What are the major misconceptions that students have when it comes to calculating the area of a triangle? Specifically, can students correctly determine which lengths are needed to get the right answer?
Using real life data and explanations from students all around the world from my Diagnostic Questions website, we can find out! :-)
Have a look at the question below.
What percentage of students do you think get it correct?
What is the most popular incorrect answer?
What explanations do students give for the incorrect answers?
Give the question a go yourself, and try to come up with an explanation of the correct answer that would make sense to students:
At the time of writing, this question has been answered 471 times, and has been answered correctly just an impressive 76% of the time. The incorrect choices are split evenly amongst the three other options, with each accounting for 8% of responses.
To see this data, read student explanations, and filter by things such as gender and age, just visit the Data Question page.
Now, looking at the figures, it doesn't appear that there are too many nasty misconceptions lurking around this question. After all, 76% success rate is probably the highest we have had in this Question of the Week feature.
But before we go and make a cup of tea, and relax in the knowledge that the majority of the world can work out the area of a triangle, there is a,ittle more going on here than meets the eye.
I have selected this question in particular because of the explanations. The way students all around the world articulate their explanations on the website (and they are getting better week by week) allows us as teacher an unprecedented insight into their thinking. And whilst the numbers getting this question wrong might seem insignificant, the reasons for these mistakes are not.
Fortunately (and this is my favourite part about the website), students with misconceptions can go along way to resolving them themselves by reading correct explanations given by students all around the world. And what wonderful explanations we have:
Correct Explanations for A
Many students explained the correct answer beautifully:
This is because the formula for the area of the triangle is base x height divide by two. 7cm is a base, 6 cm is a height so the area of the triangle is 6 x 7 /2
to work out the area of a triangle you do 1/2 x base x height. The base is 7 and the perpendicular height is 6 so it would be 7x6 and then timesed by a 1/2 or divided by 2
To find out the area of this isosceles triangle you have to times ( for this one) 7 x 6= 42 then you have to divide 42 by 2 because you found a rectangle's area ( double the sizeof this triangle) so divide by 2 and you get 21cm2.
2 of these triangles fit into a 7cm by 6m rectangle so find the area of that by multiplying 7 and 6 together then divide by 2
The base is 7 and the perpendicular height is 6. You do not need the 8cm as this is the slanted height. You do base x height and then divide your answer by 2.
All it takes is for one of these explanations to hit home with a particular student, and their misconception will be resolved. That is why it is so important that we encourage our students to review and like their favourite explanations.
Incorrect Explanations for B
B is a classic. I think it comes from the students' belief that if a number is given in a question, then it must be used in the answer, or why would it be given to them? Giving students regular exposure to situations where some of the information they are given is not needed may be the key to helping resolve this issue.
its the method you use for finding out the are of shapes only some other shapes you use a different method but not a triangle
I think this is the answer because you times all of the numbers together
the other three answers simply don't have enough information.
on the other answers where does the 2 come from
Incorrect Explanations for C
Answer C will catch out students who do not know the importance of the height being perpendicular. Or, as it also likely, they have heard the word perpendicular, but do not actually understand what it means
I think this is the right answer because you can't just times them without using a division because that would make a rectangle. I think you times the 8 and 7 because that would make like a parallelogram which fits two triangles in it that's why you divide it by 2?
as its not a right angled triangle you cant use base times height so you have to use base times length divided by 2
I think it's C because you make this isosceles triangle into a square then you time 8 and 7 and divide by two to get the area.
Because the formula is base times height divided by 2
Incorrect Explanations for D
Answer D was designed to identify those students who do not know, or who have forgotten the importance of dividing by two, or halving. Their explanations reveal some interesting insights:
you do times the perpendicular line by the base line
You dont worry about the diagonal lines.
split the triangles into 2 so there is two right angled triangles so you have two equation of 6x7/2, but these cancel out to make 6x7
You do not halve this sum as the triangle is made of two right angled triangles
So, we have a whole bundle of important misconceptions concerning the Area of a Triangle that we need to identify in our students and then resolve. Hopefully Diagnostic Questions can help you do this.
This question can also be found in an Area of Simple Shapes quiz that I put together. Try it below and click here to assign it to your students.
Below is a video showing you how to access all the pages and information discussed above: