To find the mistakes each student made, let's first recall the correct formula for the area of a triangle:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Given:
- Area = 24 square centimeters
- Height (h) = 4 centimeters
We need to find the base (b).
### Analyzing Jon's Equation
Jon wrote:
[tex]\[ 24 = 4b \][/tex]
To identify Jon's mistake, compare his equation with the correct area formula:
[tex]\[ 24 = \frac{1}{2} \times b \times 4 \][/tex]
Jon's error:
- He only included the base and height but failed to include the [tex]\(\frac{1}{2}\)[/tex] factor in his equation.
Jon's mistake: Jon did not multiply by 1/2 in the formula for the area of the triangle.
### Analyzing Annie's Equation
Annie wrote:
[tex]\[ b = \frac{1}{2} \cdot 24 \cdot 4 \][/tex]
To identify Annie's mistake, examine her approach:
According to the area formula, we should rearrange the formula to solve for b:
[tex]\[ b = \frac{2 \cdot \text{Area}}{\text{height}} \][/tex]
[tex]\[ b = \frac{2 \cdot 24}{4} = \frac{48}{4} = 12 \][/tex]
Annie's equation was:
[tex]\[ b = \frac{1}{2} \cdot 24 \cdot 4 \][/tex]
[tex]\[ b = 12 \][/tex]
She should have divided the area by [tex]\(\frac{1}{2}\)[/tex] and the height, rather than multiplying.
Annie's mistake: Annie incorrectly multiplied the area by the base and height instead of dividing the area by 1/2 and height.
### Analyzing Antonio's Equation
Antonio wrote:
[tex]\[ 24 = 2 \cdot 4b \][/tex]
To identify Antonio's mistake, compare with the correct equation:
[tex]\[ 24 = \frac{1}{2} \times b \times 4 \][/tex]
Antonio’s error:
- He incorrectly used [tex]\(2 \times 4b\)[/tex] instead of [tex]\(\frac{1}{2} \times 4b\)[/tex].
Antonio's mistake: Antonio incorrectly multiplied by 2 and missed the 1/2 factor.
### Summary of Mistakes
- Jon's mistake: Jon did not multiply by 1/2 in the formula for the area of the triangle.
- Annie's mistake: Annie incorrectly multiplied the area by the base and height instead of dividing the area by 1/2 and height.
- Antonio's mistake: Antonio incorrectly multiplied by 2 and missed the 1/2 factor.
