Math 132 - Up To Test 1

Front 1

What is a fraction?

Back 1

A number that can be written a/b where a and be are integers and b≠0

Front 2

What are the parts of a fraction a/b?

Back 2

a is the numerator
b is the denominator

Front 3

How can you use a diagram to understand the fraction a/b?

Back 3

Consider a unit and divide it into b parts of equal size

consider a of the parts

Front 4

What are equivalent fractions?

Back 4

For any fraction a/b and any interger k where k≠o, a/b = ka/kb

Front 5

What can we say about a fraction a/b where a and b have a common factor greater than 1?

Back 5

Both a and b can be divided by the factor to obtain a fraction with a smaller numerator and denominator.

Front 6

How do we get a fraction a/b to simplest form (lowest terms)

Back 6

Divide the numerator and denominator by the GCD of the numerator and denominator

Front 7

What is a common denominator?

Back 7

A common multiple of both denominators

Front 8

What are two mehtods to find a common denominator?

Back 8

We can multiply the denominators together.

We can also find the LCM of the denominators - this is the smallest common denominator.

Front 9

For any two fractions a/b and c/d when is a/b=c/d?

Back 9

a/b = c/d if and only if ad=bc

Front 10

How do we compare fractions a/b and c/d?

Back 10

Replace the two fractions with fractions having a common denominator and compare their numerators.

Front 11

For any two fractions a/b and c/d with b,d≠0 when is a/b < c/d?

Back 11

a/b < c/d if and only if ad < bc

Front 12

For any two fractions a/b and c/d with b,d≠0 when is a/b > c/d?

Back 12

a/b > c/d if and only if ad > bc

Front 13

How do we add the two fractions a/b + c/d?

Back 13

a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd

Front 14

What is a mixed number?

Back 14

A mixed number is a combination of a fraction and a whole number.

Ex. 2 1/2 = 2 + 1/2

Front 15

What is a proper fraction?

Back 15

A fraction a/b where 0≤|a|<b

Front 16

What is an improper fraction?

Back 16

A mixed number can be converted to a fraction of the form a/b. This is an improper fraction because |a| > b

Front 17

How do we subtract two fractions with a common denominator?

Back 17

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.

Front 18

How do we subtract two fractions without a common denominator using fraction strips?

Back 18

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.

Front 19

How do we subtract two fractions without a common denominator without using fraction strips?

Back 19

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

Front 20

How do you multiply a whole number and a fraction using fraction strips?

Back 20

We can think of this as repeated addition.

Ex. C x a/b

Draw C fraction strips representing a/b. add them together.

Front 21

How do we multiply a fraction times a whole number using fraction strips?

Back 21

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

Front 22

How do we multiply a fraction times a fraction (a/b x c/d) using a rectangular array?

Back 22

We can illustrate this by taking a figure that has a/b sections shaded and then taking c/d of the shaded amount.

Front 23

How do we multiply two fractions algebraically?

Back 23

a/b x c/d = ac/bd.

Front 24

How do we divide two fractions using subtraction?

Back 24

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

Front 25

How do we know if c/d divides a/b?

Back 25

given two fractions a/b , c/d a/b ÷ c/d if and only if c/d x e/f = a/b

Front 26

How do we divide two fractions using fraction strips?

Back 26

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)

Front 27

What is an inverted fraction?

Back 27

We invert the fraction c/d to get the reciprocal d/c.

Front 28

How do we divide two fractions algebraically?

Back 28

a/b ÷ c/d = a/b x d/c = ad/bc

Front 29

What is a rational number?

Back 29

It is a number that can be represented as a fraction of the form a/b.

Front 30

What are the properties of Addition of Rational Numbers?

Back 30

Closure, Associative, Commutative, Identity, Additive Inverse

Front 31

What is the Closure Property under Addition of Rational Numbers?

Back 31

Given any two rational numbers a/b and c/d, a/b + c/d is a unique rational number.

Front 32

What is the Associative Property under Addition of Rational Numbers?

Back 32

Given any two rational numbers a/b and c/d, a/b + c/d = c/d + a/b

Front 33

What is the Commutative Property under Addition of Rational Numbers?

Back 33

Given any three rational numbers a/b, c/d, and e/f (a/b + c/d) + e/f = a/b + (c/d + e/f)

Front 34

What is the Identity Property under Addition of Rational Numbers?

Back 34

Given any rational number a/b a/b + 0 = 0 + a/b = a/b

0 is the additive identity element

Front 35

What is the Additive Inverse Property under Addition of Rational Numbers?

Back 35

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

Front 36

What is the property of Subtraction under Rational Numbers?

Back 36

It is closed.

Front 37

What are the properites of multiplication under Rational Numbers?

Back 37

Closure, Associative, Commutative, Identity Property, Distributive, Zero, Mulitplicative Inverse

Front 38

What is the Closure Property under Rational Numbers?

Back 38

Given any two rational numbers a/b and c/d, a/b x c/d is a unique rational number

Front 39

What is the Associative Property under Mulitplication of Rational numbers?

Back 39

Given any two rational numbers a/b and c/d, a/b x c/d = c/d x a/b

Front 40

What is the Commutative property under Multiplication of Rational Numbers?

Back 40

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)

Front 41

What is the Identity Property under Multiplication of Rational Numbers?

Back 41

Given any rational number a/b, a/b x 1 = 1 x a/b = a/b

Front 42

What is the Zero Property under Multiplaction of Rational Numbers?

Back 42

Given any rational number a/b, a/b x 0 = 0 x a/b = 0

Front 43

What is the Distributive Property under Multiplication of Rational Numbers?

Back 43

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)

Front 44

What is the Multiplicative Inverse under Multiplication of Rational Numbers

Back 44

Given a rational number a/b a/b x b/a = 1

b/c is the multiplicative inverse of a/b

Front 45

What is the property of Division of Rational Numbers?

Back 45

It is closed.

Front 46

What are the properties of Order Relations of Rational Numbers?

Back 46

Trichotomy, Transitive, Addition, Multiplication.

Front 47

What is the Trichotomy Property under Order Relations?

Back 47

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

Front 48

What is the Transitive Property under Order Relations?

Back 48

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

Front 49

What is the Addition Property under Order Relations?

Back 49

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)

Front 50

What is the Multiplication property under Order Relations?

Back 50

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)

Front 51

What is the density property?

Back 51

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

Front 52

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?

Back 52

Find equivalent fractions having a common denominator for a/b and c/d and find a fraction that falls between.

Front 53

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?

Back 53

1/a x 1/a ....1/a = 1/(a to the power of n)

thus 10^-1 = 1/10, 10^-2 = 1/100 etc.

Front 54

What is a decimal most of ten?

Back 54

A number that can be expressed with a decimal point.

Front 55

North America, England, Europe, Scandinavia use different notations for the decimal. What are they?

Back 55

NA: 1.5
England: 1⋅5
Europe: 1,5
Scandinavia: uses 1.5 but the 5 is superscript

Front 56

What are the number of digits to the right of the decimal point called?

Back 56

The number of decimal places.

Front 57

What do the positions to the left of the decimal point represent?

Back 57

Place values that are increasing powers of 10.

Front 58

What do the positions to the right of the decimal point represent?

Back 58

Place values that are decreasing (negative) powers of 10.

Front 59

How do you order two positive decimals?

Back 59

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.

Front 60

How do we convert a fraction with a power of 10 in the denominator to a decimal?

Back 60

Ex. 64/100 = 0.64
64/1000 = 0.064

Front 61

How do we convert a fraction without a power of 10 in the denominator to a decimal?

Back 61

We check if there is an equivalent fraction with a power of 10 in the denominator, and then convert.

Front 62

What is a terminating decimal?

Back 62

A decimal with a finite number of digits.

Front 63

What is a non-terminating decimal?

Back 63

A decimal that has an infinite number of digits.

Front 64

When can a fraction a/b in simplest form be written as a terminating decimal?

Back 64

If b has only 2 and 5 as the prime factorization

Front 65

What is a repeating decimal?

Back 65

A decimal that does not terminate and contains a repeating pattern of digits.

**Every non-terminating repeating decimal can be written as a fraction**

Front 66

How do we write a repeating decimal as a fraction?

Back 66

ex. 0.345...

let x = 0.345345...
1000x = 345.345...
1000x-x=345.345...=0.345345...
=999x=345
x = 345/999

Front 67

What is an irrational number?

Back 67

A number that can not be represented as a fraction.

**Every non-terminating non-repeating decimal is irrational**

Front 68

How do we add decimals?

Back 68

Align the decimals point and the place values. Add and make sure you carry over.

Front 69

How do we subtract decimals?

Back 69

Align the decimal points and the place values. Exchange as necessary.

Front 70

How do we multiply decimals?

Back 70

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.

Front 71

How do we divide decimals?

Back 71

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.

Front 72

How do we multiply a decimal by a power of 10?

Back 72

Move the decimal point one place to the right for each power of 10.

Front 73

How do we divide a decimal by a power of 10?

Back 73

Move the decimal point one place to the left for each power of 10.