Ex. you have the following variable expression:
5x + (-2) - 7x + 3 + x
What are the terms?
5x,-2,-7x,3,x
What are the like terms?
First set --> 5x,-7x,x
Second set --> -2,3
What are the constant terms?
-2,3
What are the coefficients? (the number part of a variable term)
5,-7,1
How would I simplify:
5x + (-2) - 7x +3 + x
First change the expression to all addition:
5x + (-2) + (-7x) + 3 + x
Next combine like terms, in this case I'll combine the variable terms first:
-x + (-2) + 3
Then I'll combine my constant terms:
-x + 1
So I've simplified 5x + (-2) - 7x + 3 + x to -x + 1
Thursday, September 24, 2009
For periods B, F, and G - distributive property
Examples of the distributive property using variables a, b, and c:
a(b + c) = ab + ac
(b + c)a = ba + ca
a(b - c) = ab - ac
(b-c)a = ba - ca
Examples using numbers:
4(-7 + 4) = 4(-7) + 4(4) = -28 + 16 = -12
5(6 - 2) = 5(6 + -2) = 5(6) + 5(-2) = 30 + -10 = 20
a(b + c) = ab + ac
(b + c)a = ba + ca
a(b - c) = ab - ac
(b-c)a = ba - ca
Examples using numbers:
4(-7 + 4) = 4(-7) + 4(4) = -28 + 16 = -12
5(6 - 2) = 5(6 + -2) = 5(6) + 5(-2) = 30 + -10 = 20
Tuesday, September 22, 2009
For periods A and D, tips on formulas and tables
Tips:
If you are figuring out a formula from a table plug in other values from the table to see it works. For example say you have x values = 1,2,3,4 and corresponding y values = 2,4,6,8. Let's say you look at the first x and y value and see x = 1 and y = 2 and conclude:
y = x + 1,
You would then check this with the next x and y values, x = 2, y = 4 and see that x + 1 only equals 3 and therefore y = x + 1 is an incorrect formula for this table. You could then come up with the formula y = 2x, check this formula with all values, see that it works with all values and conclude this is the correct formula. NOTE: always check your answers.
Another tip:
If the value is being increased, as in the above example from x to y, you know you need to either add something to the x variable (add a number greater than zero) or multiply the x variable (by a number greater than 1).
If the value is being decreased, then you need to know the variable being decreased must either be subtracted from (subtract a number greater than zero) or divided by (a number greater than 1).
If you are figuring out a formula from a table plug in other values from the table to see it works. For example say you have x values = 1,2,3,4 and corresponding y values = 2,4,6,8. Let's say you look at the first x and y value and see x = 1 and y = 2 and conclude:
y = x + 1,
You would then check this with the next x and y values, x = 2, y = 4 and see that x + 1 only equals 3 and therefore y = x + 1 is an incorrect formula for this table. You could then come up with the formula y = 2x, check this formula with all values, see that it works with all values and conclude this is the correct formula. NOTE: always check your answers.
Another tip:
If the value is being increased, as in the above example from x to y, you know you need to either add something to the x variable (add a number greater than zero) or multiply the x variable (by a number greater than 1).
If the value is being decreased, then you need to know the variable being decreased must either be subtracted from (subtract a number greater than zero) or divided by (a number greater than 1).
For periods A and D, formulas and variables
For the quiz you will need to know how to:
take a formula, for example A = lw and substitute in values given to you to find area. For example if you are given:
l = 9 cm
w = 5 cm
You can plug those into the formula A = lw = (9 cm)(5 cm) = 45 square centimeters
You may also encounter a formula dealing with distance, speed, and time. You will need to know to substitute the given values into the formula and solve the formula as shown above in the area formula.
take a formula, for example A = lw and substitute in values given to you to find area. For example if you are given:
l = 9 cm
w = 5 cm
You can plug those into the formula A = lw = (9 cm)(5 cm) = 45 square centimeters
You may also encounter a formula dealing with distance, speed, and time. You will need to know to substitute the given values into the formula and solve the formula as shown above in the area formula.
Sunday, September 20, 2009
Periods A and D - order of operations
For periods A and D, notes on order of operations see post For periods B, F, and G - section 1.3 Order of Operations
For periods B, F, and G - commutative and associative properties
The commutative property of addition:
a + b = b + c
ex. -4 + 7 = 7 + (-4)
The commutative property of multiplication:
ab = ba
ex. -4(7) = 7(-4) (remember having two numbers right next to each other only separated by a parenthesis indicates multiplication)
The associative property of addition:
(a + b)+ c = a + (b + c)
ex. (1 + 4) + 7 = 1 + (4 + 7)
The associative property of multiplication:
(a x b) x c = a x (b x c)
ex. (5 x 6) x 8 = 5 x (6 x 8)
a + b = b + c
ex. -4 + 7 = 7 + (-4)
The commutative property of multiplication:
ab = ba
ex. -4(7) = 7(-4) (remember having two numbers right next to each other only separated by a parenthesis indicates multiplication)
The associative property of addition:
(a + b)+ c = a + (b + c)
ex. (1 + 4) + 7 = 1 + (4 + 7)
The associative property of multiplication:
(a x b) x c = a x (b x c)
ex. (5 x 6) x 8 = 5 x (6 x 8)
All periods - a look ahead.
This is just a quick note to all periods that you will be having a quiz either Tuesday or Wednesday this week. Details to follow in Monday's class.
Sunday, September 13, 2009
For periods B, F, and G - notes on 1.4
Things to know on section 1.4:
Integer - Any whole number that is negative, positive, or zero. Ex.
-1 is an integer, 1.3 is not an integer.
You must know how to order integers on a number line from least to greatest. Negative numbers are to the left of zero on a number line and positive numbers are right of zero on the number line.
You must know how to figure out absolute value. Ex.
|-6| = 6, since -6 is 6 units away from zero on the number line.
You must find the opposite of a number Ex.
The opposite of -6 is 6.
The opposite of 4 is -4
The opposite of b is -b
Integer - Any whole number that is negative, positive, or zero. Ex.
-1 is an integer, 1.3 is not an integer.
You must know how to order integers on a number line from least to greatest. Negative numbers are to the left of zero on a number line and positive numbers are right of zero on the number line.
You must know how to figure out absolute value. Ex.
|-6| = 6, since -6 is 6 units away from zero on the number line.
You must find the opposite of a number Ex.
The opposite of -6 is 6.
The opposite of 4 is -4
The opposite of b is -b
Friday, September 11, 2009
For periods B, F, and G - section 1.3 order of operations
Rules pertaining to the order of operations (from page 16 in your text):
Order of Operations:
1. Evaluate expressions inside grouping symbols.
2. Evaluate powers.
3. Multiply and divide from left to right.
4. Add and Subtract from left to right.
Also you can remember PEMDAS - parenthesis, exponents, multiplication, division, addition, and subtraction.
Note: when remembering PEMDAS take into account multiplication and division are treated equally and should be performed from left to right. Similarly addition and subtraction are treated equally and should be performed from left to right as well.
Order of Operations:
1. Evaluate expressions inside grouping symbols.
2. Evaluate powers.
3. Multiply and divide from left to right.
4. Add and Subtract from left to right.
Also you can remember PEMDAS - parenthesis, exponents, multiplication, division, addition, and subtraction.
Note: when remembering PEMDAS take into account multiplication and division are treated equally and should be performed from left to right. Similarly addition and subtraction are treated equally and should be performed from left to right as well.
For periods B, F, and G - section 1.2 vocabulary
Words to know:
A power - the result of repeated multiplication of the same factor. For example 3 raised to the second power is 3 x 3.
A power can be writtern with a numerical base and a number raised above the base called an exponent.
A power - the result of repeated multiplication of the same factor. For example 3 raised to the second power is 3 x 3.
A power can be writtern with a numerical base and a number raised above the base called an exponent.
For periods B, F, and G - section 1.1 vocabulary
Words to know:
numerical expression - consisting of numbers and operations. For example 3 x 8 is a numerical expression
variable - a letter used to represent one or more numbers.
variable expression - consisting of numbers, variables, and operations. Ex. 2y is a numerical expression (2y means 2 times y).
evaluate - to solve an expression
Common words and phrases for addition:
plus
the sum of
increased by
total
more than
added to
Common words and phrases for subtraction:
minus
the difference of
decreased by
fewer than
less than
subtracted from
Common words and phrases for multiplication:
times
the product of
multiplied by
of
Common words and phrases for division:
divided by
divided into
the quotient of
numerical expression - consisting of numbers and operations. For example 3 x 8 is a numerical expression
variable - a letter used to represent one or more numbers.
variable expression - consisting of numbers, variables, and operations. Ex. 2y is a numerical expression (2y means 2 times y).
evaluate - to solve an expression
Common words and phrases for addition:
plus
the sum of
increased by
total
more than
added to
Common words and phrases for subtraction:
minus
the difference of
decreased by
fewer than
less than
subtracted from
Common words and phrases for multiplication:
times
the product of
multiplied by
of
Common words and phrases for division:
divided by
divided into
the quotient of
For periods B, F, and G - tips on multiplying and dividing integers.
Rules:
Multiplying same signs :
1. Multiply.
2. The answer is always positive.
Multiplying different signs:
1. Multiply.
2. The answer is always negative.
Dividing same signs:
1. Divide.
2. The answer is always positive.
Dividing different signs:
1. Divide.
2. The answer is always negative.
Ex. -3 x 9
First you multiply 3 x 9 to get 27, then since you are multiplying different signs the answer is negative, so the answer is -27.
Ex. -9 / -3
First you divide 9 by 3 to obtain 3, then since both signs are the same the answer is positive, so the answer is 3.
Multiplying same signs :
1. Multiply.
2. The answer is always positive.
Multiplying different signs:
1. Multiply.
2. The answer is always negative.
Dividing same signs:
1. Divide.
2. The answer is always positive.
Dividing different signs:
1. Divide.
2. The answer is always negative.
Ex. -3 x 9
First you multiply 3 x 9 to get 27, then since you are multiplying different signs the answer is negative, so the answer is -27.
Ex. -9 / -3
First you divide 9 by 3 to obtain 3, then since both signs are the same the answer is positive, so the answer is 3.
Thursday, September 10, 2009
For periods A and D, section 1-7 vocabulary
Words you need to know in this section:
Trend line- a trend line can be drawn when sets of data show either a positive or negative relationship. You can make predictions about data based upon this trend line. When drawing a trend line through the data points it is important to remember to draw the line so that there are roughly an equal set of data points above and below the line.
Trend line- a trend line can be drawn when sets of data show either a positive or negative relationship. You can make predictions about data based upon this trend line. When drawing a trend line through the data points it is important to remember to draw the line so that there are roughly an equal set of data points above and below the line.
For periods A and D, section 1-6 some vocabulary
For this section you need to know:
Scatterplot - a plot showing a relationship between two sets of data. Each set of data is represented by an axis with its own scale. Each pair of values is represented by a point.
Positive relationship - when two sets of data increase at the same time. A scatterplot that slants upward to the right shows a positive relationship.
Negative relationship - when one set of data increases as the other set of data decreases. A scatterplot that slants downward to the right shows a negative relationship.
No relationship - When two sets of data neither increase or decrease together they show no relationship.
Scatterplot - a plot showing a relationship between two sets of data. Each set of data is represented by an axis with its own scale. Each pair of values is represented by a point.
Positive relationship - when two sets of data increase at the same time. A scatterplot that slants upward to the right shows a positive relationship.
Negative relationship - when one set of data increases as the other set of data decreases. A scatterplot that slants downward to the right shows a negative relationship.
No relationship - When two sets of data neither increase or decrease together they show no relationship.
For periods A and D, vocabulary to know for section 1-3.
Vocabulary to know:
Line plot - a line plot shows how many times each data value occurs. A line plot uses a number line and X marks to indicate data points.
Outliers - data values on a line plot that are separated from the rest.
(Note: line plots are NOT line graphs)
Stem and leaf diagram - an organizational system for displaying data. In this system data is displayed horizontally. Each data value is split into a stem and leaf. For a two digit value, the tens digit is the stem and the ones digit is the leaf. Single digit values have stems of 0. For a three digit value, the first two digits make up the stem.
Line plot - a line plot shows how many times each data value occurs. A line plot uses a number line and X marks to indicate data points.
Outliers - data values on a line plot that are separated from the rest.
(Note: line plots are NOT line graphs)
Stem and leaf diagram - an organizational system for displaying data. In this system data is displayed horizontally. Each data value is split into a stem and leaf. For a two digit value, the tens digit is the stem and the ones digit is the leaf. Single digit values have stems of 0. For a three digit value, the first two digits make up the stem.
For periods A and D, more tips on graphs (sections 1-1 and 1-2)
Vocabulary you need to know regarding graphs:
Sector - each section of a circle graph is called a sector
Vertical Axis - the axis of the graph that travels up and down.
Horizontal Axis - the axis of the graph that travels left and right.
Scale - the "ruler" that measures the height of the graph. Ex. If you were constructing a bar graph to measure population of various countries and the population of the countries you were sampling ranged from 5 million to 95 million you might use a scale of 0 to 100 (million).
Interval - the amount of space between the values on the scale. To use the above example I might choose an interval of 10 (million).
Sector - each section of a circle graph is called a sector
Vertical Axis - the axis of the graph that travels up and down.
Horizontal Axis - the axis of the graph that travels left and right.
Scale - the "ruler" that measures the height of the graph. Ex. If you were constructing a bar graph to measure population of various countries and the population of the countries you were sampling ranged from 5 million to 95 million you might use a scale of 0 to 100 (million).
Interval - the amount of space between the values on the scale. To use the above example I might choose an interval of 10 (million).
website currently under construction ...
Hello,
This website is currently under construction. There are notes regarding tomorrow's quiz during periods A and D. (see the post "Study material the quiz this Friday, periods A and D")
-Mr. Unkert
This website is currently under construction. There are notes regarding tomorrow's quiz during periods A and D. (see the post "Study material the quiz this Friday, periods A and D")
-Mr. Unkert
For Periods B, F, and G - tips on subtracting integers
To subtract integers you need to know one basic rule, that rule is:
"Add the opposite."
For example let's try the following problem 4 - (-4).
First you would change this to 4 + (+4), then by following your rules for adding integers you would simply add 4 + 4 to obtain the answer 8.
Let's try another one -20 - (-5).
First I would change this so I am "adding the opposite."
So we change -20 - (-5) to -20 + (+5).
Next we follow our rules for adding integers and obtain the answer which is -15.
(if you need clarification on adding integers see post "Periods B, F, and G - notes on adding integers" which was posted on Sept. 9th)
"Add the opposite."
For example let's try the following problem 4 - (-4).
First you would change this to 4 + (+4), then by following your rules for adding integers you would simply add 4 + 4 to obtain the answer 8.
Let's try another one -20 - (-5).
First I would change this so I am "adding the opposite."
So we change -20 - (-5) to -20 + (+5).
Next we follow our rules for adding integers and obtain the answer which is -15.
(if you need clarification on adding integers see post "Periods B, F, and G - notes on adding integers" which was posted on Sept. 9th)
Wednesday, September 9, 2009
Reminder All Periods!!!
This is just a friendly reminder that you will be having your first test Tuesday, September 15th.
Periods B, F, and G - notes on adding integers.
To add two integers that are the same sign, whether it be negative or positive, take their absolute values, add them together and keep the sign of the numbers.
Ex. -9 + (-10)
To perform this problem I first take the absolute values of each sign and add them. The absolute value of -9 is 9 and the absolute value of -10 is 10 so I add 9 + 10 to yield 19. Since both integers have a negative sign I place the negative sign in front of 19. So the answer to
-9 + (-10) is -19.
To add two integers that are different signs I take the absolute values of both integers and subtract the smaller absolute value from the larger absolute value. For my answer I take the sign associated with the larger absolute value and place it in front of my answer from the subtraction.
Ex. -11 + 5
Step 1: Take the absolute values of -11 and 5, which are 11 and 5. Subtract 5 from 11 since 11 is the larger value --> 11-5 = 6
Step 2: Since the larger absolute value is 11 I take the negative sign associated with the 11 and place it of the 6, therefore our answer is
-6.
Ex. -9 + (-10)
To perform this problem I first take the absolute values of each sign and add them. The absolute value of -9 is 9 and the absolute value of -10 is 10 so I add 9 + 10 to yield 19. Since both integers have a negative sign I place the negative sign in front of 19. So the answer to
-9 + (-10) is -19.
To add two integers that are different signs I take the absolute values of both integers and subtract the smaller absolute value from the larger absolute value. For my answer I take the sign associated with the larger absolute value and place it in front of my answer from the subtraction.
Ex. -11 + 5
Step 1: Take the absolute values of -11 and 5, which are 11 and 5. Subtract 5 from 11 since 11 is the larger value --> 11-5 = 6
Step 2: Since the larger absolute value is 11 I take the negative sign associated with the 11 and place it of the 6, therefore our answer is
-6.
Study material for the quiz this Friday, periods A and D
Some key points you will need to know:
Mean - the average of a set of data. To find the mean first add the sum of the data values. Then divide the sum by the number of data values.
Median - the middle value when the values are arranged in order. If there is no single middle value then the median is the mean of the two middle values.
Mode - the most common data value. If no value occurs more than once, there is no mode. If two or more values occur more than once and equally, there are two or more modes.
Range - the difference between the highest and lowest data values.
Ex. what are the mean, median, mode, and range of the following set of data values: 5,13,7,8,7?
To find the mean first add the values 5 + 13 + 7 + 8 + 7 = 40.
Then divide the sum by 5 since there are 5 values--> 40 / 5 = 8.
Therefore the mean is 8.
To find the median first arrange the data values from least to greatest: 5,7,7,8,13.
Then find the middle value, in this case the middle value is 7 so the median is 7.
To find the mode find the data value that occurs the most often. In this data set 7 occurs the most often so 7 is the mode.
To find the range subtract the smallest value from the largest, in this case you would subtract 5 from 13 ---> 13 - 5 = 8, so the range would be 8.
Mean - the average of a set of data. To find the mean first add the sum of the data values. Then divide the sum by the number of data values.
Median - the middle value when the values are arranged in order. If there is no single middle value then the median is the mean of the two middle values.
Mode - the most common data value. If no value occurs more than once, there is no mode. If two or more values occur more than once and equally, there are two or more modes.
Range - the difference between the highest and lowest data values.
Ex. what are the mean, median, mode, and range of the following set of data values: 5,13,7,8,7?
To find the mean first add the values 5 + 13 + 7 + 8 + 7 = 40.
Then divide the sum by 5 since there are 5 values--> 40 / 5 = 8.
Therefore the mean is 8.
To find the median first arrange the data values from least to greatest: 5,7,7,8,13.
Then find the middle value, in this case the middle value is 7 so the median is 7.
To find the mode find the data value that occurs the most often. In this data set 7 occurs the most often so 7 is the mode.
To find the range subtract the smallest value from the largest, in this case you would subtract 5 from 13 ---> 13 - 5 = 8, so the range would be 8.
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