Monday, September 17, 2018

Week of September 19,2018


Properties of Light

Video Clip Hubble Decodes Colors of the Galaxy

Watch video Crash Course Light

    • TED ED Light waves, visible and invisible

§  ThougtCo. Article: What is Blackbody Radiation?”


Foundations of Algebra

Here is a practice link for solving slope (word problems)

Monday, September 10, 2018

Week of September 10,2018


Complete your study guide and Newton's law lab.

Foundation of Algebra

Coordinate graphing sounds very dramatic but it is actually just a visual method for showing relationships between numbers. The relationships are shown on a coordinate grid. A coordinate grid has two perpendicular lines, or axes, labeled like number lines. The horizontal axis is called the x-axis. The vertical axis is called the y-axis. The point where the x-axis and y-axis intersect is called the origin.

The numbers on a coordinate grid are used to locate points. Each point can be identified by an ordered pair of numbers; that is, a number on the x-axis called an x-coordinate, and a number on the y-axis called a y-coordinate. Ordered pairs are written in parentheses (x-coordinate, y-coordinate). The origin is located at (0,0). Note that there is no space after the comma.
The location of (2,5) is shown on the coordinate grid below. The x-coordinate is 2. The y-coordinate is 5. To locate (2,5), move 2 units to the right on the x-axis and 5 units up on the y-axis.

The order in which you write x- and y-coordinates in an ordered pair is very important. The x-coordinate always comes first, followed by the y-coordinate. As you can see in the coordinate grid below, the ordered pairs (3,4) and (4,3) refer to two different points!

Wednesday, September 5, 2018

Week of September 4, 2018


Newton's Laws of Motion

The laws that govern motion eluded scientists, philosophers and other great thinkers until the 17th century. Then, in the 1680s, Isaac Newton proposed three laws that explained how inertia, acceleration and reaction influence the motion of objects. Along with Newton’s law of gravitation, these laws formed the basis of classical physics.

The Law of Inertia

Stones will remain at rest until a force causes them to move.
Newton's first law of motion, also known as the law of inertia, states that objects neither move nor cease to move on their own. An object only changes its state of motion when acted on by an outside force. A ball at rest, for example, will remain at rest until you push it. It will then roll until friction from the ground and the air brings it to a halt.

The Law of Acceleration

A horse accelerates the cart's movement. The cart's mass slows the horse.
Newton's second law explains how external forces affect the velocity of an object in motion. It states that acceleration of an object is directly proportional to the force that causes it, and inversely proportional to the object's mass. In practical terms, this means that it takes more force to move a heavy object than a light one.
Consider a horse and cart. The amount of force the horse can apply determines the cart's speed. The horse could move faster with a smaller, lighter cart in tow, but its maximum speed is limited by the weight of a heavier cart.
In physics, deceleration counts as acceleration. Thus, a force acting in the opposite direction of a moving object causes an acceleration in that direction. For example, if a horse is pulling a cart uphill, gravity pulls the cart downward as the horse pulls upward. In other words, the force of gravity causes a negative acceleration in the horse's direction of motion.

The Law of Reaction

Pushing away with the pole generates the reaction of forward motion.
Newton's third law states that for every action in nature, there is an equal and opposite reaction. This law is demonstrated by the act of walking or running. As your feet exert force down and backward, you are propelled forward and upward. This is known as "ground reaction force."
This force is also observable in the motion of a gondola. As the driver presses his punting pole against the ground beneath the water's surface, he creates a mechanical system that propels the boat forward along the water's surface with a force equal to that which he applied to the ground.


Multiplying Polynomials

Multiplying polynomials involves applying the rules of exponents and the distributive property to simplify the product. This multiplication can also be illustrated with an area model and can be useful in modeling real world situations. Understanding polynomial products is an important step in factoring and solving algebraic equations.

The Product of a Monomial and a Polynomial

The distributive property can be used to multiply a polynomial by a monomial. Just remember that the monomial must be multiplied by each term in the polynomial. Consider the expression 2x(2x2 + 5x + 10).

This expression can be modeled with a sketch like the one below. This model is called an area model because the rectangular pieces represent the area created by the multiplication of the monomial and the polynomial.






We can see that the product of the width, 2x, and the length, 2x2 + 5x + 10, is the area of the entire shaded region. The area can be split into three smaller pieces. Each of those pieces has a width of 2x and a length represented by one of the terms of the polynomial.

Area models are a helpful way to visualize a multiplication problem. But we can also find the product of two polynomials algebraically, by applying the distributive property. Remember that the distributive property says that multiplying a sum by a number is the same as multiplying each addend by the number and then adding: a(b + c) = ab + ac. It doesn't matter how many terms there are: a(d) = ab ac + ad.

Let's try one:


5x3(4x2 + 3x + 7)

Distribute the monomial to each term of the polynomial

Add the products

Product of Two Binomials

Now let's explore multiplying two binomials. Once again, we can draw an area model to help us make sense of the process. We'll use each binomial as one of the dimensions of a rectangle, and their product as the area.

The model below shows (x + 4)(2x + 2):














Each binomial is expanded into individual variables and numbers, x + 4 along the top of the model and 2x + 2 along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, which is also the final product of multiplying the binomials. If we combine all the like terms, we can write the product, or area, as 2x+ 10x + 8.

We can also use algebra to determine the product of two binomials. Just multiply each term in one binomial by all the terms in the other binomial as shown below:

(x + 4)(2x + 2)

x(2x + 2) + 4(2+ 2)

Multiply each term in one binomial by each term in the other binomial

2x2 + 2+ 8+ 8

Rewrite to group like terms together

2x2 + 10+ 8

Combine like terms
2x2 + 10+ 8

Look back at the rectangle and see where each piece of 2x2 + 2x + 8x + 8 comes from. Can you see where we multiply x by 2x + 2, and where we get 2x2 from x(2x)?

Because multiplication is commutative, the terms can be multiplied in either order. The expression (2x + 2)( + 4) has the same product as ( + 4)(2x + 2), both having a product of 2x2 + 10+ 8. (Work it out and see.) The order in which we multiply binomials does not matter. What matters is that we multiply each term in one binomial by each term in the other binomial.

The last step in multiplying polynomials is to combine like terms. Remember that a polynomial is simplified only when there are no like terms remaining.

Monday, August 27, 2018

Week of August 27, 2018


Kepler's Three Laws

In the early 1600s, Johannes Kepler proposed three laws of planetary motion. Kepler was able to summarize the carefully collected data of his mentor - Tycho Brahe - with three statements that described the motion of planets in a sun-centered solar system. Kepler's efforts to explain the underlying reasons for such motions are no longer accepted; nonetheless, the actual laws themselves are still considered an accurate description of the motion of any planet and any satellite.
Kepler's three laws of planetary motion can be described as follows:
  • The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)
  • An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
  • The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)


The Law of Ellipses

Kepler's first law - sometimes referred to as the law of ellipses - explains that planets are orbiting the sun in a path described as an ellipse. An ellipse can easily be constructed using a pencil, two tacks, a string, a sheet of paper and a piece of cardboard. Tack the sheet of paper to the cardboard using the two tacks. Then tie the string into a loop and wrap the loop around the two tacks. Take your pencil and pull the string until the pencil and two tacks make a triangle (see diagram at the right). Then begin to trace out a path with the pencil, keeping the string wrapped tightly around the tacks. The resulting shape will be an ellipse. An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. The two other points (represented here by the tack locations) are known as the foci of the ellipse. The closer together that these points are, the more closely that the ellipse resembles the shape of a circle. In fact, a circle is the special case of an ellipse in which the two foci are at the same location. Kepler's first law is rather simple - all planets orbit the sun in a path that resembles an ellipse, with the sun being located at one of the foci of that ellipse.

The Law of Equal Areas

Kepler's second law - sometimes referred to as the law of equal areas - describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. For instance, if an imaginary line were drawn from the earth to the sun, then the area swept out by the line in every 31-day month would be the same. This is depicted in the diagram below. As can be observed in the diagram, the areas formed when the earth is closest to the sun can be approximated as a wide but short triangle; whereas the areas formed when the earth is farthest from the sun can be approximated as a narrow but long triangle. These areas are the same size. Since the base of these triangles are shortest when the earth is farthest from the sun, the earth would have to be moving more slowly in order for this imaginary area to be the same size as when the earth is closest to the sun.


The Law of Harmonies

Kepler's third law - sometimes referred to as the law of harmonies - compares the orbital period and radius of orbit of a planet to those of other planets. Unlike Kepler's first and second laws that describe the motion characteristics of a single planet, the third law makes a comparison between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets. As an illustration, consider the orbital period and average distance from sun (orbital radius) for Earth and mars as given in the table below.
Distance (m)
3.156 x 107 s
1.4957 x 1011
2.977 x 10-19
5.93 x 107 s
2.278 x 1011
2.975 x 10-19

Observe that the T2/Rratio is the same for Earth as it is for mars. In fact, if the same T2/Rratio is computed for the other planets, it can be found that this ratio is nearly the same value for all the planets (see table below). Amazingly, every planet has the same T2/R3 ratio.
Distance (au)
(NOTE: The average distance value is given in astronomical units where 1 a.u. is equal to the distance from the earth to the sun - 1.4957 x 1011 m. The orbital period is given in units of earth-years where 1 earth year is the time required for the earth to orbit the sun - 3.156 x 107 seconds. )

Kepler's third law provides an accurate description of the period and distance for a planet's orbits about the sun. Additionally, the same law that describes the T2/R3 ratio for the planets' orbits about the sun also accurately describes the T2/R3 ratio for any satellite (whether a moon or a man-made satellite) about any planet. There is something much deeper to be found in this T2/R3 ratio - something that must relate to basic fundamental principles of motion. In the next part of Lesson 4, these principles will be investigated as we draw a connection between the circular motion principles discussed in Lesson 1 and the motion of a satellite.

How did Newton Extend His Notion of Gravity to Explain Planetary Motion?

Newton's comparison of the acceleration of the moon to the acceleration of objects on earth allowed him to establish that the moon is held in a circular orbit by the force of gravity - a force that is inversely dependent upon the distance between the two objects' centers. Establishing gravity as the cause of the moon's orbit does not necessarily establish that gravity is the cause of the planet's orbits. How then did Newton provide credible evidence that the force of gravity is meets the centripetal force requirement for the elliptical motion of planets?
Recall from earlier in Lesson 3 that Johannes Kepler proposed three laws of planetary motion. His Law of Harmonies suggested that the ratio of the period of orbit squared (T2) to the mean radius of orbit cubed (R3) is the same value k for all the planets that orbit the sun. Known data for the orbiting planets suggested the following average ratio:
k = 2.97 x 10-19 s2/m3 = (T2)/(R3)
Newton was able to combine the law of universal gravitation with circular motion principles to show that if the force of gravity provides the centripetal force for the planets' nearly circular orbits, then a value of 2.97 x 10-19 s2/mcould be predicted for the T2/Rratio. Here is the reasoning employed by Newton:
Consider a planet with mass Mplanet to orbit in nearly circular motion about the sun of mass MSun. The net centripetal force acting upon this orbiting planet is given by the relationship
Fnet = (Mplanet * v2) / R
This net centripetal force is the result of the gravitational force that attracts the planet towards the sun, and can be represented as
Fgrav = (G* Mplanet * MSun ) / R2
Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force are equal. Thus,
(Mplanet * v2) / R = (G* Mplanet * MSun ) / R2
Since the velocity of an object in nearly circular orbit can be approximated as v = (2*pi*R) / T,
v2 = (4 * pi* R2) / T2
Substitution of the expression for v2 into the equation above yields,
(Mplanet * 4 * pi* R2) / (R • T2) = (G* Mplanet * MSun ) / R2
By cross-multiplication and simplification, the equation can be transformed into
T/ R= (Mplanet * 4 * pi2) / (G* Mplanet * MSun )
The mass of the planet can then be canceled from the numerator and the denominator of the equation's right-side, yielding
T/ R= (4 * pi2) / (G * MSun )
The right side of the above equation will be the same value for every planet regardless of the planet's mass. Subsequently, it is reasonable that the T2/R3 ratio would be the same value for all planets if the force that holds the planets in their orbits is the force of gravity. Newton's universal law of gravitation predicts results that were consistent with known planetary data and provided a theoretical explanation for Kepler's Law of Harmonies.



Scientists know much more about the planets than they did in Kepler's days. Use The Planetswidget bleow to explore what is known of the various planets.


Check Your Understanding

1. Our understanding of the elliptical motion of planets about the Sun spanned several years and included contributions from many scientists.

a. Which scientist is credited with the collection of the data necessary to support the planet's elliptical motion?
b. Which scientist is credited with the long and difficult task of analyzing the data?
c. Which scientist is credited with the accurate explanation of the data?

Foundations of Algebra


Distributive property explained

The distributive property tells us how to solve expressions in the form of a(b + c).  The distributive property is sometimes called the distributive law of multiplication and division.
Normally when we see an expression like this …
distributive property format
we just evaluate what’s in the parentheses first, then solve it:
distributive property in practice
This is following the official “order of operations” rule that we’ve learned in the past.
With the distributive property, we multiply the ‘4’ first:
distributing values
We distribute the 4 to the 8, then to the 3.
Then we need to remember to multiply first, before doing the addition!
We got the same answer, 44, with both approaches!
Why did we do it differently when we could have easily worked out what was in the brackets first? 
This is preparation for when we have variables instead of numbers inside the parentheses.
Another example before we start to use variables:
Example of the distributive property using variables:
More examples
  • We usually use the distributive property because the two terms inside the parentheses can’t be added because they’re not like terms
  • Make sure you apply the outside number to all of the terms inside the parentheses/brackets