Learning Activity 2-B-4: Building a Personal Vocabulary with Blogs

Transformation (Informal Definition): A change in the position of a shape.  This definition is based on my prior knowledge of knowing what it means to transform, and mathematical knowledge pertaining to geometric shapes.

Transformation (Formal Definition): An operation that moves a shape to a different position without changing its size, area, angles and line lengths. 


 

 

 

 

 

http://www.mathsisfun.com/geometry/transformations.html

http://www.mathwarehouse.com/transformations/

http://www.learningtoday.com/player/swf/Geometry_Transformations_L4_V1_T1a.swf

Volume (Informal Definition): The amount of space within a 3-D figure.  This definition is based on my prior knowledge of 3-D dimensions and capacity.

Volume (Formal Definition): The number of cubic units needed to fill a solid figure.

 

 

 

 

 

 

 

 

 

http://www.helpingwithmath.com/by_subject/geometry/geo_volume.htm

http://www.mathsisfun.com/definitions/volume.html

Learning Activity 2-B-3 (Introduction)

Hello, all.  My name is Wanda Dickson; however, I prefer to be called Nichol.  I have taught in New Jersey’s Public Schools for 16 years, and I currently teach 6th & 7th grade mathematics; my 7th grade class is an ESL class, which I adore.  My school district is very unique because we have a large Caribbean and Hispanic population.  Thus, I’ve had the privilege of teaching students from a plethora of diverse backgrounds and cultures.  I am confident I’ve learned more from them, than they could have ever learned from me.

Ironically, I have a graduate degree in Criminal Justice, although I’ve never worked in the field.  I aspired to be  Agent, so my master plan was to get case management experience as a parole or probation officer, then apply for a position as a FBI Special Agent.  Well, to make a long story short, I took every Civil Service Exam under the sun; nonetheless, while I awaited the test results, I was broke & unemployed.  At the urging of a friend, I took the National Teacher’s Exam, and began to teach.  I planned to teach until I received my Civil Service results, and landed my high-paying government job.  However, to my dismay, that high-paying government job wasn’t paying as much as I had anticipated, nor could it have ever been as rewarding as my teaching career.  Fortunately, sixteen years later, I can honestly say, I have no regrets, and I was destined to teach.

As a result of taking this class, I look forward to learning new and insightful ideas, strategies and concepts that will enhance my teaching, as well as my student’s learning.  Additionally, I also look forward to engaging in meaningful, mathematical dialogue with my fellow classmates.  All of the teachers in the middle school math department have different schedules, so we rarely have the opportunity to share our ideas, accomplishments and challenges.  Thus, it is very therapeutic to have the opportunity to converse with others who have similar math-specific experiences.

Learning Activity 8-B-2: Reflections on Blogging

Overall, I enjoyed my blogging experience, although the initial stages of setting-up the blog were challenging.  As a novice blogger, everything was trial, error and experimentation for the first couple of weeks.  However, I eventually got the hang of the basics, and the process wasn’t as daunting.  In hindsight, the site is actually very user-friendly.  I learned a lot from reading my classmate’s blogs, and from the other math blogs I began to follow, after perusing the entire site.  I am grateful I was encouraged to explore the world of blogging; the experience was very beneficial as a student and a teacher.

If it weren’t a requirement, I would have never started a blog, and it is very unlikely that I will continue to post information any time soon.  However, I do plan to continue to follow others blogs, and use them as ancillary resources.  My primary reason for not continuing to post blogs is time.  For instance, this year my school district is mandating us to create websites, and post our grades, lesson plans and homework on-line.  Considering I am not very technologically savvy, I am still trying to adapt to the new requirements listed above.  Therefore, blogging, for me, would just be another item added to a very long list of “technological things to do.”  I do believe I can up-load my blog to my website, however.  If this is true, I am definitely inclined to up-load my current blog, although I don’t anticipate up-loading a new post.

As a result of my blogging experience, I learned that writing about math was more challenging than “doing math.”  For example, sometimes the writing process was time consuming, because I am not accustomed to articulating procedures in writing, and formulating my own definitions.  Since my mathematical education involved rote memorization, I was never asked to summarize my mathematical knowledge, and construct my own meanings; therefore, this was a new experience, and I never imagined it would take me as long as it did.

I entered this course confident I was adept at Middle School Math.  However, the blogging experience made the math more interesting, because it gave me the opportunity to explore, analyze and communicate in my own way.  For example,  whether I used blogging to explore Pascal’s Triangle, write a review about the Puma Site, or merely paraphrase vocabulary, the math became more interesting, because the requirements extended beyond merely “doing math.”  As a result, I am now inspired to allow my students to conduct more internet explorations, so they can discover something new and interesting that extends beyond the textbook and the classroom.

For example, one of the most interesting concepts I learned about in this course was fractals, and other non-linear patterns.  Although I’ve heard of fractals before, I never explored the concept to gain an understanding of what they really were.  However, once I conducted my own research, I was pleasantly surprised to see the plethora of fractals that can be found just about anywhere, including inside our own homes, as well as in nature.  So much time is spend trying to get students to understand linear patterns, tables and equations, that non-linear patterns become secondary.  Therefore, it was intriguing to see all the non-linear patterns we overlook, although we’re surrounded by them every day.

In the near future, I would love to incorporate blogging into my classroom.  My students are already required to journal bi-weekly; however, I would love to introduce blogging to provide them with a different alternative to writing in a marble notebook.  Since students love to work on the computer, although they loathe writing, I think blogging is a unique way to pique their interest, and increase their writing and participation.  It’s also a great way to have students share and exchange knowledge with their peers.  I wouldn’t be surprised if my students know more about blogging than I do; I am confident they will be eager and motivated to began the process.  Their motivation may actually inspire me to continue my own blog.

Learning Activity 8-B-1: Factoring Quadratics – In Your Own Words

Factoring Quadratic Equations: Paraphrased

Example: x2 + 6x + 8

Step 1: Locate the third term in the quadratic, and find all the factor pairs associated with that term.

Step 2: Identify the factor pairs that will give you a sum that’s equal to the number in the middle, or the second term.

Step 3: Factor the first term.  If the first term is X², both binomials will start with X.  For example, it would be (X  )(X  ) because X(X) = X².

Step 4:  Since the factor pair (2, 4) has a sum of six, two will be added in the first binomial, and four will be added in the second binomial.

For example, (X + 2)(X + 4).  Two and four are added because all of the terms in the original quadratic are positive.

By paraphrasing the steps, I was able to internalize the concepts more, because I had the opportunity to interpret and communicate the procedures in my own personal way.  Decoding language is an integral part of the learning process; thus, summarizing and paraphrasing allowed me to express my understanding in a unique way that made sense to me.  This type of lesson could be applied in class by having students paraphrase procedures that are difficult for them to understand; therefore, students will have the opportunity to “confront” difficult concepts and make sense out of them.  Since communication is essential to student’s understanding, if they are given the opportunity to paraphrase & construct meaning for themselves, the they will enhance their ability to learn and understand the math.  Even if students understand procedures without paraphrasing them, I would require them, for example, to explain the factoring process, so I could assess their understanding by having them answer questions similar to the ones below:

  • Why are both binomials X?
  • Why did you use the factor pair (2, 4)?
  • Where there other options besides (2, 4)?
  • How did you know that (2, 4) needed to be added?

Learning Activity 5-D-2: Applets

I like the Shape Sorter game on the illuminations website, as indicated below.  I was immediately drawn to the game because I am currently teaching a geometry unit; thus, the game has current relevance.  One of things I like most about the game is that it is very comprehensive.  For example, the game encompasses everything from polygon characteristics, to symmetry, angles, and Venn Diagrams.  Students select their own Venn Diagram categories from a long list of options; therefore, I also like the fact that the students are in control of their own learning.  I can definitely see myself using the applet as an assessment tool in the near future.  Since the applet displays the correct & incorrect responses at the end of each round, I would also have my students write about why some of their answers were wrong.

http://illuminations.nctm.org/ActivityDetail.aspx?ID=34

 

Learning Activity: 5-A-4 Evaluating our Definitions: Equations and Functions

After reviewing my classmate’s posts on functions and equations, I noticed that many of the definitions and examples were similar.  Other than having different semantics, we all pretty much created the same definitions.  If I were to alter one thing about my definition, however, I would write more about the patterns that are created by functions, and mention that equations don’t have to have variables on each side of the equation.  Although I provided examples that illustrated this, I didn’t mention it in my definition.

To evaluate whether my students grasp the difference between the two, I would have them write examples of equations and functions in their journals.  Subsequently, I would have them create a Venn Diagram to organize their data, and require them to compare and contrast the two.  I would allow them to use any resource they would like to create a multitude of examples.

Learning Activity 5-3-A: My Definition of Equations & Functions

Equation: An equation is a numeric or algebraic number sentence that is made up of two expressions: one on each side of an equal sign.  Equations must maintain their balance on each side of the equal sign.

Examples of Equations:

Y = 2r + 7                   76 = 14/X                      7(4 + 5) =

Function: A function is an algebraic equation that has two variables or missing values: one on each side of an equal sign.  When one of the variables is replaced with a numeric value, the equation can be solved, and there is an output for the other variable.

The links and activities below are supplementary resources that reinforce equations and functions.

 

 

 

 

 

 

 

 

Examples of Functions:  

f(x) = 3 + 6x       f(x) = 54x + 17       f(x) = 87 – 54x

Defines Functions; Provides Examples & On-line Practice.

http://www.studyzone.org/testprep/math4/d/functiontable4l.cfm

Defines Functions; Provides Examples; Discusses Variables and Domains & Ranges; Graphing Functions & Function Tables.

http://www.ehow.com/about_5431722_math-function-table.html

Comprehensive Review of Linear Equations, Including On-Line Practice, Tests & Illustrations.

http://www.mathsisfun.com/equation_of_line.html

Comprehensive Video That Reviews  A Variety Of Different Forms Of Simple Linear Equations.

http://player.discoveryeducation.com/index.cfm?guidAssetId=9D45C18B-ECE5-4B1B-BFB9-D8A56F3CCCFA&blnFromSearch=1&productcode=US
                                                                        

Journal Activities

1.  Write about a real-world situation that has a constant rate of change and a Y-intercept.  For example,  NYC cabs charge .45 per mile and an initial fee of $3.00.

1a. Create an equation that represents your example.

2. Create your own pattern of six numbers, along with a concrete visual of your pattern.  Describe the rule associated with your pattern.

Learning Activity 5-B-1: The Magic of Proportions

In 2010 I had the opportunity to travel to Moscow, Russia where the currency is the Russian Ruble.  29.93 Russian Rubles is equivalent to 1 US Dollar; therefore, every time I made a purchase, I converted the Russian Rubles to US Dollars to understand the real value of my purchases.  For example, if a large bag of potato chips had a price tag of $89.73, I used my knowledge of proportions to convert the price to US Dollars.  The example below illustrates how I used proportional reasoning to convert the prices.

Example #1

If 1 US Dollar = $29.93 Russian Rubles, what is the dollar value of a $89.73 bag of potato chips?

Step 1

Create a ratio, or part to part relationship.

$1 US Dollar/$29.93 Russian Rubles

The ratio above means that for every $1 US Dollar, there are $29.93 Russian Rubles. 

Step 2

Create another ratio, or part to part relationship, that includes the missing value.

C/$89.73 Russian Rubles

The ratio above asks that question: how many US Dollars are there for every $89.73 Russian Rubles?

Step 3

Make the two ratios equivalent so they become a proportion.

$1 US Dollar/$29.93 Russian Rubles = C/$89.73 Russian Rubles

The proportion above asks the question: if $1 US Dollar represents $29.93 Russian Rubles, how many US Dollars represent $89.73 Russian Rubles?

Step 4

Cross multiply on both sides of the proportion.  For example, $29.93 Russian Rubles times C, and  $1 US Dollar times $89.73 Russian Rubles.  The cross multiplication produces the following equation:

$29.93(C) = $89.73

Step 5

To solve the equation, isolate the variable, C, by performing the inverse, or opposite, operation.  The inverse of multiplying by $29.93 is dividing by $29.93.

$29.93(C)/$29.93 = 1(C)

Step 6

Since we divided by $29.93 on the left side of the equation, we must also divide by $29.93 on the right side of the equation, because equations must maintain their balance.

$89.73/$29.93 = $2.99

Final Answer

1C = $2.99

The final answer means that $89.73 Russian Rubles is equivalent to 2.99 US Dollars.  Thus, a large bag of potato chips that are $89.73 Russian Rubles have the same value as 2.99 US Dollars.

Example #2

The school’s planning committee has decided to make and purchase Country Time Lemonade for the student’s Valentines Day party.  One 82.5 oz canister of Country Time serves 136 people.  How many canisters of lemonade are needed to serve approximately 300 people/students?

Step 1

Create a ratio, or part to part relationship.

One Canister (82.5 oz)/136 Students (Servings)

The ratio above means that for every one canister of 82.5 oz lemonade, 136 students can be served.

Step 2

Create another ratio, or part to part relationship, that includes the missing value.

C/300 Students

The ratio above asks that question: how many canisters of lemonade are needed to serve 300 students?

Step 3

Make the two ratios equivalent so they become a proportion.

1 Canister/136 Students = C/300 Students

The proportion above asks the question: if 1 Canister of lemonade serves 136 students, how many canisters are needed to serve 300 students?

Step 4

Cross multiply on both sides of the proportion.  For example, 136 Servings times C, and  1 Canister times 300 students.  The cross multiplication produces the following equation:

136(C) = 300

Step 5

To solve the equation, isolate the variable, C, by performing the inverse, or opposite, operation.  The inverse of multiplying by 136 is dividing by 136.

136(C)/136 = 1(C)

Step 6

Since we divided by 136 on the left side of the equation, we must also divide by 136 on the right side of the equation, because equations must maintain their balance.

300/136 = 2.20

Final Answer

1C = 2.2 of a Canister, or 3 Whole Canisters

The final answer suggests that 3 Canisters (82.5 oz) of Country Time Lemonade are needed to serve 300 students.  Since the final answer, 2.20, has a remainder, it means that 2 Canisters are not enough; therefore, an additional Canister is need.

Learning Activity 4-C-1: Non-Linear Pattern Web Quest

For this Learning Activity, I conducted a web-quest on “Fractals and Nature.”  I chose this particular web-quest because, although I’ve heard of fractals before, all I really knew about them was that they were patterns.  I didn’t know the formal definition of the word, nor did I know how the patterns were created.  Since I started this process with a clean slate, everything I learned was entirely new to me.  For instance, one of the most fascinating things I learned was that fractals can be computer-generated with the appropriate software and mathematical formula.  I also found it interesting that some artists make a living from drawing fractals, whereas others are paid to create computer-generated fractals and set them to music.  Initially, many of the websites I visited provided very detailed and mathematically confusing definitions of fractals.  However, I found the links below to be very helpful and simplistic.

http://player.discoveryeducation.com/index.cfm?guidAssetId=710F2EBB-27F0-439A-AB92-ECFD1F6879D1&blnFromSearch=1&productcode=US

http://tiger.towson.edu/~gstiff1/fractalpage.htm

Here are a few fractal pictures I found particularly striking.  From left to right, the first illustration is a mountain range found in nature, the second is a fractal that was created on someone’s back as a result of being struck by lightning, and the last fractal was found by slicing a head of cabbage.

At my school we have a large student population of Caribbeans, Caribbean Americans, and African-Americans.  Thus, I think it would be nice to have students identify fractal patterns within the various braided hair styles many of them wear to school.  Student’s could also find braided fractal patterns on-line, or in print.

In my home, I have non-linear patterns on my curtains, counter-top, and within a crocheted bed spread I am in the process of making.  In my classroom, there are non-linear patterns on my bulletin board, based on the way the student’s work is staggered and displayed, and there are graphing posters that are displayed throughout the room that have non-liner patterns, as well.

In addition to conducting a web-quest on “Fractals and Nature,” I also conducted a web-quest on “Fibonacci and Prime Numbers.”  Although I was quite familiar with prime numbers, I was not so familiar with Fibonacci’s Sequence, although I’ve heard about it before.  Initially, I didn’t know how to solve the sequence, until I analyzed it thoroughly.  However, I did eventually notice that two consecutive numbers could be added together to  obtain the third number.  Another concept I learned was that within any group of three numbers, the first and the third number could be multiplied, and the middle number could be squared; the difference between the two numbers is always one.  With specific regard to prime numbers, I learned that every number in the sequence is either prime, or is the product of the prime factorization of the number.  For example, although the number eight is not prime, it is the product of four and two, and two is a prime factor of eight.  The websites that assisted me with understanding this new insight are listed below.

http://player.discoveryeducation.com/index.cfm?guidAssetId=80B45D3D-F904-4560-BE00-BA479DF7511E&blnFromSearch=1&productcode=US

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#primefactor

Within the school setting, students could identify Fibonacci’s Sequence on the piano keys in the music department, or they could use manipulatives to re-create the production of the rabbit pairs.  Within my home, I could probably find Fibonacci’s sequence applied in some of the knitting & crochet books I have; especially those patterns that have petals and spirals.  The photos below are striking photos of Fibonacci’s Sequence found in architecture, nature, needle work and the human body.

 


Learning Activity 4-A-2: Working with the Definition of Linear Patterns

Non-Traditional Patterns (Formal Definition): Patterns that do not follow a repetitive format.

Linear Patterns (Kid-Friendly Version): Patterns that have constant rates of change.  The patterns may constantly increase, constantly decrease, or constantly stay the same.  For example,  the amount of money one makes for babysitting may constantly increase by $10 every hour; the temperature may constantly fall/decrease by -5 every hour; Kim may constantly receive an allowance of $20 each week, no matter how many chores she has or has not completed.

Linear Patterns (Formal Definition): Linear patterns are patterns that have the same distance between each term; the pattern repeats indefinitely along a line.

Definition Comparisons:

  • My definition references constant rates of change, while the formal version references distances between terms.
  • My definition mentions that the constant rate may be positive, negative, or the same; the formal definition has no mention.
  • We both mention that linear patterns look like straight lines.
  • The formal definition makes reference to the line being indefinite; mine does not.
  • Both definitions make reference to the pattern being repetitive or constant.
  • I provided examples of real-world constant rates of change, even although it wasn’t a requirement.

I think students can learn the formal definition of linear patterns by providing them with visual representations first.  For example, if students were provided visuals of linear patterns that demonstrated positive rates of change, negative rates of change, and a zero rate of change for X and Y, I am confident they could work backwards to  come up with their own definition of a linear pattern.  Once they analyze the data and see the patterns for their self, the formal definition would make more sense, and it wouldn’t have to be memorized.  Rather, the abstract definition would be  more easily understood as a result of the prior visual analysis and informal interpretation.

 

 

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