Comparing Freezing Points

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:

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.