During a study, the temperature, in degrees Celsius (°C), of the air in a chamber was recorded to the nearest...

1. TRANSLATE the problem requirements

The question asks for 'the average rate of change, in °C per minute, from x = 5 to x = 7'

This tells us:

• We need to find how temperature changes per minute
• We're focusing on the interval from x = 5 to x = 7
• We need to find two specific points on the scatterplot

2. TRANSLATE the graph information into coordinates

Look carefully at the scatterplot to identify the exact coordinates:

• At x = 5 minutes: The point is at y = 14°C → coordinate (5, 14)
• At x = 7 minutes: The point is at y = 24°C → coordinate (7, 24)

Take your time reading these values - the grid lines help you count accurately.

3. Apply the average rate of change formula

The average rate of change between two points is:

\(\frac{\mathrm{y_2} - \mathrm{y_1}}{\mathrm{x_2} - \mathrm{x_1}}\)

Substituting our values:

\(\frac{24 - 14}{7 - 5}\)

4. SIMPLIFY the calculation

\(\frac{24 - 14}{7 - 5} = \frac{10}{2} = 5\)

Since temperature is measured in °C and time in minutes, the rate is 5 °C per minute.

Answer: 5 (or 5.0, or 5 °C per minute)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misreading the y-coordinate at x = 7

Some students might read the point at x = 7 as (7, 22) instead of (7, 24) because they miscount the grid lines or confuse it with a nearby value.

If they use (7, 22) instead of (7, 24):

• Calculation becomes: \((22 - 14)/(7 - 5) = 8/2 = 4\)
• This leads to an incorrect answer of 4

Second Most Common Error:

Weak TRANSLATE skill: Reading coordinates but using the wrong pair of points

A student might accidentally read values from x = 5 and x = 6 (which is (6, 16)) instead of x = 5 and x = 7:

• Calculation becomes: \((16 - 14)/(6 - 5) = 2/1 = 2\)
• This leads to an incorrect answer of 2

Alternatively, some might use x = 6 and x = 7:

• Calculation becomes: \((24 - 16)/(7 - 6) = 8/1 = 8\)
• This leads to an incorrect answer of 8

The Bottom Line:

This problem tests your ability to carefully extract precise information from a visual display. The mathematics (using the rate of change formula) is straightforward once you have the correct coordinates. The challenge is in the careful, accurate reading of the scatterplot - take your time to locate each point exactly using the grid lines as guides.