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73 Cards in this Set

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What is a fraction?
A number that can be written a/b where a and be are integers and b≠0
What are the parts of a fraction a/b?
a is the numerator
b is the denominator
How can you use a diagram to understand the fraction a/b?
Consider a unit and divide it into b parts of equal size

consider a of the parts
What are equivalent fractions?
For any fraction a/b and any interger k where k≠o, a/b = ka/kb
What can we say about a fraction a/b where a and b have a common factor greater than 1?
Both a and b can be divided by the factor to obtain a fraction with a smaller numerator and denominator.
How do we get a fraction a/b to simplest form (lowest terms)
Divide the numerator and denominator by the GCD of the numerator and denominator
What is a common denominator?
A common multiple of both denominators
What are two mehtods to find a common denominator?
We can multiply the denominators together.

We can also find the LCM of the denominators - this is the smallest common denominator.
For any two fractions a/b and c/d when is a/b=c/d?
a/b = c/d if and only if ad=bc
How do we compare fractions a/b and c/d?
Replace the two fractions with fractions having a common denominator and compare their numerators.
For any two fractions a/b and c/d with b,d≠0 when is a/b < c/d?
a/b < c/d if and only if ad < bc
For any two fractions a/b and c/d with b,d≠0 when is a/b > c/d?
a/b > c/d if and only if ad > bc
How do we add the two fractions a/b + c/d?
a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd
What is a mixed number?
A mixed number is a combination of a fraction and a whole number.

Ex. 2 1/2 = 2 + 1/2
What is a proper fraction?
A fraction a/b where 0≤|a|<b
What is an improper fraction?
A mixed number can be converted to a fraction of the form a/b. This is an improper fraction because |a| > b
How do we subtract two fractions with a common denominator?
We can use the missing addend model. Ex: 3/6 - 1/6 is the same as 1/6 + ? = 3/6

We can also use the take away model using fraction strips. Draw a fraction strip and indicate 3/6. We then take away 1/6 of the 3/6 to obtain 2/6.
How do we subtract two fractions without a common denominator using fraction strips?
Draw a fraction strip for each fraction. You would then further divide the fraction strips so they have equivalent amount of sections. Take away the smaller fraction from the larger fraction.
How do we subtract two fractions without a common denominator without using fraction strips?
Find two fractions having a common denominator and compute their difference by subtracting the numerators and retaining the denominator.

a/b - c/d = ad/bd - bc/bd = (ad-bc)/bd
How do you multiply a whole number and a fraction using fraction strips?
We can think of this as repeated addition.

Ex. C x a/b

Draw C fraction strips representing a/b. add them together.
How do we multiply a fraction times a whole number using fraction strips?
For a/b x C we can think of it as a/b of C. (ex. 1/3 of 4)

We use a modified fraction strip (a whole fraction strip but it is divided into equal parts that represent a whole number - for example, 4).

We then divide the fraction strip into equal parts (for example 3)

In this example, we would take one of those thirds to represent our answer - 4/3
How do we multiply a fraction times a fraction (a/b x c/d) using a rectangular array?
We can illustrate this by taking a figure that has a/b sections shaded and then taking c/d of the shaded amount.
How do we multiply two fractions algebraically?
a/b x c/d = ac/bd.
How do we divide two fractions using subtraction?
We can think of it as repeated subtraction.

Ex. 3/5 ÷ 1/10 can be thought of as how many 1/10 can we subtract from 3/5
How do we know if c/d divides a/b?
given two fractions a/b , c/d a/b ÷ c/d if and only if c/d x e/f = a/b
How do we divide two fractions using fraction strips?
ex. 5/6 ÷ 1/3

Create a fraction strip which illustrates 5/6

Partition the strip into thirds.

How much of the shaded area is represented in the thirds? (5/2)
What is an inverted fraction?
We invert the fraction c/d to get the reciprocal d/c.
How do we divide two fractions algebraically?
a/b ÷ c/d = a/b x d/c = ad/bc
What is a rational number?
It is a number that can be represented as a fraction of the form a/b.
What are the properties of Addition of Rational Numbers?
Closure, Associative, Commutative, Identity, Additive Inverse
What is the Closure Property under Addition of Rational Numbers?
Given any two rational numbers a/b and c/d, a/b + c/d is a unique rational number.
What is the Associative Property under Addition of Rational Numbers?
Given any two rational numbers a/b and c/d, a/b + c/d = c/d + a/b
What is the Commutative Property under Addition of Rational Numbers?
Given any three rational numbers a/b, c/d, and e/f (a/b + c/d) + e/f = a/b + (c/d + e/f)
What is the Identity Property under Addition of Rational Numbers?
Given any rational number a/b a/b + 0 = 0 + a/b = a/b

0 is the additive identity element
What is the Additive Inverse Property under Addition of Rational Numbers?
Given any rational number a/b, there is a unique number -a/b such that a/b + (-a/b) = 0

-a/b is the additive inverse of a/b
What is the property of Subtraction under Rational Numbers?
It is closed.
What are the properites of multiplication under Rational Numbers?
Closure, Associative, Commutative, Identity Property, Distributive, Zero, Mulitplicative Inverse
What is the Closure Property under Rational Numbers?
Given any two rational numbers a/b and c/d, a/b x c/d is a unique rational number
What is the Associative Property under Mulitplication of Rational numbers?
Given any two rational numbers a/b and c/d, a/b x c/d = c/d x a/b
What is the Commutative property under Multiplication of Rational Numbers?
Given any three rational numbers a/b, c/d and e/f (a/b x c/d) x e/f = a/b x (c/d x e/f)
What is the Identity Property under Multiplication of Rational Numbers?
Given any rational number a/b, a/b x 1 = 1 x a/b = a/b
What is the Zero Property under Multiplaction of Rational Numbers?
Given any rational number a/b, a/b x 0 = 0 x a/b = 0
What is the Distributive Property under Multiplication of Rational Numbers?
Given rational numbers a/b, c/d, e/f a/b x (c/d + e/f) = (a/b x c/d) + (a/b x e/f)

OR

a/b (c/d - e/f) = (a/b x c/d) - (a/b x e/f)
What is the Multiplicative Inverse under Multiplication of Rational Numbers
Given a rational number a/b a/b x b/a = 1

b/c is the multiplicative inverse of a/b
What is the property of Division of Rational Numbers?
It is closed.
What are the properties of Order Relations of Rational Numbers?
Trichotomy, Transitive, Addition, Multiplication.
What is the Trichotomy Property under Order Relations?
Given any two rational numbers a/b and c/d one of three things can occur:
a/b < c/d OR
a/b > c/d OR
a/b = c/d
What is the Transitive Property under Order Relations?
Given three rational numbers a/b, c/d, e/f

if a/b < c/d AND
c/d < e/f THEN
a/b < e/f
What is the Addition Property under Order Relations?
Given three rational numbers a/b, c/d, and e/f

if a/b < c/d then (a/b + e/f) < (c/d + e/f)
What is the Multiplication property under Order Relations?
Given rational numbers a/b, c/d, and e/f

if a/b < c/d and e/f > 0 then (a/b x e/f) < (c/d x e/f)

if a/b < c/d and e/f ≤ 0 then (a/b x e/f) > (c/d x e/f)
What is the density property?
For any rational numbers a/b, c/d with a/b < c/d, at least one fraction exists such that a/b < e/f < c/d
Given two fractions a/b and c/d where a/b < c/d how do we find the fraction e/f that falls inbetween the two?
Find equivalent fractions having a common denominator for a/b and c/d and find a fraction that falls between.
Remember that a to the power of n means a x a x a n times.

How do we define a to the power of -n?
1/a x 1/a ....1/a = 1/(a to the power of n)

thus 10^-1 = 1/10, 10^-2 = 1/100 etc.
What is a decimal most of ten?
A number that can be expressed with a decimal point.
North America, England, Europe, Scandinavia use different notations for the decimal. What are they?
NA: 1.5
England: 1⋅5
Europe: 1,5
Scandinavia: uses 1.5 but the 5 is superscript
What are the number of digits to the right of the decimal point called?
The number of decimal places.
What do the positions to the left of the decimal point represent?
Place values that are increasing powers of 10.
What do the positions to the right of the decimal point represent?
Place values that are decreasing (negative) powers of 10.
How do you order two positive decimals?
Compare the digits in the corresponding place values to the right of the decimal and determine the first place that differs.

The decimal with the greater digit is the greater decimal.
How do we convert a fraction with a power of 10 in the denominator to a decimal?
Ex. 64/100 = 0.64
64/1000 = 0.064
How do we convert a fraction without a power of 10 in the denominator to a decimal?
We check if there is an equivalent fraction with a power of 10 in the denominator, and then convert.
What is a terminating decimal?
A decimal with a finite number of digits.
What is a non-terminating decimal?
A decimal that has an infinite number of digits.
When can a fraction a/b in simplest form be written as a terminating decimal?
If b has only 2 and 5 as the prime factorization
What is a repeating decimal?
A decimal that does not terminate and contains a repeating pattern of digits.

**Every non-terminating repeating decimal can be written as a fraction**
How do we write a repeating decimal as a fraction?
ex. 0.345...

let x = 0.345345...
1000x = 345.345...
1000x-x=345.345...=0.345345...
=999x=345
x = 345/999
What is an irrational number?
A number that can not be represented as a fraction.

**Every non-terminating non-repeating decimal is irrational**
How do we add decimals?
Align the decimals point and the place values. Add and make sure you carry over.
How do we subtract decimals?
Align the decimal points and the place values. Exchange as necessary.
How do we multiply decimals?
Initially ignore the decimal point and multiply as whole numbers.

After completing the algorithm, you place the decimal point. Add the number of digits to the right of the decimal point in both the quotients - this tells you how many digits are to the right of the decimal in your answer.

ex. 2.37 x 1.6901 will have 6 digits to the right of the decimal point.
How do we divide decimals?
The divisor can not contain a decimal. Move the decimal place in the divisor to the right so that it becomes a whole number - however many places you move it, you must move it that many in the dividend as well.

Use long division, and remember to put the decimal point in the quotient.
How do we multiply a decimal by a power of 10?
Move the decimal point one place to the right for each power of 10.
How do we divide a decimal by a power of 10?
Move the decimal point one place to the left for each power of 10.