Sure, let's carefully examine the two students' solutions and identify any errors made:
### Alejandro's Solution:
Alejandro's problem:
[tex]\[ 5.7 - 9.6 \][/tex]
Alejandro rewrote the problem using the property of adding the opposite:
[tex]\[ 5.7 + (-9.6) \][/tex]
Then Alejandro found the result:
[tex]\[ -3.9 \][/tex]
First, let's understand whether Alejandro's transformation and solution are correct. For the purpose of verification:
- Subtracting a larger positive number (9.6) from a smaller positive number (5.7) means we will move left on the number line, past zero into the negative numbers.
- This can be done by keeping the sign of the larger number (9.6) and subtracting the smaller number (5.7) from it:
[tex]\[
9.6 - 5.7 = 3.9
\][/tex]
Since 9.6 is larger (and initially negative in the transformed problem), the result is
[tex]\[
-3.9
\][/tex]
Alejandro's process seems correct. There is no error in Alejandro's solution.
### Rose's Solution:
Rose's problem:
[tex]\[ 45.80 - 60 \][/tex]
Rose rewrote the problem using the property of adding the opposite:
[tex]\[ -45.80 + 60 \][/tex]
Then Rose found the result:
[tex]\[ 14.2 \][/tex]
Let's check Rose's steps:
1. First, we verify the initial transformation:
[tex]\[ 45.80 - 60 \][/tex]
Rewriting as an addition:
[tex]\[ 45.80 + (-60) \][/tex]
2. Now, let's calculate:
- Subtracting a larger positive number (60) from a smaller positive number (45.80) results in moving left on the number line, similarly to Alejandro's case.
- We compute:
[tex]\[
60 - 45.80 = 14.20
\][/tex]
Since 60 is larger (and initially negative in the transformed problem), the result is
[tex]\[
-14.20
\][/tex]
Rose wrote the result as [tex]\( 14.2 \)[/tex], which has the incorrect sign. The correct result should retain the negative sign since 60 is larger than 45.80 and was initially negative.
### Summary:
- Alejandro: Correct. The problem [tex]\( 5.7 - 9.6 \)[/tex] was correctly transformed to [tex]\( 5.7 + (-9.6) \)[/tex], and resulting in [tex]\( -3.9 \)[/tex].
- Rose: Incorrect. While the transformation [tex]\( 45.80 - 60 \)[/tex] to [tex]\( 45.80 + (-60) \)[/tex] was correct, the mistake was in the final calculation. The correct answer is [tex]\( -14.2 \)[/tex].
### Helping Rose:
To help Rose correct her error, I would explain:
1. How to correctly handle the subtraction operation by understanding the position and movement on the number line.
2. To pay attention closely to the signs, especially when dealing with the result of operations involving positive and negative numbers.
3. That after finding the value of [tex]\(|60 - 45.80| = 14.2\)[/tex], we should observe that since [tex]\(60\)[/tex] was initially negative in the adjusted equation [tex]\( 45.80 + (-60) \)[/tex], our result should reflect this by being negative.
Thus, the correct solution for Rose’s problem would be [tex]\(-14.2\)[/tex].
