CHAPTER 1 Notes
1-1 Using Variables
A variable is a symbol, usually a letter, that represents one or more numbers
An expression is a statement that uses addition and/or subtraction.
Example: 5 + 8 is an expression.
This happens to be a numerical expression because it only contains numbers.
4x – 5 is a variable expression because it contains at least one variable.
A variable expression is also called an Algebraic expression.
Equations vs Expressions
Equations are mathematical sentences that contain an equal sign. Expressions do NOT contain an equal sign. As a result, equations can be solved, expressions can only be simplified.
Terminology:
Indicate Addition:
More than
Greater than
Increased by
Sum
Indicate Subtraction:
Difference of
Decreased by
*Less Than
Minus
Indicate Multiplication:
Product
Of
Twice
Doubled
Indicate Division:
Quotient
Divided by
Per
Miscellaneous:
Any form of “Is” means equals
Any time you see two operator words in a row, it indicates the use of parenthesis is required.
Example: Three times the sum of x and 12 is 32
Times the sum is two operators in a row. Thus the equations should read:
3(x + 12) = 32
1-2 Exponents and Order of Operations
To simplify a numerical expression, you replace a value with a simpler equivalent value.
Example: 5 + 8 – 6 can be replaced with 13 – 6 since 13 is a simpler equivalent value for 5 + 8. This can then be replaced with 7 since 13-6 is equivalent to 7.
Exponents are used to quickly multiply. They indicate the number of times a base (the number the exponent sits on) is used as a factor.
Example: 63 = 6 * 6 * 6 = 36 * 6 = 216
When simplifying, you follow the Order of Operations, usually referred to as “PEMDAS”
Parenthesis, Exponents, Multiplication and Division (in the order they appear) and Addition and Subtraction (in the order they appear)
Example: Simplify the expression 25 – 8 * 2 + 32
Following PEMDAS, you square the 3 first to get 25 – 8 * 2 + 9
Then do the multiplication to get 25 – 16 + 9
Finally the addition/Subtraction 9 + 9
To end up with 18
To evaluate means to substitute in a number for a variable and then simplify
Example: Evaluate 2x – 5y for x = 2 and y = 3
2(2) – 5(3)
4 – 15
-11
1-3 Exploring Real Numbers
The smallest number group is the Natural Numbers {1, 2, 3, 4, 5, 6, 7…} they are typically the numbers we count with.
The next smallest is the Whole Numbers {0, 1, 2, 3, 4, 5, 6…} they are the Natural Numbers with 0.
The next group is the Integers {…-3, -2, -1, 0, 1, 2, 3, …} They are the Whole numbers with the addition of the negatives of the same.
Next we come to the Rational Numbers. These are any number that can be written as a fraction (which includes all the Integers).
A separate group from all of those include the Irrational Numbers. Those are all the numbers that are non-ending non-repeating decimals. In other words, they can not be written as a fraction (ratio)
All of these numbers are part of what we call the Real Numbers. There is also a set of Imaginary Numbers, which we may or may not get to later this year.
If you are trying to prove something is not true, you can do so by using a Counterexample. This is just an example that does NOT work in the given situation, thus indicating the situation must not be true.
An inequality is a mathematical statement indicating a comparison of two or more expressions. (Usually it indicates that one expression is greater than the other)
Opposites are numbers which are the same distance from zero. For example, 7 is the same distance from 0 as -7, thus 7 and -7 are opposites. The distance they are form zero is called the Absolute Value. Since distance is measured in positive values, the Absolute Value of a number is always positive, regardless of the side of zero it is on.
The Coordinate Plane
The Coordinate Plane (Cartesian Plane) is formed by a Horizontal Axis (X-Axis which goes left to right) and a Vertical Axis (Y-Axis which goes up and down) meeting at a right angle. The point where the two axes meet is called the Origin and has ordered pair (0, 0) This breaks the Coordinate Plane in to 4 Quadrants. Any point in the coordinate eplane can be identified by it’s location, which is given as a (x, y) ordered pair.
Any segment in the coordinate plane has a midpoint which can be found by taking the sum of the two X coordinates and dividing by 2. The same for the Y coordinates.
1-4 Patterns and Functions
A function is a relationship that assigns exactly one output for every input. A “Function Rule” is the equation that describes the function.
To write a rule for a set of data, you need to determine the change between dependent points related to the change between independent points. This is what gets multiplied by the variable. Additionally, what is the value when the variable is zero? This is the initial value and needs to be added.
The change in the dependent variable (Y) is 25 for every change of 5 in the independent variable (X) 25/5 = 5 so the variable multiplier is 5.
Start with the function rule y = 5x. Do we know the value of the function when x = 0? In this case we do not so we will need to find it. Since I know the value of x changes by 5, I need to make 4 changes backwards to get to 0, (because I’m at 20 right now and need to get to zero going by 5 each time) I then need to make 4 changes also backwards on y to get to my starting value. Four changes at 25 each change is 100 less than where I am no (which is 15) so I get a starting value of -85. I add this to the equation I have to get
Y = 5x + (-85) or simply y = 5x – 85.
I check my ordered pairs to make sure they work.
Y = 5x – 85
40 = 5(25) – 85
40 = 125 – 85
40 = 40 Yup! It works! Thus I know my function rule is correct.
It is always a good idea to check the domain (x values) and range (y values) to make sure they make sense. If asked to identify the domain and range of a function rule that describes your spending when you only have $10, it would not make sense to include values that had you spending more than $10.
1-5 Scatter Plots
A Scatter Plot is a graph that relates TWO SETS of data to each other. Each data point represents an item’s relationship in two variables. If the scatter plot has a look that appears to be a line going up from left to right, we say it has a positive correlation. If it appears to go down from left to right, we say it has a negative correlation. If it seems to be all over the place with no real trend apparent, we say it has no correlation.
The trend line is a straight line that could be drawn through the data as evenly as possible (the same number of points above and below the trend line). We will be using our calculators to generate these lines.
1-6 Mean, Median, Mode and Range
The Measures of Central Tendency are all ways to describe data.
Mean – The sum of all the data divided by the number of data items
The mean is best used when there is no outlier (or piece of data that is much higher or lower than the rest of the data points) We usually refer to the mean as rhe average
Median – The middle most data point when the data is arranged in order form least to greatest. If there is an even number of data points, the median is the mean (or average) or the two middlemost data points. The median is best used when the data set DOES have an outlier.
Mode – the data item that occurs most often. A data set can have one, more than one, or no mode. It is best used when the data set is non-numeric, or when looking for the “most popular” such as with votes.
Range – The difference between the greatest and least values. It tells you how spread out the data is.
Stem and Leaf Plots
Create a stem using the first digit and a leaf using the remaining digits. Make sure the stem represents the same place value for all numbers.
A variable is a symbol, usually a letter, that represents one or more numbers
An expression is a statement that uses addition and/or subtraction.
Example: 5 + 8 is an expression.
This happens to be a numerical expression because it only contains numbers.
4x – 5 is a variable expression because it contains at least one variable.
A variable expression is also called an Algebraic expression.
Equations vs Expressions
Equations are mathematical sentences that contain an equal sign. Expressions do NOT contain an equal sign. As a result, equations can be solved, expressions can only be simplified.
Terminology:
Indicate Addition:
More than
Greater than
Increased by
Sum
Indicate Subtraction:
Difference of
Decreased by
*Less Than
Minus
Indicate Multiplication:
Product
Of
Twice
Doubled
Indicate Division:
Quotient
Divided by
Per
Miscellaneous:
Any form of “Is” means equals
Any time you see two operator words in a row, it indicates the use of parenthesis is required.
Example: Three times the sum of x and 12 is 32
Times the sum is two operators in a row. Thus the equations should read:
3(x + 12) = 32
1-2 Exponents and Order of Operations
To simplify a numerical expression, you replace a value with a simpler equivalent value.
Example: 5 + 8 – 6 can be replaced with 13 – 6 since 13 is a simpler equivalent value for 5 + 8. This can then be replaced with 7 since 13-6 is equivalent to 7.
Exponents are used to quickly multiply. They indicate the number of times a base (the number the exponent sits on) is used as a factor.
Example: 63 = 6 * 6 * 6 = 36 * 6 = 216
When simplifying, you follow the Order of Operations, usually referred to as “PEMDAS”
Parenthesis, Exponents, Multiplication and Division (in the order they appear) and Addition and Subtraction (in the order they appear)
Example: Simplify the expression 25 – 8 * 2 + 32
Following PEMDAS, you square the 3 first to get 25 – 8 * 2 + 9
Then do the multiplication to get 25 – 16 + 9
Finally the addition/Subtraction 9 + 9
To end up with 18
To evaluate means to substitute in a number for a variable and then simplify
Example: Evaluate 2x – 5y for x = 2 and y = 3
2(2) – 5(3)
4 – 15
-11
1-3 Exploring Real Numbers
The smallest number group is the Natural Numbers {1, 2, 3, 4, 5, 6, 7…} they are typically the numbers we count with.
The next smallest is the Whole Numbers {0, 1, 2, 3, 4, 5, 6…} they are the Natural Numbers with 0.
The next group is the Integers {…-3, -2, -1, 0, 1, 2, 3, …} They are the Whole numbers with the addition of the negatives of the same.
Next we come to the Rational Numbers. These are any number that can be written as a fraction (which includes all the Integers).
A separate group from all of those include the Irrational Numbers. Those are all the numbers that are non-ending non-repeating decimals. In other words, they can not be written as a fraction (ratio)
All of these numbers are part of what we call the Real Numbers. There is also a set of Imaginary Numbers, which we may or may not get to later this year.
If you are trying to prove something is not true, you can do so by using a Counterexample. This is just an example that does NOT work in the given situation, thus indicating the situation must not be true.
An inequality is a mathematical statement indicating a comparison of two or more expressions. (Usually it indicates that one expression is greater than the other)
Opposites are numbers which are the same distance from zero. For example, 7 is the same distance from 0 as -7, thus 7 and -7 are opposites. The distance they are form zero is called the Absolute Value. Since distance is measured in positive values, the Absolute Value of a number is always positive, regardless of the side of zero it is on.
The Coordinate Plane
The Coordinate Plane (Cartesian Plane) is formed by a Horizontal Axis (X-Axis which goes left to right) and a Vertical Axis (Y-Axis which goes up and down) meeting at a right angle. The point where the two axes meet is called the Origin and has ordered pair (0, 0) This breaks the Coordinate Plane in to 4 Quadrants. Any point in the coordinate eplane can be identified by it’s location, which is given as a (x, y) ordered pair.
Any segment in the coordinate plane has a midpoint which can be found by taking the sum of the two X coordinates and dividing by 2. The same for the Y coordinates.
1-4 Patterns and Functions
A function is a relationship that assigns exactly one output for every input. A “Function Rule” is the equation that describes the function.
To write a rule for a set of data, you need to determine the change between dependent points related to the change between independent points. This is what gets multiplied by the variable. Additionally, what is the value when the variable is zero? This is the initial value and needs to be added.
The change in the dependent variable (Y) is 25 for every change of 5 in the independent variable (X) 25/5 = 5 so the variable multiplier is 5.
Start with the function rule y = 5x. Do we know the value of the function when x = 0? In this case we do not so we will need to find it. Since I know the value of x changes by 5, I need to make 4 changes backwards to get to 0, (because I’m at 20 right now and need to get to zero going by 5 each time) I then need to make 4 changes also backwards on y to get to my starting value. Four changes at 25 each change is 100 less than where I am no (which is 15) so I get a starting value of -85. I add this to the equation I have to get
Y = 5x + (-85) or simply y = 5x – 85.
I check my ordered pairs to make sure they work.
Y = 5x – 85
40 = 5(25) – 85
40 = 125 – 85
40 = 40 Yup! It works! Thus I know my function rule is correct.
It is always a good idea to check the domain (x values) and range (y values) to make sure they make sense. If asked to identify the domain and range of a function rule that describes your spending when you only have $10, it would not make sense to include values that had you spending more than $10.
1-5 Scatter Plots
A Scatter Plot is a graph that relates TWO SETS of data to each other. Each data point represents an item’s relationship in two variables. If the scatter plot has a look that appears to be a line going up from left to right, we say it has a positive correlation. If it appears to go down from left to right, we say it has a negative correlation. If it seems to be all over the place with no real trend apparent, we say it has no correlation.
The trend line is a straight line that could be drawn through the data as evenly as possible (the same number of points above and below the trend line). We will be using our calculators to generate these lines.
1-6 Mean, Median, Mode and Range
The Measures of Central Tendency are all ways to describe data.
Mean – The sum of all the data divided by the number of data items
The mean is best used when there is no outlier (or piece of data that is much higher or lower than the rest of the data points) We usually refer to the mean as rhe average
Median – The middle most data point when the data is arranged in order form least to greatest. If there is an even number of data points, the median is the mean (or average) or the two middlemost data points. The median is best used when the data set DOES have an outlier.
Mode – the data item that occurs most often. A data set can have one, more than one, or no mode. It is best used when the data set is non-numeric, or when looking for the “most popular” such as with votes.
Range – The difference between the greatest and least values. It tells you how spread out the data is.
Stem and Leaf Plots
Create a stem using the first digit and a leaf using the remaining digits. Make sure the stem represents the same place value for all numbers.