Parabola
Definition: all points whose perpendicular distances from them to the directrix equals the distances from them to the focus.
We'll use these properties from the … more
CONIC SECTIONS APPLICATIONS
The parabola has an electromagnetic signal reflection property. Four signals are shown in green and blue. These signals are shown with arrows on both ends to indicate … more
Circle
In the following figure we are passing a plane (red cross hatch) from the front to the back of the cone. And we are doing this parallel to the base … more
Hyperbola
The hyperbola is created when the intersecting plane is not parallel to an edge of the cone. The following diagram illustrates this intersection, the hyperbola. Although the diagram … more
Introduction to Conic Sections
Conic Sections
A conic section or more simply, conic, is the intersection of a plane and a right circular conical surface. This intersection is a curve (or curves) … more
Parabolas
The intersection of a plane parallel to the edge of a nappe of a cone is called a parabola.
The red dashed lines are parallel, one is along the edge of the cone and the other is in the … more
Distance, Rate and Time
Distance is a measurement that answers “how far?”
Time measures the length of time to travel that distance and answers “how long?”
Rate is a … more
Angles
Definition: An angle is the shape created in a plane when two rays share the same endpoint. Below we see the plane P that contains two rays, a and b, that share the same endpoint V.& … more
Cartesian Coordinate System in Geometry
Coordinate Systems
Suppose we have these points in the plane:
We can see that they are spatially related. First of all they are all in the same … more
Derivation of the Ellipse Equation
Study the following diagram for an ellipse:
Recall the definition of an ellipse requires that d₁ + d₂ = 2a.
Let (x,y) be the coordinates for the point P. In … more
Solving Conic Intersections
1) Plot:
x2 -6x + y2 +4y + 4 = 0
Complete the square to get standard form. … more
This figure shows a plane slicing a right circular cone at an angle, β, greater than 900 measured from the the axis of the cone.
When β starts at 900 the plane slices the cone parallel … more
Definition: The ability for a student in 1st grade to recognize the proper sequence of base 10 numbers.
Basics:
What is base 10 math?
Answer: Numbers 0-9.
What are the order … more
Decimals
Decimals are numbers written using 10 as the base. Each position of the decimal number represents a power of 10 and the decimal point, the period '.', is used to … more
Question:
Can anyone take the first and second derivatives of y=3x2/3 – x2 ?
Answer:
1st and 2nd derivatives by repeated application of the exponent rule
& … more
Equations of the Line
For pre-algebra students, focus on the slope intercept form; the others will be addressed in Algebra 1 and Algebra 2.
… more
Exploring Circles
This circle explanation is the basis for teaching advanced 6th grade Geometry concepts including Radius, Diameter, pi (π), Perimeter, and Circumference. … more
Exploring Circles - Explanation of radians and π
Right click on the image below and save it to your desktop, then print off the circle teaching resource for your students.
& … more
A mathematical theorem is a statement that can be proved using deductive reasoning.
This theorem is stated within a mathematical model that has
& … more
Line Segment, Ray, and Line Definitions
Line Segment: the shortest path between two points Q and R. We visualize this segment as a trace without bends, that is drawn straight between Q and R. … more
Two triangles are similar if:
1) the corresponding angles of the first in the second are the same, or
2) the ratios of corresponding sides of the two triangles are equal
… more
Properties of Integers
The set of Integers obey certain properties that are used heavily in algebraic manipulations. We'll investigate each now.
A. Well Ordered Set
Each … more
Operations with Integers:
Remembering the debit/credit analogy for negative and positive numbers, it would make sense that debits should add to a larger debit, and credits should add to a larger … more
Introduction to IntegersThe Natural numbers are those numbers with which we first became aware, "I have 1 marble, 2 jacks, 9 pennies..." We then had to add the place holding number "0& … more
Line Properties
Line property 1: two distinct lines in a plane either intersect in exactly one point in the plane or do not intersect and are parallel.
Line property 2: through any given point … more
Line Property Orthogonal Lines
Line property 4: if two lines are parallel (notation: L1 || L2), then from a given point on one line there exists a shortest line segment whose other endpoint lies … more
Money
Teaching your child about money reinforces the basic number facts of the base 10 number system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. At first, for younger … more
Multiplication
The signs used for multiplication are x and ·
2 x 3 means to add 2 to itself 3 times: 2 + 2 + 2 = 6
-3 · 2 means to add -3 to itself 2 times: -3 + -3 = -6
4 x -2 … more
Point Definition
Point: Pythagoras defined a point as "unity with position." For the purposes of planar geometry, a point is location in the plane. More practically, a point is a dot on … more
Operations with Integers:
Remembering the debit/credit analogy for negative and positive numbers, it would make sense that debits should add to a larger debit, and credits should add to a larger … more
Pythagorean Theorem
In a right-angle triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.
a² + b² = c²
Various … more
Rational Numbers
A rational number is any number that can be represented by the ratio of two integers.
Examples:
, , ,
The horizontal bar is … more
Exploring Circle Rotation
Question: Two circles(r1, r2) of radius 10, and 25 are touching each other and spins without slippage. when r1 is spinning at 50 rpm, at what rpm is r2 … more
Areas of Circles, Triangles and Rectangles
(Review Measurement before proceeding.)
The area of a rectangle, shown below, is its length multiplied by its width. For example if its … more
Exterior and Interior Angle Relationships
Every exterior angle is equal to the sum of the opposite interior angles.
Exterior angles: Any angle that is adjacent to an interior angle of a … more