Algebra Honors ( Periods 6 & 7)

The Pythagorean Theorem 11-6

Here is an interesting demonstration illustrating the Pythagorean Theorem.

We watched the following in class. Did you catch all the mistakes made?  

What about Homer Simpson?

Algebra Honors (Periods 6 & 7)

The Pythagorean Theorem 11-6

The Pythagorean Theorem can be used to find the lengths of a right triangle. The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side. The other two sides of a right triangle are called the legs of the triangle.

In any right triangle the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs.  a² + b² = c²

Example:

The length of one side of a right triangle I 28 cm the length of the hypotenuse is 53 cm. Write and solve an equation for the length of the unknown side.

a² + b² = c²       a² = c² –b²            

Pythagorean Triples

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

Constructions:

To draw a line segment with a length of √2 , draw a right triangle with legs of length 1 unit . Using that length You can construct a segment  √3 units long… and so on

Converse of the Pythagorean Theorem

If the sum of the squares of the lengths of the two shorter sides of a triangle is equal to the square of the length of the longest side of the triangle, then the triangle is a right triangle.

The Distance Formula

The distance between two pints on the x-axis 
(or a line parallel to that axis)  is the absolute value of the
difference between their x-coordinates.

The distance between two points on the y-axis
(or a line parallel to that axis) is the absolute value of the
difference between their y-coordinates.

To find the distance between two points NOT on an axis or a line parallel to either axis, use the Pythagorean Theorem.

This can be generalized as the Distance Formula

For any points P₁(x₁, x₂) and P₂ (x₂, y₂)

Math 6A (Periods 1 & 2)

Changing a Decimal to a Fraction 6-6

As we have seen, every fraction is equal to either a terminating decimal or a repeating decimal. It is also true that every terminating or repeating decimal is equal to a fraction.

To change a terminating decimal to a fraction in lowest terms, we write the decimal as a fraction whose denominator is a power of 10. We then write this fraction in lowest terms.

Change 0.385 to a fraction in lowest terms

0.385 = 385/1000 = 77/200

Change 3.64 to a mixed number in simple form

3.64 = 3+ 64/100 = 3 + 16/25 =  

We discovered that some fractions became repeating decimals.

For example

4/9 = 0.444444...

31/99= 0.31313131...

275/999= 0.275275275...
243/999 = 0.243243243...

and we found that you could write these with a vinculum.

so

We talked about the 9th's family.... and found the simple rule for changing a single digit decimal with the vinculum over it...

and then we saw that 

and

but we discovered that using our divisibility rules we could simplify 45/99  to 5/11

We then talked about the wonderful 11th's family and found the simple rule for changing those special fractions.

That that point we talked about

Again, using our divisibility rules we found we could simplify 243/999  because we saw that both the numerator and the denominator were divisible by 9. At that point we find that 243/999 = 27/111 But then we realize that both the numerator and the denominator are still divisible by 3. so we can simplify the fraction to 9/37.

This discussion continues with our next class! I can't wait! How about you?

Math 7 (Period 4)

Factoring Numbers & Expressions 5.1

A Prime number has exactly two factors—itself and 1

A whole number greater than 1 that has factors other than 1 and itself is called a composite number.

The number 1 is neither prime nor composite

Zero is neither prime nor composite.

You can use factor tress to find the prime factorization

and we found that there were many ways to use factor trees to arrive at the same prime factorization

So multiple Factor Trees could result in how you factor out number.

The product of prime numbers is called the prime factorization of that number.

You can also use something a called INVERTED DIVISION. It is similar to Factor Trees but you must begin with the smallest  prime factor that goes into the number and keep using it until it no longer works. Then go to the next higher prime factor… and keep going…

No matter which method you use,
you MUST write your PRIME FACTORIZATION from the smallest prime number to the biggest prime number.

54= 2∙3∙3∙3 That’s in expanded notation

54 =2∙3³ That’s in exponent notation

Negative numbers can be factored by using (-1) as a factor

What happens with variables?

-63a³ becomes (-1)∙3∙3∙7∙a³= (-1)∙3∙3∙7∙a∙a∙a

-27r3s = (-1)∙3∙3∙3∙r∙r∙r∙s

We could use exponents and when asked for prime factorization using exponential notation it would be:

(-1)∙3³∙r³∙s 

Math 6A (Periods 1 & 2)

Changing a Fraction to a Decimal 6-5
There are two methods that can be used to change a fraction to a decimal.

1) find an equivalent fraction whose denominator is a power of 10. (this method does not always work but when it does it becomes really easy to change to a decimal)

13/25 multiply the numerator and the denominator by 4 to get 52/100 and then just close your eyes and see Chapter 3... and 0.52


2) divide the numerator by the denominator. It's a great way to determine your score out of 100 and then figure out your percent.

If you got 67/75 on the last test

divide 67 by 75 carefully 0.8933333... you earned a B+

take 3/8 and divide 3 by 8 8 goes into 3 0.375 times

so 3/8 = 0.375

If the numerator is smaller than the denominator we know our number must be between 0 and 1--> it must be a decimal.


When the remainder is 0 as in the case of dividing 3 by 8, it is called a terminating decimal.


By examining a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal.

If the denominator has no prime factors other than 2 or 5, the decimal representation will terminate.

7/40

looking at 40 we notice the prime factorization ( oh no, it's Chapter 5)
40 = 2³·5 Since the only prime factors are 2 and 5

7/40 must terminate.

What about 5/12 ?

12 = 2²· 3 Since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal.


What about 9/12 ? At first it looks the same as the one above, but look carefully and realize 9/ 12 = 3/4

Since 4 = 2² , 4 has no other prime factors except 2, this can be expressed as a terminating decimal.


Let's look at 15/22
Since 22 has the prime factor of 11 we know that this will not terminate. In fact when you divide 15 by 22 you end up with 0.681818181...


We write this as ( Please check page 196) Notice that the bar is only over the 81 and represents a block  numbers that continues to repeat indefinitely and is called a repeating decimal.


EVERY FRACTION CAN BE EXPRESSED AS EITHER A TERMINATING DECIMAL OR A REPEATING DECIMAL.


Let's look at 4/9 = 0.4444444....

5/9 =

7/9 =


31/99 = 0.3131313131...

8/ 11 = 72/99 = .72727272...