My last post focused on three major mistakes I made in my first semester of skill-focused, mastery-based assessment: separating skills into chunks that were impractically small, choosing some skills that were too simple (almost trivial), and neglecting to plan for the end of the semester. This post will focus on my process creating the skills list for semester two (for Algebra 2). I’d love to hear your opinions or your own process – leave a comment or a link below!

The place to start is your curriculum map, whether that’s a list of topics, a set of state standards, a final exam, some chapters of a textbook, or whatever. Find or create the document that describes what you hope to teach this semester. The list I used last semester I got directly out of my textbook. I knew what chapters I was going to teach, and I just ripped concepts out of the table of contents. This process got me a list that I was moderately happy with; click here to download it. Your list will almost certainly need to be different, since I was planning for 36 class periods.

For semester two, I set out in much the same way. I went through the chapters in my text book I was planning on studying, and every time something popped up that seemed like it would be a good candidate for a skill, I wrote it down. This gave me the following list of 36 skills:

Evaluating functions |

Analyzing the domain and range of functions |

Modeling relationships with functions |

Modeling arithmetic sequences with functions |

Modeling geometric sequences with functions |

Distinguishing between arithmetic and geometric sequences |

Recognize exponential growth from situations, tables, graphs, or equations |

Understand multiple representations of exponential functions |

Represent exponential functions algebraically |

Using basic laws of exponents to simplify expressions |

Use exponential functions to solve problems involving growth or decay |

Find equations of exponential functions through two given points |

Identify graphs of quadratic, cubic, square root, absolute value, etc functions |

Transform a graph by stretching, shifting, or flipping it |

Write a general equation for a family of functions |

Use the “completing the square” technique |

Model physical situations with quadratic functions |

Write equations in graphing form |

Invert functions analytically and graphically |

Form compositions of functions |

Express the relationship between a function and its inverse |

Understand logarithms and transform their graphs |

Use properties of logarithms |

Use logarithms to solve exponential equations |

Count possibilities in situations that require a particular order |

Count possibilities in situations in which order does not matter |

Draw a tree diagram to represent and calculate probabilities |

Draw an area diagram to represent and calculate probabilities |

Calculate expected value |

Using the fundamental principle of counting |

Calculate conditional probabilities |

Find the value of arithmetic series of arbitrary length |

Find the value of geometric series of arbitrary length |

Find the value of geometric series with infinite length |

Writing a series with summation notation |

Using mathematical induction |

The next step in the process is to look at each skill from the brainstorm and ask,

- How will I test this skill?
- Is this skill big enough to be its own skill?
- Is this skill small enough to be a single skill?
- Does this skill have multiple levels, so that intro level tests will be significantly different from master level skills?
- If a student does not understand this skill at all, am I willing to flunk him? (My grades are set up that each student must get a minimum of 3/5 in every skill to pass the class. If you’re using a simple average, you can ignore this question).

Consider the first skill, “Evaluating Functions.”

- How will I test this skill? What leaps to mind is showing a kid f(x)=3x+6, and asking for f(2). Maybe f(f(2)) – or is that composition? They also need to be able to evaluate functions from graphs and tables. Maybe the question should be a three-parter?
- Is this skill big enough to be its own skill? Hmm… it’s pretty small, isn’t it?
- Is this skill small enough to be a single skill? Yes, I am confident that it is.
- Does this skill have different levels? So, I could ask them to evaluate f(x)=3x+6, or f(x)=3x-2/x+(x+5)^-3, but those aren’t different levels of evaluating functions, those are different levels of order of operations or something. I could ask them to find g(f(2)), but maybe that’s composition.
- Is this skill a requirement of passing the course? Absolutely. I am not letting anyone who can’t evaluate a function out of Algebra 2.

So this first skill has some complications. I really like testing f(g(2)) because it requires students to understand the input/output aspect of functions where I feel like a simple g(2) might let them slip by without it. Since this skill is so essential, I’m leaving it in, though it might be a little bit small. It may end up conflicting with the composition skill, but there might be flexibility in that skill. “Evaluating functions” seems like a solid requirement.

Let’s take another skill, “Modeling arithmetic sequences with functions.”

- How will I test this skill? I’m looking for students to be able to come up with functions that describe arithmetic relationships, like, “write a function that outputs the number of gloves that x people will need,” or something. I could show a table of inputs 1, 2, 3, 4, and outputs 8, 11, 14, 17, and have them write this function.
- Is this skill big enough to be on its own? You know, I think it’s possible it could be combined with the skill before it, “modeling relationships with functions,” and the one after it, “modeling geometric sequences with functions.” These three skills are so closely related, with the only difference being the arithmetic skills required. I’m not trying to test those arithmetic skills – I hope the kids already have them – so I’m going to combine these three skills into just “modeling relationships with functions.” I’ll answer the rest of these questions for the new skill.
- Is this skill small enough to be on its own? Clearly the title can involve arbitrarily complex functions and relationships, but I think the kinds of simple relationships we’ll study in class can all be combined under one roof. This skill may be a little bit too big. If I was the organized man I wish I were, I’d note somewhere that I should revisit this skill after we touch on it to see how I felt.
- Are there different levels for this skill? For intro level questions I can ask the students to model the relationship between Celsius and Fahrenheit, or some other linear relationship. For master level questions I could ask a geometric volume question which includes an extra level of abstraction.
- Is this skill absolutely required for every student that passes the class? Yes, I think so. Who wants to teach precalc to students who can’t create their own functions?

Now, I don’t have time to write an essay about each of these questions for each of these skills, and neither do you. In this post and in my brain I’m deciding to move quicklky here. I only have so many hours to get this done, and it’s not going to be perfect. I hope that you can make sacrifices like this – it’s taken me a long time to accept the impossibility of perfection in a finite time frame. I spent about 30 minutes considering this list, eventually deciding to cut 7 skills and reword several. Click here for my final checklist.

I hope this post has showed you how easy it can be to come up with a list of representative skills to assess. It’s an unglamorous process, and the hardest part is coming up with the rough list, but once you do that you can have an effective list in less than an hour. I don’t recommend using my skills lists wholesale. I am in the process of trying out several different textbooks and my order is wonky. My school is exempt from most standardized tests and if you have specific objectives you need to hit you’ll need to take them into consideration, obviously. That said, here are my skills lists for Algebra II and Calculus this year:

Also, Dan Meyer has posted his suggestions for Algebra 1, Geometry, and Precalculus (imagine my chagrin when I saw he covered exactly everything but what I needed!). So, if you’re considering getting started with this system, you at least have a launching point for your class. If you have comments about my lists or process, I’d love to hear about them! I’m especially interested what you think of the questions I use on each skill. What else needs to be asked?

I’m interested in how you assess. Is it similar to Dan’s method? How many concepts do you assess at once? How often do you assess? How do the students prove they have mastered the material? What kind of assignments are given where concepts are combined?

What is hardest for me is differentiating between a section of the book and a specific skill. My skill list is basically the objective for each section which is so huge. I can’t seem to narrow it down into the appropriate skill.

My method is very similar to Dan’s method indeed. The complete description is at http://larkolicio.us/blog/?p=3 . Mastery is tested by a skills test with interacting nuances of a skill, given at a later date than the initial “intro” test. Concepts are only combined in class, where we do lots of group work and “real world” kind of things.

My text book doesn’t have a single objective for each section. Instead of saying, “The student will understand logarithms,” for example, it says something like, “this section will help you understand how to 1) calculate logarithms, 2) simplify logarithmic expressions, 3) graph logarithmic equations, 4) solve logarithmic equations using exponents.” If you post something you’re having trouble breaking down, maybe we could help break it up!

Very cool…I’m going to have to think about this more. Thanks for sharing your process.

On the topic of evaluating functions, plugging in to x+5 is definitely easier than plugging into x^2+x (because there are two x’s). Also, plugging in 3 is easier than plugging in -3 which is about the same as plugging in b which is easier than plugging in 2b which is easier than plugging in b+1.

Thanks for pointing out the important difference between x+5 and x^2+x. I like that difference for this skill much more than I like the difference between 3 and -3, which may point out different problems with arithmetic but is unlikely to highlight a problem with the process of evaluating functions. I might just START with -3!

On the other hand, I completely missed f(b+1). That’s a doozie, but an important doozie. What about this:

intro-level test:

Part 1: If f(x)=x^2+x, evaluate f(-3)

Part 2: Given the graph below of g(x), evaluate g(3)

master-level test:

Part 1: If f(x)=x^2+x, evaluate f(b+1)

Part 2: If f(x)=x^2+x, evaluate f(f(2))

Does this cover everything? Is the intro-level appropriate to be required? Is the master-level significantly different from the intro?

Or even:

master, part 2: Given the graph below of f(x), evaluate f(f(2))

Riley,

Thanks for posting about this. My colleague and I have spent a few hours tearing apart our CA state standards for Algebra 2, reconciling them with the flow of our outdated book, all the while being sure to address the topics included on the district mandated final exam and the California Standards Test (CST). It’s exhausting work. Everything I’ve read on SBG definitely states that the concept list is a work in progress. ActiveGrade on the horizon!

Good luck! I just found your blog – keep us updated!

A colleague and I stumbled upon Dan’s blog/method very recently and we are looking to implement this year next year. I teach Algebra 2 and Calculus. Your concept lists look great! Thanks for posting them.

Good luck! Let us know how it goes