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Increasing or Decreasing? Variation 2
Task
Consider the expression $$\frac{R_1 + R_2}{R_1R_2}$$ where $R_1$ and $R_2$ are positive.
Suppose we increase the value of $R_1$ while keeping $R_2$ constant. Find an equivalent expression whose structure makes clear whether the value of the expression increases, decreases, or stays the same.
IM Commentary
The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.
Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.
The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.
Solution
We rewrite
$$
\frac{R_1+R_2}{R_1R_2}
$$
in the form
$$
\frac{R_1}{R_1R_2} + \frac{R_2}{R_1R_2} = \frac{1}{R_2} + \frac{1}{R_1}.
$$
Now the bigger a (positive) number is, the smaller its reciprocal is, and so the value of this expression decreases as $R_1$ increases.
Comments
Log in to commentLisa says:
over 4 yearsThe statement "you can see that the expression decreases" assumes that all readers of the task can indeed see that. An explanation of the statement would improve this commentary. I wouldn't accept "it's obvious" from students. It seems the demand on students for justification should be met here in the commentary for all the tasks.
Cam says:
over 4 yearsAgreed! I've made a first stab at including such language, but would happily entertain further suggestions. Thanks for your comments.