Basic algebra and graphing for young students

1. Basic Algebra and
Graphing for
Kaya and Zoey

2. Elementary Algebra is Easy!
There is only one new idea
and four things to learn

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One New Idea
Algebra is way to manipulate unknown numbers.
These are sometimes call variables because they can have different values
The Romans called them “unknown number”
Sometime people use the symbol
A better way was discovered by the Arabs
They used letters like x,y,z to represent unknown numbers
This idea was adopted by the Europeans.
?
The Arabs also gave us our number system 0,1,2,3.. to replace Roman numerals I,V..

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Four Things to Learn
1. Substitution
2. Combining
3. Solving
4. Graphing

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Definitions
1. Unknown numbers, Variables, x, y, z
2. Expressions 2*x, x + y +3, 3*y +4
3. Equations 2*x =10 , x + y+ 3 = 3*y + 4

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Substitution

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Substitution
Substitution is when you learn the value of an unknown number
and replace the letter with a number
Take x + 3
If x = 5 then x + 3 = 8
If x= 7 the x + 3 = 10
If x = 0 then x + 3 = 3
Take 2 * y ( 2 times y) usually written as 2y
If y = 5 then 2 * y= 2y = 10
If y = 7 then 2 * y = 2y = 14
If y = 0 then 2 * y = 2y =0
Take z * z called z squared usually written as z^2
If z= 5 then z^2 = 25
If z = 7 then z^2 =49
If z = 0 then z^2 =0

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Substitution - Negative Numbers
Think of negative numbers as losses
and positive numbers as gains
Take x + 3
If x = - 5 then x + 3 = -2
Lose 5 and win 3 = -2
If x= - 7 the x + 3 = - 4
Take 2 * y ( 2 times y) usually written as 2y
If y = - 5 then 2 * y= 2y = -10
Lose 5 two times = - 10
If y = - 7 then 2 * y = 2y = -14
Take z * z called z squared usually written as z^2
If z= - 5 then z^2 = +25
Trickiest Don’t lose 5 five times = 25
If z = - 7 then z^2 = +49

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More Substitution
Take z * z called z squared usually written as z^2
If z= 5 then z^2 = +25
If z = 7 then z^2 = +49
The opposite of squaring is called square root (sqrt or )
If x = 25 then sqrt (x)= 5
If x = 49 then sort (x) = 7
More square roots
Note: Google will calculate squares (8^2) and square roots (sqrt 64)

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More Substitution
Take y * y * y called y cubed usually written as y^3
If y = 2 then y^3 = 2 * 2 *2 = 8
If y= 3 then y^3 = 27
If y=4 then y^3 = 4*4*4 = 64
The opposite of cubing is called cube root or
If x = 27 then the cube root (x)= 3
If x = 64 then the cube root (x) = 4
3
Higher powers
x * x * x * x = x^4
If x = 2 then x^4 = 2*2*2*2 =16
If x = 3 then x^4 = 3*3*3*3 = 81
x * x * x * x * x= x^5
If x =2 then x^5 = 2*2*2*2*2 = 32
Note: Google will calculate cubes (8^3), cube roots (64), and higher powers (5 ^4)

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Bigger Substitutions
If x = 2 then x^2 + 3x + 4 = 2*2 + 3*2 +4 =14
If x = 5 then x^2 + 3x + 4 = 5*5 +3*5 +4 = 44
If x = 3 and y =4 then x + y =7
If x = 3 and y =4 then x*y =xy = 12
If x = 3 and y =4 then y^2 - x^2 = 16 - 9 =7
If x = 3 and y =4 then x^3 +y =3*3*3 +4 =27 + 4 =31
If x = 3 and y =4 then x^2 +5xy + 2y = 9 +5*3*4 + 2*4= 77

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Combining

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Combining Terms
x + x = 2x
x + y = x+ y
2x + 3x = 5x
X^2 + x = x^2 +x
5X^2 + 2x^2 +1 + 3x +8x +9 = 7x^2 + 11x + 10
x*x = x^2
x*(x^2) = x* (x*x) =x^3
x *y = xy
(x^2)*y *x *y = x^3 * y^2
All of these combinations can be done on https://www.wolframalpha.com/

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Solving

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Solving
Solving is the opposite of substitution
These equations can be solved on https://www.wolframalpha.com/
One equation in one unknown variable
Solve x + 3 = 9 is the same as find x if x + 3 = 9 . Answer is x=6
Solve x^2 = 25. Answer = 5
Solve sqrt( x) = 4. Answer = 16
Solve 2x + 5 = 9 . Subtract 5 from both sides of equation.
It becomes: Solve 2x = 4 . Answer x = 2
Solve 3x + 10 = 5x + 4. Subtract 4 from both sides of equation.
It becomes: Solve 3x + 6 = 5x. Subtract 3x from both sides.
It becomes 6 = 2x. Answer = 3
Check answer by substituting 3 for x in original equation
3*3 + 10 = 5*3 + 4 = 19

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Solving
Solving is the opposite of substitution
Two equation in two unknown variables
Solve for x and y
x + y = 10
x - y = 4
Add the two equations
2x =14 and x = 7
Substitute in first equation
7 + y = 10 and y = 3
Check in second equation
7 - 3 = 4
Two equations in two unknowns can be solved on https://www.wolframalpha.com/

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Graphing

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Graphing
Graphing combines algebra and geometry
Most important advance in mathematics
Unified all previous mathematics
Invented in 1600’s by Descartes
Descartes also revolutionized philosophy
“I think therefore I am” is the basis of reality

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Graphing Points
Basic idea: consider 2 unknowns: x and y
Put x on a number line (x axis) from left to right
Put y on a number line(y axis) from bottom to top
Number lines meet at x=0 and y=0
Let x = 2 and y= 3 for example
Plot (point x = 2 and y = 3 )
“Plot” is another word for graphing
x=2 and y=3 or (2,3)
Plotting can be done on https://www.wolframalpha.com/
x axis
y axis

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Graphing Points
Plot point x=-2 and y = 4 (-2, 4)
Note both number lines (axis) goes from negative to positives
This is only a partial pictures
Plotting can be done on https://www.wolframalpha.com/
x axis
y axis
____________
|
|
x = -2 and y=4 (-2, 4)

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Graphing Points
Plot point (-3, -1) or x=-3 and y =- 1
Plotting can be done on https://www.wolframalpha.com/
x axis
y axis
(-3, -1)

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Graphing Lines
Key idea : An equation in two variables (x,y) can give many points to plot
For example : x + y = 6 gives x =0, y= 6 or (0,6) and (1,5), (2,4), (3,3),(4,2), (5,1), (6,0) and negative numbers (7, -1), (-2, 8)
and many more including fractions like x = 2 1/2 and y = 3 1/2 (2.5, 3.5). Decimals $2.50 + $3.50 = $6
Putting them all together on a graph gives the straight line below
Plot x +y = 6
Plotting can be done on https://www.wolframalpha.com/
x axis
y axis
(0,6)
(6,0)
(3,3)
(-1,7)
(5,1)
(1,5)

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Graphing Two Lines
Plotting can be done on https://www.wolframalpha.com/
It is possible to graph two equations at the same time.
The points where the graphs intersect are solutions to the equations
You can solve equations by algebra or geometry
For example, Plot x + y =10 and x - y = 4
The two lines intersect at x=7 and y =3 (7,3)
Note that the picture below starts at x=6 and y=2

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Graphing Useful Equations
Plotting can be done on https://www.wolframalpha.com/
d axis
Distance = Speed * Time or d=s*t
If you run at s = 5ft/sec for t = 4 sec, your distance traveled = 20 ft
If your speed is always 5ft/sec then d = 5 * t
This can be plotted as a straight line
t axis

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Graphing with Squares
Plotting can be done on https://www.wolframalpha.com/
x axis
y axis
The equation x^2 =y has a square
It can be graphed but is not a line (called parabola)
Some points are x= 0, y =0 (0,0) and x =2, y =4 (2,4) also (1,1),(3,9), (4, 16)
Also x = -2 and y =4 (-2,4) since -2 * -2 = 4 and (-1,1),(-3,9),(-4,16)
Plot y = x^2

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Graphing with Squares
Plotting can be done on https://www.wolframalpha.com/
x axis
y axis
The equations y = x^2 (parabola) and y = x (line) can be graphed
Plot y = x^2 and y = x
The parabola and the line meet in two points
x = 0, y = 0 (0,0) and x=1, y=1 (1,1)

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Graphing with Squares
Plotting can be done on https://www.wolframalpha.com/
x axis
y axis
The equation x^2 + y^2 =25
x = 3 and y = 4 (3,4) Check 3^2 + 4^2 9 + 16 =25
x = 0 and y = 5 (0,5) Check 0^2 + 5^2 =25
Plot x^2 + y^2 = 25. Result is a circle
(3,4)
(0,5)

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Useful Equation with Squares (Gravity)
Plotting can be done on https://www.wolframalpha.com/
t axis
d axis
Objects fall down from gravity with distance = 16*time^2 or d= 16t^2
If you jump off a building 2000 ft high after 1 sec , you fall 16 *1 =16 ft,
after 2 sec you fall 16 *4= 64 ft Faster after 3 sec you fall 16 *9 =144 ft
after 10 sec you fall 16 *100 =1,600 ft. Plotting d= 2000 - 16 * t^2

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Useful Equation with Squares (Gravity)
Plotting can be done on https://www.wolframalpha.com/
t axis
d axis
Objects fall down from gravity with distance = 16*time^2 or d= 16t^2
if you start on the ground and throw a ball with speed = s then
the distance would d = s * t without gravity. With gravity it becomes d = s *t - 16 * t^2
Throw ball up with speed 80 ft per second, d = 80 *t - 16*t^2 = 80t-16t^2
Plotting this shows how high the ball will go and when it hits the ground

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Graphing with Cubic Powers
Plotting can be done on https://www.wolframalpha.com/
x axis
y axis
Plot x^3 - x +2 =y gives a curve that has two bends
Plotting y=2 gives a straight line
The intersection of the line with the curve give three points (-1, 2), (0,2) and (1,2
Check x=1 and y=2 : Substitute in x^3 - x +2 =y gives 1^3 -1 +2 = 1-1 +2 = 2
(-1,2)
(0,2)
(+1,2)

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Graphing with Three Variables
Plotting can be done on https://www.wolframalpha.com/
It is possible to graph equations with three variables x,y,z
The z axis is perpendicular to the x and y axis
If the x and y axis are on a sheet of paper, the z axis goes up and down
For example, plotting x = 1 y = 3 and z =5 (1,3,5). plot3d point (1,3,5)
y axis
x axis
z axis

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Graphing with Three Variables
Plotting can be done on https://www.wolframalpha.com/
It is possible to graph equations with three variables x,y,z
For example, plot x^2 +y^2 +z^2 =9
Some solutions are (0,0,3) (1,2,2), (-2,-2, +1)