Lily made 36 cups of jam. Lily then filled x small containers and y large containers with all the jam...

1. TRANSLATE the problem information

• Given information:
  • Lily made 36 cups of jam total
  • She used x small containers and y large containers
  • The equation \(\mathrm{4x + 6y = 36}\) represents this situation

• What this tells us: The equation shows how all 36 cups are distributed between small and large containers

2. INFER what each part of the equation represents

• Looking at the structure \(\mathrm{4x + 6y = 36}\):
  • The right side (36) is the total cups of jam
  • The left side shows how jam is split up
  • Since \(\mathrm{x}\) = number of small containers, what does \(\mathrm{4x}\) represent?
  • Since \(\mathrm{y}\) = number of large containers, what does \(\mathrm{6y}\) represent?

3. TRANSLATE the meaning of each term

• Breaking down the terms:
  • \(\mathrm{4x}\) = 4 cups per small container × \(\mathrm{x}\) small containers = total cups in small containers
  • \(\mathrm{6y}\) = 6 cups per large container × \(\mathrm{y}\) large containers = total cups in large containers

• Therefore: \(\mathrm{6y}\) represents the total number of cups of jam in the large containers

4. Verify by checking other answer choices

• Choice A: "Number of large containers" would just be \(\mathrm{y}\), not \(\mathrm{6y}\)
• Choice B: "Number of small containers" would be \(\mathrm{x}\), not \(\mathrm{6y}\)
• Choice D: "Total cups in small containers" would be \(\mathrm{4x}\), not \(\mathrm{6y}\)

Answer: C. The total number of cups of jam in the large containers

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the variable \(\mathrm{y}\) (which represents the number of large containers) with the expression \(\mathrm{6y}\) (which represents the total cups in those containers).

They see \(\mathrm{y}\) in the expression \(\mathrm{6y}\) and think "y represents large containers, so \(\mathrm{6y}\) must also represent large containers." They miss that the coefficient 6 changes what the expression represents - it's no longer counting containers, it's counting cups.

This may lead them to select Choice A (The number of large containers Lily filled).

Second Most Common Error:

Inadequate INFER reasoning: Students don't connect the coefficient 6 to the capacity of each large container, so they can't determine what \(\mathrm{6y}\) represents as a total quantity.

Without understanding that 6 represents "cups per large container," they can't see that \(\mathrm{6y}\) represents "total cups in large containers." This leads to confusion about what any of the terms mean.

This causes them to get stuck and guess among the answer choices.

The Bottom Line:

Success requires distinguishing between variables (which count things) and expressions with coefficients (which represent totals or products). The key insight is that \(\mathrm{6y}\) isn't just about containers anymore - it's about the cups that fill those containers.