A rubber ball bounces upward 1/2 the height that it falls each time it hits the ground. If the ball...

1. TRANSLATE the problem information

• Given information:
  • Ball dropped from 20.0 feet initially
  • Each bounce reaches one-half the height it fell from
  • Need maximum height between 3rd and 4th ground hits

• What this tells us: We need to track the ball through multiple bounces and find the height after the 3rd hit.

2. INFER the bouncing pattern

• This creates a sequence where each bounce height = \(\mathrm{(previous\ height)} \div 2\)
• "Between 3rd and 4th hit" means the maximum height after the 3rd bounce
• We need to calculate three consecutive bounces

3. SIMPLIFY through the sequence

• After 1st hit: \(20.0 \div 2 = 10.0\) feet
• After 2nd hit: \(10.0 \div 2 = 5.0\) feet
• After 3rd hit: \(5.0 \div 2 = 2.5\) feet

The maximum height between the 3rd and 4th ground hits is 2.5 feet.

Answer: 2.5 (acceptable forms: 2.5, 5/2, 2½)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students misinterpret "between the third and fourth time it hit the ground" and think they need to find some intermediate height during the fall, rather than recognizing this refers to the bounce height after the 3rd hit.

This confusion about the timing leads them to get stuck and guess randomly among potential bounce heights.

Second Most Common Error:

Poor SIMPLIFY execution: Students lose track of which bounce they're calculating or make arithmetic errors in the division sequence, potentially calculating \(20 \div 2 \div 2 = 5.0\) instead of continuing to the third bounce.

This may lead them to incorrectly answer 5.0 or stop their calculation prematurely.

The Bottom Line:

This problem tests whether students can track a geometric sequence through multiple steps while correctly interpreting the timing language about "between hits."