Ratio, Rate and Proportion Series - Post 2

I wrote an epistle reflecting on our past three sessions but somehow it did not publish. Now, I am trying to reconstruct what I wrote.

On Tuesday, we introduced fraction as number and fraction as ratio on the Cartesian coordinate plane. Both operate differently due to the continuous nature of fraction as number and the discrete nature of fraction as ratio. What they both have in common however is the visual representation that allows us to see what it means to have a common denominator. Too often that is an abstract concept that students can determine but which many just do not understand. The way in which we introduce division can set up our students for success or for reliance on cutesy sayings rather than helping them understand we are simply looking to see how many groups of something we have or how many items each member of a group will get.

Using the graphs to convert ratios to percents is also a valuable tool for students especially when thinking about slope as percent…think about a 40% grade on a hill. Students need to be able to move seamlessly within the ratio, decimal/percent representations and this is yet one more model to use.

On Thursday, we really worked hard solving problems on the Cartesian coordinate plane. It is really challenging to use a model for the first few times. None of us were taught this as students yet the CCSS progressions insist that our students use it as a problem solving strategy. This model really emphasizes the concept of ratio as being discrete and answering the question "How Many?"  But, what I found most interesting is how hard it is for some of us to move to a “relative” representation rather than an exact numerical one. This is really a stretch since we have not been trained to think that way. What an advantage your students will have from their exposure to thinking like this!

The highlight of the week was our field trip to the Science Museum. I was really impressed with the depth of conversation around the different exhibits especially the Mt Everest exhibit. I was listening to a news show this morning and they mentioned that the base camp is 17,000 ft above sea level. Imagine!!! Tom mentioned how cold it must be, but I was thinking about the thin air. I had trouble breathing atop Pike’s Peak which is a measly 14,000+ feet above sea level. Of course today hikers bring portable oxygen tanks, but imagine Sir Hillary doing it au natural.

The other interesting point for me was seeing the difference between the pencils made to a scale of 12:1 versus a scale of 10:1. The end products were amazing.

I look forward to reading your posts.

Anne

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.

Anne

Functions and Algebra I Series - Post 1

I don’t know about all of you but I was exhausted at the end of our marathon Friday evening -all day Saturday immersion into the deeper look at algebra. Notice that although we were doing algebra we did very little paper and pencil procedural skills or practice. That was by design. It is my belief that as we enter an algebra course we discard the notion that algebra is about manipulating symbols and embrace the fact that algebra is a generalization of arithmetic and is a logical, sequential, representation of the data that surrounds us.

If you think about how algebra was invented back in ninth century by the Arabic mathematician Al-Khwarizmi (whose word al jabr, describing a common process that we use in algebra, gave us the word algebra) do you picture a man sitting around simplifying expressions and solving equations, or do you perceive this gentleman experimenting and investigating the phenomena he saw around him, a man who tried to make logical sense of those phenomena and develop a structure that would always hold true?  He most certainly “played” with the mathematics as he worked out a systemic process by making conjectures, testing them, revising them, and testing them again. 

What I hope we accomplished this weekend is that need to model the phenomena whether it was through the water vases or the slinkies. The inclusion of Hooke’s Law was a more formal method for discovering a general equation that will always work. The equation y = 0.07x which most of you discovered illustrates the k constant rate of change to be 0.07 which is a ratio, the slope, and as someone in class stated the rise over run. The problem that we worked on with the texting options formalized how an equation with a constant rate of change can look in a table, graph, and equation. Then the big question arose…what do you do with the data? How do you interpret the information to make the best decision about which text plan is the best FOR YOUR NEEDS? There really was no right answer until the question became more specific and asked which plan was the least expensive plan for 55 text messages?

In order to be successful with integers, it is necessary to understand how they operate.  I really like the algeblocks because they can be used for all for operations, for solving equations (something to look forward to), modeling multiplication on the quadrant grid and actually illustrating a binomial times a trinomial. The “aha” moments students have when they first get that three dimensional object is heart- warming.  We will continue working with the algeblocks next weekend as we work with solving equations and develop an understanding of function.

I hope you found the class interesting and engaging.

Anne