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/

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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.

Unit 1: Real Number Relationships

Standard

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

REAL NUMBERS

In this unit we went over Real Numbers and the classification system that is set up with Real Numbers. Here is the Rational Numbers Poster we created in class

VennDiagram

We used Nesting boxes to demonstrate how the sub groups are inside each other.

Description of Each Set of Numbers
Natural The natural numbers (also known as counting numbers) are the numbers that we use to count. It starts with 1, followed by 2, then 3, and so on.

1, 2, 3, 4…

Whole The whole numbers are a slight “upgrade” of the natural numbers because we simply add the number zero to the current set of natural numbers. Think of whole numbers as natural numbers and the number zero.

0, 1, 2, 3, 4…

**Remember a hole in the ground is shaped like the number zero!**

Integers The integers include all whole numbers together with the “opposites” of the natural numbers (their negatives).

-4, -3, -2, -1, 0, 1, 2, 3, 4 …

Rational The rational numbers are numbers which can be expressed as ratio of integers. That means, if we can write a given number as a fraction where the numerator and denominator are both integers; then it is a rational number.

Caution: The denominator cannot equal to zero.

Rational numbers can also appear in decimal form. If the decimal number either terminates or repeats, then it is possible to write it as a fraction with an integer numerator and denominator. Thus, it is rational as well

Irrational The irrational numbers are all numbers which when written in decimal form does not repeat and does not terminate.
Real The real numbers includes both the rational and irrational numbers. Remember that under the rational number, we have the subcategories of integers, whole numbers and natural numbers.

Click here to do IDENTIFY REAL NUMBERS online and get automatic feedback (it grades it)! 🙂

Ordering Numbers

Watch a video about ordering number here!

To order numbers there are a couple steps to follow.

1st: Convert all you numbers to decimals

2nd: Convert these to percents

3rd: Label a number line

4th: Put your numbers on the number line

Ascending versus Descending 

Ascending Order

To put numbers in order, place them from lowest (first) to highest (last).

This is called “Ascending Order”.

Example: Place 17, 5, 9 and 8 in ascending order.

  • Answer: 5, 8, 9, 17

Descending Order

Sometimes you want the numbers to go the other way, from highest down to lowest, this is called “Descending Order”.

Example: Place 17, 5, 9 and 8 in descending order.

  • Answer: 17, 9, 8, 5

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

ascending vs descending

Scientific Notation

Scientific notation is about writing really big and really small numbers in an equivalent form.

We normally write numbers in what we call STANDARD NOTATION, but we can also write this number in SCIENTIFIC NOTATION.

For example the number three hundred and twenty.

STANDARD:   320

SCIENTIFIC NOTATION: 3.2 × 102

CONVERTING NUMBERS

When given a number in scientific notation you can easily convert the number to standard notation.

Example #1

Convert from Scientific Notation:  3.6 × 1012    

to Standard Notation

Now, since the exponent on 10 is positive, I know they are looking for a LARGE number, so I’ll need to move the decimal point in the positive direction (to the right), in order to make the number LARGER. Since the exponent on 10 is “12“, I’ll need to move the decimal point twelve places over.

First, I’ll move the decimal point twelve places over. I make little loops when I count off the places, to keep track:

3.6 _ _ _ _ _ _ _ _ _ _ _ .

Then I fill in the loops with zeroes:

3.600000000000.

In other words, the number is 3,600,000,000,000, or 3.6 trillion

Example #2

Convert 4.2 × 10–7 to standard notation.

Since the exponent on 10 is negative, I am looking for a small number. Since the exponent is a seven, I will be moving the decimal point seven places. Since I need to move the point to get a small number, I’ll be moving it to the left. The answer is 0.00000042

Click here to do Scientific Notation practice online and get automatic feedback (it grades it)! 🙂