ellenengland

Teaching Algebra in Middle School

Reflections on Blogging

 Describe your blogging experience in this course. Do you think you will continue using your blog?

Blogging was not a new experience for me.  In 2001 I took an online course which included creating a blog.  At the time no one in the course had ever heard of blogs before, so it was an adventure for all of us.  Little did we know that there would soon be a “blogosphere.”

Surprisingly,  my experience this time was very similar to my experience ten years ago.  I am not really interested in posting my ideas on the web and I find it very frustrating use the software.  I struggle to try to make something appear the way I want it to and end up giving up.  I wish I could just post my documents the way I type them with my images and fonts.  It is not likely that I will continue to use the blog after this class is over.

What did you learn about yourself and your abilities or interests in Math or Algebra?

I learned that I prefer to do math and teach math than to write about how to teach math.

Did you learn or discover anything you found particularly interesting through your course activities or your own internet research?

The Shields article about definitions was very interesting to me.   I have already started pointing out to my students this summer the “nested” aspect of many definitions.  I have also started to require my students to write their own definitions for key terminology and then revise the definitions as they learn more.   Countryman’s book also had an immediate impact on my teaching. I had my summer class freewrite their own definitions of Algebra on the first day of class.  I also read my mathography from this course to them.   Their homework on the first day of class was to write their own mathographies.  In addition, they are required to write a three to five entries in a math journal each week.

Describe one interesting discovery and why you found it fascinating.

I was intruiged by The Thirteen Factors Problem in Chapter 5 of Joan Countryman’s book.  It is the type of problem that I enjoy doing, but I was also fascinated by how her students wrote about their experiences doing this problem.  In the past when I have asked students to explain their reasoning or the process they used to solve a problem the results were much less specific and not as descriptive as the samples in this book.   I wonder if it is possible for my students to write as clearly about their math experiences.  I need to know more about how to scaffold my students to be able to write this way.  I am now determined to learn more about how to help students communicate their thinking when trying to solve math problems.  I am eager to see if they are better able reach the learning targets in my course if they are able to put their thoughts and activities into words.

Do you think you will use blogs? Why or why not?

I am not likely to use blogs because I find it difficult to learn the ropes.  I would not require my students to blog because I would prefer to collect their work on paper than to check their blogs and type responses to them.

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July 19, 2011 Posted by | Reflections on Blogging | 5 Comments

Factoring Quadratics

Please click the link to view my paraphrase of how to factor a quadratic expression.

https://ellenengland.files.wordpress.com/2011/07/factoring-a-quadratic-expression.pdf

Thanks!

Ellen

July 19, 2011 Posted by | Factoring a Quadratic | 5 Comments

Applet Review

Pan Balance – Expressions

http://illuminations.nctm.org/ActivityDetail.aspx?ID=10

This applet is an excellent dynamic tool for student exploration and discovery of the effect of changing the value of the variable in algebraic expressions. It is a visually appealing, easy-to-understand demonstration of the relationship between two expressions.  Students can enter numeric, linear, quadratic, or cubic expressions in the pans on the balance scale.  The slider at the top of screen is used to change the value of the variable in the expressions.  As the value of the variable changes, the balance scale and the graphs reflect the changes.  The expressions are graphed on the same coordinate plane and are color-coded to match the pans, so students can see which graphed line corresponds to which expression.  By observing the effects of the changes students are able to build an understanding of the concepts.

This applet would be useful when teaching and learning about the meaning of the slope and y-intercepts of linear equations and also about systems of linear equations,.  I would use this activity in a lesson about systems of linear equations.  I think it would help students understand that the intersection of the graphs of equations in a system is the solution to the system.  I would use this applet before teaching the three methods of solving systems; graphing, substitution, and linear combination.  It would be interesting to see if using the applet this way helps students master the techniques for solving systems of equations.

July 4, 2011 Posted by | Applet Review | Leave a comment

Evaluating our Definitions: Equations and Functions

 After reviewing your classmates post, would you alter your definition?

After reviewing Kristi’s definitions of equations and functions, I would change my definition of function to say that a function is written in the form of an equation because I did not connect equations and functions in my original definitions.

Would you provide different examples?

I would include a table of values for an example of a relation that is not a function and an example of a relation that is a function for students to compare in addition to the graphs I included originally.

How can you evaluate whether or not your students grasped the difference between equations and functions?

To assess if my students understand that some equations are also functions, I would ask them to write one example of an equation that is not a function and one example of an equation that is a function.

June 30, 2011 Posted by | Definitions: Equations & Functions | Leave a comment

The Magic of Proportions

Lemonade for the 4th of July Picnic

Our neighborhood has a potluck picnic on the Fourth of July.  Each family brings either a beverage or food to share with others.  I need to bring enough lemonade for 40 one-cup (8 ounce) servings.  I would like to make fresh squeezed lemonade, but I want to know how much it would cost compared to buying pre-made lemonade or frozen lemonade.

First use of proportions to solve a problem:

The recipe that I have for fresh squeezed lemonade makes 16 servings. Proportions will help me find the quantities of the ingredients needed to make 40 servings.

Second use of proportions to solve a problem:

Proportions will help me find the cost of the 40 servings of each of the three types of lemonade.

To see the solution to my problems click this link:

https://ellenengland.files.wordpress.com/2011/06/lemonadeproportions.pdf

June 30, 2011 Posted by | Proportions | 2 Comments

My definition of Equations and Functions

Equation

 Definition:

An equation is two expressions connected by an equal sign to indicate that the expressions have the same value.

Examples:

Function

 Definition:

 function is a “relation” between inputs and outputs.  Each input maps to exactly one output.

Graphs of  functions must pass the “vertical line test” to demonstrate that each input value maps to only one output value.

To do the vertical line test, hold a pencil parallel to the y-axis (vertical axis) and slide it horizontally across the coordinate plane.  The graphed relation is a function if your pencil only crosses the graph once at each x value.

Examples:   x’s are inputs and y’s are outputs         

Example: Graph of a Relation that is not a Function:

Pretend the red vertical line is your pencil.  If you slide the vertical line horizontally across the coordinate plane, for each x value greater than zero, it crosses the graphed relation in two places.  This means that the x values map to more than one y value, so this is NOT the graph of a function.  It fails the vertical line test.

June 28, 2011 Posted by | Definitions: Equations & Functions | 5 Comments

Translating Pattern Narrative into Formal Language

Narrative description of growth pattern of the triangles in Pascal’s Triangle:

Looking at the triangles that are colored blue instead of at the numbers, I saw a geometric growth pattern that starts with the smallest triangle, the three ones at the top. That triangle is duplicated twice. One of the copies is added to the bottom left of the first triangle and the other copy is attached to the bottom right. The next iteration is to copy the new, larger triangle with the three copies of the first triangle. Add a copy to the bottom left and the bottom right. Keep repeating this process. The growth is fractal-like because the same process is repeated creating larger and larger triangles, which contain more copies of the first triangle.

Translation to formal form using formal Math and Algebra vocabulary:

Each new triangle has three times the number of triangles in the previous triangle.  The growth pattern is exponential.

 an  = number of copies of the first triangle in the nth term

n

0

1

2

3

4

n

an

1

3

9

27

81

3( a n – 1)

 Recursive rule:        an= 3(a n – 1),  a0= 1

n

0

1

2

3

4

n

an

1

3

9

27

81

3n

Explicit rule:             an = 3n

June 23, 2011 Posted by | Growth Pattern in Pascal's Triangle | 1 Comment

My Reflection on Math Myths

As a younger math student I probably believed all twelve of the math myths in this activity. As a math teacher, I have seen that the majority of my students also believe most of these myths. The myth that is most prevalent among my students and their parents is:

“There is a “math mind” – some people have it and some people don’t.”

My personal transformation from a student who struggled and did poorly in math in high school to a math teacher, earning A’s in graduate level math classes, has completely dispelled this myth for me. In addition, I have worked with many students who progressed from a conviction that they were just not good at math to becoming very successful in math.

In hindsight, I see that the math teachers I had in junior and senior high school believed this myth and used methods which made it difficult to see that all kinds of minds can be good at math. Students who were quicker with answers in class also contributed to my belief that my mind was not as math-capable as their minds were. The other myths I believed served to support my  conclusion that I did not have a good math mind. I believed that memorization was more important than understanding and my mind was not good at memorization. When I reached college, I discovered that I need find ways to make math meaningful to me. I also I found out that making mistakes is part of the learning process.

In the last few years of graduate school I have learned that mathematicians use creativity, intuition, and false starts in their work. Realizing that there is usually more than one way to understand and solve a particular math problem also helped me appreciate that there are many kinds of minds that are good at math.

How can you help dispel these myths for your students?

Since I teach community college students who have done poorly on the placement test, most of my students are convinced that they do not have the right kind of mind to do math. One way I can start to dispel this belief is to tell them my own experiences learning math. I also can tell them about famous mathematicians’ struggles and failures while developing the math they are famous for.  It is important to positively reinforce the students’ abilities to think about math in different ways from each other. One way to do this is to praise the questions that they ask as strongly as I acknowledge the questions they answer correctly.  It is also important to find something “right” or good about incorrect answers that will help steer thinking in an appropriate direction, rather than turn their thinking off. Encouraging observations, exploration and investigation rather than only correct answers is important.  This requires me to ask questions rather than “tell” information, and to carefully word my questions so that they stimulate thinking.

June 23, 2011 Posted by | Math Myths | 1 Comment

Non-linear Pattern Webquest

Search results for the terms “The Golden Ratio” and“Pentagrams”
Images and ratios are from:
http://www.contracosta.edu/legacycontent/math/pentagrm.htm

Were there ideas or concepts you were not familiar with?

I was familiar with Φ and the Golden Ratio, but the relationships between the segments in pentagrams inscribed in pentagons are new to me. I was impressed by the image that shows the ratios of the lengths of the segments:

Ø = (1 + 5)/2 = 2cos(/5) = 1.61803…
(Note that b + d = 1)
The length of the black line segment is 1 unit.
The length of the red line seqment, a, is Ø.
The length of the yellow line seqment, b, is 1/Ø.
The length of the green line seqment, c, is 1, like the black segment.
The length of the blue line seqment, d, is (1/Ø)2, or equivalently,  1 – (1/Ø), as can be seen from examining the figure.

I was also impressed by the image below and the ratios of the areas:

The ratio of the area of larger pentagon to the area of the
smaller (white) pentagon is Ø4 to 1.

The ratio of the area of the larger regular pentagon to the
area of the pentagram (star – red & white combined)
is Ø3 to 2.

 

Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they?

The frets on my husband’s banjo are spaced non-linearly.

The spiral shape of the stool back at my kitchen counter is based on a non-linear pattern.

How can you adapt this webquest activity for your classroom?

My students could work in groups of two or three to find examples of non-linear images on the web.  Each student could then identify an example of a non-linear pattern in their own home, which they could photograph.  They could create PowerPoint presentations for the class with their images.

June 23, 2011 Posted by | Non-linear pattern webquest | Leave a comment

Working with the definition of linear patterns

Non-traditional Pattern
Formal definition (from Teaching Algebra to Middle School Students, Module 4-A, Key Information):

 “Non-traditional patterns are simply patterns that do not follow a repetitive format.”

Linear Pattern
My definition:
A linear pattern is one in which the difference between any two consecutive items is the same. The pattern is called linear because if you graph it on a coordinate plane it will form a straight line.
Example:

Sequence  number

1

2

3

4

5

Item

2

5

8

11

14

Formal Definition
Note: I did not find anything that could be called “the” formal definition of a linear pattern. Below are two definitions I found using a google search.

Below is a non-numeric definition of linear patterns from the interactive traveling exhibition of the Brooklyn Children’s Museum found at: http://www.brooklynkids.org/patternwizardry/pattern_linear.html

“Linear patterns repeat indefinitely in either direction along a line. Beads on a necklace, the weave of a fabric or basket, wallpaper borders, stripes on clothing, zippers, walking footprints, musical rhythms, the meter of poetry, the passage of a day, and the changing seasons are all examples of linear patterns that are created or extended by the regular repetition of units, sounds, or events.”

Here is a more mathematical definition of linear patterns from: http://www.mathsteacher.com.au/year8/ch15_graphs/02_linear/patterns.htm

“If the plotted points make a pattern, then the coordinates of each point may have the same relationship between the x and y values. In such a case, the x and y values are connected by a certain rule.
A linear pattern is said to exist when the points examined form a straight line.”

Differences between my definition and the more mathematical formal definition

The formal definition starts out referring to plotted points, but I started out with a number. The formal definition refers to the coordinates of each point, while I explained how to get the next number in the pattern. The formal definition states that the relationship between the x and y values are the same, while I did not mention x and y values. The formal definition says that the x and y values are connected by a certain “rule.” I did not use the word rule at all. Both my definition and the formal definition state that a straight line will be formed by points.

How I could help students learn the formal definition without having them memorize it

To help students learn the formal definition of a linear pattern without memorizing I would first have them practice plotting points on a coordinate plane. Then I would provide them with tables of values for the x and y coordinates of points. Some of the tables would have values in a linear relationship and some would not. I would have the students graph the values from each table on a separate coordinate plane and connect the points. Next they would work in groups to examine the tables with values that plot as straight lines. Each student would write an explanation of how they can tell from the table of values if a set of points is linear or not.

June 21, 2011 Posted by | Linear Patterns | 1 Comment