Ikeyn uses the ratio and proportion method to teach direct variation. It is
correct, but it is not the way variation is taught in schools, nor is it the
way it should be taught. That's because there are more than just one kind
of variation. There are several kinds of variation, and all these types are
covered together, so the approaches to them all should show as much similar
as possible.
Ikleyn should strive to stick to the methods taught in schools rather than
methods we teachers and tutors with more mathematical maturity might think
of. But a struggling student would not think of them.
Ikleyn should watch videos such as these to learn the standard way variation
is taught in schools.
https://www.youtube.com/watch?v=mwPHQLtKf1Q
https://www.youtube.com/watch?v=nZO5GEaunr4
https://www.youtube.com/watch?v=FpsS0jyZCgA
https://www.youtube.com/watch?v=hCOVwaEoqwA
DIRECT VARIATION:
y varies directly as x
y varies as x
y is directly proportional to x
These begin with y = kx
-----------------------
INVERSE VARIATION:
y varies inversely as x
y is inversely proportional to x
These begin with y = k/x
-----------------------
JOINT VARIATION:
y varies jointly as x and z
y is jointly proportional to x and z
These begin with y = kxz
-----------------------
COMBINED VARIATION:
y varies directly as x and inversely as z
y is directly proportional to x and inversely proportional to z
These begin with y= kx/z
y varies jointly as x and z and inversely as u and v
These are y = kxz/(uv)
---------------------------------------------
The key words for direct variation are:
"varies as",
"varies directly as",
"is proportional to",
"is directly proportional to"
Direct variation problems are of the form:
---------------------------------
The key words for inverse variation are:
"varies inversely as",
"is inversely proportional to"
Inverse variation problems are of the form:
---------------------------------
The given problem contains the words "varies directly as",
so it is direct variation
Choose single alphabet letters for the words:
Substitute the case when BOTH D and W are given
Solve for k, which is to
divide both sides by 25:
Go back to the original equation
Substitute this and the case where only W is given
Edwin
.
Since the problem says "The dosage usually varies directly as the weight",
there is much simpler/shorter way to get the solution/answer.
1. These words mean that the dosages and the weights of patients are proportional.
So, you can write a proportion
= , (1)
or
= , (2)
where x is the unknown dosage under the problem's question.
From proportion (2), x = = 160*3 = 480 mg. ANSWER
2. You can also write the base proportion (1) in other form
= , (3)
or
= (4)
From the proportion (4), you get the same answer
x = = 160*3 = 480 mg.
3. Instead of using proportions, you can use common sense/reasoning, which tells you that if in the second case
the body weight is 3 times as much as in the first case (as you have = 3), then the dosage in the second case
should be 3 times as much as in the first case.
Solved.
By doing in this way, you avoid calculating an intermediate coefficient of proportionality "k",
which you do not need and which is simply an auxiliary value. You directly obtain then the requested/required answer,
making your calculation shorter and the solution more straightforward.
These straightforward approaches work always and every time,
when the problem contains key words "one value varies directly as the second value".
The terms
- "one value varies directly as the second value",
- "the values are in direct proportion" and
- "the values are proportional"
are synonymous (i.e. EQUIVALENT by their meaning).
-----------------
On proportions, see the lessons
- Proportions
- Using proportions to solve word problems
- Using proportions to solve word problems in Physics
- Using proportions to solve Chemistry problems
- Typical problems on proportions
- Using proportions to estimate the number of fish in a lake
- HOW TO algebreze and solve these problems using proportions
- Problems on proportions for mental solution
- Selected problems on proportions from the archive
- OVERVIEW of lessons on proportions
in this site.
These lessons were written specially for you.
Read them and become an expert in this area.
Start from the very first lesson.
Consider these lessons as your textbook, handbook, tutorials and (free of charge) home teacher.
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After reading the comment by Edwin.
Edwin, thank you for your comment which you sent to me.
Be sure, I always read very attentively every your post . . .
Edwin, I think that it is great as you presented your post and your approach, which is a DOMINANT in school teaching.
I know it VERY WELL, so it is not something new to me.
But I think that if the student knows BOTH APPROACHES, he (or she) will have GREATER BENEFITS.
Also, let me add THIS: I think that if the student knows your approach (=standard), but does not know
my interpretation, it means that he (or she) does not know and does not understand ANYTHING in the subject.
So, what I explained in my post, is actually a prerequisite for that problem.