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




• 
  

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

• 
  
  
• 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".

October 20, 2010

What have we learned?  Things to make sure I am familiar with for the content test at the end of this class....

• Part Part Whole -
• Commutative Propery
• Associative Property
• Distributive Property
• Identity Property
• Zero Property
• Concrete - Representational - Abstract
• Set/Measurement Models
• Base Ten
• Standard
• Expanded
• Word
• Shortened
• One to one correspondent
• Terminology
• "Nice" numbers
• Regrouping
• Inverse operation
• Operations
• +
• -
• x
• divide
• Hops, jumps, skip counting
• Array
• Area
• Compensation Method
• Algorithm
• Minuend
• Addened
• Difference
• Factor
• Sum
• Product
• Compatable numbers
• Rounding
• Estimating
• Shortened form
• Partitive
• Quotative

Integers

• We started with counting, then there is zero, and then progressed to Integers which includes negative numbers. 
• 

• The number line is a tool kids should be familiar with.
• Where do the negative integers start?  With -1, don't include the fractions that fall between zero and 1. 

• Less than - numbers to the left of that number on number line

• Greater than - numbers to the right of that number on the number line

• Absolute Value - how far a number is from Zero 

• The point of absolute value is distance, and is used in physics
• Opposites - relationship between numbers that make the whole number Zero.

Addition = Part + Part = Whole
Subtraction = Whole - Part = Part

Readings for Next Week
Van de Wall pgs 479 - 481 - Integers 481 - 486 (problems - look at them)
Parker - pgs 185 - 189

October 12th Reflection

I feel like I didn't get very much from last weeks class.  Part of the is my fault, since I was late coming to class.  I feel like each week I leave with different strategies for teaching that particular different, and this week I left knowing about the scaffolding method for divison and the lattice mathod for multiplication.  I really liked learning the scaffolding method, I have never seen that before and can see how that could help some students with the whole division process.  I thought it was funny how long we spent talking about the different ways to remember the steps to division, Multiply, Divide, Subtract, Bring Down.  Some teachers are very creative, I was boring when I taught this to my students.  I just had a poster I created that was on my whiteboard and as we solved the division problem we would put a check mark next to each step as we finished it. 

Was there more I should have gotten from last week's class? 

October 12, 2010

Division

Help students remember the division process - D, M, S, B - Divide Multiply Subtract, Bring Down

Ideas to show students how to divide by zero and why it doesn't work

• show an equation
• give students a word problem

Scaffolding Method

The Lattice Multiplication Method

The answer is 64,470

October 6 Reflection - Better Late than Never Right?

So, I didn't do my reflection until just now but to be honest I didn't really know what to write after that class.  I feel better about the break it up strategy.  I think as a teacher I would present these different math strategies and let the students decide what works best for them.  I am not satisfied with students just getting the right answer, I like to know that students know HOW to get to the right answer.  I think it is important to show students that there are multiple ways to get to an answer.  I was always taught to solve problems using one method, and I think I would have done better in math had I been shown different strategies.

October 6, 2010

Grid Multiplication Model:

Multiplication Facts

• As math gets more challenging, kids need to make sure they know their math facts or they won't be able to move on.
• Kids can't go back in mathematics
• There is forced progression in math, THEY NEED TO KNOW THEIR MULTIPLICATION FACTS

Multiplication Fact Strategies

• 9 = 10 take away 1 
• 9 x 4 = 10 x 4 = 40 then take away one group of 4 40 - 4 = 36
• 2 = double
• 4 x 2 = (4 + 4 = 8)
• 3 = skip counting
• 4 = double, double
• 4 x4 = (4 +4 +4 +4 = 8 + 8 = 16)
• 5 = skip counting, there is a pattern
• 6 = 5 plus 1
• 6 x 5 = (5 x5 = 25 + 5 = 30)
• 7 = skip counting
• 8 = double, double, double
• 12 = ten plus 2s 
• 12 x 2 = (10 x 2 = 20 +2 + 2 = 24)

Lesson Study

• Launch - how are you going to prep those kids for the lesson
• What are the guiding questions?  Get the students talking.
• Anticipatory set - where students may make mistakes
• Interventions - how can you respond to the student errors
• Exploration
• What are the guided questions
• Anticipatory set
• Interventions
• Wrap Up/Classroom Discussion
• What are the guided questions
• Anticipatory set
• Interventions
• Assessment Piece

Readings for Next Week:

• Beckman p 115 Activity 6B #1-2
• Beckman reading 234 - 235
• Van de Wall pg 155
• 
  

Kitten Match

A second grade teacher showed me this online game.  It is a great game to have students practice addition skills.

Click on the picture to go to the game.

Ma and Pa Math

I showed this video to my 5th grade students a few years ago.  We had a great discussion about what misconceptions Ma and Pa had about division.  We also talked about how we would try to explain the correct way to divide to Ma and Pa

September 29th Reflection

Most of what was covered in this lesson was information I already knew.  One concept I am having a difficult time with is the Break it Up mental math strategy.  It looks complicated to me and not something i could do as mentally.  Isn't that the point of mental math, is to do it in your head?

Maybe that is a misconception I have.  When I think mental math, I am thinking of problems I can do in my head without writing it down on paper.  However, I find it very important for students to write down the process to which they got to an answer.  The examples in the lesson for Break it Up, weren't problems I would expect students to do "mentally," I would expect to see their work on paper.

September 29th Class - Independent Study

Multiplication and Division

Repeated Addition is used to focus on the meaning of the operation of multiplication.

5 + 5 + 5 + 5 = 4 x 5  

4 groups of 5

Models for Multiplication

Properties of Multiplication

• Associative Property: grouping doesn't matter
• 2 x (3 x 4) = (2 x 3) x 4
• 35 x 4 = (7 x 5) x 4 = 7 x (5 x 4) .....this property can also break down larger #s
• Commutative Property: order doesn't matter
• 5 x 4 = 4 x 5
• 3 x 2 x 5 = 5 x 3 x 2
• Zero Property: zero times any number is zero
• 5x 0 = 0
• 0 x 4 = 0
• Identity Property: anything times one is itself
• 7 x 1 = 7
• Distributive Property: One of two factors can be split into two or more parts and each part multiplied separately and then added.
• 5 x 32 = 5(30 +2) 
• 5 x 38 = 5(40 - 2)

Mental Math 

• Rounding and Estimating: When multiplying and estimating round numbers to the leading digit place and then multiply.
• Benefits - rounding help make the multiplying process much easier to do mentally by making the numbers more compatable
• Disadvantages - when rounding, you are either going to have an estimate that is high or low, and students need to remember that it is an estimate not an exact answer.  Some student struggle with this concept and why to even estimate.  
• Example:
• 479 x 54 = 500 x 50 ~ 25,000
• Expanding: when multiplying a single digit by a number with a two or more digits, expand the multiplication by each place value in the larger number.
• 542 x 3 = (3 x 500) + (3 x 40) + (3 x 2)
• 1500 + 120 + 6
• =1626
• Break it Up: You can break up a number and multiply by a power of ten or hundred
• 256 x 4 + (250 x 4) + (6 x 4)
• Skip counting:  Sometimes it is just easier for some kids to learn the order and count by multiples.
• 2, 4, 6, 8, 10, 12
• 5, 10, 15, 20, 25, 30
• 4, 8, 12, 16, 

Thinking Strategies for Multiplication

• Stage 1 
• Introductory stage
• develops the meaning and interpretations of multiplication
• introduced using repeated addition and rectangular arrays
• Stage 2
• lasts through grades 2 - 3
• 2 goals: increasing calculational proficiency and deepening conceptual understanding
• students see plenty of word problems, mental math and worksheet problems
• begin committing 1-digit multiplication facts to memory
• Stage 3
• goal is simple: all students should have memorized the multiplications up to 9 x 9 by the end of third grade

Reflection for September 22nd Class

1. A mathematical idea from the session that is new or important.
   This session validated how important it is to teach students different strategies when adding and subtracting.  After the previous session I left class thinking to myself, "Why on earth would I teach students compensation?  It is SO confusing!"  This class reminded me how important it is to teach the strategies, and then students will pick what strategy works for them.  When I was in the classroom, I was constantly having my kids come to board and solve the problems, but not just solve them but talk the rest of the class through what they did to get to the answer.  It is also nice to hear how the other teachers are connecting what we are doing in out math endorsement class to what they are doing with their students.  I feel that I actually have a disadvantage being an Ed Tech.  I don't get to practice what I am learning on students, but at least I can learn from my peers.
2. A question I have.
   I don't have a question right now, but I am excited for the next session when we get into multiplication and division.
3. An application of the mathematical idea for my classroom.
   My answer to this is the same as last week.

September 22, 2010

LookUsing a Ten Structure to Solve an Addition Problem

 

pg 52 of Beckman activity manualing at examples of students solving an addition problem using Ten Structure pictures

• Students #1 and 2 get it.  They understand how to group numbers and "carry" to the next place value.
• Student #3 - may not fully understand place value, and doesn't understand "changing out"  will struggle when moves on to subtraction.

Subtraction

How can we represent subtraction so students understand?

• 2+__=7 - student will subtract 2 from 7.....or count up
• Students need to understand fact families

• Number bond idea  

A misconception that a lot of student have is that addition and multiplication ALWAYS make numbers bigger.  NOT TRUE....not when you are adding or multiplying negative numbers.

Many students would say that addition is easier because they can "move" the numbers around, where they can't be moved in subtraction.

Pictorial Models for Subtracton:

Solving the same problem using Compensation

Another strategy to solve a subtraction problem  (I think its the Austrian strategy)

Pg. 54 in the Beckman activity manual

Pg. 221 of the Van de Wall book

• examples/strategies that students have used to solve the subtraction problem 73-46, using a number line.
• Add tens to get close, then ones
• Add tens to overshoot, then come back (it's good for kids to know that they can go backwards on a number line)
• Add ones to make a ten, then tens and ones
• Take tens from tens, then subtract ones
• Take away tens, then ones
• Take extra tens, then add back (compensation examples)
• Add to the whole if necessary (compensation examples)

Reading Assignment

• Van de Wall pg. 157 - 161
• Parker 1.5 pgs. 25-30