Question

III Fina wants to find the midpoint of a segment on a IIARIIA number line. She finds the length of the segment and divides by 2 . She adds this number to the coordinate of the left endpoint to find the midpoint. Kenji says she should subtract the number from the coordinate of the right endpoint to find the midpoint. Who is correct? Explain your reasoning.

Short answer

Both III Fina and Kenji are correct; their methods yield the same midpoint.

Step-by-step solution

Understand the Problem

We need to determine whether III Fina or Kenji is correct in finding the midpoint of a segment on a number line. III Fina calculates it by adding half the segment's length to the left endpoint, while Kenji suggests subtracting from the right endpoint.

Analyze III Fina's Method

III Fina finds the length of the segment, divides it by 2, and adds this number to the coordinate of the left endpoint. Mathematically, this is the formula to find the midpoint: if the endpoints are \(a\) and \(b\), the length is \(b-a\), and thus the midpoint is \(a + \frac{b-a}{2}\).

Analyze Kenji's Method

Kenji suggests subtracting half of the segment's length from the coordinate of the right endpoint. This can be expressed as \(b - \frac{b-a}{2}\). Simplifying this gives \(b - \frac{b}{2} + \frac{a}{2} = \frac{a+b}{2}\), which is also the midpoint.

Conclusion and Verification

Both III Fina's and Kenji's methods result in the same formula \(\frac{a+b}{2}\) for the midpoint. Thus, both approaches are mathematically equivalent and correct.

Key concepts

Number Line

A number line is a visual representation where numbers are placed in order along a straight path. It's an essential tool in mathematics that helps us think about the size and order of numbers intuitively. To find the midpoint of a segment on a number line, you simply find the point that is exactly halfway between the two endpoints.
To compute this mathematically on a number line, you need to know the coordinates of the endpoints of the segment. Let's say the endpoints are at points \( a \) and \( b \) on the number line. The process is straightforward:

• Subtract the smaller number from the larger number to find the segment's length, which is equivalent to \( b - a \).
• Divide this length by 2 to get half of the length.
• Finally, you can either add this half-length to \( a \) or subtract it from \( b \), as both will give you the midpoint.

The midpoint formula, \( \frac{a+b}{2} \), emerges from this operation as it captures the idea of averaging the coordinates of \( a \) and \( b \). This approach allows us to pinpoint the center without any confusion regarding direction or origin.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves analyzing geometric shapes using numerical coordinates. It's a vital part of understanding the relationship between algebra and geometry by translating problems into a coordinate system.
The concept of finding a midpoint lies at the heart of coordinate geometry. When we talk about a line segment in this context, we often refer to it as having its ends marked by coordinates on a plane. For a one-dimensional line segment, such as the one we're focusing on here, these coordinates are simply scalar values on a number line.
Using the midpoint formula \( \frac{a+b}{2} \), we can calculate the point that divides the segment into two equal halves. This operation is not just limited to number lines but extends to two-dimensional spaces as well.

• In a 2D plane, the midpoint between points \((x_1, y_1)\) and \((x_2, y_2)\) is \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
• This technique allows for precise placement of points crucial for geometric constructions and proofs.
• Coordinate geometry bridges algebraic and spatial reasoning, granting us tools to analyze and solve complex spatial problems efficiently.

Mathematical Reasoning

Mathematical reasoning is about using logic and mathematical principles to make sense of problems and conclusions. It ensures that conclusions like both Fina's and Kenji's methods are sound and trustable.
In this scenario, both Fina and Kenji use valid mathematical reasoning to arrive at the same endpoint, despite using different paths. This involved recognizing that adding to one endpoint and subtracting from another effectively yield the same result because:

• Adding half of the segment's length to the left endpoint results in \( a + \frac{b-a}{2} \).
• Subtracting half of the segment's length from the right endpoint results in \( b - \frac{b-a}{2} \), which simplifies to the same point.

Through logical simplification, we observe both expressions represent the same midpoint formula \( \frac{a+b}{2} \). This kind of duality and equivalence often appears in mathematics, demonstrating the elegance of mathematical reasoning.
Encouraging students to engage in mathematical reasoning nurtures a deeper understanding. It helps them realize that different approaches can and often do lead to the same correct outcome, reinforcing their problem-solving skills and confidence in mathematics.