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

Ratio, Rate and Proportion Series - Post 1

Many of us successfully solve ratio problems partly because we know how to represent them algebraically and can solve for the unknown, and some of us because we have innate or developed number sense and can “see” the relationships between and among quantities that are being compared. We are lucky again partly because as adults we have been working on these types of relationships for a long time and have been successful implementing the algorithms we were taught. 

BUT, think back to when you were learning about ratios and did not understand what it meant to be related to some other value. I recall being told that the ratio of two things were in a 5 : 7 ratio and there were a total of 763 items. I sat mesmerized when my math teacher proceeded to write 5x + 7x = 763. It was like magic. At no time was I shown a visual representation of what was going on nor was I encouraged to think about the fact that there was a multiplicative relationship happening. 

When I finally realized that teaching as telling is totally ineffective I became a deeper thinker, sought out ways in which to help students visualize, touch and manipulate the mathematics and relate it to things they are interested in I discovered my students, no matter how old, started to progress in their own learning.

I hope that after today’s class the inclusion of the bar model makes sense to you and it is something that you can bring into your own battery of teaching strategies. Modeling how the original problem looks and comparing it to the results after an activity has occurred allows students to make sense of a given situation and adds sense making to doing the mathematics.

Anne

Upcoming Workshop: Rational Numbers

Dr. Anne M. Collins will be leading a workshop on Saturday, March 9th on Rational Numbers: Hard to Teach, Harder to Learn.

The workshop begins at 9am at University Hall, 1815 Massachusetts Ave located in Porter Square.

In this workshop participants will explore visual representations of rational numbers that seamlessly develops an understanding of the slope of a line. We will examine bar diagrams and graphing ratios. Graphing ratios on the Cartesian coordinate plans allows participants to connect slope to linear equations. Problems involving proportions can also be solved using the Cartesian plane. This novel approach is required in the CCSS and the 2011 Massachusetts Curriculum Frameworks.

Cost is $50 per participant and there is still room available!

To register or to get more information, follow this link http://www.lesley.edu/EventDetail.aspx?id=7995.

We hope to see lots of you there!

Recognizing Effective Implementation of the Standards

We hope you all enjoyed the first weekend in February.  This past weekend was full of mathematics to bring into the classroom in discussions about Groundhog Day or the Superbowl!  In addition, we received the February issues of our NCTM journals and are excited for what's included in those as well.

Last Thursday, I participated in a class on Ratio, Rate and Proportion being held in one of our partner districts.  During this class we had a really lively discussion about the relationship between fractions and ratios.  Are ratios fractions?  Are fractions ratios?  What is the difference?  How can we denote them differently? 

This morning, I was speaking with Anne about the class, and it reminded her of a presentation she did at the MassMATE conference in 2012.  Please find a link to it below.  It will link to a Scribd site and open in a new window.

Effective Implementation of the Standards Power Point


There are some really good problems within the presentation, so I hope you enjoy it!

Katie

Congratulations Anne!

The Center for Mathematics Achievement would like to offer our congratulations to our director, Dr. Anne Collins, on being awarded the Association of Mathematics Teacher Educators (AMTE) 2013 Nadine Bezuk Award for Excellence in Leadership and Service in Mathematics Teacher Education!

From left to right: Dr. Nadine Bezuk, Dr. Anne Collins, President of AMTE Dr. Marilyn Strutchens, and Awards Committee Chair Dr. Doug Corey.

We will be posting a video of Anne's presentation soon.  Congratulations Anne!

Developing Number Sense

As our students enter middle school and beyond, too often they lose opportunities to continue developing their number sense. Students at this age tend to appreciate more and more the beauty behind numbers, our number system, and the patterns and relationships among them. For this month’s blog I am resurrecting some fun number questions from the Mathematical Digest, Term 1, 1994, Number 105. This mathematical digest has a wealth of information and challenges to be solved. Enjoy!

Anne

Match the clues with the numbers in the box.

Clues

1. An odd cube
2. The first prime 
3. The fourth triangular number
4. Srinivasa Ramanujan* said that this number was equal to (9²  + 19² ÷ 22)²⁵
5. The second perfect number
6. The smallest odd abundant number
7. 6! + 5! + 4! + 3! + 2! + 1!
8. The first number after 1 to be both a square and a triangular number
9. The ninth highly composite number
10. A three digit palindromic square number
11. G. H. Hardy’s taxi cab’s number 

There are twelve numbers in the box for the eleven clues.  Which number does not have a clue written for it?  This number is featured in a very well-known book written in 1726. What is the name of the book? 

*Srinivasa Ramanujan (1887-1920) has been described as the greatest mathematician India has produced in the last 1000 years. His work has only just started to be appreciated and understood. His formulae are being used in areas such as polymer chemistry, statistical mechanics, computers and even cancer research.

Happy New Year!

Lesley's Center for Mathematics Achievement has some exciting offerings in the upcoming year.  These include monthly Saturday workshops, a Dine and Discuss focusing on the CCSS and PARCC, graduate level math courses in Brockton, Quincy, and Springfield, collaborations with UEI and MoS, and a Summer Institute.  We are excited for the upcoming new year and continuing our work with mathematics teachers and education.  We hope that you can join us for some of these events.  If you want more information, you can find all of it at: CMA Homepage!

And to start of the new year...
How many factors does 2013 have?  How many of the factors are prime factors?