small mistake that leads to BIG mistake

not really to point out the mistakes done by students, rather to correct the mistakes that can be prevented in math.

say, if the teacher asked as simple question (on formula) such as "what is the formula to find the area of triangle?", most students would reply "half times base times height". from this, the students will then write down in their paper as 1/2 x base x height. for me, I would say that this is very incorrect. why? read on, and you will be glad that you read. this is based on real situations and discussions.

Question: A triangle has a base of 10 cm. Given that its area is 20 squared cm, calculate the height of the triangle.

Common replies/mistakes done by students:
1/2 x base x height [a]
= 1/2 x 10 x 20 [b]
= 100 cm. [c]

ok. i am not making this up. the question is given to my year 11 (form 5) students, about 2 years ago when I was doing part-time tutoring. this question is so simple that I was very confident, all 10 students in that class would get it correct. it turned out that I was WRONG! Seeing this working (as shown above), I straight away spot the mistake.

Yes they got the idea right, the question is about area of triangle, which is shown in the line [a] that they have written. HOWEVER, they made mistake in writing down the formula. I always said this to my students, "WHEN YOU ARE WRITING ANY FORMULA, WHETHER IT IS SIMPLE, SHORT OR WHATEVER IT IS, MAKE SURE YOU WRITE DOWN THE COMPLETE FORMULA!"

Back to line [a]. for me, 1/2 x base x height is meaningless. I mean, what is 1/2 x base x height? Is it a formula? I would say no, since it does not have any indication or label. It is an improper way of writing down the formula. Because of this, students would mix up the information given by the question into their own version of the formula, as shown in line [b]. this student know the value for the base (10 cm) but when it comes to the value of height, this student got confused/lost. He/she might think "what is the height?? I am supposed to find the height, but where do I put this 20 cm in this formula?" Due to this confusion, the student then bet on their luck and put the 20 cm under the height space. See what I mean?

BETTER VERSION. If and only if, the student write down the complete formula, then it is almost impossible to make the mistake mentioned above.

Area of triangle = 1/2 x base x height

To make things simple, just remember this. Take formula as a puzzle, where you need to fill in as many empty spaces according to the clue given just to solve the question. In this question, we know the area of triangle (as given in the question) is 20 squared cm, so we fill 20 in the area of triangle space.

20 squared cm = 1/2 x base x height

Then, read again the question for the next clue. Do we know the value of base? Sure we do! It is 10 cm. Again, fill in the base space with 10 cm, to get

20 cm2 = 1/2 x 10 cm x height

From this, we are left with height. We don't have any value left to be picked from the question. Moreover, we are asked to find the height, so this is the complete set already. solve this!

20 = 1/2 x 10 x height
20 = 5 x height
20/5 = height
4 = height

From this discussion, what I am trying to emphasize is the importance of writing down the complete formula in solving math. Believe me, you won't regret it if you do what I suggest here. Because, formula are meant to be written in complete set so that you won't get mixed up or lost when solving them. GOOD LUCK!