{"id":2101,"date":"2022-06-01T01:26:00","date_gmt":"2022-06-01T01:26:00","guid":{"rendered":"https:\/\/www.terc.edu\/adultnumeracycenter\/?p=2101"},"modified":"2022-06-01T11:38:56","modified_gmt":"2022-06-01T11:38:56","slug":"will-this-be-on-the-test-june-2022","status":"publish","type":"post","link":"https:\/\/www.terc.edu\/adultnumeracycenter\/will-this-be-on-the-test-june-2022\/","title":{"rendered":"Will This Be on the Test? (June 2022)"},"content":{"rendered":"\n

by Sarah Lonberg-Lew<\/p>\n\n\n\n

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Welcome to the latest installment of our monthly series, \u201cWill This Be on the Test?\u201d Each month, we\u2019ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually.<\/em><\/p>\n\n\n\n

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Welcome back to our continuing exploration of how to bring real conceptual reasoning to questions students might encounter on a standardized test. In April<\/a>, we looked at a question that mixed contextualized and decontextualized representations of a relationship. Sometimes, though, we encounter questions that are purely decontextualized:<\/p>\n\n\n\n

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Note: This question is lightly adapted from one found in <\/em>Tools of Algebra: Expressions, Equations, and Inequalities Part 2<\/em><\/a>, a free resource from CUNY designed for adult learners.<\/em><\/figcaption><\/figure><\/div>\n\n\n\n
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How many ways can you think of to approach this task? Do you have a memorized procedure? How would you tackle this if you didn\u2019t? What skills and understandings do students really <\/em>need to be able to answer this question? How could you use visuals to help you reason about this?<\/strong><\/p>\n<\/div><\/div>\n\n\n\n

As I\u2019ve said earlier in this series<\/a>, not every problem is one for every student to take on. This one is very abstract and for students who are not ready to work with these kinds of relationships at this level of abstraction, it may be best to just make a guess and move on. The progression from concrete to representational to abstract is real and important. Just because a student can reason about a certain algebraic concept concretely does not necessarily mean they are prepared to reason about it abstractly. However, we can help students build that bridge and bring their reasoning to more abstract levels. (By the way, please don\u2019t confuse \u201cabstract\u201d with \u201chigh level\u201d. The progression of concrete to representational to abstract is one that happens over and over with each new concept and at every level. Students reasoning more concretely can still take on problems involving systems of equations. See the December 2020<\/a> post in this series for an example.)<\/p>\n\n\n\n

Also, there are a couple of notational\/vocabulary details students need to be comfortable with to solve this problem. Specifically, students need to know what it means for a value to \u201csatisfy\u201d an equation (the equation will be true when the variable is replaced with that value), and they need to understand that the ordered pairs in the answer choices give the values of x and y in that order. These are understandings that can be taught as when they become relevant. When students are ready to abstract their thinking, they will have an intellectual need for these details.<\/p>\n\n\n\n

With these understandings, let\u2019s press on to making sense of the problem and persevering in solving it. Here are some possible approaches:<\/p>\n\n\n\n

1. Turn it into a story. <\/strong>It could be quite a bit of work to come up with a realistic story that the system of equations models, but a student can still take something that looks like alphabet soup and make it more accessible by telling the story in words. For example, a student might retell the story of the system this way:<\/p>\n\n\n\n

There are two numbers. One thing I know about them is that if you double one of them and subtract the other from it, you\u2019ll get 11. The other thing I know is that together they add up to 10. I want to figure out what the two numbers are.<\/em><\/p>\n\n\n\n

This is probably not the most compelling story you\u2019ve ever read, but it is a way of making sense of the problem that makes it feel more accessible. Framing this \u201cproblem\u201d as a game or a puzzle, even a quest, can free a student from feeling stuck trying to remember the right procedure. What ideas about playing with the numbers come to you from reading this story that you might not think of when just looking at the equations?<\/p>\n\n\n\n

2. Draw a picture.<\/strong> It can be challenging to draw pictures of subtraction, but the visualizing the relationship in the second equation could really help students get a handle on what kinds of numbers might be solutions to the system. A student who has had opportunities to work with concrete manipulatives like counters, might imagine the second relationship as a group of ten counters in two different colors like this:<\/p>\n\n\n\n

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3. Reframe subtraction.<\/strong> One thing that makes it challenging to draw pictures of subtraction is that it often feels like an action \u2013 the action of taking away. However, a student who has developed operation sense with subtraction may be able to frame the first equation in one or more of the following ways:<\/p>\n\n\n\n