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Study Guides > Prealgebra

Notation and Modeling Division of Whole Numbers

Learning Outcomes

  • Use words and symbols to represent division
  • Model division of whole numbers using base-10 blocks
 

Use Division Notation

So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the 1212 cookies and want to package them in bags with 44 cookies in each bag. How many bags would we need? An image of three rows of four cookies to show twelve cookies. You might put 44 cookies in first bag, 44 in the second bag, and so on until you run out of cookies. Doing it this way, you would fill 33 bags. An image of 3 bags of cookies, each bag containing 4 cookies. In other words, starting with the 1212 cookies, you would take away, or subtract, 44 cookies at a time. Division is a way to represent repeated subtraction just as multiplication represents repeated addition. Instead of subtracting 44 repeatedly, we can write

12÷412\div 4

We read this as twelve divided by four and the result is the quotient of 1212 and 44. The quotient is 33 because we can subtract 44 from 1212 exactly 33 times. We call the number being divided the dividend and the number dividing it the divisor. In this case, the dividend is 1212 and the divisor is 44. In the past you may have used the notation 4)124\overline{)12} , but this division also can be written as 12÷4,12/4,12412\div 4,12\text{/}4,\frac{12}{4}. In each case the 1212 is the dividend and the 44 is the divisor.  

Operation Symbols for Division

To represent and describe division, we can use symbols and words.
Operation Notation Expression Read as Result
Division\text{Division} ÷\div ab\frac{a}{b} b)ab\overline{)a} a/ba/b 12÷412\div 4 124\frac{12}{4} 4)124\overline{)12} 12/412/4 Twelve divided by four\text{Twelve divided by four} the quotient of 12 and 4\text{the quotient of 12 and 4}
  Division is performed on two numbers at a time. When translating from math notation to English words, or English words to math notation, look for the words of and and to identify the numbers.

example

Translate from math notation to words. 1. 64÷864\div 8    2. 427\frac{42}{7}    3. 4)284\overline{)28} Solution
  • We read this as sixty-four divided by eight and the result is the quotient of sixty-four and eight.
  • We read this as forty-two divided by seven and the result is the quotient of forty-two and seven.
  • We read this as twenty-eight divided by four and the result is the quotient of twenty-eight and four.
  In the video below we present more examples of how to express division with words, and how to translate those words into math notation. https://youtu.be/WxJxY4aJ9Vk  

Model Division of Whole Numbers

As we did with multiplication, we will model division using counters. The operation of division helps us organize items into equal groups as we start with the number of items in the dividend and subtract the number in the divisor repeatedly. Doing the Manipulative Mathematics activity Model Division of Whole Numbers will help you develop a better understanding of dividing whole numbers.

example

Model the division: 24÷824\div 8.

Answer: Solution To find the quotient 24÷824\div 8, we want to know how many groups of 88 are in 2424. Model the dividend. Start with 2424 counters. An image of 24 counters placed randomly. The divisor tell us the number of counters we want in each group. Form groups of 88 counters. An image of 24 counters, all contained in 3 bubbles, each bubble containing 8 counters. Count the number of groups. There are 33 groups. 24÷8=324\div 8=3

 

try it

Model: 24÷624\div 6. No Alt Text Model: 42÷742\div 7. No Alt Text
In the video below we show another way to model division using area. https://youtu.be/jKHAsIcEolM

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