The graph shown models the number of candy bars a certain machine wraps with a label in x seconds. According...

1. TRANSLATE the question

• The question asks: "what is the estimated number of candy bars the machine wraps with a label per second?"
• Key phrase: "per second" - this means rate (how many candy bars for each second of time)

2. INFER what mathematical concept matches this question

• When we see "per second" in a graph where:
  • x-axis = time (seconds)
  • y-axis = quantity (candy bars)
• We need to find: How much does y change when x increases by 1?
• This is the slope of the line!
• The slope tells us the rate of change - exactly what "per second" is asking for

3. TRANSLATE coordinates from the graph

Looking at the graph, identify two clear points the line passes through:

• Point 1: \(\mathrm{(0, 0)}\) - at 0 seconds, 0 candy bars
• Point 2: \(\mathrm{(2, 80)}\) - at 2 seconds, 80 candy bars

You could also use \(\mathrm{(4, 160)}\) or other visible points, but \(\mathrm{(0, 0)}\) and \(\mathrm{(2, 80)}\) are easiest to read.

4. Apply the slope formula and SIMPLIFY

• Slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Substitute the points:
  \(\mathrm{m = \frac{80 - 0}{2 - 0}}\)
  \(\mathrm{m = \frac{80}{2}}\)
  \(\mathrm{m = 40}\)
• This means 40 candy bars are wrapped per second

Answer: B. 40

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not connecting "per second" to the concept of slope

Students might think "per second" means they should look at what happens at \(\mathrm{x = 1}\) second or \(\mathrm{x = 2}\) seconds. Looking at the graph, at \(\mathrm{x = 2}\), the y-value is 80. A student might think this is the answer because they read the coordinates but didn't understand that "per second" requires calculating the rate (slope), not just reading a single y-value.

This may lead them to select Choice D (80).

Second Most Common Error:

Incomplete SIMPLIFY execution: Reading coordinates incorrectly or using the wrong points

If a student misreads the graph and thinks the line passes through \(\mathrm{(2, 78)}\) instead of \(\mathrm{(2, 80)}\), they would calculate:

• \(\mathrm{m = \frac{78 - 0}{2 - 0} = \frac{78}{2} = 39}\)

Since 78 is an answer choice and close to their calculation, this may lead them to select Choice C (78).

Third Error Path:

TRANSLATE confusion: Mixing up which variable represents rate

A very confused student might think the answer is how many seconds per candy bar rather than candy bars per second, leading to calculations like \(\mathrm{\frac{2}{80}}\) or similar backwards reasoning. However, since none of the decimals appear in the choices, this would cause confusion and guessing.

The Bottom Line:

The key challenge is recognizing that "per second" on a graph means finding the slope (rate of change). Students who treat this as a simple reading-the-graph problem rather than a rate-calculation problem will select wrong answers that represent y-values at specific points rather than the slope of the line.