Magic of Proportions
April 13, 2008 at 1:14 am | In algebra | Leave a CommentTags: Module 6, Proportions
Problem #1:
Julie is creating a scale model of her bedroom to help her decide how to rearrange her furniture. Her actual bedroom is 12 feet by 15 feet. She has a shoebox that is 4 inches wide that she can cut to be the correct length. She plans to use the following proportion to help her figure out how long to make the shoebox:
Feet 12 = 15
Inches 4 x
Then Julie uses cross multiplication to find the equation 12x = 60. She solves it by dividing both sides by 12 which produces x = 5. She should cut the shoebox to be 5 inches long.
Problem #2:
Mrs. Boyer has to drive to Lancaster to visit her grandmother. She has a map with scale 1 cm: 10 miles. She measures that it is approximately 10 centimeters from Lewisburg to Lancaster. She uses the following proportion to help her figure out how far it is:
Centimeters 1 = 10
Miles 10 x
Then she cross-multiplies to get the equation x = 100 which tells her that it is 100 miles to Lancaster. Then she knows that her car gets 23 miles per gallon, and gas is currently $3.35 per gallon. She wants to estimate how much it will cost her to drive to Lancaster, so she uses the following proportion:
Miles 23 = 100
Gallons 1 x
She cross-multiplies to get the equation 23x=100. She divides both sides by 23 to find that she will use about 4.3 gallons of gasoline. Then she uses the following proportion:
Dollars 3.35 = x
Gallons 1 4.3
She cross-multiplies to get x=14.405, so she knows it will cost about $14.41 to buy gas for her trip to Lancaster.
My Definition of Equations and Functions
April 11, 2008 at 8:49 pm | In algebra | 1 CommentTags: Module 6
Equation – a mathematical sentence where two expressions are set equal to each other
For example: 3 + 2 = 5
(3)(2) = (6)(1)
x + 6 = 12
x + y = 12
Function – a relationship between two sets of numbers, called the domain and range. Each element in the domain is paired with exactly one element in the range.
For example: set of ordered pairs: {(-2, 1), (5, 2), (3, 4)}
Table:
x y
-2 1
5 2
3 4
* Functions can also be represented as mappings and graphs, but these wouldn’t copy from my microsoft word document
Functions as equations – equations that represent functions can be written as function rules. Function rules can be written with function notation.
For example: a function rule: y = 2x + 3
Function notation: f(x) = 2x + 3
My Reflection on Math Myths
April 4, 2008 at 7:50 pm | In Goals, Module 5, algebra | 4 CommentsI’ve encountered several of these myths throughout the years. The one that stands out the most is “Math requires a good memory, and memorizing formulas and rules is the best way to learn it.” My high school geometry teacher made us memorize everything! While my friends in the other section got to have a cheat sheet for tests, my section had to memorize all of the theorems and postulates! I don’t think any of us understood them any better because we had memorized them. Many people failed her tests because they hadn’t memorized them even though they probably could have USED them correctly to complete the proofs if they were given a list. This experience has definitely had an impact on how I teach. There are some formulas that students should KNOW because they are so common, but a student should never have to sit and memorize a formula because they might need it on a test. I always supply students with a formula sheet or individual formulas as needed. Even for the PSSA, students are given a formula sheet and not expected to memorize everything! I think (hope) my general attitude towards this is apparent to students and will help dispel this myth.
The other myth that hits home with me is “There is a math mind – some people have it and some don’t.” I think many people to this day believe this myth. I know my friends thought this in school, many students still do today, and most parents believe it. I even hear teachers saying it. This is one of the myths that most effects girls, because “girls don’t have the math mind.” For me, some of the most logical concrete thinkers are the ones who struggle most with math when it really counts, because they can’t problem solve. I always tell my kids that someday a potential employer won’t care that you can solve a page filled with equations if when it comes to writing and applying an equation she has to give you the equation before you can solve it. If she has to give you the equation, she might as well solve it herself! Some of my students who are the most creative are the best problem solvers in math even though they are stereotypically thought of as “non-math people.” I think I was fooled by this myth when I was younger because I had many friends who despised math. However, being a teacher has really helped me to see otherwise. I think that I help my students avoid this myth be encouraging problem solving and applications of skills instead of rote memory. I can further help by encouraging and praising all students, including girls!
Non-Linear Pattern Web Quest
April 4, 2008 at 6:14 pm | In Module 5, Patterns, algebra | Leave a CommentI started by searching “Fractals” and “Nature” and “Patterns”. I like fractals because they can be so complicated yet start with such a basic concept. I love the phrase “pattern of chaos” because it just seems so ironic to me. This first site I liked because it goes through many types of fractals: simple, natural, etc. You can click the links at the topic to see the different categories of fractals.
http://www.miqel.com/fractals_math_patterns/visual-math-iterative-fractals.html
I liked the Wikipedia entry for Fractals because it provides animations for both the Sierpinksi Triangle and Koch snowflake.
http://en.wikipedia.org/wiki/Fractal
This third site I enjoyed because it looks into many topics in Science that involve Fractals. These topics would be a great way for math and science teachers to collaborate.
http://kluge.in-chemnitz.de/documents/fractal/node2.html
Then I searched “Fibonacci” and “Phyllotaxis” and “Prime Numbers”. This search did not lead me to nearly as many useful sites as the one above, but I did really like this Prime Factorization Machine, because students could use it to check their answers.
http://britton.disted.camosun.bc.ca/jbprimefactor.htm
I also found this site helpful for Phyllotaxis since I wasn’t familiar with that term really at all.
http://en.wikipedia.org/wiki/Phyllotaxis
Questions:
1. Were there ideas or concepts you were not familiar with? What were they? I really had not heard of Phyllotaxis before, so that was the most foreign to me. Nature is so interesting!
2. What images did you find particularly striking? I liked the interactive images of fractals, because they help students visualize what’s going on. I also really enjoyed the fractals in nature, because they are so amazing!
3. Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they? Our schedule this week at school for PSSA testing was definitely nonlinear! Additionally, the activity in my bank account lately has been nonlinear with preparing for a baby.
4. How can you adapt this webquest activity for your classroom? In order to do a web quest with my seventh graders, I would have to provide specific sites for them to visit and specific questions or guidelines for them to explore. I would probably also let them work in partners.
My Definition of Linear Patterns
April 3, 2008 at 1:13 am | In Module 5, Patterns, algebra | Leave a CommentA non-traditional pattern is one that does not have consistent recurring set of events. It does not have to be repetitive or symmetrical. It may have segments that have patterns, but most are irregular. These include family trees and concept maps.
A linear pattern is one that has a repetitive format. It has the same rate of change between any two events. The pattern may be increasing or decreasing but must increase or decrease the same amount in each interval. Linear equations form straight lines when graphed because they have a constant rate of change between points.
Return to Earth
March 29, 2008 at 12:29 pm | In Activities, Module 4, Ratios, algebra, seventh grade | Leave a Commenthttp://www.nasa.gov/audience/foreducators/topnav/materials/listbytype/Space.Shuttle.Glider.html
This link is to a hands-on activity where students create their own Space Shuttle Glider that is a scale model to the actual U.S. Space Shuttle Orbiter. This activity caught my attention because it is hands on and seventh graders love to build things! I also think it’s really neat to learn something about the orbiter while practicing math skills, specifically ratios and proportions. Project materials are included in the downloadable PDF file. Some additional materials must be supplied.
The project challenges students to follow directions to build an accurate glider that will “take flight.” Then students have to apply ratios and proportions to find the scale factor between their glider and the real orbiter. Next students find the glide ratio (height to distance) of their glider as they launch it horizontally and it makes a safe landing on the ground. This will lead students to research and compare the glide ratios of other aircraft. Lastly, students use a fishing line guidance system to simulate the landing of an orbiter on the airstrip.
Inverse Properties
March 28, 2008 at 1:59 pm | In Module 4, Vocabulary, algebra | Leave a CommentInverse Property of Addition
The inverse property of addition says that if you add a number and its opposite, the sum is 0.
For example: (-6) + 6 = 0 or 2x + (-2x) = 0
In general: a + (-a) = 0 or (-a) + a = 0
Inverse Property of Multiplication
The inverse property of multiplications says that if you multiply a number and its multiplicative inverse, the product is 1.
For example: (8)(1/8) = 1 or (3/4)(4/3) = 1
In general: a(1/a) = 1 or (1/a)(a) = 1
How does this work?
March 21, 2008 at 3:11 am | In Module 3, algebra | 1 CommentAt the beginning of the week, I read this assignment and worried that I would have nothing to write about. How embarrassing would that be that a math teacher didn’t encounter any math throughout the week? Now that it is time to do the assignment, I find myself choosing from several events.
A teacher at my school sent out an email with the subject, “How does this work?” It was a link to this page: http://www.milaadesign.com/wizardy.html. She seemed to think that it really might be magic, and that her computer really could read her mind! I took a quick look and figured there was math behind it, and later that day, when I had time, returned to the site for a closer examination. The site asks you to pick on 2-digit number. Then you add together the digits and subtract the sum from your original number to get your answer. Then it shows you an assortment of pictures paired with numbers. You are supposed to stare at the image that matches with your answer for 10 seconds. While staring, your image shows up and enlarges in the center of the screen; just as if the computer might be reading your mind! You might want to check out the site to see for yourself.
That afternoon I emailed her back with the following explanation. It turns out that you can let xy represent the 2-digit number. The tens-digit is represented by x, and the ones-digit by y. This value of this number would be 10x + y. Then you subtract from that (x + y), giving you (10x + y) – (x + y) which simplifies to 9x. Thus, your “answer” from above will always be a multiple of 9. All multiples of 9 such as 9, 18, 27, 36, 45, etc. are paired with the same symbol, so no matter what multiple of 9 you are looking at, it will be the same image that appears on your screen! Then, so you don’t get suspicious when you try it again, they mix up the symbols so you won’t get the same symbol each time! I don’t know who has enough time on their hands to design a website like that, but I thought it was pretty cool. The next chance I have to use my mobile lab with my students, I am going to have them try the website for sure!
Welcome to my Weblog!
March 15, 2008 at 1:59 am | In Goals, Introduction, algebra, seventh grade | 2 CommentsHi! I am Mrs. Boyer and I am a 7th grade math teacher at DHEMS. I teach Algebra A and Algebra I. I have a husband, Mr. Boyer, a dog, Sawyer, and two cats, Samson and Delilah. This summer I will be having my first child, a baby GIRL! I enjoy scrap booking, activities outside, watching movies, and reading.
Throughout this course, I hope to learn some strategies to help learning algebra more interesting and engaging to 7th grade students. I am looking for some realistic and easy to implement activities. These strategies should help me to better teach and relate concepts to students. Secondly, I would really like to learn ways to implement writing and journal writing into my algebra curriculum.
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