SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 2)

29 Sep

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

I can reason abstractly and quantitatively.


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?


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?


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.  


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


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


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?

Screen Shot 2014-09-24 at 8.43.43 AM Screen Shot 2014-09-24 at 8.46.55 AM Screen Shot 2014-09-24 at 8.43.55 AM 


[Cross-posted on Experiments in Learning by Doing]


1 Comment

Posted by on September 29, 2014 in Standards for Mathematical Practice


Tags: , ,

One response to “SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 2)

  1. howardat58

    September 29, 2014 at 7:45 pm

    Just looking at the three calculus example solutions: score is for de/con-textualising
    1: Knows what they are doing, keeps track of the units, but doesn’t make it to the end. Score-good
    2: I can’t follow this one at all. Score – bad.
    3: Correct result, difficult to follow, score – bad.
    Far worse than letting the units take care of themselves is the complete absence of explanatory text.

    More use than following each line of a solution development with an account of the units etc is the ability to do a dimensional analysis of the final formula, before the numbers go in, as a check on the units.
    I did like the way that number 3 saw a better way than using the formula given at the bottom – probably get zero as a result (my cynical side speaking).


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