Parents: Please be certain that your student is studying a minimum of 30 minutes each
night and is completing homework assignments (given Monday - Thursday). Always check to make sure that the homework assignments are completed. IF your students informs you that I DID NOT give homework, please click on the Course Attachment tab and print them an assignment for extra practice. The elements that we are currently working on is below. In order to master elements in mathematics, one must practice continuously. In addition to the assigned assignments, please click on the Course Attachment and Helpful Resources tabs for additional resources. Students are encouraged to use these resources for additional support and practice.
night and is completing homework assignments (given Monday - Thursday). Always check to make sure that the homework assignments are completed. IF your students informs you that I DID NOT give homework, please click on the Course Attachment tab and print them an assignment for extra practice. The elements that we are currently working on is below. In order to master elements in mathematics, one must practice continuously. In addition to the assigned assignments, please click on the Course Attachment and Helpful Resources tabs for additional resources. Students are encouraged to use these resources for additional support and practice.
Unit 1 - Equations - Test
Variable - a letter that is used to represent an unknown number
Constant - a number that does not change in value
Coefficient - a constant that is in front of a variable
Expression - a collection of constants, variables, grouping symbols, and one or more of the four basic operation signs
Term - the individual parts of an expression between addition or subtraction signs
Equation - shows that two expressions are equal by using the equal sign
To solve equations, use the inverse operations to isolate the variable on one side of the equation.
When an equation contains fractions on both sides, it is helpful to multiply both sides by a common denominator of the fractions. This process is called clearing fractions.
To solve an equation with the same variable on both sides of the equal sign, you must first add or subtract to eliminate the variable term from one side of the equation.
The distributive property is handy to help you get rid of parenthesis.
a(b+c) = ab + ac
Combining Like Terms is a process used to simplify an expression or an equation using addition or subtraction of the coefficient of terms.
7x + 5x + 4 - 2 is 12x - 2
These ARE like terms: 6x, -10x, x they have the same single variable
-17ab, ab, they have are ab with a coefficient
7, 0.13, -24 they are constants
These are NOT like terms: 5x, 5y they do not have the same variable
We will solve one-step, two-step, and multi-step equations. We will also discuss and solve word problems. Please refer to the attachments under Course Attachments and Helpful Resources for additional support and extra practice!
Unit 2 - Radicals and Integer Exponents -
Exponents
Multiplying with Exponents - if the base is the same, add the exponents and keep the base
Dividing with Exponents - if the base is the same, subtract the exponents and keep the base
Power of an Exponent - keep the base and multiply the exponent by the power
Negative Powers - rewrite as a fraction and that base now has a positive exponent
Zero Power - Any base raised to the zero power has a value of 1
Adding and Subtracting Exponents - Follow the order of operations (PEMDAS)
Scientific Notation
Scientific notation is a way of using exponents to write numbers. A number in scientific notation has two factors:
* The first factor is a number less than 10 and greater than or equal to 1
* The second factor is a power of 10 (10 with an exponent).
Please click on the Course Attachments tab to find practice problems related to scientific notation!
Square Roots and Cube Roots
Every positive number has two square roots, one positive and one negative. One square
root of 16 is 4, since 4x4=16. The other square root of 16 is -4, since (-4)(-4)=16. The numbers 16, 36, and 49 are examples of perfect squares. A
perfect square is a number that has integers as its square roots. Other perfect squares include 1, 4, 9, 25, 64, and 81.
To cube a number, just use it in a multiplications 3 times... 4 cubed = 4x4x4 = 64
There is an attachment of perfect squares, square roots, and cube roots that your student
needs to study nightly. They should have one of these forms in their composition notebook. They can be made into flash cards in order to check for mastery. Knowing these square roots and cube roots will assist your student in estimating square roots/cube roots and placing them in the appropriate place on a number line.
The Real Numbers
Real Numbers - all rational and irrational numbers
Rational Numbers - can be written as a fraction, integers, counting numbers, whole numbers, terminating decimals, perfect squares, repeating decimals
Irrational Numbers - non perfect squares, non terminating decimals (without pattern), pi
Decimal Expansion - changing repeating decimals to fractions, fractions to decimals
Unit 3 - Pythagorean Theorem and Volume - Test
Pythagorean Theorem
Please refer to the Volume 2 Worktext for examples of the Pythagorean Theorem. They can be found from pgs. 339-376.
hypotenuse - in a right triangle, the side opposite the right angle, the longest side
leg - in a right triangle, the sides that include the right angle; in an isosceles triangle, the pair of congruent sides
Volume of Cylinders, Cones, and Spheres
Please refer to the Volume 2 Worktext for examples of finding the volume of cylinders, cones, and spheres. It can be found in Chapter 6, pgs. 383-410. There are more practice problems under Course Attachments.
PARENTS: PLEASE HAVE YOUR CHILD PRACTICE WITH MULTIPLYING 3 DIGIT BY 3 DIGIT NUMBERS (including decimals) AND DIVIDING FOUR DIGIT BY ONE DIGIT (including decimals). CALCULATING THE VOLUME OF THESE FIGURES REQUIRES MASTERY OF THESE SKILLS. PLEASE DO NOT ALLOW YOUR CHILD TO USE A CALCULATOR (OTHER THAN TO CHECK ANSWERS) WHEN COMPLETING ASSIGNMENTS. IT IS A MUST THAT THEY SHOW THEIR WORK! THEY WILL NOT HAVE THAT RESOURCE ON QUIZZES, TESTS, OR THE GA MILESTONES. THERE ARE PRACTICE SHEETS UNDER COURSE ATTACHMENTS.
Unit 4 - Similarity and Congruence - (Performance Task instead of test due to finals)
Congruent figures are the same size and the same shape.
Similar figures are the same shape, not the same size. The corresponding angles (matching angles) are congruent.
Transformations
translations - slide
reflections - flip
rotations - turn
These three transformations are congruent. They have the same shape and
size.
Translation Rules
up: add to the y-coordinate
down: subtract from the y-coordinate
right: add to the x-coordinate
left: subtract from the x-coordinate
Reflection Rules
reflect over the y-axis (x, y) - - > (-x, y)
reflect over the x-axis (x, y) - - > (x,-y)
reflect over y = x (x, y) - - > (y, x)
Rotation Rules
rotating 180 degrees about the origin clockwise or counterclockwise (x, y) - - > (-x,-y)
rotating 90 degrees about the origin counterclockwise (x, y) - - > (-y, x)
rotating 90 degrees about the origin clockwise (x, y) - - > (y,-x)
rotating 270 degrees about the origin counterclockwise (x, y) - - > (y,-x)
rotating 360 degrees about the origin clockwise or counterclockwise (x, y) - - > (x, y)
dilations - change size, but not shape.
When dilating a figure, multiply the coordinates by the scale factor to find the vertices of the image
scale factor - a number 0-1 will make the figure smaller
a number greater than 1 will make the figure larger
Dilations are NOT congruent.
Dilations are similar because of the change in size. The interior angles are the same.
Adjacent angles are next to one another.
Supplementary Angles = 180 degrees
Complementary Angles = 90 degrees
The sum of the interior angles of a triangle equal 180 degrees.
The sum of the exterior angles of a triangle equal 360 degrees.
Special Angle Pairs when Parallel Lines are Cut by a Transversal
Alternate Exterior Angles - congruent
Alternate Interior Angles - congruent
Corresponding Angles - congruent
Vertical Angles - congruent
Same Side Exterior Angles - supplementary
Same Side Interior Angles - supplementary
Linear Pair - supplementary
Unit 5 - Functions -
A function is a rule that relates two quantities so that each input value corresponds to exactly one output value. The domain is the set of all possible input values, and the range is the set of all possible output values.
Functions can be represented in many ways, including tables, graphs, ordered pairs, and equations. If the domain of a function is infinitely many values, it is impossible to represent them all in a table, but a table can be used to show some of the values and to help in creating a graph.
If a relationship is a function, each input has exactly one output. When the relationship is graphed, use the vertical line test. Place a vertical line on the graph. If the line intersects the graph at only one point, then the relationship is a function. If the line intersects the graph at more than one point, then the relationship is not a function.
Linear functions are functions whose graph of solutions form a straight line.
Nonlinear functions are functions whose rates of change are not constant. Therefore, their graphs are not straight lines.
Slope-Intercept Form of an equation is y = mx + b
m is the slope and b is the y-intercept (where the line intersects the y-axis)
another term for y-intercept is initial value
The slope of a line refers to how steep a line is. Slope is also defined as the rate of change. When we graph a line using ordered pairs, we can easily determine the slope.
Steps for writing an equation from a table:
Step 1: Pick a point from the table (x, y)
Step 2: Find the slope of the table (change in y/change in x)
Step 3: Substitute in slope-intercept form of an equation and solve for b (y-intercept)
Step 4: Rewrite the equation in slope-intercept form using the appropriate slope and y-intercept
Direct Proportions
A direct proportion is a special kind of linear equation. A direct proportion may be written in the following form: y = mx. If the graph of an equation is a non-vertical, straight line that passes through the origin, the graph shows a direct proportion. So a direct proportion is linear. The constant of variation, m, is the slope of the line, and the y-intercept is (0, 0).
Unit 6 - Linear Models and Bivariate Data
Standard Form of an equation is Ax + By = C
The x-intercept is the point where the graph of a line crosses the x-axis.
The y-intercept is the point where the graph of a line crosses the y-axis.
To find the x-intercept - set y = 0
To find the y-intercept - set x = 0
An equation that contains two variables, each to the first degree (exponent 1), is a linear equation. The graph for a linear equation is a straight line. To put a linear equation in slope-intercept form, solve the equation for y. This form of the equation shows the slope and the y-intercept. Slope-intercept form
follows the pattern of y = mx + b. The "m" represents the slope, and the "b" represents the y-intercept. The y-intercept is sthe point at which the line crosses the y-axis.
When the slope of a line is not 0, the graph of the equation shows direct variation between y and x. When y increases, x increases in a certain proportion. The proportion stays constant. The constant
is called the slope of the line.
Interpreting Graphs
A continuous graph is a graph made up of connected lines or curves. An example of a situation that will result is a continuous graph: The volume of water in a swimming pool steadily decreases by 20 gallons per minute for a period of 3 hours. (you will connect the line because you can determine how much water is being taken out at any given time... such as 30 seconds)
A discrete graph is a graph made up of distinct, or unconnected, points. An example of a situation that will result in a discrete graph: A market sells pumpkins for $5 each. (you cannot sell half of a pumpkin so the line is not connected)
bivariate data - a data set involving two variables
clustering - a condition that occurs when data points in a scatter plot are grouped more in one part of the graph than another
frequency - the number of times the value appears in the data set
outlier - a value much greater or much less than the others in a data set
relative frequency - the frequency of a data value of range of data values divided by the total number of data values in the set
scatter plot - a graph with points plotted to show a possible relationship between two sets of
data
line of best fit - a straight line that comes closest to the points on a scatter plot
two-way table - a table that displays two-variable data by organizing it into rows and columns
Unit 7 - Systems of Equations -
You can find the solution of a system by graphing the two lines and identifying the coordinates of the point(s) of intersection. There may be no solution, one solution, or infinitely many solutions.
If the lines do not intersect (parallel lines), there is no solution.
If the lines intersect at one point, that point (ordered pair) is the solution.
If the lines are identical (same line), there are infinitely many solutions.
Systems can be solved algebraically using substitution and elimination. Systems may also be used to solve word problems.