April 6, 2011 - Graphing

A Maths Dictionary for Kids

What is linearity?  The study of lines.

• Directionality
• Ordered Pairs
• Graph
• Equation

Linear

Coordinate Plane

Quadrants of Coordinate Planes

Ordered Pairs

Slopes

• Slope (or rate of change) - the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline.  
• Slope = rise/run (rise over run)
• always treat slope in the simplest form

Types of Slopes:

Positive slope - rises left to right

Negative slope - goes down - left to right

Zero slope - never rises, never a change in y

Undefined slope - perfectly up and down line  - x never changes
Always in the form of x = 

Examples:

3/2 (think fraction)

4/3

-2/5

-3 or -3/1

Slope Triangle

The red line is the slope I am starting with.  The white lines are helping me determine the slope.

Up 3 over 3 = 3/3...or in simplest for 1/1

Up 2 over 3 = 2/3

Negative slope
Down 4 over 5 = -4/5

The slope is 3/4.  The red and blue lines represent the slope triangle.

Formula for the slope intercept of a line  

• y = mx + b 
• the m represents the slope - how fast that line is changing
• y and x are the variables
• b = y intercept
• intercept is when a line crosses the axis. The y intercept is where the line crosses the y axis

Drawing Slopes

If your slope has a negative like - 3/5....assume that the numerator (in this case the 3)
is the negative #.  You could assume that the 5 is negative number....if you
do, you will get the same slope.

Stories to graphs

Using Coordinates to determine slope

March 30th - Homework

March 30, 2011

Coefficient

Variable:

• Unknown
• Symbols/Letters
• Changed
• Live in a variable world....things constantly change

The word equation comes from the word "equality"

• 4x + 2 = 6x + 4
• A common misconception by students "part of an equation".  Each equation has at least two sides.  If you add something to one side of the equal sign, you have to do the same to the other sign, to keep the sides equal.

Hands on Equations System

For this activity you need 8 blue pawns, 8 white pawns 4 green blocks and 4 red blocks.  Two of each of those blocks need to be low numbers and the other two need to be high.....for each color.  The white pawns are negative variables, and the red cubes are negative numbers....although in my pictures the red cubes are being used as positive numbers.

This represent the equation 2x = 10.  The blue pawn is the variable.  2 x 5 = 10.  x = 5

This picture represent x + 2 = 9.  The blue pawn is the variable.  What plus 2 = 9? x = 7.

This picture represents 2x = x + 3.  x = 3.

Picture below shows the solution

Picture below shows the solution

 

This is the solution of the above two pictures

No picture to go with this explanation.  But it will show you the process.

This example should have been done with the pawns and blocks.  The x is the blue pawns.

2(x+3) = 10.......the 2 tells us how many times we are going to build (x+3).  So is becomes (x+3) (x+3) = 10.  2x + 6 = 10.  x = 2.  See picture above

Types of Equations

1. One step equations
1. Adding/subtracting equations
2. multiplying/dividing equations
3. Do the opposite of whatever is happening to the x (or variable)
2. Two step equations
1. Either add/subtract and then mutiply/divide
2. Do the opposite of whatever is happening to the x (or variable), but do it twice.
3. Multi step equations
1. Variables on both sides
2. Distributive property 
3. Combine like terms
4. Applying the distributive property...... a(b + c) = ab + ac 

 One Step Equations

• x + 4 = 6
• x hates people, wants to be alone. He hates things that are adding or subtracting to  him.  He also hates when things are multiplying and dividing to him.  We need to get x by himself.  In this problem 4 is being added to x.  apply -4 to each side.  X is now happy all by himself.  x = 2
• x - 12 = 4
• Get x by himself.  12 is being subtracted from the x.  What is the opposite of -12?  +12.  Add 12 to each side.  x = 16
•  - 4 = x +6
• What is the opposite of +6?  -6. Subtract 6 to each ide.  -10 = x.
• 3x = 15
• Get x by himself to make him angry.  3 is being multiplied to x.  Opposite of multiplication is division.  Divide each side by 3.  x = 5.
• 2/3x = 4 (two-thirds)x = 4.  
• How do you undo division? Multiply.  Multiply each side by 3.  3(2/3)x = 4(3).  The 3's cross out.  2x = 12.  Divide two to each side. x = 6.

Sorry the picture is sideways.  I'm too lazy to fix it.

Two Step Equations

• 3x + 4 = 22
• Step one:  deal with the addition or the subtraction first.
• What is the opposite of +4?  Subtract the 4 from each side.
• We now have 3x = 18
• Step 2: deal with the multiplication or division
• 3 is being multiplied to x.  What is the opposite of multiplication?
• Divide 3 ....both sides
• x = 6
• Two step equation examples:



I don't know why my pics keep coming in sideways.






Multi Step Equations:








Combining Like Terms






Putting it ALL together...."the Pinnacle"






What students need to know to be successful

• Integers
• Order of Operations
• Equality (What the = means)
• Distributive Property
• Like terms
• Companion/Inverse Operations

March 23, 2011

The Golden Rectangle

The Golden Spiral
Take the rectangle and "spiral" it counter-clockwise



Fibionacci Number Sequence

Link to Video: TeacherTube Videos - The Fibonacci Sequence-Rabbit Story





Pascal's Triangle

Sierpinski Triangle



Triangular Numbers

Square Numbers

Exponential Growth

Link to Video:  Exponential Growth