A straight pipe 1 yard in length was marked off in fourths and also in thirds. If the pipe was...

1. Translate the problem requirements: A 1-yard pipe is marked at both fourth intervals (\(0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1\)) and third intervals (\(0, \frac{1}{3}, \frac{2}{3}, 1\)), then cut at every marking. We need to find all possible piece lengths.
2. Identify all cutting positions: Combine the marking positions from both systems to find every location where the pipe will be cut.
3. Calculate distances between consecutive cut points: Measure the length of each piece by finding the difference between adjacent cutting positions.
4. Identify unique piece lengths: List all distinct piece lengths that result from the cutting pattern.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have: a pipe that's 1 yard long (think of it like a ruler or stick). We need to make marks on this pipe in two different ways.

First, we mark it in fourths. This means we divide the pipe into 4 equal parts, so we make marks at the \(\frac{1}{4}\) point, the \(\frac{1}{2}\) point (which is \(\frac{2}{4}\)), and the \(\frac{3}{4}\) point. Including the ends, our marks are at positions: \(0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1\).

Second, we mark it in thirds. This means we divide the pipe into 3 equal parts, so we make marks at the \(\frac{1}{3}\) point and the \(\frac{2}{3}\) point. Including the ends, our marks are at positions: \(0, \frac{1}{3}, \frac{2}{3}, 1\).

Now we cut the pipe at every single mark we made. We need to find all the different lengths of pieces we get.

Process Skill: TRANSLATE - Converting the problem description into clear mathematical positions

2. Identify all cutting positions

Let's combine all our marking positions and put them in order from left to right:

From fourths: \(0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1\)
From thirds: \(0, \frac{1}{3}, \frac{2}{3}, 1\)

Combining these and removing duplicates: \(0, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, 1\)

But wait - let's make sure we have the right order. To compare fractions like \(\frac{1}{4}\) and \(\frac{1}{3}\), let's convert to a common denominator of 12:

• \(\frac{1}{4} = \frac{3}{12}\)
• \(\frac{1}{3} = \frac{4}{12}\)
• \(\frac{1}{2} = \frac{6}{12}\)
• \(\frac{2}{3} = \frac{8}{12}\)
• \(\frac{3}{4} = \frac{9}{12}\)

So our cutting positions in order are: \(0, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, 1\)

Process Skill: VISUALIZE - Arranging all cut points in proper sequence

3. Calculate distances between consecutive cut points

Now we find the length of each piece by calculating the distance between consecutive cutting positions. Let's work with the common denominator of 12 to make this easier:

Piece 1: From 0 to \(\frac{1}{4} = \frac{1}{4} - 0 = \frac{3}{12} = \frac{1}{4}\)
Piece 2: From \(\frac{1}{4}\) to \(\frac{1}{3} = \frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}\)
Piece 3: From \(\frac{1}{3}\) to \(\frac{1}{2} = \frac{1}{2} - \frac{1}{3} = \frac{6}{12} - \frac{4}{12} = \frac{2}{12} = \frac{1}{6}\)
Piece 4: From \(\frac{1}{2}\) to \(\frac{2}{3} = \frac{2}{3} - \frac{1}{2} = \frac{8}{12} - \frac{6}{12} = \frac{2}{12} = \frac{1}{6}\)
Piece 5: From \(\frac{2}{3}\) to \(\frac{3}{4} = \frac{3}{4} - \frac{2}{3} = \frac{9}{12} - \frac{8}{12} = \frac{1}{12}\)
Piece 6: From \(\frac{3}{4}\) to \(1 = 1 - \frac{3}{4} = \frac{12}{12} - \frac{9}{12} = \frac{3}{12} = \frac{1}{4}\)

So our piece lengths are: \(\frac{1}{4}, \frac{1}{12}, \frac{1}{6}, \frac{1}{6}, \frac{1}{12}, \frac{1}{4}\)

4. Identify unique piece lengths

Looking at our list of piece lengths: \(\frac{1}{4}, \frac{1}{12}, \frac{1}{6}, \frac{1}{6}, \frac{1}{12}, \frac{1}{4}\)

The unique (different) lengths are: \(\frac{1}{12}, \frac{1}{6}, \frac{1}{4}\)

Let's double-check this makes sense: we have some pieces that are \(\frac{1}{12}\) of a yard long, some that are \(\frac{1}{6}\) of a yard long, and some that are \(\frac{1}{4}\) of a yard long.

Process Skill: CONSIDER ALL CASES - Ensuring we account for all pieces and identify only unique lengths

Final Answer

The different lengths of pieces are \(\frac{1}{12}, \frac{1}{6}, \frac{1}{4}\) of a yard.

This matches answer choice D: \(\frac{1}{12}, \frac{1}{6}, \frac{1}{4}\).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "marked off in fourths and thirds" means
Students may think this means the pipe is divided into 4 + 3 = 7 equal parts, rather than understanding that these are two separate marking systems that create different division points along the same pipe.

2. Forgetting to include both endpoints when determining cut positions
Students may only consider the internal marking points (\(\frac{1}{4}, \frac{1}{2}, \frac{3}{4}\) for fourths and \(\frac{1}{3}, \frac{2}{3}\) for thirds) and forget that the endpoints at 0 and 1 are also boundaries that affect piece lengths.

3. Misinterpreting "different lengths" in the question
Students may think they need to list every single piece length (including duplicates) rather than understanding that the question asks for all the unique/distinct piece lengths that occur.

Errors while executing the approach

1. Incorrectly ordering the fraction cut points
When combining marks from fourths and thirds, students may struggle to correctly order fractions like \(\frac{1}{4}\) and \(\frac{1}{3}\) without converting to a common denominator, leading to wrong piece length calculations.

2. Arithmetic errors when subtracting fractions
When calculating piece lengths by finding differences between consecutive cut points (like \(\frac{1}{3} - \frac{1}{4}\)), students may make errors in fraction subtraction, especially when working with different denominators.

3. Missing cut points or including non-existent ones
Students may accidentally skip legitimate cutting positions or incorrectly add positions that don't exist, leading to wrong piece count and lengths.

Errors while selecting the answer

1. Including duplicate piece lengths in the final answer
After calculating all piece lengths correctly, students may list repeated lengths (like including \(\frac{1}{6}\) twice because two pieces have this length) rather than identifying only the unique lengths as required.

2. Not simplifying fractions to match answer choices
Students may calculate piece lengths correctly but express them in unsimplified form (like \(\frac{2}{12}\) instead of \(\frac{1}{6}\)), causing them to not recognize their answer among the given choices.