We are right around a month into the school year, so I thought it might be a good time to do a general check in. A few of my recent posts have been pretty long winded (I just have a lot of feelings okay!?), and I have a few random notes and ideas to share that are not necessarily worth a whole post. So check out the headings for ideas that might interest you.
What We Are Up To
In 7th grade Pre-Algebra we have reviewed fractions (quickly) and integers (very-quickly). We also looked at some arithmetic properties and started to generalize patterns using variables. This week we have been working on representing relationships generally with equations.For example, we looked at a basketball scores. Since there are only three ways to score, we can represent their total score as 3*three-pointers made + 2* two-pointers made...etc. This really threw the kids. The representation was just too foreign. Things clicked better when I had them figure out a teams score based on specific values -- 6 three-pointers, 4 two-pointers, and 7 free-throws for example. That helped them see how the variables were going to help determine the final score. We have also been working on simplifying expressions after substituting in values for variables which necessitated talking about order of operations. Up next is combining like terms and solving equations.
In 6th grade we have begun the CMP3 unit on fraction operations (Let's Be Rational). The unit begins by asking students to estimate fraction sums. This was great for returning students who had strong conceptual models for fractions, but some of the new kids who were more used to just being told the algorithms have been challenged by the curriculum. I have been really impressed, though, by their willingness to adjust to a math class that "feels really different." They have been working so hard! It helps too that all of the returning students really know my routines and can serve as wonderful models for the new students.
What Is Great
While it has been a ton of work to organize two new courses, I am really enjoying the intellectual challenge. Pre-Algebra is a whole new course for me, and I am loving figuring out what models or examples are clearest for the kids. Teaching sixth grade is great too. I have really enjoyed re-teaching ideas and content that I taught in my second year when I had almost no idea what was going on. It has helped me see that ways in which my teaching has grown.
Documenting my lessons and organizing my course documents is big goal for me this year. This summer, while working with the teacher that is taking over my 5th grade course, I realized that more of that course exists only in my brain than I would like. My goal this year is to ensure that is not the case for my new courses. I am tracking my lessons (with links to relevant documents) in a google sheet and I have been using a google presentation instead of notes under a document camera.
The presentation has been great in a few ways. The most significant way is that it has allowed me to build good (computer drawn) visual models ahead of time as I am planning. I can then adjust these after one class meeting so that they are better for the next. Another benefit is that, by screen-capturing the textbook, I can draw directly on top of the questions or diagrams. This has really helped our summary discussions at the end of class. Of course, the presentation also means I have a record of what I did in each class which I can share with the kids.
What I Am Struggling With
I am struggling with figuring out what justification I should expect from the students about WHY common denominators are necessary for adding fractions. The kids see that common denominators make adding and subtracting fractions easier but that does not justify why it is a necessity. Here are two excerpts from the mathematical background section of the CMP3 Teachers' Guide:
These are great models -- they are represented more appropriately and in greater detail in the student text -- and really help the students understand what is happening when fractions are added. However, I still don't think they provide a clear reason as to why common denominators are a necessity for adding fractions.I think many of my students are still
waiting hoping for an easier way that doesn't involve making common denominators.
I am leaning towards not expecting them to justify the necessity of common denominators. Instead, I will just ask them problems that require them to reason with different denominators like tupelo township.