SMP2: Reason Abstractly and Quantitatively #LL2LU (Take 2)

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We want every learner in our care to be able to say

I can reason abstractly and quantitatively.
(CCSS.MATH.PRACTICE.MP2)

But…What if I think I can’t? What if I have no idea how to contextualize and decontextualize a situation? How might we offer a pathway for success?

We have studied this practice for a while, making sense of what it means for students to contextualize and decontextualize when solving a problem.

Students reason abstractly and quantitatively when solving problems with area and volume. Calculus students reason abstractly and quantitatively when solving related rates problems. In what other types of problem do the units help you not only reason about the given quantities but make sense of the computations involved?

What about these problems from The Official SAT Study Guide, The College Board and Educational Testing Service, 2009. How would your students solve them? How would you help students who are struggling with the problems solve them?

There are g gallons of paint available to paint a house. After n gallons have been used, then, in terms of g and n, what percent of the pain has not been used?
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A salesperson’s commission is k percent of the selling price of a car. Which of the following represents the commission, in dollar, on 2 cars that sold for $14,000 each?
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In our previous post, SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 1), we offered a pathway to I can reason abstractly and quantitatively. What if we offer a second pathway for reasoning abstractly and quantitatively?

Level 4:
I can create multiple coherent representations of a task by detailing solution pathways, and I can show connections between representations.

Level 3:
I can create a coherent representation of the task at hand by detailing a solution pathway that includes a beginning, middle, and end.   

Beginning:
I can identify and connect the units involved using an equation, graph, or table.

Middle:
I can attend to and document the meaning of quantities throughout the problem-solving process.

End:
I can contextualize a solution to make sense of the quantity and the relationship in the task and to offer a conclusion.

Level 2:
I can periodically stop and check to see if numbers, variables, and units make sense while I am working mathematically to solve a task.

Level 1:
I can decontextualize a task to represent it symbolically as an expression, equation, table, or graph, and I can make sense of quantities and their relationships in problem situations.

What evidence of contextualizing and decontextualizing do you see in the work below?

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[Cross-posted on Easing the Hurry Syndrome]

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