ALGEBRA

Counting  Arithmetic  Fractions  Graphing  Algebra  Geometry

Variables and Terms

Linear Equations: y=f(x)

Quadratic Equations: y=f(x)²

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ALGEBRA

Variables and Terms: Zevy Like Terms

Start with Combine Like Terms, first linear terms then exponential terms, each tool increasing in difficulty. An Invisible Math graphic and Inventory Activity are included.


Linear Equations y = f(x)

Examples showing how to use reciprocals and distribute to solve for X get students started in this section. The equations increase in difficulty requiring 2-step solutions, then distributions, fractional and bivariate equations. All equations in this section have detailed step-by-step solutions that animate to facilitate teaching and learning.


Quadratic Equations y = f(x²)

Standard Equation is y=x² when A=1, B and C = 0.

Central Parabola and Transitions

The Quadratics section Features a tool generating equations in Standard format. In Easy mode A=1 and factoring requires finding numbers which add to B and multiply to C. In Hard mode when A>1, we use "Slide and Divide" to first simplify the A term, then Factor, Divide and Solve for X. Detailed solutions are generated for all equations. Students re-visit this tool using the Quadratic Equation to solve for X at the end of Level/3.

Examples showing Standard Parabolas with Transitions, Roots, Symmetry, Vertex, Factors and Solutions are great teaching and learning aids. Drop Rock examples adds real world context to the mathematics. Students learn about completing the square and solving any Quadratic Equation.

Play Video Part 1

Play Video Part 2


Use the Reciprocal to Solve for X

25 41X = 4 52 * 52 *

Rewrite Equation

1010 X = 202

Simplify

X = 10

reset

Invisible Math

• Every term is positive, unless it has a negative sign.

3 = +3


• Multiplication is implied for terms next to each other.

abc = a•b•c = (a)(b)(c)


• Every term has a 1 in front of it.

X = 1X


• Every term can be written as a Fraction

3 = 3/1     X = X/1


• Every whole number can be written as a decimal

5 = 5.0 = 5.00 ...


• Every term has a default Exponent of 1

9 = 91     X = X1




Expand Terms

Reset

Exponential Terms : Multiply & Divide

X3•X8=X(3+8)=X11 X•X•X  •  X•X•X•X•X•X•X•X = X•X•X•X•X•X•X•X•X•X•X X3÷X8=X(3 - 8)=X-5 X • X • X X • X • X • X • X • X • X • X = 1 X • X • X • X • X X-N=1XN

Attempt/Correct
X:0/0   XY:0/0
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LINEAR TERMS
Xterm  X,Yterm
  New Answer

How to Simplify

X   Y

Make Your Own Inventory Example

Think of an example that makes use of terms similar to the Zevy example, cosmetics, clothing, shoes...

Code a sample inventory and a sample of new activity, write the current inventory and the new inventory.

Zevy Motor Company Sells Cars

Z : is a Zorvette       M : is a Zamero

Cars may have one or both upgrades

(8) → V8 Engine Performance Package

(L) → Premium Leather and Sound

A sample inventory item is shown below

Z(8)(L) → Zorvette with Performance and Leather

In this activity students will:

• Define and Code 8 Inventory Items

• Apply Initial Quantity to Inventory

• Code and Apply One Days Activity

• Record Final Inventories and Check Answers

Continue Online

Print Activity Sheet


Show Inventory

Reset

Write 8 Inventory Codes

codes inventory description initialone daysfinal quantityactivityinventory Zorvette Standard Zorvette V8 Engine Zorvette Leather Interior Zorvette V8 and Leather Zamero Standard Zamero V8 Engine Zamero Leather Interior Zamero V8 and Leather Z Z(8) Z(L) Z(8)(L) M M(8) M(L) M(8)(L) 413 951 3-1-3 -1 4 33 65

Apply Initial Inventory 4 Zorvettes 1 Zorvette with Leather 3 Zorvette with V8 and Leather 9 Zameros 5 Zameros with V8 Engine 1 Zamero with V8 and Leather Apply One Days Activity Purchased: 3 Zorvette with V8 Engine Sold: 3 Zorvette w/V8 and Leather 3 Zamero 1 Zamero w/V8 and Leather

Show Final Inventory

Print Answer Key

To solve Any Standard Quadratic Equation

y = Ax²+ Bx + C

We may use a technique called Completing the Square. First move all the numbers to the right of the equal sign, by subtracting C and dividing by A → See below

y = x²+   x +    = -  abac b²4a² + b²4a²

Recall Factoring when A=1, finding terms that add to B and Multiply to C. Well here we will create the C term and add it to both sides of the equation. We need the number when added to itself = b/a. The answer is one half b/a, or b/2a.Complete Square

ab =½ab+½ ab and½ ab *½ab= b²4a²

Perfect Square Equation

(x +    )² = -b2a ca +b²4a² √

Take Square Root of Above then Solve for X Below


x +b 2a= -ca +b²4a² √

Now X is on the left side of the equation and is to the first power. The radical part of the quadratic equation is in the number.

Show ALL Steps to Complete the Square

Full Solution

Step-by-Step Solution

Ax² + Bx + C = 0

Divide by a Subtract c/a both sides Complete the Square How to Complete the Square Factor this side Combine these terms Take the square root of the equation Simplify Subtract b/2a from both sides ax² + bx + c = 0  x² + bx + c = 0      a    a  x² + bx     = -c      a         a x² + bx + b²  = b²  - c     a    4a²   4a²   a (x+b2a)² = b² - 4ac4a²( )4a4a x + b  =  b² - 4ac    2a       2a x =-b ± b² - 4ac2a

!! Put this Equation in your Notes !!

Use Quadratic Equation to Solve for X


Roots are where Y=0

It is often important to know the X value when Y=0. The Drop Rock Activity is a real world example.   Also, if we know the roots we can determine the Axis of Symmetry because it is the midpoint of the roots and if we know the AOS we may substitute that value for X and find the maximum or minimum value of the parabola called Vertex.

Before continuing with the Roots Examples and Worksheet, Please see the Factoring Equations tutorial

Roots Worksheet / Key

Factoring Equations


• ROOTS ↙ ↘ • Two Roots↑ Y = 0 Y-Axis → • • ↑ X-axis One Root• • Vertex Axis of Symmetry → ↗ No Roots

x²-3x-4=0

Show Solution

(x-4) (x+1)=0

(x-4)=0   or   (x+1)=0

X = 4 or X = -1

Y-Axis X-Axis ↖ ↗ Roots X = ← Vertex ← AoS = 3/2 Data TableX     Y -1012340-4-6-6-40 (-1,0) (4,0) ← Midpoint between Roots (-1 + 4) 2 To get Vertex use 3/2 for X and solve for Y Y = -25/4 (3/2,-25/4) • • ••••••••

Roots when Y=0

Reset


x²+2x+5=0

Show Solution

No numbers add to 2 and multiply to 5

Never = 0

Data TableX    Y-3-2-10185458 •• •• •AoS: X = -1 Vertex: (-1,4) X-AxisY-Axis

Roots when Y=0

Reset


x²-8x+16=0

Show Solution

(x-4) (x-4)=0

So X must equal 4


X-AxisY-Axis • ↗Root(4,0)↖ Vertex(4,0) Data TableX     Y 23456 41014 Axis of Symmetry → X = 4

Roots when Y=0

Reset


How long for a rock dropped off a building to hit ground?

(Hint: The Answer depends 3 things)

Discuss Ideas then Continue

1. Gravitational Force → How fast the rock falls, on Earth Gravity = 9.8 Meters / Second²

2. Initial Velocity → Just drop the Rock or throw it

3. Height of Building → Ground=0, Height is positive

Each point above is directly related to one of the coefficients in the Standard Quadratic Equation

Ax² + Bx + C

Draw a Picture and Set Up Equation

Perfect Squares have 1 Factor

(x + d)²(x + d) (x + d) x² + 2dx + d²

Difference of 2 Perfect Squares


One of the easiest equations to factor and solve is the difference of two perfect squares such as:

x² - 144 = 0A and C are Perfect Squares and B = 0(x + 12) (x - 12) = 0x² - 12x + 12x - 144 = 0x = 12   or   x = -12

Graphical Representation

Solve for X Worksheet / Key

Y-AxisX-Axis X² - 1 = 0 (X + 1) (X - 1) = 0 Roots: X=1 or X=-1 X² - 4 = 0 (X + 2) (X - 2) = 0 Roots: X=2 or X=-2 X² - 9 = 0 (X + 3) (X - 3) = 0 Roots: X=3 or X=-3 Axis of Symmetry : Y = 0 Roots : (0, √C )   (0, -√C ) Vertex : (0, C)

Algebraic Equations

In the Drop Rock Example the building was chosen because it is exactly 490 meters tall and the equation is a Perfect Square. When changes are made to A, B or C, we need different techniques to solve for X.

Solve Any Quadratic Equation

Perfect Square Equations

Transition A: Drop Rock on Moon where Gravity = 1.6 m/s².

Transition B: Throw Rock so Initial Velocity is not equal 0.

Transition C: Use a different building with different height.

Transitions Worksheet / Key


y = Ax² + Bx + C

X = Time → How long it takes to hit ground → Solve for X

FirstSecondThird

←-- Positive ------------- Negative --→ rock ☁Ground = 0 ←---- + 4 9 0   m e t e r s -------→

Let Y=0 the ground and C=Height of the Building. We may choose any building to, "Drop the Rock", so let's use the Joint Venture TV Building in Bithlo Florida, which is 490 meters tall → C = +490

B=0   Let's just drop the rock, not throw it

A = -4.9 → A is Negative because gravity pulls down ↓ recall from physical science:
Distance=(½)(Acceleration)(Time²) and Gravity or acceleration is 9.8 and half of that is 4.9

Plug in Numbers and Solve for X (Time)

0 = -4.9 X² + 0X + 490

4.9X² = 490

X² = 100

X = 10 Seconds

Transitions of A, B and C

Reset

Changing A requires a different gravity. How long would it take for the Rock to hit the Ground on the Moon where Gravity = 1.6m/s² ? Show SolutionReset

-----------------------------→←---- 4 9 0   m e t e r s --------→rock ☁ Ground Reset 0   =  (-0.8) X²   +   (0) X   +  490 ↑GroundY=0 ↑(½)GravityA = -0.8 ↑Init VelocityB = 0 ↖HeightC = 490 Use Quadratic Equation to solve for X X = X = = X = 0 (0)² - (4)(-0.8)(490) ± √ (2)(-0.8) 1568 ± √ -1.6 -39.6 -1.6 24.75 Seconds

Back to Transitions

What if B is not equal to 0

If we threw the rock rather than just drop the rock, there would be initial velocity and B would not equal 0.

How long would it take for the Rock to hit the ground, if we threw it up 20meters/second?   Show Solution

-----------------------------→←---- 4 9 0   m e t e r s --------→rock ☁ Ground Reset 0   =  (-4.9) X²   +  (20) X   +  490 ↑GroundY=0 ↑(½)GravityA = -4.9 ↑Init VelocityB = 20 ↖HeightC = 490 Use Quadratic Equation to solve for X X = X = = X = = -20 (-20)² - (4)(-4.9)(490) ± √ (2)(-4.9) -20 10004 ± √ -9.8 -20 ± 100 -9.8 -120 -9.8 12.24 Seconds

Back to Transitions

What if C changes

What if used a different building that was 700 meters tall, then C would = 700. How long would it take for the Rock to hit the ground? Show Solution

-----------------------------→←---- 7 0 0   m e t e r s --------→rock ☁ Ground Reset 0   =  (-4.9) X²   +   (0) X   +  700 ↑GroundY=0 ↑(½)GravityA = -4.9 ↑Init VelocityB = 0 ↖HeightC = 700 Use Quadratic Equation to solve for X X = X = = X = 0 (0)² - (4)(-4.9)(700) ± √ (2)(-4.9) 13720 ± √ -9.8 -117.1 -9.8 11.95 Seconds

Back to Transitions

Reset

Slide and Divide is a common technique used to factor quadratic equations when A is not equal to 1:

First Slide: multiply C by A and make A=1

Ax² + Bx + C x² + Bx + AC

Next Factor: Find two numbers that add to B and multiply to AC

(x + d) (x + e)

B = d + e     AC = de

Then Divide: each factor by the original A term

(x +   ) (x +   ) deAA

!! SIMPLIFY !!


Multiply by Zero Rule

(x + b) (x + d) = 0one of these two termsmust be equal to zero (x + b) = 0   or   (x + d) = 0 thereforex = -b   or   x = -d

Factoring is one method of solving equations. It breaks the Standard Quadratic Equation into two linear terms, which are easy to solve, see below.

F.O.I.L. → First Outer Inner Last

First Outer+Inner Last

x² + Bx + C(x + b) (x + d) A=1B=d+bC=bd

When A=1
Find 2 Numbers that Add to B and Multiply to C


Factor when A != 1

The Last Step


F.O.I.L. → First Outer Inner Last

First Outer+Inner Last

Ax² + Bx + C(ax + b) (cx + d) A=acB=ad+bcC=bd

When A != 1 we use a technique commonly known as Slide and Divide to make A=1 and then we factor.

Factor when A = 1

Slide and Divide



Put these in your Notes

!! or remember them !!

♦ Parabolas are not exactly "U" shaped. Parabolas continue to get wider, in fact parabolas continue to infinity in both X and Y directions.

♦ Parabolas may be wide or narrow.

♦ Parabolas may open up or down.

♦ Parabolas have a maximum or minimum called Vertex.

♦ Parabolas are symmetrical with an Axis of Symmetry going through the Vertex.

♦ Parabolas may have 0, 1 or 2 Roots, when they cross the X-Axis, at Y=0.

♦ Any parabola may be graphed using the Standard Quadratic Equation:   y = ax² + bx + c

♦ The simplest parabola is the Central Parabola Y=X² when a=1, b=0, c=0.


Quadratic Equations are Parabolas

Example Parabolas

Hover A, B and C below to see Transitions

y = Ax² + Bx + C

Y-AxisX-Axis Central ParabolaY = X² A < 0Flips (-) 0 < A < 1Narrow A > 1Wide C > 0Move Vertex Up Move Vertex DownC < 0 ↑ ↓ C+ C- Move Vertex straight Up orstraight Down exactly C Units Changing B causes the Axis ofSymmetry and Vertex to move ← Right / Left →Axis of Symmetry is the line: X = -B / 2A Use -b/2a for Xand solve for Yto get Vertex.

Terms and Definitions

The Central Parabola


The Central Parabola

 Y = X²

(A = 1   B = 0   C = 0)

VertexAxis of SymmetryPointsRoots

Vertex ↗ • 0 •••••••• 1234 5678 9-3-2-1 123Data Table XY -39-24 -1100 1124 3900 Roots are where Y=0and GRAPH touchesthe X-Axis To find Roots for anyParabola - solve for Xwhen Y = 0.0 = Ax² + Bx + C Parabola has 1 Rootcalled a Double Root •↖Root Y-AxisX-Axis Vertex is maximumor minimum point Axis of Symmetry X = -B/2A ← Parabolas reflect acrossthe Axis of Symmetrylike it is a mirror↖ The Vertex is on the Axis of SymmetryAxis of Symmetry → Vertex↗ •

Standard Parabola

Print Worksheet


Solve for X

aX² + bX + C = 0

This tool operates in 2 modes

Easy Mode: A=1, students must find factors that add to b and multiply to c to solve for X.How to Factor

Hard Mode: A>1, a "slide and divide" technique used to solve for X.

Attempt / Correct
E:0/0   H:0/0
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A=1  A≠1
New     Answer

Parabolas and Equations

Drop Rock Activity

Factors and Roots

Transitions of A, B and C

Overview Video

All Numbers Below

Variables Stand For Things

When Numbers and Variables combine into Terms there is some Invisible Math to know about.

Numbers define how much there is or how many there are. Variables define what it is.

In the Zevy Inventory Example, variables stand for different kinds of cars they have and numbers stand for how many they have. Inventory means keeping track of how many of each car they have.


Numeric Terms Are Quantities

√36015783.141593(2+4)6.02x1023

Above is the Number 5 but 5 What?

Variables Represent Things

Click Expressions to Simplify

3X - 9Y + 4X² - 2X - 3Y + Y³

X - 12Y + 4X² + Y³


3X - 9 + 4X + 12 - 3 - 7X

0


3X - 9Y + 3Y³ + 8Y - 3X - 2Y³

-Y + Y³


RESET

Easy mode:
generates 3 terms with coefficients between 1 and 5

Medium mode:
generates 5 terms with coefficients between -6 and 6

Hard mode:
generates 7 terms with coefficients between -9 and 9

Attempt / Correct
3: 0/0   5: 0/0   7: 0/0
hide reset

  EXPONENTIAL TERMS
  3 Term   5 Term  7 Term
  NewAnswer

Invisible Math Zevy Inventory Activity

 

DISTRIBUTION: is different for
Addition and Multiplication

;'; 3 ( 5x + 2 ) = 15x + 6 3 ( 5x • 2 ) = 3•5x•2 = 60x

Clear Answers

Show All Answers

2(2x+3+5)=4x+16
2(2x•3•5)=60x

3x(1+1+1x)=3x²+6x
3x(1•1•1x)=3x²

5x(2+0+5)=35x
5x(2•0•5)=0

3(2x+3+5x)=21X+9
3(2x•3•5x)=33x

Identity Fractions = 1

Numerator = Denominator

5/5   3/3   .1/.1   2X/2X   Y²/Y²

Any Fraction • Reciprocal = Identity

A/B • B/A = AB/BA = 1

Use Reciprocals to Solve

simplify reset 57 3535 75 75 75 75 ()() X = 1X= X =

Flip the Fraction

The Product Equals 1

2662← Reciprocals →

when reciprocals are multiplied the product
is always 1 because numerators
and denominators equal each other

2662X=1212= 1

Reciprocals and Algebra

Fractional Equations

Bivariate Equations

Slope / Intercept in Graphing Section in Hard Mode


Linear Algebra features a comprehensive set of Tools, Examples and Activities

Overview Video

The Equality Property states any Operation performed on one side of an equal sign must be performed on the other side.

(same operations) ← = → (same operations)

Review Properties Below

Distributive Identity Equality

 Attempt / Correct
E:0/0   M:0/0   H:0/0
resethide

ax=b  ax+b=c  d(ax+b)=c
Easy Med  Hard
NewAnswer

Arithmetic Properties

Advanced Equations

Overview Video

X =   /

Step-by-Step Solution

Full Solution

Easy Mode: Coefficients are 1, 2 or 3 and Constants are positive.

Hard Mode: Coefficients between -5 and 5 but ≠ 0, Constants may be Negative.

One way to solve equations like these is to:
1. Combine the equations and eliminate one variable
2. Solve for the remaining variable
3. Substitute the value into one equation and solve for
the other variable, consider these equations:

2x + y = 10

8x + 2y = 4

To eliminate the X variable, multiply the first equation by -2, then add the equations.

-4x - 2y = -20

8x + 2y = 4

4x = -16;
X = -4;

Substitute -4 into the first equation for X and solve for Y

(2)(-4) + Y = 10

-8 + Y = 10;
Y = 18;

The point (-4,18) on the graph is where these 2 lines intersect

Attempt / Correct
E:0/0   H:0/0
resethide

Easy   Hard
 NewAnswer

How to Solve for X and Y using Elimination

a1X + b1Y = C1
a2X + b2Y = C2

X =     Y =


dn/dd ( an/ad X + b) = c

Solve for X

1: Distribute the Fraction

2: Combine the Constants

3: Simplify the Coefficient

4: Simplify Answer, if Necessary

Attempt / Correct
0/0
hidereset

NewAnswer

 

X = /

Full Solution

Step-by-Step Solution

; '; ( X+ ) = Distribution first X = Combine constants Simplify constants () X = = Simplify the X term () () Rewrite equation X = =