students

Preparation: Be sure to provide at least one set of base-ten blocks for each pair of students.

  • Ask: How can we write 276 divided by 6?
    276 ÷ 6 or two-hundred seventy-six divided by six will probably be the two notations cited by the students.
  • Ask: Which notation will you use to find the quotient of 276 divided by 6?
    Preference should be two-hundred seventy-six divided by six to enable students to use the division algorithm.
  • Say: Use your base-ten blocks to represent 276.
    base-ten blocks
  • Ask: Let's begin with the hundreds. Since we are dividing by 6, we need to make 6 groups. Can this be done if we have only 2 hundreds?
    No. When you cannot make groups, you need to regroup.
  • Ask: If we regroup the 2 hundreds as tens, how many tens would we get? (20) If we include the 7 tens, how many tens would that be altogether?
    If we combine the 20 tens with 7 tens, we get 27 tens.
  • Ask: Since we are working with tens now, how many groups of tens can we put in 6 equal groups? Be sure to use your base ten blocks.
    base-ten blocks

    Notice that there are 6 groups of 4 tens with 3 tens left over.

  • Say: Since we have 6 groups of 4 tens, we place a 4 over the tens place in 276.
    division problem
  • Ask: If one group of 4 tens is 40, what is 4 groups of 6 tens worth?
    240. Encourage students to use their base-ten blocks if necessary to count the value.
  • Say: Remember that we began with 276 and want to divide it by 6. Since we have made six groups of 4 tens, we can take 240 away from 276.
  • Ask: How many tens and ones are left over when we take away the 6 groups of 4 tens?
    3 tens and 6 ones are left over.
  • Say: We can do this by writing 240 below 276 in our division problem and subtracting.
    division problem
  • Ask: What is 276 − 240? (36) What is the value of the base-ten blocks that you have left over? (36) What do you notice about the two values?
    They are the same. This allows students to see the connection and validation between using the base-ten blocks and the algorithm they are learning to use.
  • Ask: Since we cannot make any more groups of 6 tens with the remaining base-ten blocks, we can regroup the 3 tens as how many ones? (30 ones)
    Be sure to show students the regrouping of 3 tens as 30 ones.
  • Ask: How many ones do we have now?
    We have 36 ones.
  • Ask: How many ones can we put in each of the six equal groups?
    We can put 6 ones in each group.
  • Ask: Where do you think we will write the 6 that represents the 6 ones?
    The 6 is written above the ones place in 276.
  • Ask: Are there any ones left over? (no)
  • Ask: What is the quotient of 276 ÷ 6? (46)
  • Continue this activity, using different numbers. Be sure to use problems that will not have remainders at first. Remember to have the students check their answers, using multiplication.

Wrap-Up and Assessment Hints
Students need a great deal of practice when learning to divide multidigit numbers. Do not be in a rush for students to put away their manipulatives when learning this difficult concept. This can be a trying time in many students' mathematical development. As a teacher, do not be discouraged with slow progress. Remember, this is the first time most of the students have encountered the concept. Your task is to take the time and effort needed to encourage students to learn this process.

It is important to continually assess students' progress when they're learning division. Extra practice and assessment later in the year also ensures that students have mastered this sometimes difficult algorithm. When assessing students, try to remember what it was like when you learned how to divide large numbers.

Houghton Mifflin Math Grade 4