Ice Cream

Task

After a container of ice cream has been sitting in a room for $t$ minutes, its temperature in degrees Fahrenheit is $$ a - b\cdot2^{-t} + b, $$ where $a$ and $b$ are positive constants. Write this expression in a form that

1. Shows that the temperature is always less than $a + b$.

2. Shows that the temperature is never less than $a$.

IM Commentary

This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard A-SSE.B.3. Students are provided with an expression giving the temperature of a container at a time $t$, and have to use simple inequalities (e.g., that $2^t>0$ for all $t$) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.

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 is linked to Standard for Mathematical Practice #7, “Look for and make sense of structure.”  This expression can convey different meanings based on the form it takes on, therefore putting its own structure to use.  By grouping terms, and thinking about whether terms are positive or negative or how the value of t affects a term, students can understand the expression better.  This gives meaning to the algebraic manipulations that students are asked to make, giving students a reason to write the expression in different forms.  By looking closely at the structure of different versions of the expression, students can recognize properties of the expression. 

Solution

1. To begin, we can first rearrange this expression into:

   $$ (a+b)-b\cdot2^{-t} $$

   We can now see that we have an $a + b$ together on the left, and our last term is $b\cdot2^{-t}$, and this term will dictate if the temperature is greater or less than $a + b$. Since $b$ is a positive constant, and since $2^{-t}$ is positive regardless of the value of $t$, we know that $b\cdot2^{-t}$, is positive. So, we have

   $$ a+b-b\cdot2^{-t} \lt a+b $$

2. We can rearrange the expression in the following way:

   $$ a+b-b\cdot2^{-t} = a+b \left( 1 - \frac{1}{2^t} \right). $$

   We see that now the term $b(1 - \frac{1}{2^t})$ is going to dictate if the temperature is greater or less than $a$. Since $t$ is the number of minutes that the ice cream has been sitting in the room, we know that $t$ will always be greater than zero. Therefore, $2^t > 1$, so $\frac{1}{2^t} \lt 1$, so $(1 - \frac{1}{2^t}) \gt 0$. From this we can conclude that since $b > 0$, we have $b(1 - \frac{1}{2^t}) \gt 0$.Therefore,

   $$ a+b\left(1 - \frac{1}{2^t}\right) > a $$