Math 6H (Friday)

Word Problems Review
Remember to use the following plan for solving word problems.
5-Step Plan
for Solving Word Problems
1. Read the problem carefully. Make sure you understand what it says. You may need to read it more than once.

2. Use questions like these in planning the solution
a. What is asked for?
b. What facts are given?
c. Are enough facts given?
If not, what else is needed?
d. Are unnecessary facts given?
e. Will a sketch or diagram help?

3.Determine which operation or operations can be used to solve the problem.

4.Carry out the operations carefully.

5.Check your results with the facts given in the problem. Label your answer.

Algebra Period 3 (Thursday & Friday)

USING THE PYTHAGOREAN THEOREM - WORD PROBLEMS: 11-8
There are many real life examples where you can use the Pythagorean Theorem to find a length.

EXAMPLE: HOW HIGH A 10 FOOT LADDER REACHES ON A HOUSE
A 10 ft ladder is placed on a house 5 ft away from the base of the house.
Find how high up the house the ladder reaches.
The ladder makes a right triangle with the ground being one leg, the house the other, and the ladder is the hypotenuse ( see drawing in #1 on p. 515)
You need to find the distance on the house, so you're finding one leg.

ANOTHER EXAMPLE:
You're flying your kite for the kite project and you want to know how long the kite string must be so that it can reach a height of 13 ft in the air if you're standing 9 feet away from where the kite is in the air.
The string represents the hypotenuse.
You know one leg is the height in the air (13 ft) and the other leg is how far on the ground you are standing away from where the kite is flying (9 ft)
You need to find the hypotenuse.

EQUATIONS WITH RADICALS: 11-9
When you have an equation where the variable is under the √ sign,
simply square both sides to solve for the variable.

This is actually similar to regular equation balancing!
The goal is still to find out what the variable is.
But to find it, you need to ISOLATE the √ with the variable first
Then you square


Example #1 on p. 521
√x = 5
Square both sides and you'll get x = 25

Example # 9 is much harder
3 + √(x - 1) = 5
move the 3 to other side first √(x - 1) = 5 - 3
Now square both sides x - 1 = 22
Now add 1 to both sides: x = 4 + 1 x = 5

Look at Examples #15 & 16 ---- In both cases, there is no possible value for x because the square root of a number CAN NEVER BE NEGATIVE IN THE REAL NUMBER SYSTEM!

Try Example #17 yourself, and see what happens (you should also end up with no value, but why?)

Need to review how to estimate the value of square roots--Check out the blog post on March 16th.

Math 6 Honors Periods 6 & 7 (Thursday)

Multiplication and Division of Mixed Numbers 7-5

One method of finding the product of two mixed numbers is to first change the mixed numbers into improper fractions and then multiply

6 X 3 1/12 becomes
6/1 X 37/12
Simplify using the methods taught and reviewed this week
6/1 X 37/ 12 = 37/6 = 18 1/2

5 3/4 X 4 2/3
23/4 X 14/3 becomes
23/2 X 7/ 3 = 161/6 = 26 5/6

To divide one mixed number by another, we change the mixed numbers into improper fractions and use the methods from the previous lessons

2 2/3 ÷ 10 2/3 first change to improper fractions
8/3 ÷ 32/3 Remember. that dividing by a fraction is the same a multiplying by its reciprocal
8/3 X 3/32
Using the GCF to simplify first
1/1 X 1/4 = 1/4

What about
1 7/15 ÷ 5 1/2
22/15 ÷ 11/2
22/15 X 2/11 = 4/15

What if we have equations such as the book shows
n X 2 1/3 = 6 5/12
First rewrite it as (2 1/3)n = 6 5/12
THen change the mixed numbers to improper fractions
(7/3)n = 77/12
Now, we know to isolate the variable, we must use the inverse operation and in this case we would multiply 7/3 by its reciprocal 3/7 to both sides
(3/7)(7/3) n = (77/12)(3/7)
n = 11/4
n = 2 3/4

Algebra Period 3 (Wednesday)

PYTHAGOREAN THEOREM 11-7
(an old friend) -
FOR RIGHT TRIANGLES ONLY!
2 legs - make the right angle - called ‘a’ and ‘b’
(doesn't matter which is which because you will add them and adding is COMMUTATIVE!)
hypotenuse - longest side across from the right angle - called ‘c’
You can find the third side of a right triangle as long as you know the other two sides:
a² + b² = c²
After squaring the two sides that you know, you'll need to find the square root of that number to find the length of the missing side (that's why it's in this chapter!)

EASIEST - FIND THE HYPOTENUSE (c)
Example #1 from p. 510
8² + 15² = c²
64 + 225 = c²
289 = c²
c = 17

A LITTLE HARDER - FIND A MISSING LEG (Either a or b)
Example #5 from p. 510
5² + b² = 13²
25 + b² = 169
b² = 169 - 25
b² = 144
b = 12

ONE THAT YOU WOULDN'T HAVE HAD IN PRE-ALGEBRA:
One of the legs = √5
√5² + b² = 13²
5 + b² = 169
b² = 169 - 5
b² = 164
√ b2 = √164
b = √4•41
b = 2√41

DISTANCE FORMULA
(based on the Pythagorean Theorem):
see p. 513 in book
The distance between any two points on the coordinate plane (x y plane)
The distance is the hypotenuse of a right triangle that you can draw using any two points on the coordinate plane (I'll show you how to draw it in class).

The formula is:
distance = √[( difference of the two x's)² + (difference of the two y's)²]
distance = √(x₁-x₂)² + (y₁- y₂)²
The difference between the 2x’s is the length of the leg parallel to the y axis and
the difference between the 2y’s is the length of the leg parallel to the x axis
The difference of the two x's is length of one of the two legs
The difference of the two y's is the length of the other leg
The distance is the length of the hypotenuse
So you could actually rewrite the distance formula to look like the Pythagorean Theorem:
c = √[a² + b²]

EXAMPLE: What is the distance between (3, -10) and (-7, -2)?
d = √[(3 - -7)² + (-10 - -2)²]
d = √[10² +( -8²)]
d = √(164)
Simplifying:
2√41
Check out the following website- with its 81 different proofs
Pythagorean Theorem and its many proofs. See if you can find the proof that is attributed to one of our US Presidents
Practice your skills with the Pythagorean Theorem using this Shodor Interactive Site at
Pythagorean Explorer

Math 6 Honors Periods 6 & 7 (Tuesday)

Division of Fractions 7-4

Certain numbers when multiplied together have the product 1
5 X 1/5 = 1
3/4 X 4/3 = 1

Two numbers whose product is 1 are called reciprocals of each other.
Thus 3/4 is the reciprocal of 4/3.
Zero does not have a reciprocal


Look at the following:
We know 18 = 3 X 6 and we know 18 ÷ 6 = 3 as well as 18 X 1/6 = 3
Dividing a number by a fraction is the same as multiplying the number by the RECIPROCAL of the fraction
a/b ÷ c/d = a/b ÷ d/c
Remember- you are using the reciprocal of the divisor... that is , as students want to say "You FLIP the 2nd number!!"

42/ 55 ÷ 36/11
you must rewrite the problem using the reciprocal of the 2nd number
42/55 X 11/36
Now using your skills of observing GCF simplify before you multiply ( MUCH EASIER and FASTER)
42/ 5 X 1/36 which becomes 7/5 X 1/ 6 = 7/30

Algebra Period 3 (Tuesday)

ADDING AND SUBTRACTING RADICALS 11-6
Radicals function like variables, so you can only COMBINE LIKE RADICALS!
You cannot add √2 to √3!!!!
However, you may add 3√2 to 5√2 and get 8√2:
(3 + 5)√2 = 8√2

Make sure you simplify all radical expressions before trying to combine them!
Sometimes, it looks like they are not like radicands, but then after simplifying they are.

EXAMPLE #34 from p. 505:
√( x2y) + √( 4x2y) + √(9y) - √( y3)

Simplify each term first!!!!!!!!

x√y + 2x√y + 3√y - y√ y

NOW THEY ARE ALL LIKE TERMS BECAUSE ALL HAVE SQRT y!
(x + 2x + 3 - y)√y
(3x - y + 3)√y (final simplified answer)

Math 6 H Periods 1, 6 & 7 (Tuesday)

Multiplication of Fractions 7-3

If a rectangle is divided into 4 equal parts, each part is ¼ of the whole. If each of these parts is then divided into 3 parts, that is into thirds, then there are 12 equal parts and each is 1/(3 ∙4) or 1/12 of the whole.

That is 1/3 of 1/4 is 1/(3 ∙4) or 1/12 and 1/3 ∙ 1/4 = 1/12 is

so another example 2/3 of 4/5 is 2∙4 /(3∙8) or 2/3 ∙4/5 = 8/15


Notice, that the numerator of the product, 8, is the product of the numerators 2 and 4. The denominator of the product, 15, is the product of the denominators 3 and 5

Rule
If a, b, c, and d are whole numbers with , then

a/b(c/d) = a∙c/(b∙d)


When multiplying two fractions, you can simplify the multiplication by dividing either of the numerators and either of the denominators by common factors

6/35 ( 7/3) we can simplify first because both 6 and 3 are divisible by 3
2/35 (7/1) and then both 35 and 7 are divisible by 7 so 2/5 (1(1) = 2/5

Try the following

25/6 ( 42/5) What can we do there?

7/8(20/21) How about with these two sets of fractions?

19/20 ( 25/38) … and these fractions?

Algebra Period 3 (Finding SQ RTS)

FINDING SQUARE ROOTS

Many textbooks seem to think that since calculators can find square roots, that students don't need to learn how to find square roots using any pencil-and-paper method. But learning at least the "guess and check" method for finding the square root will actually help the student UNDERSTAND and remember the square root concept itself!

Practice at least the first method presented here. This method, "guess and check", actually works around what the square root is all about.
The square root of a number is just the number which when multiplied by itself gives the first number. So 2 is the square root of 4 because 2 * 2 = 4.

Method 1: Guess, Divide & Check
Start with the number you want to find the square root of. Let's use 12. There are three steps:

1. Guess
2. Divide
3. Average.

... and then just keep repeating steps 2 and 3.

First, start by guessing a square root value. It helps if your guess is a good one but it will work even if it is a terrible guess. We will guess that 2 is the square root of 12. ( Which does not really make sense because we all know that 3 * 3 = 9) However, this is a great example of how this method works—even if you pick a number that isn’t near.

In step two, we divide 12 by our guess of 2 and we get 6.
In step three, we average 6 and 2: (6+2)/2 = 4
Now we repeat step two with the new guess of 4. So 12/4 = 3
Now average 4 and 3: (4+3)/2 = 3.5
Repeat step two: 12/3.5 = 3.43
Average: (3.5 + 3.43)/2 = 3.465

We could keep going forever, getting a better and better approximation but let's stop here to see how we are doing. 3.465 * 3.465 = 12.006225


Method 2- Estimating your Square root

This method requires you to know your perfect squares. You should know them up to 400 by now. Start with the number you want to square root. Let’s say √183. We know that 183 is between two perfect squares 169 and 196 – or 13² and 14² So our SQ RT must also be between 13 and 14.
196
183
169

Next, find the difference between 196 and 169 ( 196-169) = 27
find the difference between 196 and 183 ( 196-183) = 13
and the difference between 183 and 169 ( 183-169) = 14

Looking at those three numbers, you notice that 183 is almost right in the middle or half way between the two perfect squares—so we can approximate
√183 ≈ 13.5

What happens if it isn’t quite in the middle, figure out the ratio and decide what a good approximation would be.

Years ago, teachers taught an algorithm for finding square roots, but these two methods are much easier and serve to approximate square roots accurately for middle school students.

If you have any questions or comments about these two ways, post a comment here.

Algebra Period 3

DIVIDING RADICALS 11-5
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.

HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)

EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3 = √3
√16 4


√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25= 5
√36 6

EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3

Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27
√ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3
√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!

AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!

RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)

If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.

RATIONALIZING THE DENOMINATOR EXAMPLE:
√7
√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.

You need to multiply both numerator and denominator by √3 to get it out of there:
√7 = √7 • √3 = √(7• 3) = √21
√ 3 √3 √ 3 3 3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!

Approximating Pi Day

Since Pi Day fell on a Saturday this year, we approximated Pi Day and celebrated on Monday instead... So What did we learn about pi?

Today we celebrate Π (Pi), a very cool number. Π is a comparison between the measurements of the circumference to the diameter of a circle—any circle. Pi is an IRRATIONAL number. That means it has no pattern and never terminates. It CANNOT be written as a fraction with an integer in the numerator and denominator. We use 3.14 and 22/7 as APPROXIMATIONS of pi. These are not the exact values. The only symbol that tells the exact value is Π. Pi is a ratio, a comparison between two numbers. You will be able to discover many interesting facts about pi—even finding it on your own.

After measuring 3 different circular objects and completing the table...
Think about your results and answer these questions:

How does the measurement of the circumference compare to the measurement of the diameter? Is it twice as large? Is it three times as large or more than three times as large?

Compare your values to 3.14 Were your calculations greater or less than the 3.14 values of pi?

Are you values of pi consistent?Explain why or why not?

What reasons do you think would account for these differences?

On your computer, read about the history of pi at this website
http://ualr.edu/lasmoller/pi.html
Record 5 new or interesting facts about pi that you learned

Finding Yourself in Pi
Pi is an irrational number. That means that it is a non-terminating, non-repeating decimal. Since the number order keeps changing you will eventually find any group of numbers in a sequence, somewhere in the never-ending list. Through the influence of high speed computers that can process large amounts of data, we can examine this aspect of pi much more easily. There are even websites that will instantly search pi for any string of numbers.
Your assignment is to locate a specific string of numbers in pi. Pick a string of 7 or more numbers that is meaningful or significant to you, such as your phone number or birthday (for example: if your birthday is February 9, 1993, then your number string would be 02091993). On your computer, use the Pi Searcher at http://www.angio.net/pi/piquery to find where in pi your number string occurs. Record the string of numbers and what position it holds in the list of numbers.
There are a number of great Pi day songs.. but I think the best is from Fort Vancouver High School .. Check out this video and rap they created...