Let's break down Ethan's study hours and understand why he chose to model [tex]\(2 \frac{2}{3}\)[/tex] hours using [tex]\(\frac{1}{6}\)[/tex] strips.
### Understanding the Fraction
First, let's express [tex]\(2 \frac{2}{3}\)[/tex] in a way that will help us see why [tex]\(\frac{1}{6}\)[/tex] strips were chosen.
1. Original Fraction: [tex]\(2 \frac{2}{3}\)[/tex]
2. Convert to Improper Fraction:
[tex]\[
2 \frac{2}{3} = 2 + \frac{2}{3}
\][/tex]
We want a common denominator to combine the whole number and the fraction. For convenience, we'll use the denominator 6.
3. Express Whole Number using Denominator 6:
[tex]\[
2 = \frac{12}{6}
\][/tex]
So,
[tex]\[
2 \frac{2}{3} = \frac{12}{6} + \frac{2}{3}
\][/tex]
4. Convert [tex]\(\frac{2}{3}\)[/tex] to a Fraction with Denominator 6:
[tex]\[
\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}
\][/tex]
Thus,
[tex]\[
2 \frac{2}{3} = \frac{12}{6} + \frac{4}{6} = \frac{12 + 4}{6} = \frac{16}{6}
\][/tex]
5. Simplify (Optional):
Using mixed number format, it's
[tex]\[
2 \frac{4}{6} \text{ or } 2 \frac{2}{3}
\][/tex]
### Reason for Choosing [tex]\(\frac{1}{6}\)[/tex] Strips
When representing a fraction, using a common reference or smaller units can make visualization and calculation easier. Here's why [tex]\(\frac{1}{6}\)[/tex] strips are helpful:
1. Divisibility and Common Denominator:
- [tex]\(\frac{1}{6}\)[/tex] is a smaller unit than [tex]\(\frac{1}{3}\)[/tex], which means it provides a finer breakdown.
- When breaking down fractions, having a common denominator simplifies the arithmetic. [tex]\(\frac{1}{6}\)[/tex] is directly compatible with both [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] because both can be easily expressed as a multiple of [tex]\(\frac{1}{6}\)[/tex].
2. Representing the Hours:
- Using [tex]\(\frac{1}{6}\)[/tex] strips, we can represent [tex]\(2 \frac{2}{3}\)[/tex] as follows:
- [tex]\(2\)[/tex] whole hours = [tex]\(\frac{12}{6}\)[/tex] = 12 strips,
- Plus [tex]\(\frac{2}{3}\)[/tex] hours = [tex]\(\frac{4}{6}\)[/tex] = 4 additional strips.
- Total: [tex]\(12 + 4 = 16 \)[/tex] strips.
This makes [tex]\(2 \frac{2}{3}\)[/tex] hours represented by [tex]\(16\)[/tex] strips of [tex]\(\frac{1}{6}\)[/tex] each.
### Visual Representation:
Imagine having a set of strips where each strip represents [tex]\(\frac{1}{6}\)[/tex] of an hour. It's much easier to piece together [tex]\(16\)[/tex] strips to represent [tex]\(2 \frac{2}{3}\)[/tex] hours than to mix [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] strips. The consistency helps in understanding and calculating the total hours studied more intuitively.
By modeling [tex]\(2 \frac{2}{3}\)[/tex] hours with [tex]\(\frac{1}{6}\)[/tex] strips, Ethan made it easier to work with matching the common denominators effectively and efficiently. This way, when adding or comparing hours, everything aligns perfectly.
