# 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 0.75/1 = 75/100 × 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 75/100 = 15/20 = 3/4 ÷5 ÷ 5

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 ÷ 5 625/1000 = 25/40 = 5/8 ÷ 25 ÷ 5

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 35/100 = 7/20 ÷ 5

Bring back the 2 (to make a mixed fraction)

### ►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!

## ******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 0.333.../1 = 0.999.../3 × 3

And 0.999... = 1

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 3/4 = 75/100 ×25

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

### ►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 3/16 = 1,875/10,000 ×625

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

### ►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 1/3 = 333/999 ×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 1 = 2 = 4 2 4 8 × 2 × 2

And visually it looks like this:

 1/2 2/4 4/8 = =

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

 ÷ 3 ÷ 6 18 = 6 = 1 36 12 2 ÷ 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:

𝙸𝚗𝚝𝚎𝚐𝚎𝚛 𝚁𝚞𝚕𝚎𝚜 𝚏𝚘𝚛 𝙼𝚞𝚕𝚝𝚒𝚙𝚕𝚒𝚌𝚊𝚝𝚒𝚘𝚗 & 𝙳𝚒𝚟𝚒𝚜𝚒𝚘𝚗:

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.

• # Square / Square Roots

Posted by Tiffany Norman on 7/16/2017

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.

• # Cube / Cube Root

Posted by Tiffany Norman on 7/17/2017

🅰🅿🅿🆁🅾🆇🅸🅼🅰🆃🅸🅽🅶 🅲🆄🅱🅴 🆁🅾🅾🆃🆂:

# 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░:░

░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

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

# Thinking Exponentially

• Exponent Rules

Vocab:

-Base

-Power

-Exponent

-Exponential Form

-Expanded Form

-Standard Form

Notes:

# Scientific Notation

•

【Ｓｃｉｅｎｔｉｆｉｃ　Ｎｏｔａｔｉｏｎ】

꧁꧁Vocab:꧂꧂

-Standard Form

- Scientific Notation

-Coefficient

-Base

-Exponent

-power

# 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
• cubic function

Proportional vs Non-Proportional Relationship:

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

Function vs Not a Function:

Linear Function vs Non-Linear Function:

Intervals:

Slope:

Y-intercept

Graphing Linear Functions (Equations):

Writing an Equation of a Line in Slope Intercept Form:

Comparing Functions:

# 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:

# 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:

# 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:

Sιɱιʅαɾ Tɾαɳʂϝσɾɱαƚισɳ:

# Lines and Angle Relationships

• 𝐿𝒾𝓃𝑒 𝒶𝓃𝒹 𝒜𝓃𝑔𝓁𝑒 𝑅𝑒𝓁𝒶𝓉𝒾🍪𝓃𝓈𝒽𝒾𝓅𝓈

VӨᄃΛB:

• Angle

• Vertex

• Complementary

• Supplementary

• Complement

• supplement

• 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

• Volume

Vocab:

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

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

Volume:

Volume of a cylinder:

Volume of a Cone:

Volume of a Sphere:

Volume of Hemisphere:

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

# Pythagorean Theorem

• Pythagorean Theorem

Vocab:

-right triangle

-hypotenuse

-leg

-Pythagorean Theorem

Notes:

# 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

Example #2:

• ★彡 Proportional Reasoning 彡★

VσƈαႦ:

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

• Writing Ratios

We use ratios to make comparisons between two things. When we express ratios in words, we use the word "to"--we say "the ratio of something to something else." Ratios can be written in several different ways: as a fraction, using the word "to", or with a colon.

Examples:                 3/6               3 to 6                          3:6

Which way you choose will depend on the problem or the situation.

There are other ways to make the same comparison, such as using equal ratios. To find an equal ratio, you can either multiply or divide each term in the ratio by the same number (but not zero).

For example, if we divide both terms in the ratio 3:6 by the number three, then we get the equal ratio, 1:2. If we multiply 1:2 by 4, then we get the equal ratio 4:8. Do you see that these ratios both represent the same comparison (equivalent fractions)?

Short cut: You can cross multiply the diagonals of the fractions. If they are equal to each other you have equivalent ratios.

• Finding Unit Rate

A Rate is a ratio comparing two different units. Example: A bike travels 54 miles in 4 hours. When written as a rate: You can write equivalent rates by multiplying or dividing the numerator and denominator by the same number.

Unit Rate is a ratio of two different measurements where the 2nd measurement is 1. In “Kid Language” it’s the amount (cost, miles, # of objects, etc) for 1 (1 item, 1 mile, 1 hour, 1 ounce, etc).

Let's Practice!

Example #1: Rate of 180 miles in 3 hours. To turn this into a unit rate, we need the denominator to be 1. In other words, we need to find out how many miles were traveled in 1 hour. Let’s divide!

Example #2: You can buy a 40 oz jar of peanut butter for \$5.25, or you can buy a 15 oz jar for \$2.10. Which is the better deal? If we find out the cost per ounce for each jar, then we can determine which has a better price. To do this, we need to divide. ALWAYS put “money on top.”

complex fraction is a fraction where the numerator, denominator, or both contain a fraction.

Another way to put it is to say that a complex fraction is a ratio of two fractions. Use your rules for multiplying rational numbers.

The steps are:

1. Leave the first fraction/numberin the equation alone.
3. Flip the second fraction/numberover (find its reciprocal).
4. Multiply the numerators (top numbers) of the two fractions
5. Multiply the denominators (bottom numbers) of the two fractions

Example #1:

Example #2:

Two quantities are in a Proportional Relationship if all of the ratios that relate the quantities are equivalent.

A proportion is an equation that represents equal ratios.

To solve a proportion...

Constant of Proportionality is the constant multiple that relates proportional quantities x and y. It is the value of the ratio  and is represented by k.

Constant of Proportionality exists when the ratio of two quantities in a table, graph, or ordered pairs simplify to the same unit rate. To check if there is a constant of proportionality: From Ordered Pairs (x, y)/Table: make a ratio (fraction) for all ordered pairs. Then find the unit rate (divide: y/x). The unit rate must be the same for all pairs.

It is also called the unit rate.
• It is what you multiply times x to get y.
• It is typically represented by the letter k.  (y = kx   or  y/x = k)

• 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:

# Inequalities

• ιηєqυαℓιтιєѕ

Example #2: Solving multi-step inequalities

# 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