Monday, March 18, 2013
Functions and Algebra I Series - Post 2
Working with the algeblocks again to solve equations hopefully facilitated an effective way of helping students articulate the mathematical procedures they are doing when solving equations. Students in grade 5 on up should have these kinds of experiences. I think you agree that if you don’t understand the fact that the product of any two factors is a rectangle can cause angst in an algebra course when working with binomials. Obviously, given a trinomial in which like terms have already been calculated is a lot more difficult then when all the terms are nicely represented in the rectangle.
Factoring a variety of trinomials is crucial in helping all students understand the differences the signs of the trinomials represent. For instance if all the signs are positive then the factors are all positive, x2 + 4x + 4 or (x + 2)(x+2) for example. Or if the first operator is negative and the second positive then the factors have two negatives x2 – 4x + 4 or (x – 2)(x - 2). But the tricky part is when you have a trinomial where you have two negatives such as x2 – x – 2 which indicates like terms have been computed from the binomial multiplication (x-2)(x+1) . Not easy to visualize unless you are using manipulatives.
I enjoyed the excitement you displayed with the CBRs when modeling graphs. I was impressed that many of you made up more and more challenging graphs to model. This engenders the spirit of inquiry that we are trying to foster with our students as well as helping to visualize what distance graphs actually represent.
Two hands on activities hopefully broke up the paper and pencil work we have to do as well. I realize how hard some of us found the toothpick problem and how important it is to hear from everyone as each time one of you presented the way you solved the problem it helped others who were stuck in their own thinking and could not change that vision in their head or the values in the tables.
The inclusion of functions and function notation is a critical component of algebra and algebraic reasoning. So much so it permeates the Common Core. The idea of function machines is so appropriate in the lower grades that the transition to the notation in algebra will be seamless. We did some work with functions but will solidify it next weekend. I felt that if I pushed up to compositions of functions Saturday, it would have been overload so we will begin with functions at our next session.
I appreciate the emails some of you send asking for clarifications…always happy to oblige. Your homework assignment is very clear and Katie is sending the blogging info for those of you who still need it.
In terms of giving the homework on Sat or Sun, I need to make sure the homework reflects what we did in class so I do need the extra day to make it appropriate. On Sun, I am so exhausted and like you work all week and need some family time I have to give myself permission to get the homework to you on Monday. I hope you understand.
Another exhilarating weekend for me.