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.

    Cardboard boxes

    Photograph of 11 large cardboard boxes, 8 on bottom layer and 3 on top layer.


    Photograph of a stack of bricks.


    Photograph of a stack of 7 notebooks.


    Photograph of a stack of 15 pennies.
  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.

    • \(72 \boldcdot 2\)
    • \(72 - 2\)
    • \(2 \div 72\)
    • \(72 \div 2\)
  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

Your teacher will give you two sets of papers with division expressions.

  1. Without computing, estimate the quotients in each set and order them from greatest to least. Be prepared to explain your reasoning.

    Pause here for a class discussion.

    Record the expressions in each set in order from the greatest value to the least.

    1. Set 1
    2. Set 2
  2. Without computing, estimate the quotients and sort them into the following three groups. 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

Write 10 expressions of the form \(12 \div ?\) in a list ordered from least to greatest. Can you list expressions that have value near 1 without equaling 1? How close can you get to the value 1?


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.