Skill: Critique another solution

When you critique a solution, do more than say “right” or “wrong.” Check:

1. What the problem is asking
2. Whether the representation matches the situation
3. Whether the calculations are correct
4. Whether the conclusion makes sense

Example

Problem:
One batch of cookies uses $\frac{3}{4}$ cup of flour. How much flour is needed for $2$ batches?

A student writes:

“I calculate $\frac{3}{4}+2=2\frac{3}{4}$ cups.
So, $2\frac{3}{4}$ cups of flour are needed.”

Step 1: Check the student’s representation

The recipe uses $\frac{3}{4}$ cup for each batch. For $2$ batches, we need two groups of $\frac{3}{4}$:

$$\frac{3}{4}+\frac{3}{4}$$

This is also written as:

$$2 \times \frac{3}{4}$$

The student added $2$ to $\frac{3}{4}$, but the $2$ represents the number of groups, not an amount of flour to add. So the representation does not match the situation.

Step 2: Check the calculation

Calculate using repeated addition:

$$\frac{3}{4}+\frac{3}{4}=\frac{6}{4}=1\frac{1}{2}$$

Therefore, the correct amount is:

$$\boxed{1\frac{1}{2}\text{ cups}}$$

Step 3: Check whether the answer makes sense

Two batches should use twice as much flour as one batch. Since $\frac{3}{4}$ cup is less than $1$ cup, twice that amount should be less than $2$ cups. The answer $1\frac{1}{2}$ cups makes sense, but $2\frac{3}{4}$ cups is too large.

Final critique

The student’s conclusion is incorrect because the number of batches was added instead of used as the number of groups. This is a misunderstanding of the relationship, not just a small calculation mistake. The correct equation is

$$2 \times \frac{3}{4}=1\frac{1}{2}.$$