1.1: Number Talk: Size of Dividend and Divisor
Find the value of each expression mentally.
$5,\!000 \div 5$
$5,\!000 \div 2,\!500$
$5,\!000\div 10,\!000$
$5,\!000\div 500,\!000$
Let’s explore quotients of different sizes.
Find the value of each expression mentally.
$5,\!000 \div 5$
$5,\!000 \div 2,\!500$
$5,\!000\div 10,\!000$
$5,\!000\div 500,\!000$
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.
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. $72 \div 2$
Record the expressions in each set in order from largest to smallest.
Set 1
Set 2
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
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.