Increasing or Decreasing? Variation 1

Task

Consider the expression $$\frac{1}{\displaystyle\frac1{R_1} + \frac1{R_2}}$$ where $R_1$ and $R_2$ are positive.

 

Suppose we increase the value of $R_1$ while keeping $R_2$ constant. Does the value of the expression above increase, decrease, or stay the same? Explain in terms of the structure of the expression.

IM Commentary

It may prove instructive to have students try a few numerical values to get a feeling for this problem. For example, if $R_1$ and $R_2$ are both equal to 1, then $\displaystyle \frac{1}{R_1} + \frac{1}{R_2} = 1 + 1 = 2$, so $$\frac{1}{\displaystyle\frac1{R_1} + \frac1{R_2}} = \frac12.$$ If $R_1$ is increased to $2$, while $R_2$ is kept constant at 1, then $\displaystyle \frac{1}{R_1} + \frac{1}{R_2} = \frac12 + 1 = \frac32$, so $$\frac{1}{\displaystyle\frac1{R_1} + \frac1{R_2}} = \frac23,$$ which is greater than $1/2$.

Students might infer the answer from numerical trials like this, and such trials could help students think about the structure of the expression. To truly engage with the task, students should reason quantitatively about the structure of the expression, as in the solution.

The expression arises in physics as the combined resistance of two resistors in parallel. However, the context is not explicitly considered here. The task is good general preparation for problems more specifically aligned to either A-SSE.1 or A-SSE.2.

Variation 2 of the task presents a simpler expression (the reciprocal of the combined resistance), but in a different form, and so focuses more on the ability to transform an expression for a given purpose.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This task helps illustrates Standard for Mathematical Practice 7, Look for and make sense of structure. For many students, algebraic expressions are just a jumble of letters, but students should be able to see meaning in both pieces of an expression and the expression as a whole. In this case, there are two layers of structure: the fact that the entire expression is a rational expression, and the fact that the denominator also contains rational expressions. Students should be able to reason that as a positive number gets larger, the reciprocal will get smaller, and vice versa. Then students need to reason about the denominator inside the denominator, and in the process, decode the complex structure of the expression.

Solution

An increase in $R_1$ causes $\displaystyle \frac{1}{R_1}$ to decrease. Since the denominator of the expression is the sum of $\displaystyle \frac{1}{R_1}$ and a constant, $\displaystyle \frac{1}{R_2}$, the denominator also decreases. A decrease in the denominator results an increase in the value of the expression.