If a is the mean and b is the median of nine consecutive integers, what is the value of |a...

1. TRANSLATE the problem information

• Given information:
  • Nine consecutive integers
  • a = mean of these integers
  • b = median of these integers
  • Need to find |a - b|

2. TRANSLATE consecutive integers to mathematical form

• Let's represent nine consecutive integers as: \(\mathrm{n, n+1, n+2, n+3, n+4, n+5, n+6, n+7, n+8}\)
• This covers all possible sets of nine consecutive integers

3. SIMPLIFY to find the mean (a)

• \(\mathrm{Sum = n + (n+1) + (n+2) + (n+3) + (n+4) + (n+5) + (n+6) + (n+7) + (n+8)}\)
• \(\mathrm{Sum = 9n + (0+1+2+3+4+5+6+7+8)}\)
  \(\mathrm{Sum = 9n + 36}\)
• \(\mathrm{Mean\ a = \frac{9n + 36}{9}}\)
  \(\mathrm{Mean\ a = n + 4}\)

4. INFER the median (b) using properties of ordered data

• Since we have 9 values (odd number), the median is the middle value
• When arranged: \(\mathrm{n, n+1, n+2, n+3, n+4, n+5, n+6, n+7, n+8}\)
• The middle position is the 5th value: \(\mathrm{n+4}\)
• So \(\mathrm{b = n+4}\)

5. SIMPLIFY to find |a - b|

• \(\mathrm{|a - b| = |(n+4) - (n+4)|}\)
  \(\mathrm{|a - b| = |0|}\)
  \(\mathrm{|a - b| = 0}\)

Answer: 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students try to use specific numbers instead of variables, such as choosing 1, 2, 3, 4, 5, 6, 7, 8, 9 as their nine consecutive integers.

While this would give the correct answer for this specific case, students often don't realize they need to prove this works for ALL possible sets of nine consecutive integers. They calculate \(\mathrm{mean = 5}\) and \(\mathrm{median = 5}\), get \(\mathrm{|a - b| = 0}\), but don't understand why this is always true. This approach works but misses the general mathematical insight.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the fundamental property that for any symmetric set of numbers (like consecutive integers), the mean and median are always equal.

Instead, they get bogged down in complex calculations or think they need to consider different cases. This leads to confusion about whether the answer depends on which specific consecutive integers are chosen, causing them to second-guess the answer or abandon the systematic approach and guess.

The Bottom Line:

This problem tests whether students can work with variables to represent general cases rather than specific examples, and whether they understand the relationship between mean and median for symmetric data sets.