Is 1/4 of the circle shaded? Most students get this wrong

Summary

This video discusses a math problem that stumped year 8 students in New Zealand. The problem involves determining whether 1/4 of a circle, divided into three equally spaced vertical sections, is shaded. Surprisingly, 81% of year 8 students answered incorrectly. The video explains the correct solution, which involves calculating the area of the shaded segment using trigonometry. It highlights the broader struggle students face with fractions, even in simple textbook settings, and explores how context, such as pizza slices, can influence problem-solving. The video also touches on the practical application of this concept in determining fluid levels in cylindrical tanks.

Takeaways

• 🧮 The question involves determining whether 1/4 of a shaded circle divided by three vertical slices is shaded. The correct answer is no.
• 📊 Many students struggled with this problem: 86% of year 4 students and 81% of year 8 students gave the wrong answer, believing 1/4 was shaded.
• 🍕 When presented in a more familiar context like a pizza, students can easily recognize which slice is larger, indicating they understand fractions better in practical situations.
• 🧩 Students continue to struggle with fractions, as demonstrated by the fact that in 2022, only 32% of year 8 students could solve a simple fractions addition problem.
• 🔄 Students who got the problem correct recognized that the shaded area was not divided into equal parts, unlike a rectangle or a circle divided into four equal slices.
• 📐 Calculating the area of a circular segment requires trigonometry, specifically using the formula for the area of a circular segment: 1/2 r²(θ - sin θ).
• ✏️ For the given problem, the central angle is 120 degrees or 2π/3 radians, and the shaded area is approximately 19.6%, close to 1/5 of the total circle, not 1/4.
• 🚚 A real-world application of this concept involves determining fluid levels in cylindrical tanks, which uses the same principle of circular segments to measure volume.
• 🔢 To fill a tank to 1/4 of its volume, the fluid level must be around 30% of the total height, calculated using a numerical approximation for the angle and height.
• 🧠 The script encourages further exploration of similar mathematical problems and practical applications, while also pointing out how challenging fractions can be for many students.

Q & A

What is the main math problem discussed in the video?

The main math problem discussed involves determining whether 1/4 of a circle is shaded when the circle is divided by three equally spaced vertical slices.

What percentage of Year 8 students gave the wrong answer to the shading problem?

81% of Year 8 students gave the wrong answer, thinking that 1/4 of the circle was shaded.

Why do many students mistakenly believe 1/4 of the circle is shaded?

Many students mistakenly believe 1/4 of the circle is shaded because they are likely associating the equal vertical slices with equal area portions, which is incorrect for a circle divided by straight lines rather than curves.

How did the students perform on a related fractions question, and what was the trend over time?

In 1997, 45% of Year 8 students correctly answered that 1/2 + 1/4 equals 3/4, but by 2022, only 32% of students got this simple fraction question correct.

What mathematical concept is introduced to calculate the shaded area of the circle?

The concept of a circular segment is introduced, with the area calculated using trigonometry involving the radius, central angle, and sine function.

What is the approximate fraction of the circle that is shaded when divided by equally spaced vertical slices?

The shaded area of the circle in this case is approximately 19.6%, which is about 1/5 of the total circle area, not 1/4.

How can this type of problem be applied in a practical, real-world context?

A practical application of this problem is determining the fluid level in a cylindrical tank, such as a fuel or water tanker, where the cross-sectional area needs to be calculated to find the volume of fluid.

What is the 'quarter tank problem' and how is it solved?

The 'quarter tank problem' involves finding the fluid level in a horizontal cylindrical tank that corresponds to 1/4 of the total volume. The solution involves calculating the height of the fluid level using the circular segment formula and trigonometry, which is approximately 30% of the tank's diameter.

What common mistake do students make when dividing geometric shapes like circles and rectangles?

Students often assume that dividing a geometric shape with straight lines always results in equal areas, which is true for rectangles but not for circles divided into segments by straight lines.

Why might students perform better when the same problem is presented in a different context, such as with a pizza slice?

Students might perform better with a pizza slice because they have more intuitive experiences with real-world objects, like knowing which slice is bigger, while textbook problems can feel abstract and disconnected from everyday life.