A Student Says That $\frac{1}{5}$ Is Halfway Between $\frac{1}{4}$ And \$\frac{1}{6}$[/tex\]. Use A Carefully Drawn Number Line To Show That This Is Not Correct. What Fraction Is Halfway Between

Introduction

In mathematics, fractions are a fundamental concept that students often struggle to grasp. A common misconception among students is that the fraction halfway between two given fractions is the average of the two fractions. However, this is not always the case. In this article, we will explore a specific example where a student claims that $\frac{1}{5}$ is halfway between $\frac{1}{4}$ and $\frac{1}{6}$. We will use a carefully drawn number line to demonstrate that this is not correct and find the actual fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$.

Understanding the Concept of a Number Line

A number line is a visual representation of the number system, where each point on the line corresponds to a specific number. In this case, we will use a number line to represent the fractions $\frac{1}{4}$, $\frac{1}{5}$, and $\frac{1}{6}$. By plotting these fractions on a number line, we can see their relative positions and distances from each other.

Plotting the Fractions on a Number Line

To plot the fractions on a number line, we need to determine their decimal equivalents. The decimal equivalents of the fractions are:

• $\frac{1}{4} = 0.25$
• $\frac{1}{5} = 0.2$
• $\frac{1}{6} = 0.1667$

Now, we can plot these fractions on a number line, with 0.25 at the left end, 0.2 in the middle, and 0.1667 at the right end.

Analyzing the Number Line

By examining the number line, we can see that $\frac{1}{5}$ is not halfway between $\frac{1}{4}$ and $\frac{1}{6}$. In fact, $\frac{1}{5}$ is closer to $\frac{1}{4}$ than it is to $\frac{1}{6}$. This is because the distance between $\frac{1}{4}$ and $\frac{1}{5}$ is smaller than the distance between $\frac{1}{5}$ and $\frac{1}{6}$.

Finding the Actual Halfway Fraction

To find the actual fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$, we need to find the average of the two fractions. The average of two numbers is found by adding them together and dividing by 2.

$\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$

Now, we divide the sum by 2:

$\frac{5}{12} \div 2 = \frac{5}{24}$

Therefore, the fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$ is $\frac{5}{24}$.

Conclusion

In conclusion, the student's claim that $\frac{1}{5}$ is halfway between $\frac{1}{4}$ and $\frac{1}{6}$ is not correct. By using a carefully drawn number line, we were able to demonstrate that $\frac{1}{5}$ is not the average of the two fractions. Instead, we found that the fraction $\frac{5}{24}$ is the actual halfway point between $\frac{1}{4}$ and $\frac{1}{6}$. This example highlights the importance of using visual aids and mathematical reasoning to understand complex concepts in mathematics.

Frequently Asked Questions

• Q: Why is the student's claim incorrect? A: The student's claim is incorrect because $\frac{1}{5}$ is not the average of $\frac{1}{4}$ and $\frac{1}{6}$. The average of two numbers is found by adding them together and dividing by 2.
• Q: How do I find the average of two fractions? A: To find the average of two fractions, add them together and divide by 2. For example, to find the average of $\frac{1}{4}$ and $\frac{1}{6}$, add them together and divide by 2: $\frac{1}{4} + \frac{1}{6} = \frac{5}{12}$, and then divide by 2: $\frac{5}{12} \div 2 = \frac{5}{24}$.
• Q: What is the fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$? A: The fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$ is $\frac{5}{24}$.

Further Reading

• For more information on fractions and number lines, see the following resources:

• Khan Academy: Fractions and Decimals
• Math Is Fun: Fractions
• IXL: Fractions and Decimals

References

• [1] Khan Academy. (n.d.). Fractions and Decimals. Retrieved from https://www.khanacademy.org/math/fractions-and-decimals
• [2] Math Is Fun. (n.d.). Fractions. Retrieved from https://www.mathisfun.com/fractions/
• [3] IXL. (n.d.). Fractions and Decimals. Retrieved from https://www.ixl.com/math/fractions-and-decimals
  

Introduction

In our previous article, we explored a common misconception among students that the fraction halfway between two given fractions is the average of the two fractions. We used a carefully drawn number line to demonstrate that this is not always the case and found the actual fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: Why is the student's claim incorrect?

A: The student's claim is incorrect because $\frac{1}{5}$ is not the average of $\frac{1}{4}$ and $\frac{1}{6}$. The average of two numbers is found by adding them together and dividing by 2.

Q: How do I find the average of two fractions?

A: To find the average of two fractions, add them together and divide by 2. For example, to find the average of $\frac{1}{4}$ and $\frac{1}{6}$, add them together and divide by 2: $\frac{1}{4} + \frac{1}{6} = \frac{5}{12}$, and then divide by 2: $\frac{5}{12} \div 2 = \frac{5}{24}$.

Q: What is the fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$?

A: The fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$ is $\frac{5}{24}$.

Q: Can you explain why $\frac{1}{5}$ is not the average of $\frac{1}{4}$ and $\frac{1}{6}$?

A: Yes, $\frac{1}{5}$ is not the average of $\frac{1}{4}$ and $\frac{1}{6}$ because it is closer to $\frac{1}{4}$ than it is to $\frac{1}{6}$. To see this, we can plot the fractions on a number line. The number line will show that $\frac{1}{5}$ is between $\frac{1}{4}$ and $\frac{1}{6}$, but it is not equidistant from both fractions.

Q: How can I use a number line to find the average of two fractions?

A: To use a number line to find the average of two fractions, first plot the fractions on the number line. Then, find the midpoint of the two fractions by drawing a line from the midpoint of the two fractions to the right end of the number line. The point where the line intersects the number line is the average of the two fractions.

Q: Can you give an example of how to use a number line to find the average of two fractions?

A: Yes, let's say we want to find the average of $\frac{1}{4}$ and $\frac{1}{6}$. We can plot these fractions on a number line as follows:

• $\frac{1}{4}$ is at the left end of the number line
• $\frac{1}{6}$ is at the right end of the number line

To find the average, we draw a line from the midpoint of the two fractions to the right end of the number line. The point where the line intersects the number line is the average of the two fractions.

Q: What is the average of $\frac{1}{4}$ and $\frac{1}{6}$?

A: The average of $\frac{1}{4}$ and $\frac{1}{6}$ is $\frac{5}{24}$.

Conclusion

In conclusion, we have answered some frequently asked questions related to the topic of finding the average of two fractions. We have shown that the student's claim that $\frac{1}{5}$ is halfway between $\frac{1}{4}$ and $\frac{1}{6}$ is not correct and found the actual fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$. We have also provided examples of how to use a number line to find the average of two fractions.

Frequently Asked Questions

• Q: Why is the student's claim incorrect? A: The student's claim is incorrect because $\frac{1}{5}$ is not the average of $\frac{1}{4}$ and $\frac{1}{6}$. The average of two numbers is found by adding them together and dividing by 2.
• Q: How do I find the average of two fractions? A: To find the average of two fractions, add them together and divide by 2. For example, to find the average of $\frac{1}{4}$ and $\frac{1}{6}$, add them together and divide by 2: $\frac{1}{4} + \frac{1}{6} = \frac{5}{12}$, and then divide by 2: $\frac{5}{12} \div 2 = \frac{5}{24}$.
• Q: What is the fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$? A: The fraction that is halfway between $\frac{1}{4}$ and $\frac{1}{6}$ is $\frac{5}{24}$.

Further Reading

• For more information on fractions and number lines, see the following resources:

• Khan Academy: Fractions and Decimals
• Math Is Fun: Fractions
• IXL: Fractions and Decimals

References

• [1] Khan Academy. (n.d.). Fractions and Decimals. Retrieved from https://www.khanacademy.org/math/fractions-and-decimals
• [2] Math Is Fun. (n.d.). Fractions. Retrieved from https://www.mathisfun.com/fractions/
• [3] IXL. (n.d.). Fractions and Decimals. Retrieved from https://www.ixl.com/math/fractions-and-decimals