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 [latex]12[/latex] cookies and want to package them in bags with [latex]4[/latex] cookies in each bag. How many bags would we need?
You might put [latex]4[/latex] cookies in first bag, [latex]4[/latex] in the second bag, and so on until you run out of cookies. Doing it this way, you would fill [latex]3[/latex] bags.
In other words, starting with the [latex]12[/latex] cookies, you would take away, or subtract, [latex]4[/latex] cookies at a time. Division is a way to represent repeated subtraction just as multiplication represents repeated addition.
Instead of subtracting [latex]4[/latex] repeatedly, we can write
[latex]12\div 4[/latex]
We read this as twelve divided by four and the result is the quotient of [latex]12[/latex] and [latex]4[/latex]. The quotient is [latex]3[/latex] because we can subtract [latex]4[/latex] from [latex]12[/latex] exactly [latex]3[/latex] times. We call the number being divided the dividend and the number dividing it the divisor. In this case, the dividend is [latex]12[/latex] and the divisor is [latex]4[/latex]. In the past you may have used the notation [latex]4\overline{)12}[/latex] , but this division also can be written as [latex]12\div 4,12\text{/}4,\frac{12}{4}[/latex]. In each case the [latex]12[/latex] is the dividend and the [latex]4[/latex] is the divisor.Operation Symbols for Division
To represent and describe division, we can use symbols and words.| Operation | Notation | Expression | Read as | Result |
|---|---|---|---|---|
| [latex]\text{Division}[/latex] | [latex]\div [/latex] [latex-display]\frac{a}{b}[/latex-display] [latex-display]b\overline{)a}[/latex-display] [latex]a/b[/latex] | [latex]12\div 4[/latex] [latex-display]\frac{12}{4}[/latex-display] [latex-display]4\overline{)12}[/latex-display] [latex]12/4[/latex] | [latex]\text{Twelve divided by four}[/latex] | [latex]\text{the quotient of 12 and 4}[/latex] |
example
Translate from math notation to words. 1. [latex]64\div 8[/latex] 2. [latex]\frac{42}{7}[/latex] 3. [latex]4\overline{)28}[/latex] 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.
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: [latex]24\div 8[/latex].Answer:
Solution
To find the quotient [latex]24\div 8[/latex], we want to know how many groups of [latex]8[/latex] are in [latex]24[/latex].
Model the dividend. Start with [latex]24[/latex] counters.
The divisor tell us the number of counters we want in each group. Form groups of [latex]8[/latex] counters.
Count the number of groups. There are [latex]3[/latex] groups.
[latex-display]24\div 8=3[/latex-display]
try it
Model: [latex]24\div 6[/latex].
Model: [latex]42\div 7[/latex].
Licenses & Attributions
CC licensed content, Shared previously
- The Language of Division. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Division of Whole Numbers using Area (No Remainder). Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Question ID: 144450. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
