by Hannah Hagerty 4 days ago
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Balancing with Decimals in Division
Balancing with Decimals in Subtraction
Making Compatible Numbers
Breaking and Bridging
Scientific Notation
Ordering Terminating Decimals
Compare the digits, the digit with the greater face value represents the greater of the numbers
2. Start at the left and find the first place where the values are different
1. Line up the numbers by place value
Theorem
a/b is simplest form can be written as a termination decimal if prime factorization of the denominator contains no primes other than 2 or 5
numbers that can be written with a finite number of digits to the right of the decimal point
if x
if x
if x
if x
-3>-5
-5<-3
Difference of Squares
(a+b)(a-b)=a^2-b^2
Distributive Property of Multiplication Over Subtraction of Integers
(b-c)a=ba-ca
a(b-c)-ab-ac
(-a)(-b)=ab
(-a)b=-(ab)
(-1)a=-a
Zero Property
a(0)=0=0(a)
Distributive Property of Multiplication Over Addition for Integers
Identity Property
1(a)=a=a(1)
Associative Property
(ab)c=a(bc)
Commutative Property
ab=ba
Closure Property
ab is a unique integer
Pattern Model
Additive Inverse Property of Integers
a+-a=0=-a+a
-a+-b=-(a+b)
Addition Property of Equality for Integers
if a=b, a+c=b+c
-(-a)=a
Identity Property of Addition of Integers
0+a=a=a+0
Associative Property of Addition of Integers
Commutative Property of Addition of Integers
Closure Property of Addition of Integers
a+b is a unique integer
Temperature Cube Model
Charged Field
Number Line Model
Chip Model
distance between the point corresponding to an integer and 0
1, 2, 3, 4...
-1, -2, -3, -4...
Breaking Up the Dividend
division by 0 is undefined
n/0= undefined, 0/n= 0, 0/0= undefined, n/1= 1
Missing Factor Model
3c=18, 3 x 6=18, c=6
Repeated Subtraction
10-2=8,8-2=6,6-2=4,4-2=2,2-2=0 (five groups of 2 in 10)
Compatible Numbers
524 x 8= 500 x 8=4000 and 25 x 8=200 = 4200
Front-End Multiplying
524 x 8= 500 x 8= 4000 and 20 x 8= 160 = 4000+160=4160
Thinking Money
Using Compatible Numbers
Front-End Multiplying
64 x 5= 60 x 5=300 and 4 x 5=20= 300+20=320
Lattice Multiplication
Partial Products Algorithm
Distributive Property of Multiplication Over Subtraction for Whole Numbers
a(b-c)=ab-ac
Distributive Property of Multiplication Over Addition of for Whole Numbers
a(b+c)=ab+ac
Multiplication Property of 0 for Whole Numbers
a x 0=0=0 x a
Identity Property of Multiplication of Whole Numbers
a x 1= a =1 x a
Associative Property of Multiplication of Whole Numbers
(a x b) x c= a x (b x c)
Commutative Property of Multiplication of Whole Numbers
a x b = b x a
Closure Property of Multiplication of Whole Numbers
a x b is a unique whole number
Cartesion-Product Model
the number of ways objects in sets can be combined
tree diagram or table
Area Model
4-by-5 grid
Rectangular Array Model
objects arranged with the same number of objects in each row and column
8+8+8=24
drop the zeros
base ten blocks
comparison model
missing addend model
number line
take-away model
subtraction is inverse of addition
using the range
rounding
clustering
grouping to nice numbers
front-end with adjustment
making compatible numbers
using compatible numbers
sums easy to calculate mentally
trading off
breaking up and bridging
adding from left
Identity Property of Addition of Whole Numbers
a+0=a
Associative Property of Addition of Whole Numbers
(a+b)+c=a+(b+c)
Commutative Property of Addition of Whole Numbers
a+b=b+a
Closure Property of Addition of Whole Numbers
sum of two whole numbers is a unique whole number
making ten
doubles
counting on
a/b=c/d if a/c=b/d
a/b=c/d if b/a=d/c
if y=kx (or k=y/x) then y is proportional to x and k is the constant of proportionality between y and x
a/b=c/d is a proportion if ad=bc
two given ratios are equal
A comparison of two quantities
a/b, a:b
Properties of Exponents
Algorithm for Division of Fractions
(a/b)/(c/d)=(a/b)(d/c)
Multiplication Property of Zero for Rational Numbers
(a/b) 0=0=0 (a/b)
Multiplication Properties of Inequalities for Rational Numbers
If a/b>c/d and e/f<0, then (a/b)(e/f)<(c/d)(e/f)
If a/b>c/d and e/f>0, then (a/b)(e/f)>(c/d)(e/f)
Multiplication Property of Equality for Rational Numbers
(a/b)(e/f)=(c/d)(e/f)
Distributive Properties of Multiplication Over Addition and Subtraction for Rational Numbers
a/b(c/d+e/f)=(a/b)(c/d)+(a/b)(e/f) and a/b(c/d+e/f)=(a/b)(c/d)-(a/b)(e/f)
Multiplicative Inverse Property of Rational Numbers
(a/b)(b/a)=1=(b/a)(a/b)
Multiplicative Identity Property of Rational Numbers
1 (a/b)=a/b=(a/b) 1
1/2,1/3,1/4,2/3, or 1
Addition Property of Equality
If a/b=c/d, then a/b+e/f=c/d+e/f
Additive Inverse Property of Rational Numbers
a/b+(-a/b)=0=(-a/b)+a/b
3. Rewrite both fractions with a common denominator
2. Rewrite both fractions with the same least common denominator
1. Simplify
Denseness Property of Rational Numbers
If there are two different rational numbers a/b and c/d there is another rational number between the two
Fundamental Law of Fractions
If a/b is a fraction and n is a non-zero number then a/b= an/bn
Set Model
Number-Line Model
Bar Model
GCD (a, b) x LCM (a, b)= ab
Euclidean Algoritm
Number Line
Euclidean Algorithm
Intersection of Sets
Colored Rods
Each composite number can be written as a product of primes in one, and only one, way except for the order of prime factors in the product.
Factor Tree
A factorization containing only prime numbers.
Divisibility Test for 6
A whole number is divisible by 6 only if the number is divisible by both 2 and 3.
Divisibility Test for 11
A whole number is divisible by 11 only if the sum of the digits in places that are even powers of 10 minus the sum of digits in places that are odd powers of 10 is divisible by 11.
Divisibility Test for 9
A whole number is divisible by 9 only if the sum of the digits is divisible by 9.
Divisibility Test for 3
A whole number is divisible by 3 only if the sum of the digits is divisible by 3.
Divisibility Test for 8
A whole is divisible by 8 only if the last 3 digits represent a whole number divisible by 8.
Divisibility Test for 4
A whole number is divisible by 4 only if the last 2 digits represent a whole number divisible by 4.
Divisibility Test for 10
A whole number is divisible by 10 only is the units digit is 0.
Divisibility Test for 5
A whole number is divisible by 5 only if the units digit is 0 or 5.
Divisibility Test for 2
A whole number is divisible by 2 only if the units digit is even.
If d is a factor of a, d is a factor of any multiples of a.
place value based on powers of 10
numerals constructed by 10 digits
0,1,2,3,4,5,6,7,8,9
multiplicative property
bar placed over numeral to multiply by 1,000
subtractive property
IV=5-1=4
additive property
VII=5+1+1=7
groupings of 20 vertically
3 symbols (including 0)
numbers greater than 59 represented by groupings of 60
place- value system
includes additive property
grouping system based on powers of 10
