Lesson 1: Size of Divisor and Size of Quotient

Let’s explore quotients of different sizes.

1.1: Number Talk: Size of Dividend and Divisor

Find the value of each expression mentally.

5,\!000 \div 5

5,\!000 \div 2,\!500

5,\!000\div 10,\!000

5,\!000\div 500,\!000

1.2: All Stacked Up

  1. Here are several types of objects. For each type of object, estimate how many are in a stack that is 5 feet high. Be prepared to explain your reasoning.

    1. Cardboard boxes
    2. Bricks
    1. Notebooks
    2. Coins

  2. A stack of books is 72 inches tall. Each book is 2 inches thick. Which expression tells us how many books are in the stack? Be prepared to explain your reasoning.

    a. 72 \boldcdot 2

    b. 72 - 2

    c. 2 \div 72

    d. 72 \div 2

  3. Another stack of books is 43 inches tall. Each book is \frac12-inch thick. Write an expression that represents the number of books in the stack.

1.3: All in Order

  1. Your teacher will give your group two sets of division expressions. Without computing, estimate their values and arrange each set of expressions in order, from largest to smallest. Be prepared to explain your reasoning. When finished, pause for a class discussion.

  2. Record the expressions in each set in order from largest to smallest.

    Set 1

    Set 2

  3. Without computing, estimate each quotient and arrange them in three groups: close to 0, close to 1, and much larger than 1. Be prepared to explain your reasoning.

    30 \div \frac12

    30 \div 0.45

    9 \div 10

    9 \div 10,\!000

    18 \div 19

    18 \div 0.18

    15,\!000 \div 1,\!500,\!000

    15,\!000 \div 14,\!500

     

    close to 0

    close to 1

    much larger than 1

Summary

Here is a division expression: 60 \div 4. In this division, we call 60 the dividend and 4 the divisor. The result of the division is the quotient. In this example, the quotient is 15, because 60 \div 4 = 15.

We don’t always have to make calculations to have a sense of what a quotient will be. We can reason about it by looking at the size of the dividend and the divisor. Let’s look at some examples.

  • In 100\div 11 and in 18 \div 2.9 the dividend is larger than the divisor. 100\div 11 is very close to 99\div 11, which is 9. The quotient 18 \div 2.9 is close to 18 \div 3 or 6.

    In general, when a larger number is divided by a smaller number, the quotient is greater than 1. 

  • In 99 \div 101 and in 7.5 \div 7.4 the dividend and divisor are very close to each other. 99 \div 101 is very close to 99 \div 100, which is \frac{99}{100} or 0.99. The quotient 7.5 \div 7.4 is close to 7.5 \div 7.5, which is 1.

    In general, when we divide two numbers that are nearly equal to each other, the quotient is close to 1.

  • In 10 \div 101 and in 50 \div 198 the dividend is smaller than the divisor. 10 \div 101 is very close to 10 \div 100, which is \frac{10}{100} or 0.1. The division 50 \div 198 is close to 50 \div 200, which is \frac 14 or 0.25.

    In general, when a smaller number is divided by a larger number, the quotient is less than 1.

Practice Problems ▶