Equivalent Numbers

  • 𝔼𝕢𝕦𝕚𝕧𝕒𝕝𝕖𝕟𝕥 ℕ𝕦𝕞𝕓𝕖𝕣𝕤

    Vocab:

    • Repeating Decimal

    • Terminating Decimal

    • Whole Number

     • Fraction

    • Equivalent

     


     

    𝓓𝓮𝓬𝓲𝓶𝓪𝓵 𝓽𝓸 𝓕𝓻𝓪𝓬𝓽𝓲𝓸𝓷

    To convert a Decimal to a Fraction follow these steps:

    • Step 1: Write down the decimal divided by 1, like this:   decimal1
    • Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there are two numbers after the decimal point, then use 100, if there are three then use 1000, etc.)
    • Step 3: Simplify (or reduce) the fraction

     

    ►Example: Convert 0.75 to a fraction

    Step 1: Write down 0.75 divided by 1:

    0.75/1

    Step 2: Multiply both top and bottom by 100 (because there are 2 digits after the decimal point so that is 10×10=100):

    × 100
    right over arrow
    0.75/1 = 75/100
    right under arrow
    × 100

    (Do you see how it turns the top number into a whole number?)

    Step 3: Simplify the fraction (this took me two steps):

      ÷5   ÷ 5  
    right over arrow   right over arrow
    75/100 = 15/20 = 3/4
    right under arrow   right under arrow
      ÷5   ÷ 5  

     

    Answer = 3/4

     

    Note: 75/100 is called a decimal fraction and 3/4 is called a common fraction!

     

    ►Example: Convert 0.625 to a fraction

    Step 1: write down:

    0.625/1

    Step 2: multiply both top and bottom by 1,000 (3 digits after the decimal point, so 10×10×10=1,000)

    625/1000

    Step 3: Simplify the fraction (it took me two steps here):

      ÷ 25   ÷ 5  
    right over arrow   right over arrow
    625/1000 = 25/40 = 5/8
    right under arrow   right under arrow
      ÷ 25   ÷ 5  

     

    Answer = 5/8

    When there is a whole number part, put the whole number aside and bring it back at the end:

     

    ►Example: Convert 2.35 to a fraction

    Put the 2 aside and just work on 0.35

    Step 1: write down:

    0.35/1

    Step 2: multiply both top and bottom by 100 (2 digits after the decimal point so that is 10×10=100):

    35/100

    Step 3: Simplify the fraction:

    ÷ 5
    right over arrow
    35/100 = 7/20
    right under arrow
    ÷ 5

    Bring back the 2 (to make a mixed fraction)

    Answer = 2 7/20

     

    ►Example: Convert 0.333 to a fraction

    Step 1: Write down:

    0.333/1

    Step 2: Multiply both top and bottom by 1,000 (3 digits after the decimal point so that is 10×10×10=1,000)

    333/1000

    Step 3: Simplify Fraction:

    Can't get any simpler!

     

    Answer = 333/1000

     

    ******A Special Note******

    If you really meant 0.333... (in other words 3s repeating forever which is called 3 recurring) then we need to follow a special argument. In that case we write down:

    0.333.../1

    Then multiply both top and bottom by 3:

    × 3
    right over arrow
    0.333.../1 = 0.999.../3
    right under arrow
    × 3

    And 0.999... = 1 

    Answer = 1/3

     


     

    𝓕𝓻𝓪𝓬𝓽𝓲𝓸𝓷 𝓽𝓸 𝓓𝓮𝓬𝓲𝓶𝓪𝓵

    Use Long Division! 

    ►Example: here is what long division of 58 looks like:

        0.625
     8 )5.000
        0
        5.0
        4.8
          20
          16
           40
           40
            0

    In that case we inserted extra zeros and did 5.0008 to get 0.625

     

    ******Another Method******

    Yet another method you may like is to follow these steps:

    • Step 1: Find a number you can multiply by the bottom of the fraction to make it 10, or 100, or 1000, or any 1 followed by 0s. 
    • Step 2: Multiply both top and bottom by that number.
    • Step 3. Then write down just the top number, putting the decimal point in the correct spot (one space from the right hand side for every zero in the bottom number)

     

    ►Example: Convert 34 to a Decimal

    Step 1: We can multiply 4 by 25 to become 100

    Step 2: Multiply top and bottom by 25:

    ×25
    right over arrow
    3/4  =  75/100
    right under arrow
    ×25

    Step 3: Write down 75 with the decimal point 2 spaces from the right (because 100 has 2 zeros);

    Answer = 0.75

     

    ►Example: Convert 3/16 to a Decimal

    Step 1: We have to multiply 16 by 625 to become 10,000

    Step 2: Multiply top and bottom by 625:

    ×625
    right over arrow
    3/16  =  1,875/10,000
    right under arrow
    ×625

    Step 3: Write down 1875 with the decimal point 4 spaces from the right (because 10,000 has 4 zeros);

    Answer = 0.1875

     

    ►Example: Convert 1/3 to a Decimal

    Step 1: There is no way to multiply 3 to become 10 or 100 or any "1 followed by 0s", but we can calculate an approximate decimal by choosing to multiply by, say, 333

    Step 2: Multiply top and bottom by 333:

    ×333
    right over arrow
    1/3  =  333/999
    right under arrow
    ×333

    Step 3: Now, 999 is nearly 1,000, so let us write down 333 with the decimal point 3 spaces from the right (because 1,000 has 3 zeros):

    Answer = 0.333 (accurate to only 3 decimal places !!)

     


     

    𝓔𝓺𝓾𝓲𝓿𝓪𝓵𝓮𝓷𝓽 𝓕𝓻𝓪𝓬𝓽𝓲𝓸𝓷𝓼
    Equivalent Fractions have the same value, even though they may look different. These fractions are really the same.

    1/2  =  2/4  =  4/8

    Why are they the same? Because when you multiply or divide both the top and bottom by the same number, the fraction keeps it's value.

     

    The rule to remember is:

    "Change the bottom by multiplying or dividing,
    And the same to the top must be applied"

    Here is why those fractions are really the same:

      × 2   × 2  
    right over arrow   right over arrow
    1  =  2  =  4
    2 4 8
    right under arrow   right under arrow
      × 2   × 2  

    And visually it looks like this:

    1/2   2/4   4/8
    pie 1/2 = pie 2/4 = pie 4/8

     


    Here are some more equivalent fractions, this time by dividing:

      ÷ 3   ÷ 6  
    right over arrow   right over arrow
    18  =  6  =  1
    36 12 2
    right under arrow   right under arrow
      ÷ 3   ÷ 6  

    Choose the number you divide by carefully, so that the results (both top and bottom) stay whole numbers. 

    If we keep dividing until we can't go any further, then we have simplified the fraction (made it as simple as possible).

     

    ******Summary******

    • You can make equivalent fractions by multiplying or dividing both top and bottom by the same amount.
    • You only multiply or divide, never add or subtract, to get an equivalent fraction.
    • Only divide when the top and bottom stay as whole numbers.

The Real Number System

  • 🅃🄷🄴 🅁🄴🄰🄻 🄽🅄🄼🄱🄴🅁 🅂🅈🅂🅃🄴🄼

     

    𝚅𝚘𝚌𝚊𝚋:
    • Natural Numbers

    • Whole Numbers

    • Integers

    • Fractions

    • Reciprocal

    • Positive Integers

    • Negative Integers

    • Closed

    • Rational Numbers

    • Irrational Numbers

    • Terminating Decimal

    • Repeating Decimal

    • Bar Notation

    • Real Numbers

    • Venn Diagram

    • Number Line

    • Zero

    • Opposite

    • Multiplicative Inverse


    𝙸𝚗𝚝𝚎𝚐𝚎𝚛 𝚁𝚞𝚕𝚎𝚜 𝚏𝚘𝚛 𝙰𝚍𝚍𝚒𝚝𝚒𝚘𝚗 & 𝚂𝚞𝚋𝚝𝚛𝚊𝚌𝚝𝚒𝚘𝚗:


    ir

    ir

    ir


    𝙸𝚗𝚝𝚎𝚐𝚎𝚛 𝚁𝚞𝚕𝚎𝚜 𝚏𝚘𝚛 𝙼𝚞𝚕𝚝𝚒𝚙𝚕𝚒𝚌𝚊𝚝𝚒𝚘𝚗 & 𝙳𝚒𝚟𝚒𝚜𝚒𝚘𝚗:


    ir

    ir

     


     

    ns  

    Rå†ïðñål ñµmßêr§:
    Numbers can be grouped in a variety of ways according to their characteristics. Sometimes, a number may fit into multiple groupings. For example, -3/4 is both a fraction and a negative number. The number 27 can be grouped with whole numbers and with integers.

     

    The set of natural numbers, consists of the numbers that you use to count objects: {1, 2, 3, 4, 5, …}. The set of whole numbers is made up of the set of natural numbers and the number 0, the additive identity. Another set of numbers is the set of integers, which is a set that includes all of the whole numbers and their additive inverses: {..., −3, −2, −1, 0, 1, 2, 3, …}

     

    When you perform an operation such as addition or multiplication on the numbers in a set, the operation could produce a defined value that is also in the set. When this happens, the set is said to be closed under the operation. The set of integers is said to be closed under the operation of addition. This means that for every two integers a and b, the sum a + b is also an integer.

     

    A rational number is a number that can be written in the form a/b, where a and b are both integers and b is not equal to 0. A rational number can be written as either a terminating or repeating decimal.

     

    A terminating decimal is a decimal that has a finite number of non-zero digits (e.g.,1/8 = 0.125). A repeating decimal is a decimal with digits that repeat in sets of one or more. You can use two different notations to represent repeating decimals. One notation is bar notation, which shows one set of digits that repeats with a bar over the repeating digits (e.g., 1/3 = .3...). Another notation shows two sets of digits that repeat with dots to indicate repetition (e.g.,1/3 = 0.33...). You can use algebra to determine the fraction that is represented by a repeating decimal. For example, write the decimal 0.44... as a fraction.

     

    Ìrrå†ïðñål ñµmßêr§:
    All other decimals are irrational numbers, because these decimals cannot be written as fractions in the form a/b, where a and b are integers and b is not equal to 0.

    ir

     

Perfect Squares & Square Roots

  • Square / Square Roots

    Posted by Tiffany Norman on 7/16/2017

    Square 6 Square 2


     

    Square 4 Square 5

     

    sr

     


    Square 1

     


    ᴀᴘᴘʀᴏxɪᴍᴀᴛɪɴɢ ꜱQᴜᴀʀᴇ ʀᴏᴏᴛꜱ:

    What happens when you have a square root that is not a perfect square?

     

    If your square root is not a perfect square it is called an irrational square root. Irrational square roots always fall between the square roots of the perfect squares they are between. 

     

    sr

    sr

    Comments (-1)

Recent

By Month

Perfect Cubes & Cube Roots

  • Cube / Cube Root

    Posted by Tiffany Norman on 7/17/2017

    Cube 1


     pc


    Cube 2


    🅰🅿🅿🆁🅾🆇🅸🅼🅰🆃🅸🅽🅶 🅲🆄🅱🅴 🆁🅾🅾🆃🆂:

    pc

    Comments (-1)

Recent

By Month

Multi-Step Equations

  • Ⓔⓠⓤⓐⓣⓘⓞⓝⓢ

     

    ░V░o░c░a░b░:░

    ○ Coefficient

    ○ Identity

    ○ Multiplicative Inverse

    ○ Property of Equality

    ○ Null Sets


    ░S░o░l░v░i░n░g░ ░E░q░u░a░t░i░o░n░s░ ░S░t░e░p░s░:░

     

    eq


    ░E░q░u░a░t░i░o░n░ ░N░o░t░e░s░ ░w░i░t░h░ ░E░x░a░m░p░l░e░s░:░

    page 1

    page 2

    page 3

    page 4

    page 5

    page 6

    page 7

    page 8

    page 9

    page 10

    page 11

    page 12

    page 13

    page 14

    page 15

    page 16

     


    ░S░o░l░u░t░i░o░n░s░ ░t░o░ ░a░n░ ░E░q░u░a░t░i░o░n░:░

     

    sol

     

     

Thinking Exponentially

  • Exponent Rules

     

    Vocab:

    -Base

    -Power

    -Exponent

    -Exponential Form

    -Expanded Form

    -Standard Form

     

    Notes: 

    exp

    exp


    exp


    exp


    exp


    exp


    exp

Scientific Notation

  •  

    【Scientific Notation】

     

    ꧁꧁Vocab:꧂꧂

    -Standard Form

    - Scientific Notation

    -Coefficient

    -Base

    -Exponent

    -power  

     


    sn

    sn

    sn sn sn


    sn sn


    sn sn


    sn sn  

Functions

  • Functions

     

    Vocab:

    • set
    • relation
    • input
    • output
    • function
    • domain
    • range
    • scatter plot
    • vertical line test
    • linear function
    • increasing function
    • constant function
    • decreasing function
    • interval of increase
    • interval of decrease
    • constant interval
    • absolute value function
    • quadratic function
    • cubic function

     


    Proportional vs Non-Proportional Relationship:

     

    prop vs nonprop

    CoP stands for Constant of Proportionality-the "k" value
    CRoC stands for Constant Rate of Change-the "m" value
    "b" is the y-value when x is zero, also called the y-intercept because it is where the graph crosses the y-axis

    prop

     

     


    Function vs Not a Function:

    fun

    fun f vs nf f vs nf

    av

    av f vs nf

    quad

     

    f vs nf

     

     

     


    Linear Function vs Non-Linear Function:

    lin vs nonlin

    lin vs nonlin

    lin vs nonlin

     

     


    Intervals:

    int

    int

    int

    int

     

     


    Slope: 

    slope

    slope

    slope

    slope

    slope

     

     


    Y-intercept

    y

    y

    y

     

     


    Graphing Linear Functions (Equations):

    graphing

    graphing

    graphing

     

     


    Writing an Equation of a Line in Slope Intercept Form:

    eol eol eol

    eol

     

     


    Comparing Functions:

    cf

    cf

     

     

     

     

Scatter Plots

  • SCATTER PLOTS

     

    Vocab:

    • bivariate data
    • explanatory variable
    • response variable
    • association/correlation
    • linear association
    • positive association
    • negative association
    • outlier

    • Gap

    • Cluster

    • line of best fit
    • model
    • trend line
    • interpolating
    • extrapolating
    • categorical data

    • Measuremental Data
    • Scatter Plot

     

    Notes:

    Bivariate Data Notes

    bivariate 1

    associate

     

    scatter Plots

    line of best fit

    line of best fit

    interpret

    predict

     

Two-Way Frequency Tables

  • Two-way Frequency Tables

    Vocab:

    -Categorical Data

    -Bivariate

    -Two-way frequency table

    -Relative Frequency

    -Row Relative Frequency

    -Column Relative Frequency

     

    Notes:

    categorical

    frequency

    ft

    Ft

    ft relative

    Ft

    Ft

    Ft

Transformations: Similar & Congruent Figures

  • Ⓣⓡ𝐚η𝕤𝒇𝔬𝓻ᗰΔ𝔱Ꭵ𝑜Ň𝕊

    Tɾαɳʂϝσɾɱαƚισɳ VσƈαႦυʅαɾყ:

    -Corresponding sides: are sides that have the same relative position in geometric figures.

    -Transformation: is the movement of a plane and all the points of a figure on a plane according to a common action or operation.

    -Pre-image: the original figure in a transformation is called the pre-image.

    -Image: the new figure created from a transformation is called the image.

    -Translation: describes a function in geometry that moves and object a certain distance. The object is not altered in any other way.

    -Reflections: the pre-image is flipped across the line of reflection to create the image. Each point of the image is the same distance from the line as the pre-image is, just on the opposite side of the line.

    -Rotations: is a transformation that turns a figure about a fixed point called the center of rotation. Rotations may be clockwise or counter-clockwise.

    -Dilations: a transformation that produces an image that is the same shape and the original but a different size. A dilation stretches or shrinks the pre-image.

    -congruent figures:same size, same shape.

    -corresponding angles: angles that have the same relative position.

    -plane: extends infinitely in all directions in two dimensions and has no thickness.

    -rigid motion: a transformation that preserves the size and shape of a figure.

    -line of reflection: line that acts as a mirror.

    -center of rotation: fixed point of rotation.

    -angle of rotation:degrees that shape turns.

    -congruent line segments:line segments with the same length and shape.

    -congruent angles: angles with the same measurement in degrees.

    -center of dilation: a fixed point where the shape enlarged or reduced in size. 

    -scale factor: how big or small the shape becomes. 

    -enlargement:shape becomes larger and has a scale factor greater that 1.

    -reduction:shape becomes smaller and has a scale factore less that 1 but greater than 0.

    -similar: Same shape, different size.


    Cσɳɠɾυҽɳƚ Tɾαɳʂϝσɾɱαƚισɳʂ:

    con


    con con trans trans


    con refl refl


    con  

    rot rot


     Sιɱιʅαɾ Tɾαɳʂϝσɾɱαƚισɳ:

    dila reduction


    sf sf

     

Lines and Angle Relationships

  • 𝐿𝒾𝓃𝑒 𝒶𝓃𝒹 𝒜𝓃𝑔𝓁𝑒 𝑅𝑒𝓁𝒶𝓉𝒾🍪𝓃𝓈𝒽𝒾𝓅𝓈

     

    VӨᄃΛB:

    • Angle

    • Vertex

    • Complementary

    • Supplementary

    • Complement

    • supplement

    • Adjacent angle

    • Vertical angles

    • Conjecture

    • Congruent

    • Triangle Sum Theorem

    • Exterior angle of a polygon

    • Remote interior angles of a triangle

    • Exterior Angle Theorem

    • Transversal

    • Alternate interior angles

    • Alternate exterior angles

    • Same-side interior angles

    • Same-side exterior angles

    • Angle-Angle Similarity Theorem


    ast angle1

    eat


    av cs ap


    angle2 angle3

    transversal2


    angle4

  • Volume

     

    Vocab:

    • cylinder
    • right cylinder
    • radius of a cylinder
    • height of a cylinder
    • oblique cylinder
    • cone
    • height of a cone
    • sphere

    • hemisphere
    • center of a sphere
    • radius of a sphere
    • diameter of a sphere
    • great circle

     

     


    Volume: 

     volume

    volume

    volume

     

     


    Volume of a cylinder:

     

     cylinder

    cylinder

    cylinder

    cylinder

     

     


    Volume of a Cone: 

    cone

    cone

    cone

     

     

     


    Volume of a Sphere: 

     sphere

    sphere


    Volume of Hemisphere:

    Find the volume of the sphere, and then divide by 2.

     

    hemisphere

     

Pythagorean Theorem

  • Pythagorean Theorem

     

    Vocab:

    -right triangle 

    -hypotenuse

    -leg

    -Pythagorean Theorem

     

    Notes:

    PT

    pt

    PT PT PT PT

     

Systems of Linear Equations

  • 𝐬ү𝐒𝐭𝔼𝕄𝐬 ᗝ𝐅 𝓁Į𝓝ⓔค𝓇 є𝓠𝔲𝐚ᵗ𝐈𝑜ηŞ

     

    νσ¢αв:

    ♦point of intersection

    ♦break-even point

    ♦system of linear equations

    ♦solution of a linear system

    ♦consistent system

    ♦inconsistent system

    ♦standard form of a linear equation

    ♦substitution method


    gra

    Example #2: 

    gra system1 system2 system3


    system4 system5

    sub1

    sub2

    sub3


    elim sub

    elim


    system6


    solutions solutions2

  • ★彡 Proportional Reasoning 彡★

    VσƈαႦ:

    • rate
    • unit rate
    • ratio
    • units
    • complex fraction
    • proportional
    • constant of proportionality
    • rate of change

    er


    complex

    Example #1:

    kcf

    Example #2:

    kcf


    proportions

     

    To solve a proportion...

    proportions

     

    prop

     

     

     

     

     

     

     

     

     

     

     


     

    ur

    ur

    ur

     

     

     

     

     

     

     

     

     

     

     

     


    ur ur ur


    PR pr pr

     

     

     

     

     

     

     

     

     

     


    propeq propeq

    propeq

     


     

     

    propgraph propgraph propgraph

  • 𝔈𝔮𝔲𝔦𝔳𝔞𝔩𝔢𝔫𝔱 𝔈𝔵𝔭𝔯𝔢𝔰𝔰𝔦𝔬𝔫𝔰

     

    Vocab:

    • Distributive Property
    • Factor
    • Expression
    • Numerical Expression
    • Algebraic Expression
    • Verbal Expression
    • Term
    • Like Terms
    • Coefficient
    • Variable
    • Constant
    • Evaluate
    • Solve
    • Solution

     

    Order of Operations:

     

    PEMDAS


     

    ee


    ee


    we

Inequalities

  • ιηєqυαℓιтιєѕ

     

    ie

    ie ie


    wie wie


    gie gie


    sie

    Example #2: Solving multi-step inequalities

    sie

Percents

  • ρ𝑒𝔯ℂ乇𝐍Ŧ𝐒

    𝐕ό𝓬𝕒b:

    • Percent increase
    • percent decrease
    • percent of change
    • percent error
    • percent
    • markup
    • markdown
    • discount
    • regular/original price
    • retail price
    • sale price
    • simple interest
    • principal
    • deposit
    • total cost
    • commission
    • sales tax
    • tax rate
    • tip
    • base salary

    percents percents percents percents


    tax


    tip tip tip  


    si si

    si si


    com com com com


    mup sp mdown


    pin

    pdec


    error