Week of August 6, 2018

Astronomy

The Seasons and Axis Tilt


The Earth's seasons are not caused by the differences in the distance from the Sun throughout the year (these differences are extremely small). The seasons are the result of the tilt of the Earth's axis.

The Earth's axis is tilted from perpendicular to the plane of the ecliptic by 23.45°. This tilting is what gives us the four seasons of the year - spring, summer, autumn (fall) and winter. Since the axis is tilted, different parts of the globe are oriented towards the Sun at different times of the year.

Summer is warmer than winter (in each hemisphere) because the Sun's rays hit the Earth at a more direct angle during summer than during winter and also because the days are much longer than the nights during the summer. During the winter, the Sun's rays hit the Earth at an extreme angle, and the days are very short. These effects are due to the tilt of the Earth's axis.

Solstices
The solstices are days when the Sun reaches its farthest northern and southern declinations. The winter solstice occurs on December 21 or 22 and marks the beginning of winter (this is the shortest day of the year). The summer solstice occurs on June 21 and marks the beginning of summer (this is the longest day of the year).

Equinoxes
Equinoxes are days in which day and night are of equal duration. The two yearly equinoxes occur when the Sun crosses the celestial equator.

The vernal equinox occurs in late March (this is the beginning of spring in the Northern Hemisphere and the beginning of fall in the Southern Hemisphere); the autumnal equinox occurs in late September (this is the beginning of fall in the Northern Hemisphere and the beginning of spring in the Southern Hemisphere).

Source:http://www.enchantedlearning.com/subjects/astronomy/planets/earth/Seasons.shtml

Foundations of Algebra

Multiplying Fractions

Multiply the tops, multiply the bottoms.

There are 3 simple steps to multiply fractions

1. Multiply the top numbers (the numerators).

2. Multiply the bottom numbers (the denominators).

3. Simplify the fraction if needed.

Example:

12 × 25

Step 1. Multiply the top numbers:

12 × 25  =  1 × 2   =  2 

Step 2. Multiply the bottom numbers:

12 × 25  =  1 × 22 × 5  =  210

Step 3. Simplify the fraction:

210 = 15

With Pizza

Here you can see it with pizza ...

Do you see that half of two-fifths is two-tenths?
Do you also see that two-tenths is simpler as one-fifth?

With Pen and Paper

And here is how to do it with a pen and paper (press the play button):

Another Example:

13 × 916

Step 1. Multiply the top numbers:

13 × 916  =  1 × 9   =  9 

Step 2. Multiply the bottom numbers:

13 × 916  =  1 × 93 × 16  =  948

Step 3. Simplify the fraction:

948 = 316

(This time we simplified by dividing both top and bottom by 3)

The Rhyme

♫ "Multiplying fractions: no big problem,
Top times top over bottom times bottom. 
"And don't forget to simplify,
Before it's time to say goodbye" ♫

Fractions and Whole Numbers

What about multiplying fractions and whole numbers?

Make the whole number a fraction, by putting it over 1.

Example: 5 is also 51

Then continue as before.

Example:

23  ×  5

Make 5 into 51 :

23 × 51

Now just go ahead as normal.
Multiply tops and bottoms:

23 × 51  =  2 × 53 × 1  =  103

The fraction is already as simple as it can be.

Answer = 103

Or you can just think of the whole number as being a "top" number:

Example:

3 × 29

Multiply tops and bottoms:

3  × 29  =  3 × 29  =  69

Simplify:

69 = 23

 Source: https://www.mathsisfun.com/fractions_multiplication.html