Message to Lourdes Families (May 26)

I have spent the evening reading the letters written by the 6th, 7th, and 8th grade students that were put together in a binder entitled “Thank You for the Memories”.  What a special gift!  Your letters were so thoughtful and genuine.  I will treasure them always.

I also spent time reading and looking at the messages you gave to Mrs. Blaum, who put them together into a beautiful creation!  (I think she is a master at scrapbooking.)  What a touching and humbling present.  I read through my tears many, many times.  Thank you for your wonderful memories and wishes.  Lourdes Catholic School will always have a special place in my heart.

Spring Break Extra

June, July, August 2010

Have a wonderful summer, but remember to keep you math brain working!

1.  Find the tip when you go out to eat with the family.
2.  Figure out how many cubic feet of water are in your swimming pool.
3.  Calculate your gas mileage while driving on your family vacation.
4.  Figure out how long the trip will take based on speed and distance.
5.  Find your current batting average after every game.
6.  Use a pedometer, and find average steps walked.
7.  Cook and use fractions and liquid measures as you create delicious food.
8.  Visit some of the math websites that are linked to Mrs. Walsh’s blog (such as Illuminations, Project Interactivate, and the National Library of Virtual Manipulatives)

8-B-2: Reflection on Blogging

Describe your blogging experience in this course. Do you think you will continue using your blog? Why or why not?
When I first learned that I was required to blog for my class “Teaching Algebra to Middle School Students”, I was nervous about it.  Two of my daughters had created blogs and done very nice work, but they are much more comfortable with technology than I am.  Besides, my daughters no longer live at home, so I was on my own for this blogging adventure.  Needless to say, I was far more at ease reading their blogs than creating my own.  Once I got my blog up and running, I did talk to my daughters on the phone, and got some suggestions on how to import photos and link websites.  By then, I knew how to accomplish connecting “images”, and blogging became much more fun for me,…more creative than “just words”.
I am planning to continue blogging, but my walshmath blog will take on a different purpose.  I plan to have it linked to my school’s website.  It will display the weekly Math assignments for my 6th, 7th, and 8th grade students.  I am also planning to link helpful websites that relate to the topics we are covering.  That should be helpful for students who need additional practice, but also good for students who enjoy math and want to do more of it!
What did you learn about yourself and your abilities or interests in Math or Algebra?
Although I do not currently teach an 8th grade Algebra class, I realize that I do teach all of the topics presented in “Teaching Algebra to Middle School Students” in at least one of my middle school math classes.  This class also helped me appreciate my MATHThematics series, by McDougal Littell.   Although it is organized by theme, not by concept, all of the important math ideas outlined in this class are topics that I cover with my students.

Although I had already used a few applets in my math classes,  (like The Maze Game, Tessellate!, and The Factor Game), there are a lot more out there to explore.  I plan to incorporate Search and Rescue , Balance Scales,and Line of Best Fit into my curriculum.  Applets are a fun way to explore concepts and learn skills.
Did you learn or discover anything you found particularly interesting through your course activities or your own internet research? Describe one interesting discovery and why you found it fascinating.
Besides blogging, this class has required making powerpoints.  When I started my first one 8 weeks ago, I was incredibly slow and did not know any shortcuts for speeding the process along.  Now I discovered that I really enjoy creating powerpoint presentations.  I love the colors and images you can include.  They are a new tool for grabbing student interest.
Do you think you will use journals with your students? Do you think you will use blogs? Why or why not?
I am planning to incorporate some journaling in my math classes this coming year.  One journal topic that I particularly enjoyed was “My Mathography”.  I would like my students to write theirs and hope that they will be informative and insightful.  The journaling that they do in my math classes will probably be rather sporadic, but I like the idea of asking students to write a short note on the back of an assignment regarding “What was hard or easy?” or “What did you learn that was new?”  I also like the idea of students responding to prompts such as “I think calculators…” or “To study for a math test I usually…”  I probably will not have my students create a blog simply because I do not have time to read 80 of them!  Maybe in the future, I could have one class start, and take it from there.

8-B-1: Factoring Quadratics in Your Own Words

When teaching my students to factor quadratics, I first make sure that they understand what a term is.  A term is a number and/or a variable that can be connected by an addition or subtraction symbol.  For instance, there are 2 terms in the binomial 2x+1 (2x and 1).  There are 3 terms in the trinomial x^2+3x+2 (x^2 and 3x and 2).   Next, I teach them to multiply binomials.  That will help them have a feel for the process of multiplying positive and negative quantities.  Then, when it is time for factoring trinomials, I tell them to undo the process.  Think of it like solving a puzzle.  The operations symbols give lots of hints.  If the sign in front of the 3rd term is positive, you know you will have “like signs”.  If the sign in front of the 3rd term is negative, you know you will have “unlike signs”.  The symbol in front of the 2nd term of the trinomial shows you dominance.  Which had more weight, the positive or the negative numbers?
Let’s practice… look at the operation symbols in the trinomial.
•    If both operation symbols are plus signs, then both signs in the binomial factors will be positive… x^2+6x+5=(x+5)(x+1)   Do you see how the 5 and the 1 multiply to make 5 (the 3rd term in the trinomial), but add to make 6 (the 2nd term in the trinomial)?
•    If the first operation symbol is minus and the second is plus, you know you will have “like signs” in your binomial, and they will both be negative…
x^2-6x+8=(x-2)(x-4)  Again, -2 and -4 multiply to make +8, but add to make -6.
•    If both operations symbols are minus signs, then you will have unlike signs in your binomial factors, but the larger number will get the negative sign in front of it…x^2-3x-10=(x-5)(x+2)  -5 and +2 multiply to make -10, but add to make-3.
•    If the first operation symbol is plus and the second is minus, you will have unlike signs in your binomial terms, and the larger number will be a positive number…x^2+x-6=(x-2)(x+3)  -2 and +3 multiply to make -6, but add to make +1.

When the quadratic equation is more complicated, and there is a coefficient in front of the x^2, it often takes some guessing and checking to find the binomial factors.  Try this…8x^2+8x-6  You know the first terms will be 8x and x or 4x and 2x.  You also know the last terms will be 6 and 1 or 3 and 2.  You also know you will be dealing with “unlike signs”.  Now it’s just a matter of guessing and checking until you find the combination of numbers that will work…(4x-2)(2x+3).
Remember, it’s like solving a puzzle. You don’t have to get it right on the first try.  Just don’t stop until you find the binomial factors that work!

Explaining the process for factoring quadratics forced me to think through the process and explain it as clearly and simply as I could.  Having my students write out the steps for completing a mathematical process will encourage them to do the same.  After covering a complex lesson, having the students write out the process may help them internalize the procedure and better understand the method.

5-A-4: Evaluating Your Definition

After viewing the definitions for equation and function that were posted by my classmates, I would make some changes to my current definitions.   I stated, “An equation is a mathematical representation of 2 quantities that are equal in value.”  Although I like my definition because it is general enough that it could be used with objects as well as numbers, it may be more clear for my students if I added that for our purposes, ”it is a number sentence with 1 or more operation symbols that includes an equal sign.”

My definition of a function, “A function is a relationship between input and output.  For every input, there is exactly one output.  The output depends on the input.” This definition would be better if I added that the output is found by applying a certain rule.

I liked both of my examples because they were both  visual and interactive, and I think they would help anyone understand what a function or an equation is by experiencing working with them.

Understanding the difference between equations and functions is difficult because some equations are functions and some are not.  The easiest way to help my students see the difference is to put examples on the board and have the students help me sort…Which are equations?  Which are functions?  Why?  Which are both equations and functions?  I might also include some numbers and expressions so they can find some items that are neither equations nor functions.

5-D-2: Applet Review

Although I have introduced my students to a few of the available applets, the ones I have used in the past are just the “tip of the iceberg”.  There is a plethora of activities available for use in conjunction with many math lessons, and I think they could be an excellent reinforcement tool that would make practice fun!

One applet that I was very excited to find is called Search and Rescue-Part I, Applet 1 and Search and Rescue-Part I, Applet 2 at ESCOT PoW Applets through mathforum.com.  It will coordinate beautifully with a 7th Grade math unit that I currently teach entitled “Search and Rescue”.  In this unit, students work in teams to locate a missing pilot whose plane has crashed in the Australian outback.  Each lesson in the unit presents a different math concept that will help students as they proceed in their search for the missing pilot.  A new skill that I teach the students is how to expand their use of a traditional 180 degrees protractor to find headings on a map from 0 degrees to 360 degrees.  We practiced the skill in class using a protractor and a map.  This applet will allow my students to actually use this skill as they navigate a helicopter to a campsite and a hospital.  I am excited that this practical use of finding headings is available for our use.

Resources: Mathforum.com

5-B-1: The Magic of Proportions

Proportions are a very useful mathematical process that can be used any time you know 3 out of 4 numbers.  I encourage my students to set up the information they know as a ratio in fraction form, with units labeled.  It is important to be consistent when setting up the equation.  Make sure the numbers in the second ratio have labels that correspond to the units in the first ratio.  Replace the unknown number with n.  Then multiply cross products to set up the equation.  Solve the equation using inverse operations.

2 examples where you can use a proportion to solve an everyday problem:

Problem 1: Emily’s dog, Amber, eats 2 lb. of dog food every 3 days.  How many lb. of dog food will Amber eat in 31 days?

Use a proportion to solve this problem.

Dog food               Dog food
eaten in 3 days    eaten in 31 days
2 lb. =     n lb.
3 days     31 days

Multiply cross products to set up an equation.
2*31=3*n
62=3*n

Divide both sides of the equation by 3
62 =  3n
3         3

20 2/3=1n

So Amber eats 20 2/3 lb. of dog food during every 31-day month.

Problem 2: There is a large oak tree outside my classroom window that is very special to our school.  It is so special that we call it “The Giving Tree” and have incorporated it into our school logo.
My principal came to my math class one day and asked if we could tell her how tall the “The Giving Tree” is.  She made it clear that she did not want any of us to climb the tree.  Fortunately, we just finished a unit on Indirect Measurement, so we got right to work.  All 19 of my students grabbed a meter stick and out we went.
Sydney volunteered to be measured.  She is 63 inches tall, and her shadow at that time of day was 45 inches long.  Then we went to the base of the tree and started lining up the meter sticks.  We counted 10 sets of 39 inches plus 17 inches more, for a total of 407 inches of shadow for the tree.

Use a proportion to solve problems involving indirect measurement.

Height of Sydney=63 inches =   n inches=Height of tree
Sydney’s shadow=45 inches     407 inches=shadow of tree

We multiplied cross products to set up the equation.

63*407=25,641
25,641=45*n

Divide both sides of the equation by 45.

25,641 = 45*n
45          45

569.8 inches=n

Then we divided by 12 to change inches to feet and got 47.48333inches
or about 47 ½ feet tall.

5-A-3: Equations and Functions in Your Own Words

An equation is a mathematical representation of 2 quantities that are equal in value.  A nice visual for this concept is the balance scale, like the one below provided by mathforum.org.  For an equation to be true, both sides must weigh the same in order to be of equal value.  Give the balance scale a try and see if you can balance the expressions.  Don’t forget to click on “Continue” and solve for x.

A function is a relationship between input and output.  For every input, there is exactly one output.  The output depends on the input.  A great visual to reinforce the concept of a function is the “Function Machine”.   Whenever you put a number in, a specific number comes out, based on the rule (or equation) that it is applying.

Give a Function Machine a try, and see if you can figure out the rule.

Resources:  Mathforum.org

Teq Smart:  Learning for Educators

National Library of Virtual Manipulatives

Shodor Interactivate

4-C-3 Translating Pattern Narrative into Formal Math Language

The triangles created by the odd numbers within Pascal’s Triangle follow a pattern of exponential growth.
1
The lengths of their sides begin with 1 at the top, or 2°.
1
1     1
The top three 1s create a triangle with 2 numbers per side; that’s 2¹.
1
1     1
1     2     1
1     3     3     1
Moving down to the side that includes 1-3-3-1, we have a 4-unit triangle, which would be 2².  The length of the triangles’ sides continues to double, or grow exponentially as we move farther down into Pascal’s Triangle.
1
1     1
1     2     1
1     3     3     1
1     4     6     4     1
1     5     10     10     5     1
1     6     15     20     15     6     1
1     7     21     35     35     21     7     1
This triangle includes 8 numbers per side, or 2³.  In other words, the length of the side is 2ˆn, with each succeeding triangle twice as large as the previous one.   This pattern of triangles, and triangles within triangles, is an example of a fractal, an object or quantity that displays self-similarity.
I also noticed that the even numbers within Pascal’s Triangle follow a pattern, as well.  The length of the sides of triangles created by even numbers follow the pattern of (2ˆn)-1.  (They are triangles with side lengths of 1, 3, 7, 15,…)  These triangles are inverted with the point down, rather than the point up.  When odd numbers are shaded black, and even numbers are shaded white, the entire form presents itself as Sierpinski’s Triangle.

Resouces:  Ask Dr. Math

Wolfram Math World