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# Ice Cream

Alignments to Content Standards: A-SSE.B.3

After a container of ice cream has been sitting in a room for t minutes, its temperature in degrees Fahrenheit is $$a - b2^{-t} + b,$$ 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:

$$(a+b)-b2^{-t}$$

We can now see that we have an $a + b$ together on the left, and our last term is $b2^{-t}$, 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 $b2^{-t}$, is positive. So, we have

$$a+b-b2^{-t} \lt a+b$$
2. We can rearrange the expression in the following way:

$$a+b-b2^{-t} = a+b \left( 1 - \frac{1}{2^t} \right).$$

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,

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

#### Lisa says:

I'm curious from where a and b are derived. The problem says they're positive constants. What are they? Are they the initial temperature of the ice cream and the temperature of the room? In order for the context to have meaning, it would be helpful to know and would give a little more value to the context. Otherwise the context feels contrived.

#### Cam says:

Here, $a+b$ would represent the ambient temperature, and $a$ would represent the initial temperature of the ice cream (so $b$ represents the initial difference in the temperatures). While I see the merits of your main point, I think I ultimately disagree -- this isn't a modeling task where students try to write down an expression giving the temperature of ice cream as a function of time. The intent of the task is to model the process of being given an expression, and deducing consequences via the algebraic structure in the expressions. For example, knowing what $a$ and $a+b$ represent physically makes the solution to the task trivial (at least, with physical intuition). On the other hand, being able to deduce those things without being spoon-fed every detail about the context seems to me a valuable skill.

This is an interesting discussion to have, though -- I'd welcome further feedback and comments from anyone else, as well.

Instead of definitively telling students that the initial expression gives them the temperature of the ice cream, you could ask them whether the expression seems reasonable, given the definitions of a and a + b. You could tell them that, in order to do so, they need to show that the temperature of the ice cream is always less than the ambient temperature (part a) and never less than the initial temperature (part b). Or, you could just allow them to figure that out on their own (a more rigorous but more time-consuming task). Either way, this would help contextualize the scenario without trivializing the mathematics.

#### Cam says:

Certainly! I think there's ample room for discussion in the class about reasonableness of this model, either prior to or after the mathematical analysis. One risk we run into in writing these tasks is that it's often easy to see how to expand small exercises into period-long discussions, but quite hard to figure out how to encapsulate part of the core material of the standard in a brief exercise. The current item tries to do the latter with A-SSE.B.3, but I certainly agree that individual teachers could spend ample amounts of time with this context for the sake of, say, discussing modeling.

#### Robert Hawke says:

over 5 years

There is a trivial case when t=0, at the instant the system is set in motion. Because of this, it's not technically true that the expression is always greater than a.

I don't think you want to give too much away by saying "for t>0", but I don't think you lose the contextual framing by adding "after the system is set in motion" (or something similar) to the questions.

#### Cam says:

over 5 years

Good catch. My inclination would be to change part (b) to read either "is always greater than or equal to $a$" or "is never less than $a$." Objections? Preferences?

#### Robert Hawke says:

over 5 years

Good work-around. I like "never less than a" better - it sounds less technical, and pushes kids towards the line of reasoning that begins with something like "what would it take for this expression to be less than a?" It also lets them make the connection between "never less than" and "always greater than or equal to." Plus, it makes sense. The ice cream is never colder than a certain temperature.

#### Cam says:

over 5 years

Agreed, and it nicely mirrors the language of part (a) ("always less than"). Sounds good.