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It's Warmer in Miami

Alignments to Content Standards: 6.NS.C.5

One morning the temperature is -28$^\circ$ in Anchorage, Alaska, and 65$^\circ$ in Miami, Florida. How many degrees warmer was it in Miami than in Anchorage on that morning?

IM Commentary

The purpose of this task is for students to apply their knowledge of integers in a real-world context. In 6th grade, students do not need to know that the problem can be represented by finding the difference of the temperatures. In 7th grade, students will be expected to formally connect the answer to this problem with the difference of the signed numbers:

Because $65- (-28) = 93$, we know it is 93 degrees warmer in Miami, Florida than it is in Anchorage, Alaska.

Solutions

Solution: 1

The temperature in Anchorage is 28 below zero and in Miami it is 65 above zero, so the difference in temperatures is $28+65=93$ degrees.

Solution: 2

We can count from -28 up to 65. If Anchorage, Alaska was 28 degrees warmer than it is on this winter morning, the temperature would be zero degrees. If Anchorage, Alaska was 65 degrees warmer still, the temperature would be 65 degrees, the same temperature as Miami, Florida. In order for Anchorage, Alaska to be the same temperature as Miami, Florida, Anchorage would have to be $28+65=93$ degrees warmer than it is.

Thus, Miami, Florida is 93 degrees warmer than Anchorage, Alaska.

bcohen says:

almost 6 years

This is a good task, but it seems to require more than 6.NS.5, which states "Understand that positive and negative numbers are used together to describe quantities having opposite directions or values...; use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation."

This task is looking for a difference, which is not introduced in grade 6 (it is introduced in standard 7.NS.1).

Kristin says:

almost 6 years

Thanks for your careful reading of the standards and the task. While technically one can represent this problem using a difference (which is, as you say, 7th grade) it is also possible for students to answer the question simply by reasoning about the meaning of the numbers in the context. I've changed the commentary and solutions to reflect this difference in expectation between 6th and 7th grade.

Lincoln says:

almost 3 years

Here is the language for 6.NS.5: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

I find it interesting that both of the tasks for 6.NS.C.5 (Warmer in Miami and Mile High) were flagged by two different members of the community as misaligned 7th grade questions. I am now the third person that is contending that this not a 6th grade question, despite the logic that Kristin provides. Just because a scholar could apply his or her understanding of the numbers in the context of this problem, doesn't mean that the 6th grade standard calls for it. The standard asks for scholars to represent positive and negative quantities. Representation does not mean application by carrying out operations. This should be the end of the debate.

Furthermore, the focus of the standard is much more around the DIRECTION that the numbers take in a given context and and not about the MAGNITUDE. This task deals with both direction and relative magnitude to find a solution. I could see this task as some sort of bridge to the 7th grade standards but then there should be a number of actual 6.NS.5 tasks from Illustrative Mathematics that cover this standard as it is described. Until then, I suggest that both of these tasks get properly filed under the 7th grade content standards.

While I really do find IM tasks to be rigorous, interesting and a valuable resource to the math education community, there should be more work done by the IM team to be responsive to repeated alignment concerns.

Dev Sinha says:

almost 3 years

Associate Director, careful alignment is indeed crucial. Why I view this differently is that coherence implies that special cases of "later grade" mathematics which are accessible with earlier tools can and in fact should be used - assuming they have good merit as tasks and they are used with care. For example, some problems which used to be first approached in "prealgbra" as they lead one to solving equations like 3x + 5 = 17 are now readily done as early as third grade with tape diagrams and the meanings of addition, subtraction, multiplication and division.

In the case at hand, representation on the number line gives a reasonable way to solve a problem which could also be solved by the more formal subtraction at seventh grade. This type of solution is common among proficient people, visualizing the problem in a way which immediately calls for addition rather than "setting it up" as subtraction. Access to this approach is arguably more important. While I see as valid a curricular choice which delays such "visual" solutions until the formal subtraction of integers in seventh grade, in which case the visual approach would be an alternate path, I for one would prefer having this work at sixth grade. In the end, proficiency with representation is much deeper if it is used to solve reasonable problems. I agree there is a danger of misunderstanding here, where things are taught by introducing 7th grade ideas, but with appropriate care this is a good activity for the sixth grade standard.
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