Area of Triangle/Square (Grade 5 Singapore Question)

[senseimichael]

The figure is made up of a triangle and a square. The area of the triangle is 1/3 the area of the square. After the overlapping part of 24 cm² is cut out, the remaining area of the triangle is 1/4 the remaining area of the square. Find the area of the triangle.

We cannot use algebra, because Grade 5 kids have not learnt this yet. They have already learnt the area of triangle and area of square, as well as fractions and ratios.

How I started working this out:

1. I got the ratio of T:S to be 1:3.
2. After cutting out the 24cm² from both, we get a ratio of 1:4.
3. So 4 units = 5 units - 24cm², so 1 unit = 24cm²
4. So the triangle should be 24cm², which is (logically speaking) TOTALLY WRONG!

Every bit of my mathematical fibre is telling me this is wrong!

The answer key for this question is 72 cm².

 

[lev888]

The figure is made up of a triangle and a square. The area of the triangle is 1/3 the area of the square. After the overlapping part of 24 cm² is cut out, the remaining area of the triangle is 1/4 the remaining area of the square. Find the area of the triangle.
View attachment 18728
We cannot use algebra, because Grade 5 kids have not learnt this yet. They have already learnt the area of triangle and area of square, as well as fractions and ratios.

How I started working this out:

1. I got the ratio of T:S to be 1:3.
2. After cutting out the 24cm² from both, we get a ratio of 1:4.
3. So 4 units = 5 units - 24cm², so 1 unit = 24cm²
4. So the triangle should be 24cm², which is (logically speaking) TOTALLY WRONG!

Every bit of my mathematical fibre is telling me this is wrong!

The answer key for this question is 72 cm².

Please explain 3.

 

[senseimichael]

Please explain 3.

I'm quite sure my thinking process is wrong anyway. The total units of the triangle + square is the same as the total units of the second set of ratios plus the parts taken away. I just sense it is wrong, but cannot put a pulse to it.

 

[lev888]

I'm quite sure my thinking process is wrong anyway. The total units of the triangle + square is the same as the total units of the second set of ratios plus the parts taken away.

Yes, this allows us to set up an equation. Is single variable equation allowed?

 

[Dr.Peterson]

What method do they use instead of algebra? Do they just use words as you did, or a visual method like the "tapes" or "strips" I have see as a staple of Singapore math? I can solve it in the latter way without too much trouble (which, of course, is algebra in disguise). It is harder to explain in words.

 

[senseimichael]

What method do they use instead of algebra? Do they just use words as you did, or a visual method like the "tapes" or "strips" I have see as a staple of Singapore math? I can solve it in the latter way without too much trouble (which, of course, is algebra in disguise). It is harder to explain in words.

Yes, it is algebra in disguise. Instead of calling it "x", we simply call it "units" or "parts" or "blocks".

 

[senseimichael]

Yes, this allows us to set up an equation. Is single variable equation allowed?

Yes, it is, but the kids learnt to call the variable "parts" or "units" instead of a letter x. It's like how I would explain solving it, at the start of the thread.

 

[Dr.Peterson]

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.

Algebra is more fun.

 

[senseimichael]

Algebra is more fun.

Agree!

 

[senseimichael]

Okay, to give closure to everyone in this very wonderful forum (where were you back in 2000, when I was still a teacher in an elementary school?).

I decided to calm down (after being so angry at the teacher who set this question) and rethink the question. I might have an even easier way of explaining this question, without even resorting to drawing anything.

The key to this question is the fact that, no matter what you did, the difference in area between the triangle and the square remains constant.

1. The original ratio of Triangle:Squareifference is 1:3:2
2. After cutting out, the ratio becomes 1:4:3
3. Making both Difference ratios equal (because the difference is supposed to remain constant), you get 3:9:6 at first and 2:8:6 after cutting out. We would realise that 1 unit has been cut out of both the triangle and the square. That 1 unit would be 24cm².
4. Therefore, the triangle, being 3 units, would be 3x24cm²=72cm²!

Zen, man! Zen!

 

[senseimichael]

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?

 

[Cubist]

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?

This is because the total numbers on lines 1 & 2 represent different things (despite using the same word "total").
Also, unfortunately, I think your method in post#10 is not guaranteed to work, it seems a coincidence that you got the correct answer.

Let me use algebra to prove this:-

Let t=area of triangle, s=area of square, c=area of cutout
Case 1: \( s=r_1 t \), where \( r_1=3 \)
Case 2: \( s-c=r_2 (t-c) \), where \(r_2=4 \)

Eliminate s, (Case 1)-(Case 2), and simplify giving

[math] t = c \times \left( \frac{r_2 - 1}{r_2 - r_1} \right) [/math]
Subst in the values for \( r_1, r_2, c \) you get t=72, so far so good.

--

But if we use \(r_2=7\) you get t=36. Using your method from post #10

1. The original ratio of Triangle : Square : Difference is 1:3:2
2. After cutting out, the ratio becomes 1:7:6
3. Making both Difference ratios equal (because the difference is supposed to remain constant), you get 3:9:6 at first and 1:7:6 after cutting out.
4. Therefore, the triangle, being 3 units, would be 3x24cm2=72cm2 but it ought to be 36!

 

[Cubist]

Here's a non-algebra way of solving the original, starting off similarly to @Dr.Peterson 's post...

 

[Dr.Peterson]

Okay, to give closure to everyone in this very wonderful forum (where were you back in 2000, when I was still a teacher in an elementary school?).

I personally was at Ask Dr. Math, answering questions from elementary teachers, among others ...

I decided to calm down (after being so angry at the teacher who set this question) and rethink the question. I might have an even easier way of explaining this question, without even resorting to drawing anything.

The key to this question is the fact that, no matter what you did, the difference in area between the triangle and the square remains constant.

1. The original ratio of Triangle:Squareifference is 1:3:2
2. After cutting out, the ratio becomes 1:4:3
3. Making both Difference ratios equal (because the difference is supposed to remain constant), you get 3:9:6 at first and 2:8:6 after cutting out. We would realise that 1 unit has been cut out of both the triangle and the square. That 1 unit would be 24cm².
4. Therefore, the triangle, being 3 units, would be 3x24cm²=72cm²!

I like this approach. I think @Cubist applied it incorrectly in his second case (post #12), because the difference in that case is 3-1 = 2 units and 9-7 = 12 units; so that one unit becomes only 12, and the triangle is 3 times that.

 

[senseimichael]

This is because the total numbers on lines 1 & 2 represent different things (despite using the same word "total").
Also, unfortunately, I think your method in post#10 is not guaranteed to work, it seems a coincidence that you got the correct answer.

Let me use algebra to prove this:-

IKR! With algebra, I can show anything...except the Grade 6 kids in Singapore would give me a strange look. I was so happy to have stumbled upon a way to show it with ratios, without drawing models...and then I found my answer demolished. Damned...

BUT, based on your scenario, where you get a ratio of 1:7 instead of 1:4 in my scenario, the answer *is* indeed 36 even by my method.

You get 2unit=24, and hence 1 unit=12. The triangle is 3 units, hence you get 3units=36, which fits the answer.

So my method somehow works ONLY for difference between Square and Triangle but not for total between them. Why?

 

[senseimichael]

I l

Here's a non-algebra way of solving the original, starting off similarly to @Dr.Peterson 's post...

View attachment 18789

I love this answer! Very useful for showing to those kids who have problem visualising - I can simply move the coloured blocks around! Thank you!

 

[senseimichael]

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.

Algebra is more fun.

Don't kill me, sir...I understand the picture (I would have drawn similar) but I do not understand the explanation...

 

[Cubist]

I think @Cubist applied it incorrectly in his second case (post #12)

the answer *is* indeed 36 even by my method.

I'm still struggling to see this but I'm sure that you're both correct! Doh! I'm much better with algebra (and pictures). I'll try to understand the method later.

You might be interested, I also solved the second case r2=7 using a similar method to post #13. It isn't quite as elegant (red bits show most of the changes, and note the lengths are no longer to scale)...

 

[Dr.Peterson]

Don't kill me, sir...I understand the picture (I would have drawn similar) but I do not understand the explanation...

@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.

 

[senseimichael]

I'm still struggling to see this but I'm sure that you're both correct! Doh! I'm much better with algebra (and pictures). I'll try to understand the method later.

You might be interested, I also solved the second case r2=7 using a similar method to post #13. It isn't quite as elegant (red bits show most of the changes, and note the lengths are no longer to scale)...

View attachment 18796

You could be a really popular Math teacher in a Singapore elementary school! We are not allowed to use algebra (though some argue that "units", "parts", etc are just algebra using different words) but we can use comparison ratios (like what I did with my method). 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.