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Comparing Freezing Points


Alignments to Content Standards: 7.NS.A.1

Task

Ocean water freezes at about $-2 \frac12 ^\circ C$. Fresh water freezes at $0 ^\circ C$. Antifreeze, a liquid used in the radiators of cars, freezes at $-64 ^\circ C$.

Imagine that the temperature has dropped to the freezing point for ocean water. How many degrees more must the temperature drop for the antifreeze to turn solid?

IM Commentary

This task is appropriate for assessing student's understanding of differences of signed numbers. Because the task asks how many degrees the temperature drops, it is correct to say that "the temperature drops 61.5 degrees." However, some might think that the answer should be that the temperature is "changing -61.5" degrees. Having students write the answer in sentence form will allow teachers to interpret their response in a way that a purely numerical response would not.

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 most beneficial for a teacher to use to assess a student’s understanding of differences of signed numbers.  To best assess this understanding, the student needs to describe in context what must happen for the antifreeze to solidify making sure to use clear, precise language when doing so (MP.6).  The teacher could use this task after multiple contextual and numerical rational number subtraction and addition problems using number lines, counters and the prior understanding of rational numbers developed in 6th grade. The teacher could determine from this task if the student is starting to formalize the rules for addition and subtraction operations with integers through the interpretation of the contextual answer the student gives. Notice that the commentary allows for a negative answer as long as the student gives a clear and correct interpretation of that answer in the sentence surrounding the number. 

Solution

We want to find the difference between the freezing point of ocean water and of antifreeze:

Freezing_points_numb_f9224d165c6bfe6f7ce60a74c8323de8

The difference between the temperature that ocean water turns to a solid and antifreeze turns to a solid is $$-2.5 - (-64) = 61.5$$ So the temperature must drop another $61.5^\circ C$ after ocean water freezes for the antifreeze to turn to ice.

Ken Mullen says:

almost 5 years

The Common Core places emphasis on modeling and making sense of problems. If it's easy to attend to more details, it might be worth doing so. Antifreeze is either what comes in the bottle (generally mainly ethelene glycol) or else the mixture of antifreeze and water that is in the car radiator. Pure ethelene glycol freezes about -12°C. If it's mixed in the right ratio with water, the freezing point can go down to about -51°C according to Wikapedia. Also, saying that something called antifreeze is used to cool cars might be a distraction to some students. The words seem like they go in opposite directions.That's ok if there's time to discuss that point.

Is it ok to say the antifreeze turns to ice or is "ice" limited to water?

Kristin says:

almost 5 years

Good call on the "cooling" issue and "ice"--I think I fixed both of those.

According to the same wikipedia article, it states that the mixture "70%/30% [provides] maximum freeze prevention down to −84 °F (−64 °C)." So I think we are safe with the numbers given in the task. It is true we could be more precise and say it isn't the pure antifreeze that produces this result, but given the purpose of the task I don't think that is necessary.

Karen Martinez says:

about 5 years

removed

Peter Cincotta says:

about 5 years

I like the problem. It has a good and interesting context; a context I hadn't seen before in such a problem. It only has a few words; it is easy to read. It (of course) addresses the standard. I wonder if there is a better way to ask the question to make it more clear that we are seeking a positive number answer? It is correctly stated that the temperature must "drop 61.5" degrees, but I wonder if some might think that the answer should be that the temperature is "changing -61.5" degrees. I wish I could offer a suggestion for better wording. I admit that I cannot.

Ken Mullen says:

almost 5 years

The word "change" is used in both ways. The temperature has to change by about 61.5 degrees in order to get down to the freezing point of the antifreeze (the antifreezing point?). That's a natural use of the word. I think the commentary gets the point across.

Franki Dockens says:

almost 5 years

Perhaps if the question was reworded, How many degrees more must the temperature drop for the antifreeze to turn to ice?, this would suggest an answer that is expressed as a positive number. Or, For the antifreeze to turn to ice, how much more, in degrees, must the temperature drop?

Kristin says:

almost 5 years

OK, that's a good idea. The only issue for me with that is it sounds like the temperature was already dropping, so I changed the previous sentence too. How does it look now?

Karen Martinez says:

about 5 years

Peter, this is a common problem with my students as well. They often ask if the answer is positive or negative. The number sentence, however, shows why it should be a positive result. The students need to understand that the question is asking for the difference (subtraction) between the two numbers, which is the same as finding the range between the two numbers. Writing the number sentence correctly should yield the positive result. Also, suggesting a written sentence to reflect the result is a great way to have the students express the correct result, as the commentary now states. Also, reminding the students that the answer reflects a distance between two temperatures, the result should be positive.

Kristin says:

about 5 years

Thanks, Peter; this is an excellent point. I'm also having a hard time deciding how to improve the wording of the task. I'll add something to the commentary based on your comment here and perhaps someone else who comes along will have a suggestion for improving the statement of the question.