RULES of Exponents

We have also been talking about exponents. This is one of those few things in math I enjoy, like my factor trees. Exponents is simply a shorthand way of writing multiplication. The exponent represents how many times it will be multiplied by itself.
ex.    5*5*5=5³

Just like all of math though, you have to be aware of the rules that come with exponents. There are some basic rules of exponents:

1st: To try to simplify or do any of the following, the base number must be the same. So in our examples below our base is a.
(a⁴)(b³) = aaaabbb = (a⁴)(b³). Nothing combines.

2nd: When multiplying powers with the same base we add the exponents.

3rd: When dividing powers with the same base we subtract the exponents.


4th:  When you have an exponent expression that is raised to a power, you can multiply the exponent and the power.


5th:   If the power contains a negative exponent, rewrite the expression with a positive exponent by taking the reciprical.

6th:    We said "The exponent represents how many times it will be multiplied by itself. When your base number is to the first power it is ALWAYS the base numer.

7th: Anything to the power of zero is ALWAYS  "1"


I got my clip art from: http://www.math-play.com/exponent-game.html

Factor Trees

Picture from:  http://www.gradeamathhelp.com/factor-tree.html

I feel like these should be done over and over when you have a good grasp on multiplication and division. I don't remember doing factor trees in school, but something else I wish we had done. I love reading and writing, I consider those fun. I can not normally say the same for math, but I really do like doing factor trees. Math just like any other subject should be made fun and I think the student will learn better. Our teacher is teaching us a few different ways to teach one or two things, which is how it should be. As a teacher I want to be able to help my kids and if they don't get it one way, to be able to show them another.

Problem Solving Strategies

 So I have a test coming up on Thursday. I try to keep these problem solving strategies in mind, while taking my test. G. Polya wrote a book titled 'How To Solve It' in 1957.  I look at all math problems with "What type of information is being asked for?"  I got that from word problems early on. Picking out all of the important information and seeing the problem through. Then going back through the problem to make sure I did not miss anything.
   I liked this diagram because it really gives you examples of each step. I feel like if I didn't get it the first time, I might not get it. This can be really discouraging. Don't be afraid to walk away and come back later. Sometimes things need to process. If it is in a test taking situation, skip the question come back to it.

I received my images : http://fcit.usf.edu/fcat8m/resource/fcat/strat.htm

Problem Solving Strategies

• Look for a pattern
  

  Example: Solution:	Find the sum of the first 100 even positive numbers.
  The sum of the first 1 even positive numbers is 2 or 1(1+1) = 1(2). The sum of the first 2 even positive numbers is 2 + 4 = 6 or 2(2+1) = 2(3). The sum of the first 3 even positive numbers is 2 + 4 + 6 = 12 or 3(3+1) = 3(4). The sum of the first 4 even positive numbers is 2 + 4 + 6 + 8 = 20 or 4(4+1) = 4(5).
  Look for a pattern: The sum of the first 100 even positive numbers is 2 + 4 + 6 + ... = ? or 100(100+1) = 100(101) or 10,100.

  
  
• Make an organized list

  Example:	Find the median of the following test scores: 73, 65, 82, 78, and 93.
  Solution:	Make a list from smallest to largest: 65 73 78 Since 78 is the middle number, the median is 78. 82 93

  
  
• Guess and check
  

  Example:	Which of the numbers 4, 5, or 6 is a solution to (n + 3)(n - 2) = 36?
  Solution:	Substitute each number for “n” in the equation. Six is the solution since (6 + 3)(6 - 2) = 36.

  
  
• Make a table
  

  Example:	How many diagonals does a 13-gon have?
  Solution:	Make a table: Number of sides Number of diagonals 3 0 4 2 5 5 6 9 7 14 8 20 Look for a pattern. Hint: If n is the number of sides, then n(n-3)/2 is the number of diagonals. Explain in words why this works. A 13-gon would have 13(13-3)/2 = 65 diagonals.

• Work backwards

  Example:	Fortune Problem: a man died and left the following instructions for his fortune, half to his wife; 1/7 of what was left went to his son; 2/3 of what was left went to his butler; the man’s pet pig got the remaining $2000. How much money did the man leave behind altogether?
  Solution:	The pig received $2000. 1/3 of ? = $2000 ? = $6000 6/7 of ? = $6000 ? = $7000 1/2 of ? = $7000 ? = $14,000

  
  
• Use logical reasoning
  
  

  Example:	At the Keep in Shape Club, 35 people swim, 24 play tennis, and 27 jog. Of these people, 12 swim and play tennis, 19 play tennis and jog, and 13 jog and swim. Nine people do all three activities. How many members are there altogether?
  Solution:	Hint: Draw a Venn Diagram with 3 intersecting circles.

  
  
• Draw a diagram

  Example:

  Fortune Problem: a man died and left the following instructions for his fortune, half to his wife; 1/7 of what was left went to his son; 2/3 of what was left went to his butler; the man’s pet pig got the remaining $2000. How much money did the man leave behind altogether?

  
  
• Solve a simpler problem

  Example:

  In a delicatessen, it costs $2.49 for a half pound of sliced roast beef. The person behind the counter slices 0.53 pound. What should it cost?

  Solution:

  Try a simpler problem. How much would you pay if a half pound of sliced roast beef costs $2 and the person slices 3 pounds? If a half pound costs $2, then one pound would cost 2 x $2 or $4. Multiply by the number of pounds needed to get the total: 3 x $4 = 12. Now try the original problem: If a half pound costs $2.49, then one pound would cost 2 x $2.49 or $4.98. Multiply by the number of pounds needed to get the total: .53 x $4.98 = $2.6394 or $2.64.

  
  
• Read the problem carefully
  Know the meaning of all words and symbols in the problem.

  Example:	List the ten smallest positive composite numbers.
  Solution:	Since positive means greater than 0 and a composite number is a number with more than two whole number factors, the solution is 4, 6, 8, 9, 10, 12, 14, 15, 16, 18. For example, 4 has three factors, 1, 2, and 4.

  
  
  Sort out information that is not needed.
  

  Example:	Last year the Williams family joined a reading club. Mrs. Williams read 20 books. Their son Jed read 12 books. Their daughter Josie read 14 books and their daughter Julie read 7 books. How many books did the children of Mr. and Mrs. Williams read altogether?
  Solution:	You do not need to know how many books Mrs. Williams has read since the question is focusing on the children.

  
  
  Determine if there is enough information to solve the problem.

  Example:	How many children do the Williams have?
  Solution:	There is not enough information to solve the problem. You do not know if Josie, Julie, and Jed are the only children.

  
  

Factors & Multiples

We did Factors and Multiples today in class. Teaching factors and multiples is a great way of enforcing and really checking to see if your students known and/or understand multiplication and division. I think that quizing your students on things like this are very important, for many reasons. Sometimes it is difficult to come to a teacher to tell them you do not understand. Doing Factors and Multiples also reinforces this for the student as well. If you feel as though you are having a hard time, practice, practice, practice.

You MUST  use the resources around you though, including your teacher!  I have found some great games for things that I did not understand as a kid. 

This is where I recieved my images:  http://www.mathsyear2000.co.uk/magnet/minus/multiples/index.html

I love Mnemonic Devices

Today we talked about Estimation, Order of Operations, Division, and Exponents. When I was a kid I learned best from mnemonic devices. Who remembers "My Very Excellent Mother Just Served Us Nectarines" (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune)?  It's hard memorizing things sometimes, but I had never heard this one before. This would have been beyond helpful in Pre-Algebra, which is when my problem with math started.

"My Dear Aunt Sally"- The Big Picture
	Two rules of thumb are helpful in remembering the order of operation or the prioritization which governs mathematical expressions. These are: The more sophisticated operation has priority. Work from left to right if operations have the same priority.     Multiplication and division are more sophisticated than addition and subtraction so, multiplication or division is done before addition or subtraction.     Multiplication and division have the same level of sophistication so they have the same priority. THE LEFT MOST OPERATION IS DONE FIRST.

I loved everything about this site. It is very math friendly and where I got my pictures from.
http://www.sd104.s-cook.k12.il.us/students/math/ileanazarco/aboutmath.htm