Question 1166784: Beethoven started to compose music at a very young age. Bur Mozart was even
younger when he started. The sum of their ages at that time was 17. The difference
was 7. How old was each when they started composing?
Answer here: (note: you may use GUESS and TEST)
A. Read and Understand:
What do you want to find?
B. Devise a PLAN
What is your plan?
C. Implement the PLAN
What is the solution and answer?
D. Look BACK/Check
Is the sum of the ages equal to 17?
Is the difference of the ages equal to 7?
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website! .
The given post offers me unexpected and interesting disposition.
From one side, it is targeted to children who are related to music by some way and know the names of Beethoven and Mozart.
From the other side, as it is clear from the context, the assignment is targeted to young students who are not experienced in Algebra.
Before developing further my observations, let me notice that both Algebra and music have a lot in common: they both teach students and people
in general to feel a harmony.
So, we have an Algebra problem offered to young students, who are related to music and are not experienced in Algebra.
Now the question is: can I teach such students to solve the posed problem ?
- can I teach them to something useful ?
- can I present my teaching in a way that they will understand me ?
- and can I, at last, to make my teaching attractive for them just from the first minute?
The answer is "YES, I can".
The strategy of solution to the problem is as follows.
You are given that the sum of their ages is 17, while the difference of their ages is 7.
1) Take the middle point between 7 and 17. It is 12, obviously. (Same as half the sum).
2) This middle value is nothing else as the age of the older person (Beethoven, in this case,
since we know that Beethoven was older than Mozart when started learning music).
3) Next subtract the given difference of ages 7 from the Bethowen age 12 - 7 = 5.
It is the age of Mozart when he started.
5) So the ages were 5 years for Mozart and 12 years for Beethoven. It is the ANSWER to the problem.
Solved. // Which means that the solution is completed.
Now, similar to as the composers check their writing by playing it, in Algebra we check our answers by substituting them into the problem' condition.
By doing it, we see that the sum of 12 and 5 is 17, and the difference of 12 and 5 is 7 - same as stated in the problem.
So the problem is solved and checked.
And then I'd complete my performance as a teacher by saying to my students that the method I tought them is a GENERAL method,
which works ALWAYS in solving similar problems.
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