Elementary Math

Why am I here? Hi, I'm Samantha Anderson a sophomore
at ASU studying Elementary Education with hopes of becoming
an elementary school teacher. As a kid I always taught Sunday school with my mom at our church and as I got older I grew in my love for kids and my love for educating young minds. Since I began my time at ASU as a Business major, I am looking forward to beginning my first semester with all education classes.
One more thing about me, I love airplanes! My parents were both pilots and a few years ago I received my student pilots license myself so brace yourself for a number of aviation-related examples in my homework this semester.

Week 1

This week we covered the information listed out in our syllabus. We learned about this Mind Map project and a number of other assignments and expectations for the upcoming semester.
On Thursday of our first week we covered a little bit about math. We discussed how to write out a number like 56,283,914 like Fifty-six million, two hundred eighty-three thousand, nine hundred fourteen.

Week 2

Week 2!
This week we dove right into addition, subtraction, multiplication and division. With each we learned about the different models for teaching the same thing in the event that we have a student who might need a different explanation to understand the material.

ADDITION MODELS
Counting on: where students solve the equation 5+3 by
beginning at 5 and saying the following three numbers
6,7,8 to arrive at the answer 5+3=8.
Set Model: this particular model is good when teaching word problems. For example, Sarah has 4 blocks in one pile and 3 blocks in another pile how many blocks does she have in total? This question is simply asking students to add 4+3 but we threw in some other information that isn't relevant to the question itself.
Number-Line Model: This model is helpful for word problems that have to do with length and measurements like feet, inches, or yards. A good visual for this model would be to have the students cut ribbon or string in different lengths to help them understand how to add the two parts. For example, we have 5 ft of red ribbon and 4 ft of white ribbon, how much ribbon do we have in total? All we are asking is for the students to solve 5+4 but sometimes the ribbon part of the equation can be helpful for the students to visualize that the answer is 9.

ADDITION PROPERTIES
These properties are as follows:
1. Commutative property of addition of whole numbers. This states that it does not matter the order in which one adds numbers. For example, 3+5 is equal to 5+3. In both situations, the answer is still 8.
2. Associative property of addition of whole numbers. this property states that if parenthesis are involved in the equation, it does not matter where they are placed. For example (a+b)+c is equal to a+(b+c). The answers will still remain the same.
3. Identity property of addition of whole numbers. This states that anything plus 0 is itself. For example, 5+0=5 and 1,000+0=1,000.

SUBTRACTION MODELS
Take-away model: this model explains subtraction of taking away a seton set of objects from the first.
Missing Addend Model: this model helps the students bridge the gap of an unknown model. For example 7-_____ = 4.
This example can also be explained by using addition by saying "4+ what equals 7?" When the student is able to answer that they will realize that 7-3=4.
Comparison model: this encourages the students to think about "how many more of one quantity exists than another?"
Number line Model: Moving left on the number line a given number of units. I personally can imagine this being a difficult model to explain to children. But the similarly to how the number line is used in addition, the students would count each number down the line to simulate subtracting one value from another.

MULTIPLICATION MODELS
Repeated Addition Model: This model should help students understand that multiplication is just an easier way to do lots of addition. For example 6+6=12 and 6x2=12.
Array and Area Model: Another model that I haven't entirely grasped at the moment, but once the textbook visuals came in a feel a bit better about explaining this model. This is a grid that can be drawn out best on graph paper. This shows that if a square were broken down into 20 sections one row/column would have 5 boxes and the other row/column would have 4. Thus explaining that 5x4=20.
Cartesian-Product Model: This model shows combinations that are possible. For example if you're trying to build your class schedule and you can choose from 3 different languages and 2 different history subjects you have a total of 6 combinations to choose from for your schedule.

MULTIPLICATION PROPERTIES
Much like the properties of addition multiplication has its own set of rules that help explain and simplify certain concepts.
1. Closure property of multiplication of whole numbers. This property states that if a and b are whole numbers then axb will be a unique number separate from the value of a or b.
2. Commutative property of multiplication of whole numbers. This property says that axb is equal to bxa. Previously I mentioned the Array and Area model where I mentioned rows and columns to help explain multiplication. This law really helps with that because if for some reason you can't remember which comes first rows or columns its no problem because in multiplication order doesn't matter.
3. Associative property of multiplication of whole numbers. This means that if parenthesis are involved in the equation it does not matter where they are placed. For example (a*b)*c is the same as a*(b*c).
4. Identity Property of multiplication of whole numbers. This is a quick rule to always remember and all math problems will become easy. Anything multiplied by 1 is itself. For example a*1=a and 2,000*1=2,000.
5. Zero multiplication property of whole numbers. This means that anything times 0 is 0. For example, 1*0=0. 250*0=0.

Division of Whole Numbers Models
Set (partition) model: The set of elements representing the divided is partitioned into divisor number of subsets.
Missing Factor Model: a is divided by b is a unique number, c, provided that b*c=a.
Repeated Subtraction Model: the divisor is continually subtracted from dividend until only a remainder is left.
Dividing whole numbers: For any whole number a and b with b not equal to 0 a/b=c if and only if c is a unique such as b*c=a.

Division Algorithm
Given any whole number number a and b with b does not equal 0 there exists a unique whole number which is q, the quotient. and r the remainder and such that a=bq+r with r less than or equal to 0 and also less than b.

Week 3

Addition Algorithms:
Concrete Model: counting by 10 and add what is left. For example, 14 + 23 = 37. 10+20=30 4+3=7 total sum 37.
Expanded Algorithm: standard algorithm with regrouping (which is the same as carrying the remaining).
Left to Right Algorithm: 568+757 500+700=1200 60+50=110 8+7=15 Total sum 1325.
Lattice Algorithm for Addition: organize numbers into boxes divided by diagonal lines and add diagonally to compute your answer.
Scratch Algorithm for Addition: Add numbers starting at the top when the sum is greater than 10 record that by the scratching next to the number.

Subtraction Algorithms:
Subtraction Algorithm: 243-61=182. Show this with manipulative like longs, flats, and units. For example, 2 flats, 4 longs, 3 units minus 6 longs, and 1 unit.
Equal Addend Algorithm: based on the fact that the difference between two numbers does not change if we add the same amount to both numbers.

Mental Math: Addition
1. Add from the left: 76+25=
70+20=90 6+5= 11 90+11=101
2. Breaking up and bridging: 76+25=
76+20= 96 96+5=101
3. Trading off: 76+25=
76+4=80 (to get up to 100) then 25-4=21 (subtract the 4 you just added) and then 80+21=101
4. Using comparable numbers. Comparable numbers are numbers whose sums are easily calculated mentally. For example, 130+50+70+20+50. 50+50=100. 130+70=200 and don't forget the 20!. so 100 + 200+ 20=320.
5. Making comparable numbers. Depending on the age group this is best displayed with quarters. 76+25= 75+25=100 so 100+1=101.

Subtraction:
Breaking up & bridging: 74-26= 74-20=54 54-6=48
Trading Off: 74-26= 74+4=78 26+4= 30 78-30=48
Drop the Zeros: 7400-6000. 74-6=68. 7400-6000=68000.

Computational Estimation
1. Front-end with Adjustment: 474+522+231=
a. add front ends 4+5+2=11
b. place value = 1100
c. adjust 22+31 is about 50 and 74 is about 70
d. adjusted estimate is 1100+120=1220.
2. Grouping nice numbers
23+39+32+64+49= 39+49= about 100
23+32+49= about 100. So our answer should be around 200.
3. Clustering: used when a group of numbers clusters around a common value. For the numbers 4724,5262,5206,4992,5102 the number these all revolve around is about 5,000. So we multiply the number of numbers (5) by the number these numbers cluster around (5000). so 5X5,000=25,000.
4. Rounding: 7262-3806= 7000-4000= 3,000.
5. Using the range: this would have the students round numbers up and down to create a high and low estimate. For example, 7262 + 3806. The low estimate would be 7000+3000= 10,000. and the high estimate would be 8000+4000= 12,000.

Multiplication & Division:
Theorem 3-9: For any whole number and natural number m and n then a^m*a^n=a^m+n. When the base is the same, you add the exponents. Example: 3^2*3^4=3^6.
Theorem 3-10: For any whole number a and natural numbers m and n: (a^m)^n = a^m*n. Example: (3^2)^4= 3^8.
Theorem 3-11: For any whole number and natural numbers m and n: a^n * b^n = (ab)^n
Theorem 3-12: if m, a, and n are natural whole numbers with m > n, then a^m/a^n = a^m-n.

Multiplication by 10^n: to multiply by 10, replace each piece with base 10 piece that represents the next higher power of 10. 23*10=230.
Multiplication using expanded addition: 4*367=
4(3*10^2+6*10+7)
4(3*10^2)+4(6*10) +4*7
(4*3)10^2 + (4*6)10 +4*7
1200+240+28
1468
Lattice Multiplication: similarly to addition, organize the numbers into boxes divided diagonally and begin in the bottom right corner and work up.

Week 4

Number Theory
4-1 Divisibility- an integer is even if it has a remainder of 0 when divided by 2. Otherwise it is odd.
We say that 3 divides 18 written 3|18 because the remainder is 0 when 18 is divided by 3. Likewise b divides a can be written b|a.
We say that 3 does not divide 2 written 3|\2.
If a is a nonnegative integer and b is a positive integer we say a is divisible by b or b divides a if a only if remainder is 0.
Any whole number a & b where b does not equal 0 b divides a if and only if there is a unique whole number such that a=bq.

Divisibility Rules
*A whole number is divisible by 2 if and only if the units digit is divisible by 2.
*A whole number is divisible by 5 if and only if the units digit is 5 or 0.
*A whole number is divisible by 10 if and only if units is divisible by 10 which means the units number is 0.
*A whole number is divisible by 4 if and only if the last 2 digits are divisible by 4.
*A whole number is divisible by 8 if the last 3 digits of whole numbers represent a number divisible by 8.
*A whole number is divisible by 3 if and only if the sum of its digits is divisible by 3.
*A whole number is divisible by 9 if and only if the sum of the digits of whole number is divisible by 9.
*A whole number is divisible by 6 if and only if the whole number is divisible by both 2 and 3.

Prime numbers only have 2 factors: 1 and itself
Composite numbers: have more than 2 factors
The number 1 is neither prime nor composite.
Each expression of a number as a product of factors is factorization. Factorization with only prime numbers is prime factorization.

Week 5

Greatest Common Multiple = Greatest Common Factor

Example: two marking bands are to be combined to march in a parade. The first band of 24 members will march behind the second band of 30 members. Combined they must have the same number of columns. Each column must be the same size. What is the greatest number of columns they can march. Both columns must divide 24 and 30. 1,2,3,6 and 6 is the greatest. Greatest common divisor (GCF) or greatest common factor (GCF) of two whole numbers a & b.

SUPPLEMENTAL VIDEO FOR HELP ON THE RIGHT -->

The Intersection of sets model: is the model where you list out the number of factors for each number, circle the numbers that are common between both numbers and determine which number is biggest.

Prime Factorization Method: Using the factor tree to determine all the prime numbers and finding which ones are common between the numbers.

Important Rules to Remember about LCM vs GCD

LCM--> 1. Factor Tree 2. Re-write using exponents 3. Take smallest and whatever is left 4. Multiply

GCD --> 1. Factor Tree 2. Re-write using exponents 3. Take the smallest ONLY 4. Multiply

Examples:
GCD (x,y,z)
x=2^3, 7^2, 11, 13
y= 2, 7^3, 13, 17
z= 2^2, 7
All of them have a 2 in common, what from the 2 is the smallest? 2
All of them have a 7 in common, what from the 7 is the smallest? 7
None of the numbers have anything else in common so 7*2=14

Word Problems: Hot dogs are sold 10 per package while buns are sole 8 per package. What is the least number of packages of each you must buy so you have an equal number of hot dogs and buns?
To solve this, find the lowest common multiple which is 40. So this means that you will have the same number of hot dogs and buns when you buy 4 hot dogs and 5 buns.

Additionally, these problems can also be found by using the Number Line Method. It does take longer, but is a good visual for students.

Intersection of sets method: listing off the multiples of a number and noting which ones are in common.

To Find the LCM of 2 non-zero whole numbers first find prime factor of each numbers. Then take each of the primes that are factors of each of given numbers. The LCM is the product of these primes, each raised to the greatest power of the prime that occurs in either of the prime factorizations.

Week 6

Today in class we had a workshop day of sorts where we created a number of different ways for kids to learn LCM and GCF and games for them to practice their new knowledge. There were a number of examples that were in class today but my favorites were the booklets for the children to make so that they can use on quizzes to help them out. Another one I liked was the rainbow model where the kids see colors and rainbow arches to help them understand how to complete the problem. And my favorite was a hangman game where the kids had to solve the problems in order to complete the hangman. All of these and more will be useful one day when I teach my classroom how to remember GCF and LCM.

On Thursday, we had a test.

Week 7

Today we began by reviewing the test we took on Thursday and were able to ask questions to realize what we did right and what we did wrong. This next section, 5-1 is about Addition of Integers.

The representation of an Integer is I.-4, -3, -2, -1, 0, 1, 2, 3,4
Finding the opposite of -4 is 4 and opposite of 3 is -3

Examples: Finding the opposite of both the variable AND the integer. a. x=-3 would be -x=-3 b. x=-5 would be -x=5 and x=0 would be -x=0. Because 0 is neither positive or negative.

Chip Model for integers:
Black chips represent positive integers and red represents negative integers and each neutralize the other. So if I have 4 red chips and 3 black chips that is the same as -4+3=-1. A similar model can be demonstrated by the charged field by + and - signs if I have 3 + signs and 5 - signs then I have 3+(-5)= -2.

Number line model: using the number line to show the students which way to go either left or right depending on whether the number is negative or positive. This can be used to show a temperature example.

Pattern model is good for students who are good at memorization. Begin with basic facts like 4+3=7 and 4+2=6 and help the students notice that the more the number on the right side of the + sign goes down the number on the right side of the = sign goes down as well. They will begin to recognize the pattern and it will help them with other problems.

Absolute Value is always positive or 0 because distance cannot be negative and absolute value is the distance a number is from 0.
Examples: |20| = 20
|-5|=5
|0| = 0
-|-3|=-3
|2+-5|=|-3|=3

real world problems: Omar's bank account is -$75 what is to be said about his debt to the bank?
Solution: bank balance is the |-75| so he owes $75. He would not owe -75$ because that would mean that he wouldn't owe anything. So he owes a *positive* 75$.

Properties of Addition of Integers
Closure property of Addition of Integers: a + b= a unique number that is not a or b.
Commutative property of addition of Integers: a + b = b+a it does not matter which order they are added in.
Associative property of addition of integers: (a+b)+c = a+(b+c). Does not matter where the parenthesis is place because it does not matter the order in which they are added.
Identity property of addition of integers: 0 is a unique integer such that for all integers a, 0+a=a=a+0
Additive Inverse property of Integers: For every integer there is a unique integer -a the additive inverse of a, such that a+-a=0.

Chip Model for Subtraction: to find 3 - - 2 add 0 in the form of 2 +-2 (2 black chips and 2 red chips) to the three black chips, then take away the two red chips.
Charged field subtraction: to find -3 - -5 represent -3 so that at least 5 negative charges are present then take away the negative charges.
Number line Model: while integer adding is modeled by maintaining the same direction and moving forward to backward depending on whether a positive or negative integer is added, subtraction is modeled by turning around. (hiker example)
Pattern model for subtraction: 3-2=1 and 3-3=0 and 3-4=-1. This means as the number on the right side of the - sign goes up the number on the right side of the equal sign goes down.
Missing Addend Approach: we compute 3-7 as follows. 3-7 = n if and only if 3=7+n because 7+-4=3 then n=-4.

Order of Operations:
recall that subtraction is neither commutative nor associative. An expression such as 3-15-18 is ambiguous unless you know which order to perform the subtraction. Mathematicians have agreed that 3-15-18 means (3-15)-18.
p
E
M/D -->
A/S --> whichever comes first left to right

Week 8

5-2 Integers
pattern Model: First find (3)(-2) using repeated addition (3)(-2) = -2 +-2 +-2 =-6
To find (-3)(-2) = -6
For all integers a,b,c set of integers
Closure property of multiplication of Integers
ab is a unique integer
Commutative property of multiplication of integers
ab=ba
Associative property of multiplication of integers
(ab)c=a(bc)
Identity property of multiplication
1 is unique integer such that for an integer a, 1*a=a=a*1

Distributive property of multiplication over addition for integers a(b+c)=ab+ac and (b+c)a=ba+ca
Zero multiplication property of integers
0 is unique integer such that for all integer a, 0*a=0=a*0

Distributive property of multiplication over subtraction for integers
a(b-c)=ab-ac

Examples: simplify so there is no () in answer
a. -3(x-2) equals -3x+6

Factoring
when the distributive property of multiplication over subtraction is written in reverse order as ab-ac=a(b-c) and ba-ca =(b-c)a and similarly for addition, the expressions on the right of each equation are factored in form. The common factor has been factored out.

Definition of Integer Division
If a and b are any integers, then a/b is the unique integer c, if it is exists, such that a=bc.
The quotient of 2 negative integers, if it exists, is a positive integer.
Quotient of a positive and negative integer, if it exists, is a negative number.
EXAMPLE:
12/4 = 12=-4c = c=-3

Order of operations
Evaluate the expressions as they come PEM/D A/S left to right

Ordering of Integers
Definition of less than for integers
For any integer a and b, a is less than b, written a<b if and only if there exists a positive integer k such that a+k=b

Properties of Inequalities of Integers
1. If x<y and n is any integer then x+n<y+n
2. If x<y, then -x>-y
3. If x<y and n>0 then nx<ny (ex: n=2)
4. If x<y and n<0 then nx>ny

Week 9

After Spring Break we began this week with Pi day. (March 14). Today we watched a number of videos and researched a number of resources to learn how to incorporate Pi day in our classrooms by including all different subjects like math, reading/spelling/language, and art. As requested, here is a fun fact about Pi that we did not cover in class today. The first 144 digits of pi add up to 666 (which many scholars say is “the mark of the Beast”). And 144 = (6+6) x (6+6). (found at https://www.factretriever.com/pi-facts). Thursdays class is cancelled so only one paragraph for this week.

Week 10

3-21-17 - submitted take home tests

6.1 Set of Rational Numbers
Numerator = top
Denominator = bottom

Area Model, Number line model, Set model all ways to help students conceptualize ratios.

meaning of fractions:
Proper Fraction: where the numerator is smaller than the denominator
Improper Fraction: where the numeration is larger than the denominator. This means you can turn it into a mixed number.

Fundamental law of fractions: what you do to the bottom you must do to the top (balance the equation)

Showing that one fraction is equal to another.
example: 12/42 is the same as 10/35. Just by looking at it, I wouldn't know if they are the same or not. But you can find out by using 1 of 3 methods.
Method 1 (Factor Tree Method): where you simplify fractions to their simplest form 12/42 simplifies to 2/7 and 10/35 simplifies to 2/7 which means they are the same.

Method 2 (LCM method): Find the LCM of 42 and 35 which is 210 and write the fractions as 12/42=60/210 and 10/35=60/210 which means they are the same.

Method 3: Rewrite fractions with a common denominator. So multiply the denominators by each other 12*35=420 and 10*42 is 420 so the numerators are the same, now for the denominators: 42*35=1470 and 35*42=1470 so the fractions both end up being 420/1470 which means they are equal.

Finally, ordering fractions.
Find 2 fractions between the fractions 7/18 and 1/2.
1. Find a common denominator: take 1/2 and multiply both the numerator and denominator by 9 (so that the denominators are the same and equal 18). So now we have the fractions 7/18 and 9/18. What number belongs in the middle? 8/18 (or 4/9 simplified)

Addition, Subtraction, and Estimation with Rational Numbers
If a/b and c/d are rational numbers, then a/b+c/d=a+c/b

Area Model: show the students shaded in pieces of the "pie" so they can understand 2/5+1/5=3/5
Number Line Model: Not always the best one to use, but helpful when ordering fractions.
If a/b and c/d are rational numbers, then a/b+c/d=ad+cb/cd

Remember to find the common denominator when adding and subtracting fractions.

Unless otherwise stated, change the improper fraction to a mixed number.

Mixed Numbers: Numbers that are made up of an integer and a fractional part of an integer.
A mixed number is a rational number and therefore it can always be written in the form of a/b.

Mixed number --> improper fraction
Multiply the whole number and the denominator, add the numerator and keep the denominator.

Improper Fraction --> Mixed number
divide the denominator by the numerator and the remainder is the fraction of the mixed number.

Additive Inverse Property:for any rational number a/b, there exists a unique rational number -a/b called the additive inverse of a/b, such that a/b+(-a/b)=0=(-a/b)+a/b

Integer Rational Number
1. -(-a)=a 1. -(-a/b)=a/b
2. -(a+b)=-a+-b 2. -(a/b+c/d)=-a/b+-c/d

Addition property of equality
If a/b and c/d are any rational number such that a/b=c/d if e/f is any rational number then a/b+e/f=c/d+e/f
Example: 1/2=2/4

Whatever you do, as long as you do it to both sides, it will still be equal.

Estimation with rational numbers
Many estimation and mental math techniques are used with whole numbers also work with rational fractions. Estimation plays an important role in judging the reasonableness of a problem.

1 1/8, 3 4/10, 5 7/8, 6/10 = 1,3,6,1 = 11

Week 11

6-3 Multiplication, Division, and estimation with real numbers
Multiplication of rational numbers
Multiplication as repeated addition
*When you multiply you do not need a common denominator
Remember every number has an understood denominator

IS/OF = %/100

if a/b and c/d are any rational numbers, then a/b*c/d=a*c/b*d

Properties of Multiplication
Multiplicative Identity
Everything multiplied by 1 is itself.

Multiplicative Inverse:
Anything multiplied by its reciprocal is 1

Distributive property of multiplication over addition
if a/b,c/d,e/f are any rational number then a/b(c/d+e/f)=(a/b*c/d)+(a/b*e/f)

Multiplication property of equality
If a/b>c/d and e/f>0, then a/b*e/f>c/d*e/f

Multiplication property of 0
Anything times 0 is 0

Using improper fractions:
2 1/2 * 2 1/2=5/2*5/2=25/4= 6 1/4
When you have mixed numbers, convert to an improper fraction first, then continue with the problems

Division of rational numbers
A radio provides 36 minutes for public service announcement, what part of the day is allotted for PSA's?
60/24= 1440 in a day and 35/1440=1/40. So 1/40 of the day is allotted for PSA's.

Extending Notation of exponents
If the base is the same= add exponents
Anything to the power of zero = 1
Anything to the negative exponent is 1 over that.
When your exponent is negative, write the reciprocal and it becomes positive.

Proportions and Ratios
Ratios
-Used to compare quantities
-Written alb or a/b
-Used to represent part-to-whiole, part-to-part, or whole-to-whole.

There were 7 males and 12 females in the hotel. In the game room there were 14 males and 24 females.
a. express number of males to females in the inn part - to - part. 7:12 or 7/12.
b. express number of males to females in the game room as a ratio part-to-part 14/24 or 7/12.
c. express number of males in the game room to the number of people part-to-whole. 14/38 or 7/19.

A proportion is a statement that two given ratios are equal. a,b,c,d are all real numbers and b does not equal 0, then the proportion a/b=c/d is true if, and only if ad=bc.

Caryl can type 8 pages for every 4 pages that Dan can type. If Dan has typed 12 pages, how many has Caryl typed?
*example of multiplicative relationship

If there are 3 cars for every 8 students at a high school, how many cars are there for 1200 students?
car/students --> 3/8 = x/1200 cross multiply and solve for X. The answer is x=450 cars.

Constant of proportionally
- what you do to one side you must do to another to keep the equation the same.

Scaling strategy to determine which is the better deal
Scaling strategy for solving the problem involves finding the common number of tickets, the LCM.

Unit-rate strategy: involves finding cost of 1 ticket. Using Division of Decimals to find out the price of the individual ticket to find out which deal is best.

Scale drawings:
Ratio and proportions are used in scale drawings. The scale is the ratio of the size of the drawing to the size of the objects.

Floor plan scale 1:300. Find dimensions in meters. 1cm=300cm=3m in true size. 3.7 cm represents 3.7*3=11.1 and 2.5cm represents 2.5 *3=7.5m
Dimensions of the living room are 11.1 by 7.5cm.

Week 12

Today we worked on our presentations for how to remember
different things we've learned in class. For example, there were word problems and games to help students with fractions, ratios, absolute value, and more. Ours in particular was a poster of tips and tricks about Fractions and mixed numbers and improper fractions. Different ways to help the students remember important things about fractions.
Thursday is our test so that's about all for this week's mindomo.

Week 13

7-1 Introduction to Finite Decimals
Decimal comes from the latin word meaning 10.
Anything to the right of the decimal is "the" tenths, hundredths, etc.
Each place is named by its power of 10.
Covert the following to decimals
25/10 2*10+5/10 = 2*10/10+5/10 = 2 +5/10= 2.5
56/100= .56
205/10,000= 0.0205 (two hundred five ten thousandths)

Write the following in word form
28.1902 twenty-eight and one thousand nine hundred two ten thousandths

0.436 four hundred thirty-six thousandths

-62.01 negative sixty-two and one hundredths

Rewrite each of the following as decimals
Five hundred thirty-six and seventy-six thousandths=536.0076

Three and eight thousandths = 3.008

Four hundred thirty-six millionths 0.000436 (sometimes it helps to add the 0 when there is no whole number to help visualize the number).

Five million and two tenths 5,000,000.2

Terminating Decimals
Decimals can be written with only a finite number of places to the right of the decimal.
A rational number a/b in simplest form can be written as a terminating decimal if, and only if, the prime factorization of the denominator contains no primes other than 2 or 5.

Can be written as terminating Cannot be
7/16 b/c 16=2^4 11/18 b/c 18=2*3^2
19/40 b/c 40=2^3*5 5/26 b/c 26=2*13
3/160 b/c 160=2^5*5 9/84 b/c 84=2^2*3*7

Ordering Decimals
Terminating decimals is easily located on a number line b/c it can be represented as a rational number a/b, where b does not equal 0 and b is a power of 10/

Compare 0.67643 and 0.6759
Start left to right and compare each number and until you find a number that is different. Then determine which number is bigger then you have your bigger number.

7-2 Operations on Decimals
Adding decimals - keep them aligned
2.16 + 1.73= 3.89
Keep the decimal where it needs to be and add accordingly. Same goes for subtraction.
Multiplying Decimals
Multiply like the decimal doesn't exist, then add the number of digits to the right of the decimal then move your decimal that same number to the left.
Scientific Notation
Handles very large or small numbers
Distance light travels in one year is 5,872,000,000,000 miles, called a light year. Is expressed as 5.872X10^12
Always something times 10 to the something power
Decimal has to be between the first two non zero numbers
Do not teach the students to just count numbers, teach them to count place values.
Mass electron is 0.00054875 atomic mass units, is expressed 5.4875 X 10^-4

When you move the decimal to the right, the exponent of the power of 10 is negative.

When you move the decimal to the left, the exponent of the power of 10 is positive.

Dividing decimals
On the outside of the problem, outside of the house.

We would not use a remainder, we always keep adding zeros until we have a terminating decimal

Dividing two decimals
You cannot just bring the decimal up unless you have a whole number on the "outside of the house". So to get that whole number, move the decimal to the right however many times you need to. Then the decimal for the number "on the inside of the house" needs to be moved to the right that same number of times. Then you have a whole number on the outside and can move the decimal on the inside up into your quotient.

Mental Computation
Breaking down and bridging

Whatever sticks out to you may be different than what sticks out to someone else.

Using Compatible numbers
Decimal numbers are compatible when their sum is a whole number

Making compatible numbers
Break down the problem and see how it would be easiest for you to solve.

Balancing with decimals in subtraction
Rounding up or down to make a whole number. Solve the problem then add or subtract what you rounded to get the exact answer.

Facts about rounding/estimation
Computations with rounding can be significantly different than the actual answer
Non-zero numbers are always significant
Zeros before other numbers are not significant
Zeros between non-zero numbers are significant
Zeros to the right of the decimal point are significant

Week 14

4-18-17
Repeating Decimals
If they do not terminate, they repeat
__
0.18 is repeating .18 Repeating numbers are called Repetend.

Anytime you have a repeating in the tenths place, it is that number over 9.
__ __
0.5 5/9 0.4 4/9

Just because you see that line, does not mean that the whole thing repeats, only the lined part.

Order repeating decimals in the same manner as for terminating decimals.
1.3478 and 1.347821 --> is the larger one

4-20-17
Percents and Fractions with a denominator of 100
The word percent means per centrum, which means per hundred
Write the following as a percent
a. 0.03 --> 3%
b. 1.2 --> 120%
c. 0.00042--> 0.042%
d. 1 --> 100%
e. 3/5 --> 60%
f. 2/3 --> 66.6(repeating) %
g. 2 1/7 --> 214 2/7%

Move your decimal two places to the right

Write the following as a decimal:
a. 5% --> 0.05
b. 6.3% --> 0.063%
c. 100% --> 1
d. 250% --> 2.5
e. 2/3% --> 0.006(repeating)
f. 33 1/3--> 0.3 (repeating)

Applications involving percent:
1. finding a % of a number
2. Finding what % one number is another
3. Finding a number when a percent of that number is known

IS/OF = %/100

A house sells for $92,000 and requires a 20% down payment how much is the down payment?
We need to find 20% of $92,000
what is 20% of $92,000?
x/92,000 = 20/100
x=18,400

We need to find a number where 42% of that number is 168.
168/x = 42/100 x=400

What is 6% of 34? 2.04
17 is what percent of 34? 50
18 is 30% of what number? 60
what is 7% of 49? 3.43

Use what you know and then estimate from there. Good thing to use when you have "bucks" they want to buy something. Can put it on sale 40% off or what not.