The range for a set of measurements is 24.7. If 6 were added to each measurement and then each of...

1. Translate the problem requirements: Understand what "range" means (difference between highest and lowest values) and how two transformations (adding 6 to each measurement, then dividing each by 10) affect this range
2. Analyze the effect of adding a constant: Determine how adding 6 to every measurement impacts the range
3. Analyze the effect of dividing by a constant: Determine how dividing every measurement by 10 impacts the range
4. Apply transformations sequentially: Calculate the final range after both transformations

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're dealing with. The "range" of a set of measurements simply means the difference between the highest value and the lowest value in that set. Think of it like this: if the tallest person in a group is 6 feet and the shortest is 5 feet, the range of heights is \(6 - 5 = 1\) foot.

We're told the original range is 24.7, which means the difference between the highest and lowest measurements is 24.7 units.

Now we need to understand what happens when we make two changes to ALL the measurements:
- First: Add 6 to each measurement
- Second: Divide each of those new measurements by 10

Our goal is to find the range of the final set of measurements after both transformations.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Analyze the effect of adding a constant

Let's think about what happens when we add the same number to every measurement in our set.

Imagine we have measurements like: 10, 15, 20, 25, 30
The range here is \(30 - 10 = 20\)

Now if we add 6 to each measurement: 16, 21, 26, 31, 36
The new range is \(36 - 16 = 20\)

Notice something important: the range stayed exactly the same! This makes perfect sense because when you add the same amount to both the highest and lowest values, the difference between them doesn't change.

In our problem, when we add 6 to each measurement, the range remains 24.7.

Here's why: If the original highest value was H and lowest was L, then:
- Original range = \(\mathrm{H} - \mathrm{L} = 24.7\)
- After adding 6: New highest = \(\mathrm{H} + 6\), New lowest = \(\mathrm{L} + 6\)
- New range = \((\mathrm{H} + 6) - (\mathrm{L} + 6) = \mathrm{H} - \mathrm{L} = 24.7\)

3. Analyze the effect of dividing by a constant

Now let's see what happens when we divide every measurement by the same number.

Using our previous example after adding 6: 16, 21, 26, 31, 36
If we divide each by 10: 1.6, 2.1, 2.6, 3.1, 3.6
The new range is \(3.6 - 1.6 = 2.0\)

Notice that our original range of 20 became 2.0 when divided by 10. The range got scaled down by the same factor!

This makes sense: when you divide both the highest and lowest values by the same number, their difference also gets divided by that same number.

In our problem, after adding 6, we have a range of 24.7. When we divide each measurement by 10, the range becomes \(24.7 ÷ 10 = 2.47\).

4. Apply transformations sequentially

Let's put it all together step by step:

Step 1: Original measurements have a range of 24.7
Step 2: Add 6 to each measurement → Range stays 24.7
Step 3: Divide each adjusted measurement by 10 → Range becomes \(24.7 ÷ 10 = 2.47\)

Therefore, the final range is 2.47.

Final Answer

The range of the final set of measurements is 2.47.

Looking at our answer choices:

1. 30.70
2. 25.20
3. 24.70
4. 3.07
5. 2.47

The correct answer is E. 2.47.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the definition of range
Students often confuse range with other statistical measures like mean or median. They might think range refers to the total spread of values or the middle value, rather than understanding it as simply the difference between the highest and lowest values in the dataset.

Faltering Point 2: Misinterpreting the sequence of operations
The problem requires two sequential transformations: first adding 6 to each measurement, then dividing each result by 10. Students may misread this as applying both operations simultaneously or in the wrong order, such as dividing by 10 first and then adding 6, which would lead to a completely different approach.

Faltering Point 3: Not recognizing that transformations affect all measurements equally
Students might not realize that since the same mathematical operation is applied to every single measurement in the set, they need to analyze how these uniform transformations affect the range specifically, rather than trying to work with individual data points.

Errors while executing the approach

Faltering Point 1: Incorrectly believing that adding a constant changes the range
This is a critical conceptual error. Students may think that when 6 is added to each measurement, the range becomes \(24.7 + 6 = 30.7\). They fail to understand that adding the same value to both the maximum and minimum values keeps their difference unchanged.

Faltering Point 2: Applying the division incorrectly to the range
When dividing each measurement by 10, students might incorrectly add this effect to the range (thinking \(24.7 + 10 = 34.7\)) or apply some other incorrect calculation, rather than understanding that dividing all measurements by 10 means the range also gets divided by 10.

Faltering Point 3: Computational errors in the final division
Even with the correct approach, students may make basic arithmetic mistakes when calculating \(24.7 ÷ 10\), potentially getting 2.74 instead of 2.47, or making decimal point errors.

Errors while selecting the answer

Faltering Point 1: Selecting the original range value
Students who incorrectly conclude that transformations don't affect the range at all might select choice C (24.70), thinking the range remains completely unchanged throughout both operations.

Faltering Point 2: Choosing an answer that reflects partial errors
Students who made the error of adding 6 to the range and then dividing by 10 would get \((24.7 + 6) ÷ 10 = 3.07\), leading them to incorrectly select choice D.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient concrete values for the original measurements

Let's create a simple dataset where we can easily track the range:
• Lowest measurement = 10
• Highest measurement = 34.7
• Range = \(34.7 - 10 = 24.7\) ✓

Step 2: Apply the first transformation (add 6 to each measurement)

• New lowest = \(10 + 6 = 16\)
• New highest = \(34.7 + 6 = 40.7\)
• New range = \(40.7 - 16 = 24.7\)

Notice: Adding the same constant to all values doesn't change the range.

Step 3: Apply the second transformation (divide each by 10)

• Final lowest = \(16 ÷ 10 = 1.6\)
• Final highest = \(40.7 ÷ 10 = 4.07\)
• Final range = \(4.07 - 1.6 = 2.47\)

Step 4: Verify our result

The final range is 2.47, which matches answer choice E.

This concrete approach confirms our understanding: adding a constant preserves the range, while dividing by a constant scales the range by the same factor \((24.7 ÷ 10 = 2.47)\).