Unit 7: Transformational Geometry

Standard

 Click here for a vocabulary list or here for flashcard practice.

TRANSFORAMTIONS

There are 4 types of TRANSFORMATIONS that we will be learning about.Click here to view information on all 4 transformations (translations, reflections, rotations, and dilations) or click here to play a video that explain each transformation.

Click here or here or here to practice identifying the Transformation that is taking place and get automatic feedback (it grades it)! 🙂


Translation      –>         Slide!

      

Click here  or here  or here to practice Translations and get automatic feedback (it grades it)! 🙂Translation rule


Reflection     –>      Flip!

     

 Click here or here or here to practice Reflections and get automatic feedback (it grades it)! 🙂

Click here to make some reflection art!!!

Reflections Rule


Rotation          –>      Turn Around!

 Click here or here or here to practice Rotations and get automatic feedback (it grades it)! 🙂Rotation Rule


How do I know if the shape is Congruent or Similar?

If one shape can become another using only Turns, Flips and/or Slides, then the two shapes are called Congruent.

Two shapes are Similar if you need to Resize for one shape to become another (you may also Turn, Flip and/or Slide).

 After these transformations (turn around/rotation, flip/reflection or slide/translation ), the shape still has the same size, area, angles and line lengths.  This means that in these 3 transformations congruence is preserved.


Dilation –> Enlarge/Grow or Reduce/Shrink

The other important Transformation is Dilation (also called reduction or enlargement). The new image becomes bigger or smaller than the original.

Click here or here or here  or here to practice Dilations and get automatic feedback (it grades it)! 🙂

 Click here for a vocabulary list or here for flashcard practice.

 

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Unit 6: Reading Graphs

Standard

Reading graphs is about understanding graphs that are showing 2 types of data.

The graphs are about having data for two variables (x, y) . Typically the two types of data are related in some way.

Below is a Scatter Plot with bivariate data. The two things that we are comparing are Temperature and Ice Cream Sales.

Here we have ice cream sales versus the temperature on that day. The two variables are Ice Cream Sales and Temperature.

(If you have only one set of data, such as just Temperature, it is called “Univariate Data, bivariate mean two types of data)

In this scatter plot you can see that as tempurate rises so do prices.

https://www.mathsisfun.com/definitions/bivariate-data.html

To find more information in Bivariate Data click here.

 

Click here and  here   and here and here to practice Identifying Scatter Plot Trends and Predicting with Best Fit Lines online and get automatic feedback (it grades it)! 🙂

 

Writing Linear Equations

Slope Intercept Form:      y = mx + b

The variable m is the slope, it explains the steepness of a line.

The variable b is the y-intercept, this is where the line touches or intersects at the y-axis.

http://www.shmoop.com/video/slope-intercept-form/

Stations

Standard

Station # 1: Functions

https://www.ixl.com/math/grade-8/identify-functions

Click here to practice Functions

Station # 2: Unit Rate

https://www.ixl.com/math/grade-8/unit-prices

Click here to practice Unit Rate

Station # 7: Unit Rate

Click here to open the document to practice Unit Rate

Station #8: Graphing Lines

https://drive.google.com/file/d/0B7iseoIHkzatZWVjRnV1eDNTYVk/view?usp=sharing

Click here to watch the video and follow along

Unit 5: Proportional vs Non-Proportional Functions

Standard

Functions

A Function is Special

A function is a type of equation.

A function has special rules it must follow:

In a function there can ONLY be ONE output(y) for every input (x).

Just like I stated before, a Function is Special! 🙂

  • It must work for every possible input value
  • And it has only one relationship for each input value

function      function

THIS IS A FUNCTION!                  THIS IS NOT A FUNCTION!

***Think of the birthday example. Each person can only have 1 birthday, other people might have the same birthday as you but you have only 1. You are like the x (input) value and your birthday the y (output) value.***

Example: y = x²

FUNCTION

function

Could also be written as a table:

X: x Y: x2
3 9
1 1
0 0
4 16
-4 16

It is a function, because:

  • Every element in X is related to Y
  • No element in X has two or more relationships
  • The X values DO NOT repeat

So it follows the rules.

(Notice how both 4 and -4 relate to 16, which is allowed since 4 and -4 are two different numbers.)

Example: This relationship is not a function:

NOT A FUNCTION

function

This is not a function, for these reasons:

  • Value “3” in X has no Y
  • Value “4” in X has no Y
  • Value “5” is related to more than one value in Y
  • REMEMBER THE X VALUE CANNOT REPEAT

Find more information about functions here.

Find more information about functions here.

Unit Rate

Unit rates are about the amount for 1 unit.

We use Ratio Tables to help solve for rates.
Our Ratio table looks like a tic-tac-toe table.

rt
We MUST label our table to know where to put our information

Ex. Bob drives 100 miles in 5 hours. How many miles does he travel every hour?

Ratio Table

Now we fill in the information that is given.

Bob drives 100 miles in 5 hours. How many miles does he travel every hour?

Ratio

Once we have entered what was given we need to see what the question is asking us.

Ex. Bob drives 100 miles in 5 hours. How many miles does he travel every hour?

Ask yourself, is the problem asking how many hours it takes for 1 mile or how many miles for 1 hour???????

If you said how many miles for 1 hour, you are correct!

So now we enter that into our Ratio Table.

Rat

The x is the unknown. This is what we are solving for.

Rati

To solve for x, we multiply 100 and 1. Then divide by the number left over which is 5.

We end up with x = 20.

So Bob drives 20 miles per hour.

Click here to watch a video with a different way to find unit rate.

Test yourself on unit rates here!

Unit 4: Slope and Y-Interecept

Standard

Slope Intercept Form:      y = mx + b

The variable m is the slope, it explains the steepness of a line of how slanted the line is.

The variable b is the y-intercept, this is where the line crosses or intersects the y-axis.

https://www.desmos.com/calculator

http://www.shmoop.com/video/slope-intercept-form/

Slope: m

Slope is how steep a straight line is.
When finding the slope from a line on a graph we use the method of rise over run .
  • Rise is how far up
  • Run is how far along

rise over runn

You can also think of it as the change in y over the change in x.
gradient
EXAMPLE #1:
rise over run
The slope here is 4/6 which can simplify to 2/3.
EXAMPLE #2:
Slope
In this example the slope is 3/5.

For more information on slope click here.

Practice Problems

Get your SLOPE on!

TEST YOUR SELF ON LINES HERE!!!!

https://my.hrw.com/wwtb/api/viewer.pl

Here is a video of slope as a rate of change.

Y Intercept: b

Y intercept is where a straight line crosses the Y axis of a graph.

Example:

Y intercept

In the above diagram the line crosses the Y axis at 1.

So the Y intercept is equal to 1.

For more information on intercepts click here.

Practice Problems

Get your Y-intercept on!

Slope Intercept Form

y = mx + b

m = Slope (how steep the line is)

b = the Y Intercept (where the line crosses the Y axis)\

y=mx+b graph

How do you find “m” and “b”?

  • b is easy: just see where the line crosses the Y axis.
  • m (the Slope) needs some calculation. Remember to write slope as a fraction.

Example 1)

y=2x+1 graph

The fastest and easiest thing to find first when looking at a graph is the y-intercept (b).

Here we see that the line crosses the y-axis at positive 1.

So, b = 1 .

 Now to find the slope (m) we will use rise over run:

  • Rise is how far up
  • Run is how far along

In this example the rise is 2 and the run is 1.

So, m = 2/1 .

Now that I know m = 2/1 and b = 1 I can plug them into the equation for slope intercept form y = mx + b.

y = (2/1) x + 1

^^^^^ This is the equation of the line.
For  more information on y = mx + b click here.

 Practice Problems

Get your Slope Intercept on!

Unit 3: Equations and Inequalities

Standard

Click here to go to the IXL website for all kinds of 8th grade topics and review problems.

Equations

What is an equation?

Examples: 4 + 3  = 7        or        3x + 5 = 10

An equation is a number sentence. We call it an equation because it has an equal sign.

The 5 Steps to Writing an Equation or Inequality

Step 1. Read and underline the question

Step 2. Find your  Χ  (your variable/unknown) and BOX it

Step 3. Circle the Math Words (product, quotient, each,                per, together, sum, difference, squared)

Step 4. Replace the operation words with their symbols (              • , + , – ,  ÷ , / , = , < , > , ≤ , ≥ ,√ , ≠ , ² , ³ )

Step 5. Write the equations

 Don’t forget our cool ‘dance’ we did to remember this!

WRITING EQUATIONS PRACTICE PROBLEMS:

Click here to practice Writing Equations online and get automatic feedback (it grades it)! 🙂

With Equations, Inequalities and Expressions we always want to combine like terms 1st!

Here is an example on how to do that:

3.4 Combining Like Terms

Once all like terms have been combined then we can solve.

Solving Equations with Models

To create your own equations using models click here!

MODELING EQUATIONS PRACTICE PROBLEMS:

Click here to practice Modeling Equations online and get automatic feedback (it grades it)! 🙂

Solving Equations Algebraically

Here is another example solving algebraically

Solve √(x/2) = 3

Start With √(x/2) = 3
Square both sides: x/2 = 32
32 = 9: x/2 = 9
Multiply both sides by 2: x = 18

And the more “tricks” and techniques you learn the better you will get.

Here is an example of how we solved equations in class:

3.9 Solving Equations with variables on both sides

SOLVING EQUATIONS (with variables on both sides PRACTICE PROBLEMS:

Click here or here to practice Solving Equations online and get automatic feedback (it grades it)! 🙂

Systems of Equations

For information on systems of equations click here.

Simple vs. Compound  Interest

Introduction to interest:

http://www.mathsisfun.com/money/interest.html

SIMPLE INTEREST

 I = Prt  

  • I = interest owed  [$] (this is ONLY the interest borrowed)
  • P = amount borrowed (called “Principal”)  [$]
  • r = interest rate   [%] (you have to divide the percent by 100)  For information On Percents click here!
  • t = time    [years]

Simple interest is money you can earn by investing some money (the principal). The interest (percent) is the rate that makes the money grow!

COMPOUND INTEREST

 A = P(1+r)^t  

  • A = All of it / Actual / total amount owed (this amount includes the interest and the principal)   [$]
  • P = amount borrowed (called “Principal”)    [$]
  • r = interest rate     [%]
  • t = time    [years]

Compound interest is very similar to simple interest. The difference is that compound interest grows much faster! The reason it grows faster is because the interest (percent) has an exponent.

********** MAKE SURE TO READ THE QUESTION AND SEE EXACTLY WHAT IT IS ASKING DOES IT JUST WANT THE INTEREST OR THE TOTAL (All of it) ???????? *************************

For information on compound interest click here.

SIMPLE INTEREST PRACTICE PROBLEMS:

Click here or here to practice Simple Interest online and get automatic feedback (it grades it)! 🙂

COMPOUND INTEREST PRACTICE PROBLEMS:

Click here or here to practice Compound Interest online and get automatic feedback (it grades it)! 🙂

Unit 2: Patterns in Data

Standard

 Mean

The mean is the average of the numbers.

It is easy to calculate: add up all the numbers, then divide by how many numbers there are.

Absolute Value

Absolute Value means …

… only how far a number is from zero:

“6” is 6 away from zero,
and “−6” is also 6 away from zero.

So the absolute value of 6 is 6,
and the absolute value of −6 is also 6

Go over Absolute Value and do some practice (the website checks it)

Mean Absolute Deviation

Mean absolute deviation (MAD) is about calculating the average distance from the mean.

MAD BURGER!!!
MAD Burger

Click here to watch a video from our 8th Grade Math textbook that goes in to detail and explain how to calculate the MAD.

MEAN ABSOLUTE DEVIATION:

The average/mean distance from the average/mean is the MAD.

Yes, we use “mean” twice: Find the mean … use it to work out distances … then find the mean of those!

Three steps:

  • 1. Find the mean of all values
  • 2. Find the distance of each value from that mean (subtract the mean from each value, ignore negative signs)
  • 3. Then find the mean of those distances

Like this:

Example: Find the Mean Absolute Deviation of

3, 6, 6, 7, 8, 11, 15, 16

Step 1: Find the mean:

Mean = 3 + 6 + 6 + 7 + 8 + 11 + 15 + 16   = 72   = 9
8 8

Step 2: Find the distance of each value from that mean:

Value Distance from 9
3 6
6 3
6 3
7 2
8 1
11 2
15 6
16 7

Which looks like this:

Step 3. Find the mean of those distances:

Mean Deviation = 6 + 3 + 3 + 2 + 1 + 2 + 6 + 7   = 30   = 3.75
8 8

So, the mean = 9, and the mean deviation = 3.75

It tells us how far, on average, all values are from the middle.

In that example the values are, on average, 3.75 away from the middle.

For deviation just think distance

MEAN ABSOLUTE DEVIATION Practice Problems:

Click here or here to practice Mean Absolute Deviation online and get automatic feedback (it grades it)! 🙂

Get your MEAN ABSOLUTE DEVIATION on!

Random Sample

This is where everyone has an equal and fair chance of being selected.