A laboratory experiment monitored the temperature of a heated metal object as it cooled in a controlled environment. The graph...

1. TRANSLATE the question into mathematical terms

The question asks: "What was the initial temperature of the object?"

• Given information:
  • The graph shows temperature (y-axis) vs. time since start of experiment (x-axis)
  • We need the temperature at the "initial" moment

• Key translation: "Initial" means "at the beginning" or "at the start of the experiment"
  • Since x represents time since the start, the initial moment is when \(\mathrm{x = 0}\)

2. INFER where to find this value on the graph

• In any time-based function, the starting value occurs when time = 0
• On a graph, \(\mathrm{x = 0}\) is located on the y-axis
• This point is called the y-intercept - where the line crosses the y-axis
• Therefore, we need to find the y-coordinate of the y-intercept

3. Read the y-intercept from the graph

• Locate where the line intersects the y-axis (the vertical axis on the left)
• Follow horizontally from that intersection point to read the temperature value
• The line crosses the y-axis at \(\mathrm{y = 70}\)
• Reading carefully: This point is at \(\mathrm{(0, 70)}\)

Answer: 70 degrees Celsius

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding that "initial temperature" means the temperature when \(\mathrm{x = 0}\)

Students might think "initial" means:

• The first labeled point they see on the graph
• Some other special point on the line
• The minimum or maximum temperature

Without translating "initial" to "when time = 0," students may search randomly on the graph for what seems like a starting point, potentially reading the wrong temperature value. This leads to confusion and guessing, possibly selecting a temperature from a different time value visible on the graph (like 30, 50, or 60 degrees).

Second Most Common Error:

Misreading the graph scale: Not carefully counting the gridlines on the y-axis

Students might:

• Miscount the intervals between gridlines
• Estimate incorrectly between marked values
• Read from the wrong gridline

For instance, if they misread the scale, they might think the y-intercept is at 65 or 75 instead of 70. This results in an incorrect numerical answer even though their approach was correct.

The Bottom Line:

This problem tests whether students can connect everyday language ("initial") to mathematical concepts (\(\mathrm{x = 0}\), y-intercept) and then accurately read values from a graph. The key insight is recognizing that in time-based contexts, "initial" always means "when time equals zero."