Fitting Equations to Data:7-7
(word problems)
Many real world relationships are LINEAR (or approximate a linear relationship), meaning they can be graphed with a LINE.
For example, if a candy bar costs $1.50, then 2 bars cost $3.00 etc
Think of the number of candy bars as the x value because that's the independent variable...
you decide how many candy bars you're going to buy.
The y value is the dependent variable.
How much money you owe at the register is the y value!
x y
1 $1.50
2 $3.00
3 $4.50
and so on....
If you graph this, you'll get a line because the price is constant.
That means that the slope is constant.
What is the slope????
The price!!! ($1.50)
Think of it as the change in y ($ you owe)
over the change in x (# of candy bars)
The money you owe goes up $1.50 every time you buy 1 more candy bar
This is a POSITIVE slope of 1.50!
So another meaning of a positive slope in the real world is that two types of data are changing at a constant rate IN THE SAME DIRECTION (called a POSITIVE CORRELATION in statistics)
You can reverse both directions and have the same relationship just as we did when we counted the slope going both up and down:
Every time you decrease your purchase by 1 bar, your purchase price also decreases by $1.50.
The data is still going in the SAME DIRECTION (now both going down!)
How do you find the equation of this linear relationship?
You use the Slope Intercept or Point Slope formulas of a line!
(It's just a real life mystery line!!! :)
SLOPE INTERCEPT:
y = 1.5x
(the y intercept is 0 because at 0 candy bars, you owe 0)
POINT SLOPE:
You can pick any point, which means any number of candy bars with its associated cost.
I used 1 candy bar (the x value) with its cost of $1.50:
y - 1.50 = 1.50(x - 1)
Y INTERCEPTS IN THE REAL WORLD THAT ARE NOT ZERO:
The scenario above has a y intercept value of 0 because you don't owe anything if you don't buy anything,
But often, the y intercept value will be a number.
For example, think of cell phone use.
Say you are charged $.10 per minute of use, but your monthly charge is $25.
Even if you don't use your cell phone, you still owe $25!
The linear equation would be:
y = .10x + 25 where y = what you owe
and x = number of minutes used
Plug in 0 for x, number of minutes, and you still owe $25 (y value)
NEGATIVE SLOPES IN THE REAL WORLD:
Let's think of a real life example that will give us a NEGATIVE SLOPE...
Often, the more you buy, the smaller the unit price per item.
This happens with copying or buying things like invitations.
Companies give you a "break" if you buy more.
SEE THE EXAMPLE ON PAGE 333 IN YOUR BOOK!
DATA THAT APPROXIMATES A LINEAR EQUATION:
Finally, sometimes real world data can be APPROXIMATED as a linear relationship.
In other words, it may not be exact, but a good way to understand the data is to look at it that way.
Think SCATTER PLOTS with POSITIVE or NEGATIVE CORRELATIONS.
Thursday, February 11, 2010
Tuesday, February 9, 2010
Math 6H ( Periods 3, 6, & 7)
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 2 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 = 36/99 = .36363636...
we talked about the 1/7th family.
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 2 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 = 36/99 = .36363636...
we talked about the 1/7th family.
Monday, February 8, 2010
Math 6H ( Periods 3, 6, & 7)
Word Problems 6-3 & 6-4
Read-- and read read your word problems. Really determine what the question is asking...
When in doubt, draw it out!!
Sally & Bill prepared 40 sandwiches for a picnic with 14 friends. If the sandwiches were divided equally among all of them, how many sandwiches would each person receive?
Wait, think... there are 16 total people because you must count Sally & Bill
so 40/16 = 2 1/2 sandwiches
A class has 48 minutes available to hear reports for 15 students. If the time is to be divided equally, how long may each student take?
48/ 15 = 3 1/5 minutes
An assembly period has 50 minutes for speeches by 4 candidates for class president. If two minutes are allowed for each introduction and the remaining time is to be divide equally, how long may each candidate speak?
4 introductions so 2 X 4 = 8 50 -8 = 42 minutes to be divided equally
42/ 4 = 10 1/2 minutes each
A fence is to be put up along a side of a lot measuring 114 feet. Nineteen fence posts are to be used-- one at each end and 17 equally spaced between ends. What is the distance between centers of two consecutive fence posts?
This is a Must DRAW OUT
I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I
count the spaces between and you will find that there are 18 spaces between the posts so 114/18 = 6 1/3 ft
A play consists of 3 acts of equal length with 2 twenty minute intermissions between acts. (this means there is one between the first and the second act and another between the second and the third act) If the whole production takes 3 hours, how long is each act in minutes?
When in doubt-- draw it out
l_______________l_____l_______________l_____l_______________l
First change 3 hours to 180 minutes
then deduct the two twenty minute intermissions 180 - 40 = 140
140 minutes left for the three equal acts
140/3 = 46 2/3 min
An hour TV program consists of four segments of equal length. There are 3 on-minute commercials between EACH segment. There is also a 2 minute intro and a 2 minute closing. How long is each segment?
When in doubt-- draw it out
l______l__________l__l__l__l__________l__l__l__l__________l______l
there are 4 minutes for into and closing and 3(3) minutes of commercials 4 + 9 = 13 minutes
60 -13 = 47 47/ 4 = 11 3/4 minutes
A table top is in the shape of a square 45 inches on each side. A chessboard is inlaid so that a border of 3 inches exists on all four sides. How wide is each of the 8 rows of squares of chessboard?
When in doubt... draw it out
You will see that there is a border of 3 inches all around so you need to deduct 3 inches from one side and 3 inches from the other or 6 inches
45-6 = 39. Now the chessboard has 8 rows so divide
39/8 = 4 7/8 inches
The Bears won 11 out of 16 games. The Eagles won 17 out of 24 games. Which team won a greater fraction of its games?
Compare
11/16 or 17/24
WAIT... not those two fractions again!!
The Eagles...
Did you have to change to a common denominator??? I hope not!!
There were 10 sunny days out of 28 in February and 12 sunny days out of 30 in November. (Anyone remember the poem...or the way to check with your knuckles???)
Which month had the higher fraction of sunny days?
10/28 12/30 I might simplify first but I can tell by comparing that November had more sunny days
What method did you use???
Read-- and read read your word problems. Really determine what the question is asking...
When in doubt, draw it out!!
Sally & Bill prepared 40 sandwiches for a picnic with 14 friends. If the sandwiches were divided equally among all of them, how many sandwiches would each person receive?
Wait, think... there are 16 total people because you must count Sally & Bill
so 40/16 = 2 1/2 sandwiches
A class has 48 minutes available to hear reports for 15 students. If the time is to be divided equally, how long may each student take?
48/ 15 = 3 1/5 minutes
An assembly period has 50 minutes for speeches by 4 candidates for class president. If two minutes are allowed for each introduction and the remaining time is to be divide equally, how long may each candidate speak?
4 introductions so 2 X 4 = 8 50 -8 = 42 minutes to be divided equally
42/ 4 = 10 1/2 minutes each
A fence is to be put up along a side of a lot measuring 114 feet. Nineteen fence posts are to be used-- one at each end and 17 equally spaced between ends. What is the distance between centers of two consecutive fence posts?
This is a Must DRAW OUT
I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I
count the spaces between and you will find that there are 18 spaces between the posts so 114/18 = 6 1/3 ft
A play consists of 3 acts of equal length with 2 twenty minute intermissions between acts. (this means there is one between the first and the second act and another between the second and the third act) If the whole production takes 3 hours, how long is each act in minutes?
When in doubt-- draw it out
l_______________l_____l_______________l_____l_______________l
First change 3 hours to 180 minutes
then deduct the two twenty minute intermissions 180 - 40 = 140
140 minutes left for the three equal acts
140/3 = 46 2/3 min
An hour TV program consists of four segments of equal length. There are 3 on-minute commercials between EACH segment. There is also a 2 minute intro and a 2 minute closing. How long is each segment?
When in doubt-- draw it out
l______l__________l__l__l__l__________l__l__l__l__________l______l
there are 4 minutes for into and closing and 3(3) minutes of commercials 4 + 9 = 13 minutes
60 -13 = 47 47/ 4 = 11 3/4 minutes
A table top is in the shape of a square 45 inches on each side. A chessboard is inlaid so that a border of 3 inches exists on all four sides. How wide is each of the 8 rows of squares of chessboard?
When in doubt... draw it out
You will see that there is a border of 3 inches all around so you need to deduct 3 inches from one side and 3 inches from the other or 6 inches
45-6 = 39. Now the chessboard has 8 rows so divide
39/8 = 4 7/8 inches
The Bears won 11 out of 16 games. The Eagles won 17 out of 24 games. Which team won a greater fraction of its games?
Compare
11/16 or 17/24
WAIT... not those two fractions again!!
The Eagles...
Did you have to change to a common denominator??? I hope not!!
There were 10 sunny days out of 28 in February and 12 sunny days out of 30 in November. (Anyone remember the poem...or the way to check with your knuckles???)
Which month had the higher fraction of sunny days?
10/28 12/30 I might simplify first but I can tell by comparing that November had more sunny days
What method did you use???
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