cheesemonkey wonders

cheesemonkey wonders

Saturday, October 1, 2016

Algebra 1 inequalities unit - notes on a conceptual and problem-based approach

My Algebra 1 inequalities unit is now the unit with which I am most satisfied, including a solid conceptual launch and framework, some serious problem-based learning, a deep treatment of quantitative reasoning and logic, and an excellent amount of discourse-rich practice activities designed to achieve both procedural and conceptual fluency, so I want to document here how it works. If there is useful stuff in it for you, then great! But I'm not going to be posting a lot of user-ready materials here to print off and give to students, so I want to be totally up front with you.

Inequalities are still one of the most procedurally taught of all Algebra 1 topics (see Holt Algebra 1, CPM, Exeter, or anything else you can find, if you have any doubts about this), and yet, it has seemed to me for a long time that they offer the opportunity for students to ground their new understandings in one of their most accessible, intuitive existing mathematical understandings — namely, their sense of when a quantity is "more than" or "less than" some other quantity.

As Nunes and Bryant clearly show, children have a very deep understanding of comparing quantities from a very young age — including both discrete and continuous quantities. For this reason, it has long seemed essential to me to hook Algebra 1 students' work with understanding with their own authentic and meaningful prior knowledge.

So in my unit,we start by activating prior knowledge with Talking Points and problem-based learning from the Exeter Math 1 sequence. In the initial part of the unit, we investigate whether quantities are going to be more than or less than other quantities. We go from comparing known, computable quantities (such as one sum or difference to another sum or difference) to more abstract quantities (such as, If x is greater than -1, then x + 2 is greater than 1), always pressing them to rely on their own understanding or constructed understanding.

The conversations that were sparked during the initial phase — both this year and last year (with Patrick Callahan in my room) — were fascinating to all.

My rule for this unit is this: whenever I give students a "rule" (or whenever we develop a "rule" together),  it has to be a guideline that will help them to ground their new ideas in their old understanding of something. This has led me to come up with much more sensible and conceptually-based guidelines that students both remember and rely on correctly. For example, one of our "rules" is that when you have some kind of inequality, it makes sense to put it into "Number Line Order" — in other words, always organizing statements or representations about smaller quantities to the left of statements or representations about larger quantities. This has had the benefit of getting students to realize that the real number line is a tool that they can use for their own benefit. It is not just another arbitrary math teacher whim.

Another "rule" we co-developed as a community was the idea of the subjectivity of x  namely, that in mathematical sentences and statements, we should organize our symbolic representations so that x is the subject of our sentences. Another way that students expressed this idea was that x is the hero or heroine of our story, so x should be the subject of our sentences.

This too has had a profound effect on students' understanding of inequalities. Most textbooks emphasize equivalence of the statements 2 > x and x < 2, but from the perspective of student understanding, privileging both statements equally is just stupid. What we are hoping for students is that they will see that as they are trying to represent values of x, that makes x the hero of our story. So our mathematical statements should be organized to reflect the subjective status of x — not the "correct" or "incorrect" equivalence of the statements. This gave students the idea that they have permission to change statements around to suit their own  learning, and that they do not need to distort themselves to please whatever mathematical authority decided to place given statements in a form that is backwards for them.

We have also developed a "rule" for distinguishing absolute value expressions from other groupings in equations and inequalities. The idea we have developed is that absolute value expressions need to be "isolated" and "unpacked" into all possible cases of ordinary equations or inequalities (i.e., NON-absolute value statements) before you can apply any solving or simplifying techniques. Our shorthand for this method is: 1 - ISOLATE, 2 - UNPACK, 3 - SOLVE. Then once you have solved, you should put everything into Number Line Order so you can INTERPRET and GRAPH your findings.

This idea of unpacking absolute values first into all possible cases is a strong way of getting students to reason quantitatively and abstractly rather than blindly applying rules. Since they have some guidelines that are grounded in their own logical understanding and personal experience, my students have been much more thoughtful and intentional when they approach absolute value inequalities and equations. It has also led to a deepening of students' understanding of when the distributive property can and should be applied to a grouping and when it should not. This has greatly deepened the discussions of the functioning of groupings among students to the better.

Because we have build a solid conceptual foundation for our work with absolute values, practice activities have become opportunities for investigation and application of our conceptual understanding, rather than blind shooting at a list of targets from 1 to 35 odd.  At every step of the way, we have been using Speed Dating as a discourse-rich activity in which students can apply their understanding to a wide range of different problems of varying levels of complexity. It also demands that they share their understanding and speak about it with their classmates, sometimes giving and sometimes receiving assistance.

In this way, practice activities have become a new form of conceptual investigation themselves, and are not mere mental exercises designed to enhance procedural efficiency. As I am discovering through my work on National Board Certification, our profession has a major problem with privileging "discovery" activities as the only ones that occasion and showcase conceptual understanding. As How People Learn clearly lays out, it is often the case that students do NOT have a 'Eureka' moment during the initial 'discovery' segment but rather, like the young learners they are, they have a lightbulb moment once they have experienced a problem in multiple different contexts. This leads me to one of my pet peeves about why teacher certification and National Board Certification overvalue only the initial discovery activity as a venue for showcasing student insight, but that is a complaint for a different blog post.

Another benefit of treating practice as an opportunity for applied investigation is that it avoids the contrived, dog bandana pseudo-context problems that strive to provide a real-world context, but only end up twisting themselves into senseless knots. The contexts should not be harder for students to make sense of than the absolute value problems themselves! The most accessible real-world absolute value problem we have worked with is the question, When is water NOT a liquid? This makes sense to my 9th graders. Convoluted problems about the conditions under which a value will land within a range of possible values are not helping here, people. I am finding that this is a case where treating absolute value problems as a system of equations or inequalities to be unpacked and investigated as number sense problems has been far more worthwhile — and far more rewarding for students. As Deborah Ball has said, Sometimes mathematics is its own best context.


  1. I have learned to try to do this also and am glad to see it written so clearly: My rule for this unit is this: whenever I give students a "rule" (or whenever we develop a "rule" together), it has to be a guideline that will help them to ground their new ideas in their old understanding of something. I will use your idea about x being the hero of the story next week. Thanks.

  2. Thanks for your thoughtful insights.
    "Because we have build a solid conceptual foundation for our work with absolute values, practice activities have become opportunities for investigation and application of our conceptual understanding, rather than blind shooting at a list of targets from 1 to 35 odd."
    I do have a question about your thoughts on practice. As students are beginning to really understand the content, what I think of as "regular" math problems can be quite interesting. Is your shot at 1-35odd because students are not interacting deeply with the problems but instead just trying to get them done?

    1. My shot at '1-35 odd' was actually in response to what I've been reading and working on as I do my National Board certification. Over and over again, the instructions and everybody's best guidance on Component 2 all tell you in no uncertain terms that the activities you feature should NOT be practice activities but rather "discovery" activities or applications. What bothers me about this is that I often see my students doing their deepest work when they are actively trying to complete different kinds of practice activities but instead, end up investigating subtle and important differences between questions that lead to breakthrough insights. I find it frustrating that they privilege so-called "discovery" activities over other forms of authentic discovery that happen in certain contexts for many students.

      Hope that is helpful!