Factoring in your Own Words

Let’s use x2 + 5x + 6 as the example

 1. List the factors of the last term. In this case 2 and 3 or 1 and 6.

 2. Choose the factors that can be added to equal the middle term. In this case 2 and 3 equal 5.

3. Take the square root of the first term which is x. (please note that this method will only work for trinomials that have a coefficient of 1.)

 4. FOIL it back to check.

 You may come across an example where the signs are not all positive. If you have two negatives you will have a positive and a negative sign. If you have the first a positive and the second sign a negative you will need a positive and negative sign. If you have a negative first and a positive second you will need two negative signs. One secret I tell my students is that the sign of the middle umber goes with the bigger factor.

 Questions: • Did paraphrasing the words help you internalize the concepts more? Paraphrasing the words did help internalize the concept.

 This particular concept especially with different signs takes students time to grasp. •

 How can you apply this type of exercise in a lesson for your own students?

You can have the students write out the steps they used. I also have the students wrote out the sign combinations and why they are what they are.

Reflections on Blogging

Describe your blogging experience in this course. Do you think you will continue using your blog? Why or why not?

My blogging experience was a challenging one. It took me quite awhile until I figured out how to get the blog to work and do everything that I needed it to do.  One that was all taken care of all was ok.  I do not see myself continuing this blog.  I have my own way of keeping my resources so I do not see a need to do it here.    I will have to keep blogging with my job.  At my high school, we are required to post assignments and pertinent information to a blog provided by the school on a daily basis.

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

I learned how much I really do love Algebra.  I teach mostly geometry, but realize that my true love is Algebra.

Did you learn or discover anything you found particularly interesting through your course actives or your own internet research? Describe one interesting discovery and why you found it fascinating.

There are two things that I found particularly interesting were the Applets and the Algebra tiles.  While I knew there were many Applets out there that pertained to what I teach, I never really took the time to explore them and see what I may be able to use where.  Hopefully, this coming school year I will be able to incorporate some of them into my lessons.  I also never used Algebra tiles before.  I found these particularly interesting.  I can not wait to teach combining like terms.  I think the tiles will make it much less confusing on what can be combined with what.

Do you think you will use journals with your students? Do you think you will use blogs? Why or why not?

I do not think I will use journals per say with my students.  What I intend to do this coming school year is give my students at least one writing assignment that they must respond to.  At this point I do not want to take the time to do it on a daily basis.  10 minutes out of 43 every day will be quite a bit of time.  I think I can incorporate some type of prompt maybe given on a Monday and due on a Friday.  That way my students will be writing but will not use up tons of class time.

The Magic of Proportions

Laura just got her driver’s license.  She needs to get gas in order to make it home from work.  Laura remembers when she got gas the last time with her mom it cost 15.54 for six gallons of gas.  She knows the tank holds 15 gallons.  She wants to fill the tank so her mom doesn’t have to later on.  How much will it cos her to completely fill the 15 gallon tank?

6 gallons         =        15 gallons

15.54                                 x

15 * 15.54 = 6x

233.1 = 6x

x = $38.85

For my Small Engine Vo-tech students:

A lawn mower requires a mixture of 2 cycle oil so the engine does not burn up.  The directions on the bottle of oil state that 2.5 ounces of oil are required for 1 gallon of gas.  If my gas can held 2.75 gallons, how many ounces of oil are required?

1 gallon = 2.75 gallon

2.5 0z               x

1x = 6.875 ounces

Text book reviewed “Beginning Algebra”  eigth edition copyright 2000 Lial Hornsby

Evaluating My Definition

After reviewing your classmates post, would you alter your definition? Why or why not? Would you provide different examples?

After reviewing all the blogs already posted, I would not change my definition.  I have been teaching equations and functions for some time and find that I can most times get my point across to my students using the definition I provided.  What I may do differently is add more types of examples and the use of some Applets to provided some additional experiences for them.

How can you evaluate whether or not your students grasped the difference between the two?

In order to evaluate whether or not y students grasped the difference, I would most likely do some type of activity like a venn diagram or bow tie (organizational strategy) where I would have the students compare the similarities and differences of the two.

5-D-2 Applet Review

The Applet that I chose to review is Algebra Balance Scales.  The Applet can be found at:  http://nlvm.usu.edu/en/nav/frames_asid_201_g_4_t_2.html?open=instructions&from=category_g_4_t_2.html

I chose to review this Applet because balancing equations is something that I teach very early in the school year in my Algebra classes, so I though why not find something I would be able to use.  I like the way this Applet visually shows both sides of the equations on a scale.  It makes it easy for students to see how to move constants to one side and variable to the other side and why you need to use the opposite sign.  I would most likely incoprorate this as an introduction to my lesson by letting students “discover” what is going on.  After they “discover” what is going on, I would then formally explain the concept.  I think that students would have an easier time understanding the concept after they have figured out on their own what is going on.  Another way I would use this Applet would be when I was tutoring students.  The Applet has a feature that you can enter your own problems.  If a student was having difficulty, I would have them enter the problem that they were working and solve it using the Applet.  Then I would have them take what they did with the Applet and apply it to what they were doing on their paper.

Equations and Functions in Your “Own” Words

An Equation is a number sentence that has at least one operation symbol and an equal sign.  An equation is solved by balancig both sides.

 

a scale view of this image can be found at www.tpub.com

A function is like a machine you input something, it does the work and produces an output.

a scale view of this image can be found at www.teacherschoice.com.au

Math Myths

People who are good at math do problems, quickly in their heads.

This myth I found particularly interesting due to the fact that it is discussed on an almost daily basis at my house.  When it comes time to compute sales tax, the cost of a list of items or some other math problem that needs to be solved at my house, my husband can quickly and accurately compute the answer at a much faster rate than I can.  My kids will even say, “come on Mom, you are the math teacher”.  It is not that I can do compute the answers, I just can not do it with the speed that my husband can.  Interestingly enough, when it is time to get help with Algebra or Advanced Math, who do all my kids and their friends call; me.  I am much better at mathematical concepts that my husband is.  He can computer problems mentally in his head, but can not solve problems algebraically and can not explain mathematical concepts. 

I also find with the students in my classes,  I have students who score Advanced on the PSSA  are not the first to solve the mental math calculations that they are asked to.  I often find that the students who do well at math do so because they take their time and carefully solve the problems taking the time to learn each step involved.  Is being speedy at Math a sign of Intelligence?  I do not believe so.  While it is true that some Intelligent people can compute problems quickly and accurately I do not find that to be true for all.  Some very intelligent people (myself included ) prefer to write problems down before I solve them.  Quick calculations a lot of the time result in incorrect answers. 

Yet another situation to dispel the myth; on a daily basis as a teacher of Math I when helping a student I will have to stop and write out all the steps in order to find the students mistake or help the student understand.  I find that students do not want to write their work down because they think they they don’t need to, they should be able to do it in their head.  I explain that writing things down is part of the learning process and does not show that you are any less smart because you need to write things down. It is part of the learning process and enables the teacher and/or student to find mistakes.

There is always one best way to do a problem.

The first thing that came to my mind when I read this was solving equations.  Do you move the variable to the left side or the right side.  Some students can not solve the problem if  the variable is on the right side.  The do not care if they need to perform 2 extra steps and end up with some extra negatives that will cancel out later on, they just want that variable on the left side.  Other students will look and find the easiest way that they can to solve the problem  and if the varialbe needs to be on the right side to do that, well then so be it…..then at the end they will use the reflexive property to move the variable to the left side.  There were two different methods used, one moving variable to the left, constants to the right and the other variable on right, constants to the left. 

This type of problem usually sparks some discussion in my classes.  I explain to students it does not matter and I do not care which method you use, you still end up with the same answer at the end.  I try and encourge students to use independent thinking and use what works for them. 

Sometimes when I see students approaching a problem differently I will have them both place their work on the board and start a discussion on “Who is Right?”  We usually have a good learning experience with students learning multiple approaches to solving problems.

Translating Pattern Narrative into Formal Math Language

The Triangle has a diagonal of 1 going down the right and left side of the triangle. Each number inside of the triangle is computed by taking the sum  the two numbers immediately above.  If you follow the 2^nd  diagonal inside all the way down  you will notice the counting numbers.  The 3^rd diagonal has a pattern which are called triangular numbers and the 4^th diagonal has tetrahedral numbers.  Another pattern I noticed is that each time you take the horizontal sum as it descends it doubles.  The final pattern I neticed is called a palindrome, the numbers in each row as they count  up, once they hit the middle the count back using exactly the same numers.

 

Non-Linear Pattern Web Quest

“Fibonacci” and “Phyllotaxis” and “Prime Numbers”

 My first encounter where I became interested in the Fibonacci Sequence came when I was reading “Angels and Demons and “The DiVinci Code”  by Dan Brown.  The books made referral to Fibonacci in trying to solve their puzzles.  At this point my interest was sparked, so I felt that this would be a good topic to do for my Web quest.

I was not familiar that  the original problem that Fibonacci investigated was about how fast rabbits could breed in ideal circumstances.

A newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was…

How many pairs will there be in one year?

At the end of the first month, they mate, but there is still one only 1 pair.

At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.

At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.

At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, …

Source:  http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#rabeecow

 I was also not aware that Fibonacci numbers were discovered in many natural forms. Many types of flowers have a Fibonacci number of petals. (daisies tend to have 34 or 55 petals, sunflowers have 89 or 144). The seeds of sunflowers spiral outward both to the left and the right in a Fibonacci number of spirals. The whorls on a pine cone, the numbers of rings on the trunks of palm trees, the patterns of snail shells all follow a sequence of Fibonacci numbers.

 Source:  http://www.daviddarling.info/encyclopedia/F/Fibonacci_sequence.html

 “Fractals” and “Nature” and “Patterns”

I became familiar with fractals a few years ago when I was sent by my school district to attend a mandatory in-service training for math teachers. It was a very long day and this topic was presented as the very last topic.  I sat there partially interested wondering how this pertained to what I was teaching and could not find a link.  So when I saw the topic as one to possibly explore I decided to go for it. 

 The geometry of Fractals brings us a new appreciation for the natural world and the patterns we observe in it.  Many things previously called chaos are now known to follow subtle subtle fractal laws of behavior. So many things turned out to be fractal that the word “chaos” itself (in operational science) had redefined, or actually for the FIRST time Formally Defined as following inherently unpredictable yet generally deterministic rules based on nonlinear iterative equations. Fractals are unpredictable in specific details yet deterministic when viewed as a total pattern – in many ways this reflects what we observe in the small details & total pattern of life in all it’s physical and mental varieties, too ….

 Source:  http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html

  Images I found particularly striking

  

  

Manifestations of Nonlinear Patterns

There are many nonlinear patterns found in my home.  The first would be the hard wood flooring I have in my home.  The floorboards are laid in a pattern depending on the measurement of the boards laid.  Another nonlinear pattern would be the brown eyed Susans and daisies planted in my garden. 

  Web Quest with My Students

If I were to do this web quest with my students I would give them just one topic to explore.  I would have them provide at least 3 new pieces of information that they learned as well as some images to illustrate what they learned.  I would also ask them to give one example in which they saw or could use what they learned in real life.  I would have them put all their information together in a PowerPoint so it could be easily shared with the rest of the class via presentation.

Defining Linear Patterns

Non-traditional patterns – are patterns that do not follow a repetitive format. 

 My definition of a linear pattern is a pattern that repeats in a predictable pattern.  Even numbers are an example of a linear pattern.  They count up by 2’s. The rate of change between the numbers is always the same.

 The formal definition of linear pattern according to http://www.mathsteacher.com.au/year8/ch15_graphs/02_linear/patterns.htm is a pattern which can be expressed as a linear equation, or a pattern that exists when the ordered pairs plotted form a straight line (formal definition)

 The difference between my definition and the formal definiton is that I do not include information on equations and graphing.  I start with a simpler form that the students are able to relate to through other topics that they may have discussed. 

 In helping students learn the definition of a linear pattern I would start with a simple definition that students can relate to such as the one I provided.  I would then show them how you could take that information and graph it or use it in an equation building on their prior knowledge.   Once the students notice the change in the pattern and then that the pattern can turn into a line, the concept of slope which comes  a bit is much easier for them to grasp.