A school is purchasing pencils for supply kits, and each kit needs 1 pencil.Pencils are sold only in boxes of...

1. TRANSLATE the problem information

• Given information:
  • Need: 155 pencils total
  • Available: Boxes containing 18 pencils each
  • Constraint: Cannot buy partial boxes

• What this tells us: We need to find how many complete boxes are required to get at least 155 pencils.

2. INFER the approach

• Since we need 155 pencils and each box has 18, divide to see how many boxes we need
• Key insight: If our division doesn't give a whole number, we must round UP (not down) because we can't buy part of a box and we need at least 155 pencils

3. Calculate the division

• \(155 \div 18 = 8.611...\) (use calculator)
• This means 8 boxes won't be enough, so we need 9 boxes

4. APPLY CONSTRAINTS to verify our answer

• Check 8 boxes: \(8 \times 18 = 144\) pencils ← Not enough
• Check 9 boxes: \(9 \times 18 = 162\) pencils ← This works!

Answer: 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students divide \(155 \div 18 = 8.611...\) and then round DOWN to 8 instead of up to 9, thinking "8.6 is closer to 8 than to 9."

They miss the key insight that in real-world purchasing problems, you need enough of the item, which means rounding up when you get a decimal. This leads them to answer 8, which would only provide 144 pencils (11 short of what's needed).

The Bottom Line:

This problem tests whether students understand that mathematical rounding rules don't always apply in real-world contexts. When you need a minimum quantity and can only buy in fixed packages, you must ensure you have enough, even if it means buying more than the exact amount calculated.