December 15, 2010

Homework Review

Beckman activity manuel pg 8

Representing decimals as lengths

Manipulatives to help students understand decimals and negative numbers

Decimals

• Money
• Base 10 blocks
• The big cube is 1, the square is .10, strip of 10 ones is .01, and one tiny cube is .001
• Strips
• Fraction circles
• Number lines
• Adapted base ten system




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Base 10 activity (Van de Wall pg 333) 

Activity 17.2

Decimals/fraction

Percent - the term percent comes from per centan which mean "of 100".  So a percent is of 100, it is a decimal. It is the hundredths spot in a decimal.  When the denominator is 100, it is a percent.

If a student has mastered decimals and fractons, you can teach them percents in a day.  They will pick up this concept fast.

My Groups example of finding 15% or 

Two other group examples of how to use a model to find 15% increase of $90

3 column model.  Steve took the $90 and broke it into 10 increments of 9, and divided one of the 9s in half, so he took 1 9 and the half (4.5) and the increase is 13.5

Realistic Percent Problems

• Realistic percent problems are still the best way to assess a student's understanding of percent.  Students can relate and gain a better understanding if they are solving a problem they relate to.
• Don't give students strategies when they are learning to solve percents.  Students will have to think deeper about what a percent is, and often times they will relate percents back to fractions and decimals.
• We need to teach our students how to think for themselves.

Beckman Hardcover pg 84 #1

December 8, 2010

Van de Wall activity pg. 292 - Developing Fraction Concepts

• The different tiers represent ways to differentiate instruction based on student experience and student achievement of lack there of. 

Tier 1 task:  for students who still need experience with having

question - How can 2 people share 3 brownies?

Tier 2 task: for students comfortable with halving and ready to try other strategies

question - How can 4 people share 3 brownies?

Tier 3 task: for students ready to solve tasks where students combine halving with new strategies.

question - How can 3 people share 5 brownies

Fractions Greater than 1

Something I already knew - The explanation that was given for 3 1/4.  There are 4 4ths in one, 8 4ths in 2, 12 4ths in 3 and the last 4th gets us to 13.

AHA Moment - don't use the term "improper fractions"  call them numbers greater than one. Improper doesn't mean the fraction is wrong. Improper fractions mean that you have something greater than one, and stress to students that an improper fraction and mixed numbers are equivalent ways to represent the same fraction.

Manipulatives that we can use to help with fractions

1. Unifix cubes
2. Linking cubes
3. Egg cartons - example: use a string to have students find a half, and then students can divide that have by a third with another string.
4. Paper folding, cutting or tearing
5. Graphing paper
6. Fraction strips
7. Circular fractions (pie charts)
8. 10 frames
9. Geoboards and rubber bands or a print out of a geoboard that students can draw on
10. Pattern blocks
11. Beans and cups
12. Hershey chocolate bars or other foods

Van de Wall pg 298

Figure 15.12 Examples of how to assess student fraction knowledge instead of testing the algorithm.

• the set model helps students recognize the whole 
• The rods can be hard, but having students use the rods can help students visualize the fractional parts of the whole, can easily substitute linking cubes or paper.
• This figure tests ALL MODELS.  Most teachers will just use the area model, which is a square or rectangle that you divide.  It is a good idea to have students divide with all models.
• Teach all models separately
• The area model is by far the most difficult model for students to master, especially if they are given irregular figures.
• Students had to understand fractions greater than one, what a numerator is, what a denominator is, and recognize the whole in order to solve the problems in Figure 15.12.

Mixed Numbers and Improper Fractions

Pluses about Mixed Numbers

• Separates whole fraction
• Quicker recognition 
•  Real world applications

Minuses about Mixed Numbers

• Can't do math
• Constantly have to change them to improper fractions or decimals when doing math operations
• Don't give you as much information in some cases (like algebra)

Pluses of Improper Fractions

• Easier to work with
• Preferred for algebra

Minuses of Improper Fractions

• Hard to visualize
• Real world applications, but are very technical and not applications we are going to give to our kids

The whole is always a separate piece of the fractions.

The wholes in this example are different, but the fractions are the same. The whole is always separate of the fraction.  The part that you have and how many groups you have broken your whole into are the fraction.

**Fundamental key to fractions is understanding to recognize the whole**

How to order fractions

• Benchmarks kids should be able to recognize to help them order fractons
• fractions should be close to zero,1/2 and 1
• When multiplying if the fraction is less than one the number will get smaller, if the fraction is larger than 1 it makes the number larger  
• If you multiply by zero the number will get larger
• 1/2 helps break fraction into subgroups.  
• Example 6/7 is close to 1, 1/10 is close to 0, 4/10 is close to 1/2, 5/7 goes between 1/2 and 1 (see example below - just comparing to 0, 1/2 and 1)

Homework

Van de Wall pgs 328 - 332

                     pgs 337 - 340

Problems from the Beckman hard back book - pgs 65 - 66 section 2.4 1, 5, 8

November 17, 2010

Sieve of Eratosthenes (located at the National Library of Virtual Manipulatives)

This virtual manipulative displays a grid containing numbers from 2 to 200. You can use it to explore patterns and relationships involving multiples.

Using this virtual manipulative you may:

• Remove multiples of a number
• Show multiples of numbers
• Reset the workspace
• Choose how many rows to display

Prime Factorization - Using the Cake Method

You can use the cake method and prime factorization to find the GCF and LCM

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• The cake method allows you to find the GCF and the LCM at the same time.

Finding the LCM

If there are no common factors then multiply the numbers together (For example: 10 and 3)

November 10, 2010

The Cake Method is a way to find the prime factorization and even least common multiple

It's called the cake method because it looks like an upside down layered cake.

Even numbers are divisible by 2

Is zero divisible by 2? Yes

What are odd numbers?  Numbers that are not divisible by 2.

Divisibility Rules

• 2 - if the number is an even number, then that number is divisible by 2
• 3 - the sum of the digits of a number is a multiple of 3, then the number is divisible by 3.  Example:  7491 = 7+4+9+1 = 21, 21 is a multiple of 3, so 7291 is divisible by 3
• 4 - if the last two digits of a number are divisible by 4, then that number is divisible by 4. or  Half half rule.  If you can half those last two number twice, then the number is divisible by 4. Example; 48 divided in half is 24, 24 divided in half is 12.  48 is divisible by 4. (up to 100)
• 5 - if the number ends with a zero of 5, then the number is divisible by 5
• 6 - If the number is divisible by 2 and 3, then the number is divisible by 6
• 8 - look at the last 3 digits of the number, cute them in half.....cut in half again.....cut in half again...if you can cut that number in half 3 times, the number is divisible by 8. (up to 100)
• 9 - add all of the digits, if the sum is divisible by 9 then the number is divisible by 9
• 10 - if the number ends in a zero, then that number is divisible by 10
• 11 - if the last two digits of a number are a double number for example; 44, 55 or even 66....then the number is divisible by 11.

Forehead Multiplication Game (Name is still in progress, input is welcome) 

• Needed:  one deck of cards, but take out the aces and face cards
• students get in groups of 3
• one person is the captain
• the captain draws two cards, and multiplies the two together (without showing the two cards to the other students)
• captain hands the cards to the other two players face down
• captain the says the product as the two players hold their card on the forehead.  (the two players don't see what their card is.  They can only see what the other player has.)
• Students then have to figure out what their card is.  
• The student who says their number first and accurately gets to keep the cards,
• The player with the most cards at the end wins and gets to be the captain.

October 27th Reflection

I really liked the cauldron activity.  What a great activity to help demonstrate the addition and subtraction of integers.  For example, -7 - 5 is n -7 subtract +5.  Doing this problem using the cauldron can help me demonstrate what this means.  It also helps demonstrate "zero pairs".  So when I have -7 (which is the yellow pieces in my cauldron) I can't take out 5 positives (the green pieces) so that is why I add my zero pairs and I can now take out 5 positives and when I take those out, I am left with the answer.  My question about this is, when do teachers teach about adding and subtracting integers.  As they are teaching addition and subtraction?  I guess, since I am not in the classroom, I can't picture when I would do this type of activity.

Ahhhh....teaching addition and subtraction of integers using the number bond.  I totally get the addition of integers, but for some reason I struggle with the subtraction of integers. The more I worked with the problems, I was starting to get it.  I was never taught to look at addition and subtraction problems as part part whole.  This class is challenging me to remember that, but this is good for me.

October 27

Real numbers are all numbers, the number system. Both rational and irrational
Rational numbers are numbers that can be written as fractions and decimals
Irrational numbers - numbers that you cannot find a perfect square of. "pie" is an irrational number, numbers that keep repeating
Natural numbers - the counting number not including zero,
Whole numbers - counting numbers including zero, don't include negative numbers
Integers - negative numbers, positive numbers and zero.  NOT including fractions and decimals.

3 - real, rational, integer, natural

-3 - integer, rational, real

3.33 - rational, real

1/2 - rational, real

√7 - irrational, real

pie - irrational and real

√25 - natural, whole, integer, rational, real

9/3 - natural, whole, integer, rational, real

√16/4 - natural, whole, integer, rational, real

4^{3 - natural, whole, integer, rational, real}

^{e - irrational, real}

^{Number Bonds}

^{Addition is always part plus part equals whole}

Subtraction is whole minus part to find the other part

Multiplying Integers

negative x negative = positive

positive x positive = positive

positive x negative = negative

negative x positive = negative

The first statement is the who  The second is the action (what happened)  and the = is the result.

example:

1. Mother Teresa wins the lottery = a good thing ( + x + = +)
2. Mother Teresa gets mugged = a bad thing  (+ x -- = --)
3. The Devil wins the lottery = a bad thing (-- x + = --)
4. The Devil gets mugged = a good thing (-- x -- = +)

positive - Eric Bana

action - wins an oscar

negative - Lindsay Lohan

negative action - bad accident and gets deformed

1. Eric Bana wins an oscar = a good thing - he gets more movie opportunities  (+ x + = +)
2. Eric Bana gets in a horrible car crash and gets deformed =  a bad thing no more acting career for Eric Bana  (+ x -- = -)
3. Lindsay Lohan wins an oscar = a bad thing - Why on earth would she win an oscar?  She's a horrible actress (-- x + = --)
4. Lindsay Lohan gets in a horrible car crash and gets deformed = a good thing - she can get out of the spotlight and rebuild her life. (-- x -- = +)

Distributive Property

Trichotomy - there are three relationship 2 integers can have and only three.  

1. negative < 
2. less than > 
3. equal to =

This is a transitivity statement;  a = b  b = c => a = c  

                                                6 < 8,  8 < 10 =>  6 <10

October 20 Reflection

I really liked the explanation for why a negative times a negative is a positive.  I remember getting this question from students and not being able to give them a good answer/explanation.  I like the explanation that Steve gave about positives and negatives using the example of money.  That is a great way to share w/students because money is an example they can relate to.   I like the idea of using the example, "If you take something bad away from you its "good".