Area of Triangle/Square (Grade 5 Singapore Question)

[senseimichael]

@Cubist did essentially what I did, with a fuller explanation. You really need to present it visually (by pointing to things) rather than with words, so I'll admit I didn't try very hard to explain it. I'm not going to try to improve on what I wrote, but if you were to point out where you don't follow me, I could try to answer.

The one I cannot understand is "What remains is therefore 2/3 of the whole triangle". I caught everything till there.

And I am still trying to figure out why my ratio explanation works with the difference in area of triangle and square being a constant but does NOT work with the total area of triangle and area being a constant. Was my latter assumption mathematically wrong?

 

[Cubist]

The funny thing is, the moment the kid enters your equivalent of Grade 7 (Secondary School), he is expected to ONLY use algebra and throw away the "childish" use of "units", "parts" and even comparison ratios.

I've just been wondering why this is the case. I always thought that I was introduced to algebra quite late in school in the UK. For me, personally, this was the point when maths became more interesting. Perhaps the reason is that some students will never truly grasp algebra - but at least they will have the fallback of having previously learnt a different method (which would doubtless help them in their future careers).

 

[Dr.Peterson]

The one I cannot understand is "What remains is therefore 2/3 of the whole triangle". I caught everything till there.

Let's see ... here is what I said:

Here's my attempt, as as picture:

    +---+-------+
Tri:|24 |       |
    +---+-------+
    +---+-------+-----------+-----------+
Squ:|24 |       :           :           :
    +---+-------+-----------+-----------+
    +---+-------+-------+-------+-------+
    |24 |       :       :       :       :
    +---+-------+-------+-------+-------+

The square (second strip) is 3 times the triangle (first strip)What remains of the square is 4 times what remains of the triangle (last strip); so 2 times the triangle is 3 times what remains of it. What remains is therefore 2/3 of the whole triangle, and 24 cm^2 is 1/3 of the triangle. The triangle is therefore 72 cm^2.

If 2 times x is 3 times y, then y is 1/3 of 2 times x, which is 2/3 of x.

And I am still trying to figure out why my ratio explanation works with the difference in area of triangle and square being a constant but does NOT work with the total area of triangle and area being a constant. Was my latter assumption mathematically wrong?

Here is what you said:

Interestingly, I tried using another way of thinking - the total area of the triangle and square remains constant. And the entire equation is thrown out totally once I use that assumption.

1. The original ratio of Triangle:Square:Total is 1:3:4
2. After cutting out, the ratio becomes 1:4:5
3. Making both Total ratios equal, you get 5:15:20 and 4:16:20........

I went "huh" after this. Why is this assumption wrong this time?

Because the assumption is wrong: The total area of the triangle and square does not remain constant when you remove part of each!

To take a numerical example, if I start with 10 and 20, their difference remains 10 if I subtract 5 from each (5 and 15), but their sum (30) changes to 20.

 

[senseimichael]

Let's see ... here is what I said:

If 2 times x is 3 times y, then y is 1/3 of 2 times x, which is 2/3 of x.

Always easier when you frame it in algebraic equations...got it.

Because the assumption is wrong: The total area of the triangle and square does not remain constant when you remove part of each!

To take a numerical example, if I start with 10 and 20, their difference remains 10 if I subtract 5 from each (5 and 15), but their sum (30) changes to 20.

And it is suddenly so clear. No wonder you are DR Peterson. Thank you sir!

And from your example (which is a fantastic one to show my students, if they ever ask), the total also does not work if you add part of each. So the total as an assumption is never going to work in such questions, where parts are added or removed.

 

[Dr.Peterson]

I could also have said, "If 2 times this is 3 times that, then that is 1/3 of 2 times this, which is 2/3 of this". I chose instead to go all the way to "algebraic" forms; x and y are really nothing more than more convenient demonstrative pronouns. (But then consider the fact that symbolism came after algebra; the first algebra was written in sentences like that, and not in symbols!)