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How Many Each? How Many Left?: Conceptualizing Division with Large Numbers

  • Lesson
3-5
1
Number and Operations
Laurie A. Penney
Acton, MA

Division is one of the most difficult operations for students to master, in part, because there are many steps to keep track of and students often do not understand the mathematical reasoning behind each step. In this lesson, students will apply the strategies behind division, and learn how those strategies can be translated into mathematical steps.

Prior to the lesson, separate counters into bags of 100 to 200 counters for each group of three or four students. Counters can be dried beans, paper clips, or other types of counting manipulatives. You can decide, for differentiation, if you want the same number of counters per group, and if you want none, some, or all groups to have remainders.

For the main activity, divide the class into groups of three or four students each. Tell them that you are going to give them a large number of counters, and that they need to divide them up evenly among themselves.

The rules of the activity are as follows (these can be written on the board):

  • The group may not count the large group of counters.
  • Each person may only touch counters that will eventually belong to him or her.
  • When all of the counters have been taken by the group members, every member of the group must have the exact same number of counters.

As students tackle the task, circulate around the class to observe students' strategies. While some groups may start by taking one counter at a time, they should soon realize that this method is inefficient. Then they will begin to take larger groups of two, five, or even more counters at a time. When the groups finish the task, gather the class together to discuss the strategies they used to divide the counters. As groups explain their methods, they may begin to see that other groups used similar strategies.This is also a terrific writing opportunity for students.

pdficon How Many Each? Activity Sheet 

Distribute the How Many Each? Activity Sheet to each group. Tell students that after each person in the group had taken counters, it counted as one "round" of the activity. As students complete the activity sheet, begin to address the questions they will answer with colored pencils. Introduce the mathematical terms to go with the parts of an equation represented by the questions: dividend (counters in original pile), divisor (number of people in the group), quotient (how many counters each person got) with remainder (leftover counters). Model and label an expression using specific numbers, as shown below.

 HowManyEachHowManyLeft IMAGE DivisionTerms

To complete the activity, ask students how many rounds it took them to complete the task. The team with the least number of rounds should explain the strategy they used to ensure the task was as efficient as possible.

pdficon How Many Left? Activity Sheet 

Keeping students in their groups, distribute new bags of counters. Have them repeat the process, using the same rules as before. Distribute the How Many Left? Activity Sheet to each student. Tell groups to keep track of the number of rounds and how many counters each student took during each round. Have all members of each group fill out their own sheets based on the division they just completed.

Finally, distribute a third bag to each group. This time ask students to count the number of counters in the bag before they begin. This should lead students to better determine the number of counters they would take for each round. Have students discuss how many counters they would take if they only had one round. Have students discuss their findings, using specific numbers.

This lesson allows students to kinesthetically explore division, particularly division with remainders. Ascertain that students understand the connection between the activities with the counters and the standard algorithm for division.

Ask, "If there were 8 students in a group and 60 counters, how would I represent this using a division bracket?" To help guide students, write a division bracket such as:

 HowMany-BlankPic 

on the board. [The number 8 would go on the left, because it is the number of students that is equally dividing the total number of counters. The number 60 would go below the bracket because it is the total number of counters that is being divided.]

If this is the first time students are seeing the division bracket, revisit the activity sheet and rewrite each of the division expressions symbolically.

Ask, "What is the most number of counters 8 students would have if each student had the same number of counters?" Encourage students to draw the counters and trace them to one of 8 groups if they need help answering. [There would be 7 counters for each student.]

Ask, "How many counters are left over? Where would the entire solution go on the board? [There would be 4 counters left over. The number 7 would go above the brackets; "R4" would go just to the right of that answer.]

Finally, have students refer back to their completed How Many Each Activity Sheet. Ask students how they would use the number of counters taken in each round to figure out the quotient not including the remainder. Students should understand that the sum of the number of counters taken is the quotient not including the remainder; the remainder is the number of counters left over.

Assessment Options 

  1. Use numbers that will result in no remainder for lower-level students.
  2. Have students take one of the bags of counters they did not divide in class and divide them by three different single-digit divisors. Ask students what conclusion they can draw about the relationship between the value of the divisor and the value of the quotient. [Students should conclude that the greater the divisor the lesser the quotient and the lesser the divisor the greater the quotient.] Encourage students to verbally explain why they think this inverse relationship exists.
  3. Students create illustrations (posters, comic strips) to describe the division process visually.

Extensions 

  1. Ask students to develop their own division story problems. Challenge them to write three problems, one in which they need to determine the quotient, one to determine the divisor, and one to determine the dividend. Have students who would like a challenge develop a division story problem in which there is a remainder that must be interpreted (and not left as a remainder). As a hint, tell students to come up with a scenario in which there is a reason that the quotient needs to be a whole number (such as having to buy a certain number of cans of paint).
  2. HowMany-ExtensionPicShow students a completed division problem solved with the traditional division algorithm and ask them to try to determine the steps used to solve the problem. Compare the steps to the algorithm they used in this lesson. 
  3. As students, "If we have a remainder, when must we round up in real life situations?" Use a problem such as, if we 138 students are going on a trip and each bus can hold 55 students, how many school buses do we need? 

Questions for Students 

1. What does it mean to "divide evenly"?

[Each student gets the same number of counters with none remaining.]

2. Why did you choose to take more than one counter at a time?

[Larger quantities are more efficient.]

3. How did you decide how many counters to take without counting the whole group?

[Estimated the number in the pile.]

4. What operations did you need to complete the task?

[Division, multiplication, and subtraction are needed.]

5. How does knowing the number of counters change the way you approach the division problem?

[Knowing the dividend helps you base your choice more on computation than estimation.]

6. What mathematics is involved in making good choices for efficient division?

[Knowing the multiples of your divisor and the results of multiplying by tens can help you make good choices in division. For example, in the problem 375 ÷ 6, knowing that 6 × 6 = 36 can lead you to realize that each group will contain at least 60 items because 6 × 60 = 360.]

7. Why do multiplication and subtraction play a key role in division?

[You need to multiply to determine how many counters have been divided, and subtract to determine how many remain to divide.]

8. When does a quotient contain a remainder?

[When there are not enough counters left over to give at least one more to each person.]

Teacher Reflection 

  • Were students able to recognize what they already knew about the concept of division through the activity? If not, how should the lesson plan be altered?
  • How did presenting the concept as something they already knew affect the success of the task?
  • How students demonstrate improved understanding after each round of division? Did they continually choose more efficient strategies?
  • If students struggled with being able to translate the strategy they used to divide the counters into mathematical notation, how could you differentiate this lesson plan?

 

Learning Objectives

Students will:

  • Use manipulatives to discover the steps necessary to divide three-digit dividends by one-digit divisors. 
      • Develop an understanding of the mathematical steps necessary to solve a division problem.

      NCTM Standards and Expectations

      • Understand various meanings of multiplication and division.
      • Understand the effects of multiplying and dividing whole numbers.
      • Identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems.

      Common Core State Standards – Mathematics

      Grade 3, Algebraic Thinking

      • CCSS.Math.Content.3.OA.A.2
        Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

      Grade 3, Algebraic Thinking

      • CCSS.Math.Content.3.OA.C.7
        Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

      Grade 4, Num & Ops Base Ten

      • CCSS.Math.Content.4.NBT.B.6
        Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

      Common Core State Standards – Practice

      • CCSS.Math.Practice.MP1
        Make sense of problems and persevere in solving them.
      • CCSS.Math.Practice.MP6
        Attend to precision.