Download Parent/Student Packet for Students Entering 8th Grade Math Honors

Hauppauge School District
Parent/Student Packet for
Students Entering 8th Grade
Math Honors
Created by:
Ms. Beardslee, 8th Grade Math Teacher
June 2009
Dear Parents and Students Entering 8th Grade Math Honors,
This packet is geared to students that were not in the 7th grade accelerated
math program and will be entering the 8th grade math honors class in
September 2009. The information in this packet is the curriculum that the
students would have learned in the 7th grade accelerated math class (NYS Eight
Grade Math Curriculum).
The enclosed packet contains a link to the New York State Eighth Grade
curriculum, glossary of important vocabulary, and sample tasks. Also included
is a list of commonly used vocabulary for quick reference.
Students should complete these problems in order to demonstrate their
knowledge of the curriculum. All students in the 8th grade math honors class
will be taking a New York State Regents Exam in June of 2010. An answer key
is provided at the conclusion of the document.
New York State Math Standards for grade 8 (seventh grade honors)
http://www.emsc.nysed.gov/3-8/MathCore.pdf
New York State Glossary of Math Vocabulary for grade 8 (seventh grade honors)
http://www.emsc.nysed.gov/3-8/8mathlang.doc
New York State Sample Tasks for grade 8 (seventh grade honors)
• This website explains the curriculum in more detail and gives example
problems
http://www.emsc.nysed.gov/3-8/Math8Sample.htm
Common Used Vocabulary in Grade 8 (seventh grade honors)
Acute Angle An angle whose measure is greater than 0° and less than 90°.
Adjacent Angles Two angles in a plane that share a common side and share a common
vertex but have no interior points in common (do not overlap).
Example: In the figure below, angles 1 and 2 are adjacent angles.
1
2
Algebraic Expression: An expression that is written using one or more variables
(Ex: 3x , x - 4 , 2a + 5 , a + b)
Alternate Exterior Angles: A pair of angles on the outer sides of two lines intersected
by a transversal, but on opposite sides of the transversal.
Alternate Interior Angles: A pair of angles on the inner sides of two lines intersected by
a transversal, but on opposite sides of the transversal.
Angle: A geometric figure formed by two non-collinear rays that have a common
endpoint.
A
Example:
B
C
∠ABC has its vertex at point B.
Angle Bisector: a line, segment, or ray that divides an angle into two congruent angles.
Angle Pairs: Pairs of angles with special relationships (e.g., supplementary angles,
complementary angles, vertical angles, alternate interior angles, alternate exterior angles,
corresponding angles, adjacent angles).
Area: The number of square units needed to cover a given surface.
Axes: Two perpendicular lines that intersect to form the coordinate plane
Base: A number used as a repeated factor.
Binomial: A polynomial with two terms.
Circle Graph: A graph used to compare the relationship of the parts to the whole
Circumference: The distance around a circle, C =
d
Coefficient: The number that is multiplied by the variable in an algebraic expression
such as 5b.
Commission: a percentage of a sale earned by a sales representative
Composite number: A whole number that has more than two whole-number
factors.
Complementary Angles: Two angles whose measure have a sum of 90˚
Congruent Figures: Figures that have the same size and shape; the symbol ≅ means is
congruent to
Coordinate Plane: A plane formed by two perpendicular number lines called axes;
every point on the plane can be named by an ordered pair of numbers.
Corresponding Angles: Any pair of angles on the same side of the transversal, one
interior and one exterior, formed when two parallel lines are intersected by a transversal
Cosine: the ratio of the adjacent leg from the angle divided by the hypotenuse
Counting numbers: The set of numbers {1, 2, 3, . . .} used for counting separate
objects.
Diameter: A line segment that passes through the center of a circle, with endpoints on
the circle.
Dilate/Dilation: to change the size of a figure; the only transformation that does not
preserve size
Distributive Property: if a, b and c are any numbers, then a(b + c)= ab + ac
Domain: The set of the first elements of a relation; see range.
Equation: A mathematical sentence that uses an equals sign to show that two quantities
are equal.
Equation of a Line: y = mx + b; where m represents the slope and b represents the yintercept
Equivalent: Having the same value.
Evaluate: To find the value.
Exponent: The number that indicates how many times the base is used as a factor.
Exterior Angle: the angle formed by one side of a polygon and the extension of the
adjacent side
Factor: A number that is multiplied by another number to get a product.
Function: A relation in which each element in the domain is matched with only one
element of the range
Function Rule: The horizontal equation that identifies the relationship between two
variables or other valid rule including a verbal description which can be translated into an
equation; an established standard pattern or behavior
Function Notation: A notation in which a function is named with a letter and the input is
2
shown in parentheses after the function name (e.g., f( x ) = x + 1 represents the function
y = x 2 +1, where the letter f is the name of the function, and f(x), read as f of x, stands
for the output for the input x).
Gratuity: tip
Greatest Common Factor (GCF): The largest common factor of two or more given
numbers.
Hypotenuse: In a right triangle, the side opposite the right angle
Image: The figure created when another figure, called the pre-image, undergoes a
transformation.
Inequality: A mathematical sentence that shows the relationship between quantities that
are not equal, using <, >, <, >, or
Integer: The set of whole numbers and their opposites.
Interest Rates: A rate which is charged or paid for the use of money
Inverse: Operations that undo each other
(Ex: Inverse of addition is subtraction)
Irrational Number: A number that cannot be expressed as a repeating or terminating
decimal
(Ex:
)
Law of Exponents for Division The quotient of two numbers in exponential form with
the same base is equal to that base with a power equal to the difference of the powers of
ab
each number; i.e., subtract their exponents: c = a( b −c ) .
a
5
a
Example: 3 = a2
a
Law of Exponents for Multiplication The product of two or more numbers in
exponential form with the same base is equal to that base raised to the power equal to the
sum of the powers of each number; i.e., add their exponents: ab ac = ab+c .
Example: a3 a4 = a7
Leg: In a right triangle, either of the two sides that intersect to form the right angle; in an
isosceles triangle, one of the two congruent sides
Like terms: Expressions that have the same variables and the same powers of the
variables.
Line: a set of points; a straight line is an infinite set of points extending endlessly in both
directions.
Linear Equation: the equation of a line; y =mx + b
Monomial: An expression that is a number, a variable, or the product of a number and
one or more variables.
Multiple: The product of a number and any non-zero whole number.
Natural Numbers: The set of numbers {1, 2, 3, . . .} used for counting separate objects.
Nonlinear Equation: an equation that when graphed does not form a line
Ordered Pair: A pair of numbers used to locate a point on a coordinate plane
(Ex: (3,2) represents 3 spaces to the right of zero and 2 spaces up)
Order of Operations: A set of rules for evaluating an expression involving more than
one operation. (Use PEMDAS).
Parallel Lines: lines in the same plane that do not intersect
Percent: a ratio that compares a number to 100
Percent of change (increase/decrease): the ratio of the amount of change to the original
amount
Perfect Square: A number that has an integer as its square root.
Perimeter: The distance around a polygon.
Pi: The ratio of the circumference of a circle to the length of its diameter;
.
Point: An exact location
Polynomial: A monomial or a sum of monomials.
Power: The value of a number represented by a base and an exponent
Pre-Image: a figure before transformation has been preformed
Prime number: A whole number greater than 1 that has exactly two factors, itself and 1.
Proportion: An equation which states that two ratios are equivalent
Proportional: Two ratios that are equivalent are proportional.
Pythagorean Theorem: In any right triangle, if a and b are the lengths of the legs and c
is the length of the hypotenuse, then a2 + b2 = c2
Reflect/Reflection: Flipping a figure across a line
Quadrilateral: A four-sided polygon
Quadrant: One of the four regions of the coordinate plane.
Radius: A line segment with one endpoint at the center of a circle and the other endpoint
on the circle.
Rational Number: Any number that can be expressed as a ratio
integers and b 0
where a and b are
Real Numbers: The set of all rational and irrational numbers.
Relation: A set of ordered pairs
Right Triangle: A triangle with exactly one right angle
Rotate/ Rotation : A transformation that turns a figure about a fixed point
Scientific Notation: A method of writing very large or very small numbers by using
powers of 10.
(Ex: 1,200,000 = 1.2 X 106)
Similar Figures: figures in which corresponding angles are congruent and corresponding
sides are in proportion
Sine: the ratio of the opposite leg from the angle divided by the hypotenuse
Solution: The value that makes two sides of an equation equal
Slope: The measure of the steepness of a line; the ratio of vertical change to horizontal
change.
Example: If point P is (x1,y1) and point Q is (x2,y2) the slope of PQ is
Δy y 2 − y 1
.
=
Δx x2 − x1
Slope-Intercept Form: The equation of a straight line in the form y = mx + b, where m
is the slope and b is the y-coordinate of the point where the line intercepts the y-axis.
Square Root: One of the two equal factors of a number
(Ex: 6 is the square root of 36 since 62 = 6 x 6 = 36)
Supplementary Angles: angles whose measures have a sum of 180˚
Tangent: the ratio of the opposite leg from the angle divided by the adjacent leg to the
angle
Tax: a percentage of the total cost of an item
Term: each number in a sequence
Translation: sliding a figure
Trinomial: 3 terms
Variable: A letter used to represent one or more numbers in an expression, equation or
inequality.
Vertical Angles: two angles with a common vertex whose sides are extensions of the
other; vertical angles are equal in measure.
Vertices: The point where two or more rays meet; the point of intersection of two sides
of a polygon; the point of intersection of three or more edges of a solid figure; the top
point of a cone or pyramid; in a network, a point that represents an object.
Whole Number: The set of natural numbers and 0; {0, 1, 2, 3, . . .}
x-axis: horizontal line in the coordinate plane where all y values are equal to zero
x- coordinate: the first number in an ordered pair. (x,y)
y- axis- vertical line in the coordinate plane where all x values are equal to zero.
y- coordinate: the second number in an ordered pair (x,y)
y-intercept The point when a graph of an equation crosses the y-axis.
Material taught during the eighth grade (7th grade honors) year
to be completed by the student.
Order of Operations:
P arenthesis
E xponents
M ultiplication
D ivision
A ddition
S ubtraction
Simplify all operations inside the parenthesis ( ), { }, [ ]
Simplify all exponents x²
Perform all multiplication and division working from left to right
(multiplication and division are equal; solve from left to right)
Perform all addition and subtraction working from left to right
(addition and subtraction are equal; solve from left to right)
Please Excuse My Dear Aunt Sally
EXAMPLES:
1) Using the numbers 6, 3, 91, and 60, fill in the boxes to make the sentence true.
(
+
)- (
-
) = 40
2) Simplify: 2[13 - 2( 1 + 6)]
3) Lindsay and Diego are arguing over the following problem. Lindsay says the solution
is correct. Diego says that the solution is wrong. Which student has the correct answer?
Evaluating Algebraic Expressions:
VARIABLE: A letter used to represent an unknown number
COEFFICIENT: A number touching a variable. The coefficient and variable are multiplying each
other.
5n
Coefficient
variable
Evaluating Expressions:
1. Substitute (replace) the value of each variable with given numerical equivalence
2. Follow the Order of Operations
Simplify when a= 2, b = -4 and c = 3
ab - 2c + 4b
(2)(-4) – 2(3) +4(-4)
-8 - 6 - 16
-30
_______________________________________________________________________________
EXAMPLES:
1) Evaluate 2abc – 3c2 : when a = -2, b = 9, c = -4
2) Evaluate 2(b + a) - cd + a db : when a= 3, b= -9, c= -12, d= -4
3) Evaluate the formula by substituting the given values for each variable:
5
(F - C) for F = 212 and C = 32
9
The Distributive Property:
The distributive property tells us that we can remove the parenthesis if the term that the
polynomial is being multiplied by is “distributed to”, or multiplied with EACH term inside the
parenthesis.
6(2 + 4x)
Now apply the Distributive Property:
6(2 + 4x)
6(2) + 6(4x)
12 + 24x
Combining like Terms:
Constant: a number with no variable attached Æ 2x + 5 Æ The 5 is the constant.
Variable: a symbol that stands for a number Æ x, y, z, etc.
Terms: variables or numbers separated by addition or subtracting signs:
1 term: 5x, 4, 3(2x), 56.76x²
2 terms: 5x + 67, x³ + x², 17 – 18x
3 terms: 5x + 67 + 5a
Like Terms: terms with the exact same variable and the same exponent
3x + 2y + 5x – 6y Æ 8x – 4y
_______________________________________________________________________________
EXAMPLES: Distribute and Combine where possible.
1) 2(xy + wx)=
2) –(2x + 3) =
3) -4(x + 4y) -2(x + 3y) + 6(2x – 3y) =
4) -3ab + 4ab2 – 2a2b + 12ab – 6ab2 – 4a2b
Equations:
The goal of solving an equation is to isolate (get alone) the variable on one side of the equal sign.
To get the variable alone, you use the inverse (opposite) operation (“what you do to one side, you
do to the other side”)
Check your answer when done by substituting (replacing) your answer into the original equation.
One-Step
X - 3 =5
CHECK:
X - 3 = 5 STEPS: 1. Added 3 to both sides
8 - 3 = 5
2. Check
+ 3 +3
X =8
5 = 5
________________________________________________________________________
Two-Step
Solve:
Check:
Steps: 1. Subtracted 2 from both sides
-9x + 2 = -79
-9x + 2 = -79
2. Divided both sides by negative 9
-9(9) + 2 = -79
3. Check
___- 2 - 2
= -81
-81 + 2 = -79
-9x
-9
-9
-79 = -79
________________________________________________________________________
Multi-Step
5(3c – 2) + 8 = 43
CHECK: 5(3C – 2) + 8 = 43
STEPS:
·
15c – 10 + 8 = 43
5(3 3 – 2) + 8 = 43
1. Distribute
15c – 2 = 43
5(9 – 2) + 8 = 43
2. Combine like terms on same side
5(7) + 8 = 43
3. Add 2 to both sides
+2
+2
35 + 8 = 43
4. Divide both sides by 15
15c = 45
15
15
43 = 43
5. Check
C = 3
____________________________________________________________________________
EXAMPLES:
1. Amy solved the following equation. Did she solve the equation correctly? If she
did not, explain what her mistake was.
8 – 2x = 2
+8
+8
-2x = 10
-2 -2
x = -5
________________________________________________
________________________________________________
________________________________________________
Equations (continued)
Solve and check the following problems:
2. -28 = -12 – 2w
Check
2
p+4=6
3
Check
3.
4. 5c - 4 - 2c + 1 = 8c + 2
Check
5. 2y + 1.8 = 4y - 4.4
Check
INEQUALITIES
If you multiply or divide (both sides) by a negative number while you are solving an
inequality, you MUST flip the inequality sign (>, <, <, >). Otherwise, just solve like an
equation.
-7z > 35
-7 -7
z < -5
STEPS:
1) Divide both sides by -7
(NOTE: we flipped the Sign because we divided
By a negative number)
GRAPHING THE SOLUTION OF AN INEQUALITY
We graph the answers to an inequality on a number line. Before you graph you MUST
solve.
* Place an open circle [ ] above the number the variable is being compared to if the
inequality is > or < .
* Place a closed circle [
inequality is > or <.
] above the number the variable is being compared to if the
* Draw an arrow from the circle pointing in the direction of the possible solutions.
1) x – 4 > -5
+4 +4
x > -1
2) 2x – 2 > 8
5
+ 2 +2
> 10
2x
5
5
x > 2
-2
1
-1
0
2
3
Is greater than,
Is more than,
Is larger than,
Has more than
Is less than,
Fewer than
Is small than,
Has fewer
Is greater than or equal to,
Is at least,
Has at least,
Is no less than
Is less than or equal to,
Is at most,
Has at most
Is not equal to,
Does not equal,
Cannot equal
>
X is greater than 5
X>5
<
4 5 6
X is less than 5
X<5
>
4 5
X is at least 5
X>5
<
4
5
X is at most 5
X<5
≠
Solve the following inequalities and graph on a number line:
2) -2y – 10 > 20
3) 2 - 5x > 3x – 14
6
4
5
6
X does not equal 5
Inequalities (continued)
EXAMPLES:
1) 15 ≤ 3y – 6
6
X ≠ 5
4) (y + 1) < y – 4
5) John wants to buy a new bike that costs $225.00. He works at the local hardware store
and earns $6.10 per hour. Which of the following represents the least amount of hours he
will have to work in order to have enough money for the bike?
A) $225 > $6.10x
C) $225 > $6.10x
B) $6.10x > $225
D) $225 < $6.10x
6) Which of the following represents twice a number increased by 16 is at most 30?
A) 2d + 16 > 30
B) 2d + 16 < 30
C) 2d + 16 < 30
D) 2d + 16 > 30
Proportions
1 to 2 and 3 to 4
1:2 and 3:4
1/2
= 3/4
A proportion is: 2 equal ratios or two equal fractions
How to solve a proportion: CROSS MULTIPLY
Key words to look for: rate, ratio, scale, proportion, mph
Tell whether the statement is a proportion:
1.
8 = 16
20
40
2. 50 =
250
320 = 320 YES
200
350
17,500 = 50,000 NO
_____________________________________________________________
EXAMPLES:
1. Are the following ratios proportional?
a.
5 15
=
3 9
b.
10 22
=
5 17
2. Solve the following proportions for the given variables.
a.
1.2 y
=
1.5 10
b.
4 12
=
6 b
3. Mrs. Jones spent 3 hours kayaking and traveled 17 miles.
a. If she travels 68 miles, how many hours did she spend kayaking?
b. How long is this in minutes?
Similar Figures: Figures that have the same shape, but do not have the same size, are said
to be similar. In mathematics, figures are said to be similar if their corresponding angles are
equal and their corresponding sides are in proportion. The symbol for similar figures is ~
With similar figures we will set up the following proportion:
Little = Little
Big
Big
EXAMPLES:
Rectangle ABCD and EFGH are similar. Find x.
A
B
10
E
F
5
D
Little
Big
30
C
5 = x
10 30
10x = 150
10
10
X = 15
H
x
G
Similar Figures (continued)
y
*see two triangles
*separate them
9
5
6
4
y
5
9
x
10
1) Solve for x and y:
4
2) At the same time of day, a 6 ft. tall person casts a shadow 8 ft. long and a nearby tree casts a
shadow 36 ft. long. What is the height of the tree?
6’
x
8‘
36’
Unit Pricing:
The unit price is:
$____
Amount
OR
$____
Quantity
A 12 ounce bottle of shampoo sells for $2.79. Find the unit price.
2.79 = .2325
(dealing with money, round to the hundredths)
12
.23 per ounce
_____________________________________________________________________
EXAMPLES:
1. A 15 pound bag of potatoes sells for $5.79. Find the unit price.
2. A 16 ounce can of soup sells for $3.10. Find the unit price.
Best Buy:
*To calculate the best buy, you must find the unit price of all items listed.
Which is the best buy: 5 bars of soap for $2.29 or 4 bars for $1.89?
2.29 = .46
5
1.89 = .47
4
STEPS: 1. Calculate unit price for both
2. Compare
3. Choose the cheaper
Best Buy
__________________________________________________________________________
EXAMPLES:
3. Find the best buy: 3 cans of tomato sauce for $1.49 or 5 cans of tomato sauce for $2.39.
4. Find the best buy: 24 oz. of juice for $2.25 or 40 oz. of juice for $3.79.
What I Need to Know About Working with Percents
TOPIC
WHAT IS IT? HOW DOES IT WORK?
Finding Tax
•
•
Price x tax rate as a decimal = tax
Price + tax = total
Finding Commission
•
Price x Commission (rate as a decimal)
Finding Discount
•
•
Price x discount as a decimal = discount
Price – discount = new price
Finding % of Increase
Change = %
Original 100
Finding % of Decrease
Change = %
Original 100
•
Finding Tip
Calculating Basic Percent
When working with word problem
Subtotal (total before tax) x 15% to 20% (in decimal form)
is = %
of 100
54 is 60% of what number?
54 = 60
X
100
60x = 5400
60
60
X = 90
Part = %
Whole 100
You just hired a new employee to work in your bakeshop. In one hour the
employee burned 250 chocolate chip cookies. If this represented 21% of the
production, how many cookies had you planned on producing that day
250 = 21
21x = 250
X
100
21
21
X = 1,190 cookies
EXAMPLES:
1) A jacket that originally cost $45 is on sale, 25% off. If there is a 8% sales tax, what is the final cost of
the jacket?
1.
2.
3.
4.
45 x .25 = $11.25 (discount)
45 - $11.25 = $33.75 (new price)
$33.75 x .08 = $2.70 (tax)
$33.75 + $2.70 = $36.45 (total)
Percents (continued)
1) The number 5.25 is what percent of 35?
2) Find 45% of 125
3) A math class has 30 students. Approximately 70% passed their last math test. How many
students passed the mast math test?
4) The price of a ticket went from $18 to $21 what is the percent of increase?
5) The Smith family went to dinner. Their bill came to $55.20. If they would like to leave a
20% tip for their waitress how much should they leave all together?
6) Matt earns a 5% commission on his total sales. Yesterday he sold $5850 of merchandise.
How much commission will he earn?
7) How much money did Yummy Ice Cream make from selling chocolate ice cream?
Yummy Ice Cream – Total Profit $106 million
41%
Chocolate
30%
Strawberry
29%
vanilla
8) Kelly wants to purchase a new jacket that originally costs $180 but is on sale for 25%
off. If there is an 8.625% sales tax, how much will Kelly pay for the jacket?
SCIENTIFIC NOTATION
A number is written in scientific notation if it has the form:
__
__________ x 10
This number must
Be greater than or
Equal to 1 and less the
10.
(1
9.9 )
This exponent tells us the number
of places the decimal point is moved.
There can only be one number in front of the decimal
Standard Form
Scientific Notation
635,000
6.35 x 105
_________________________________________________________________________
EXAMPLES:
Put in Standard Form:
1.
2.
3.
4.
7.35 x 102
4.992 x 104
7.9 x 10-3
4.62 x 100
Put in Scientific Notation:
1.
2.
3.
4.
357,200,000
.000452
47,320
.0987
Laws of Exponents
• MULTIPLICATION:
- Multiply the coefficients
- Keep the base
- Add the exponents
-3x2 · 4x3· 2x6 =24x11
•
DIVISION
- Divide the coefficients
- Keep the base
- Subtract the exponents
Example:
1. 10x10_ = -5x6
-2x4
Zero Exponents
Any number to the zero power equals one.
xº = 1
Negative Exponnts
All negative exponents MUST be rewritten as positive exponents.
base negative exponent =
2 -5
2 a b = 2a2_
b5
Laws of Exponents (continued)
EXAMPLES:
Simplify each expression
1. a 2b 4c 6 ⋅ a 5b 4c 3 =
2.
x4 y7 z5
=
x3 y 9 z 5
3.
p 9 q 7 r −5 ⋅ p −2 q 3r 8 =
4. a) 2 x 0 y
5.
(10a 7 b 4 )(5a 2b6 )
5a10b
b) (8 x 2 y )0
c) 5a 0 b −3 c 4
PARALLEL LINES CUT BY A TRANSVERSAL
E
Parallel lines AB and CD are cut by
A
G
C
H
B
transversal EF , intersecting at G and H
D
F
Supplementary Angles
<AGE AND <EGB are supplementary angles (together they add to 180 degrees – forming a straight
line.
Other supplementary angles:
<AGH AND <BGH
<CHG AND <GHD
<CHF AND <DHF
<FHC AND <CHG
<CHG AND <AGE
<EGB AND <BGH
<GHD AND <DHF
Adjacent Angles
Adjacent angles are angles that are next to each other
Vertical Angles
A
D
B
C
.
<A ≅ <C
<D ≅ <B
Parallel Lines (continued)
Corresponding Angles
Any pair of angles on the same side of the transversal,
one interior and one exterior, formed when two parallel
lines are intersected by a transversal
<a ≅ <e
<b ≅ <f
<c ≅ <g
<d ≅ <h
a b
d c
e f
h g
Alternate Exterior A pair of angles on the outer sides of two lines intersected by a transversal, but
on opposite sides of the transversal.
<a ≅ <g
<b ≅ <h
Alternate interior Angles: A pair of angles on the inner sides of two lines intersected by a
transversal, but on opposite sides of the transversal.
<d ≅ <f
<c ≅ <e (equal to each other)
Supplementary Angles: <a and <b, <d and <c, <e and <f, <h and <g,
<a and <d, <b and <c, <e and <h, <g and <f
Vertical Angles:
(next to each other
and add up to 180˚)
<a ≅ <c
<b ≅ < d
<e ≅ <g
<f ≅ <h (equal to each other)
Interior same side Angles: <d + <e = 180˚
<c + <f = 180˚ (add up to 180˚ degrees)
_______________________________________________________________________
EXAMPLES:
1) In the accompanying diagram, line a intersects line b.
What is the value of x?
Parallel Lines (continued)
2) In the accompanying figure, what is one pair of alternate interior angles?
(1) ∠ 1 and ∠ 2
(2) ∠ 4 and ∠ 5
(3) ∠ 4 and ∠ 6
(4) ∠ 6 and ∠ 8
suur
suur
3) In the accompanying diagram, parallel lines AB and CD are intersected by transversal
at points G and H, respectively, m ∠ AGH= x + 15, and m ∠ GHD = 2x. Find x.
4) In the accompanying diagram, line m is parallel to line p, line t is a transversal, m ∠ a
= 3x + 12, and m ∠ b = 2x + 13. Find both angles.
5) Two parallel roads, Elm Street and Oak Street, are crossed by a third, Walnut Street, as
shown in the accompanying diagram. Find the number of degrees in the acute angle
formed by the intersection of Walnut Street and Elm Street.
Graphing a Linear Equation: all linear equations must be in y = mx +b form before you start
graphing.
Graph the following Equation:
y = 2x – 3
1) Set up a 3 column Chart.
2) Substitute in the values of x into the equation and solve for y
X
-2
-1
0
1
2
2x - 3
2(-2) – 3
2(-2) – 3
2(0) – 3
2(1) – 3
2(2) - 3
y
-7
-5
-3
-1
1
3) Plot your coordinates (x,y) on the coordinate plane.
_________________________________________________________________________
EXAMPLES:
Use the linear equation to complete the table of values and graph the line. Graph each
line on a separate, full page graph.
2.
1.
y = 5x - 3
y = -2x + 5
X
-2
-1
0
1
2
-2x + 5
y
x
-3
y
7
3
-13
0
Pythagorean Theorem: In any right triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the legs.This relationship can be stated as:
a, b are legs.
c is the hypotenuse
(c is across from the right angle).
a²
+ b²
= c²
hypotenuse
legs
Example:
x=c
a² + b² = c²
a = 12
12² + 9² = x²
b=9
144 + 81 = x²
√ 225 = x
15 = x
_________________________________________________________________________
EXAMPLES:
:
1. Determine if the three given sides of a triangle can form a right triangle:
15, 20, 25
39, 36, 15
2. Find the missing side of the right triangle.
36 ft.
60 ft
Pythagorean Theorem: (continued)
27, 46, 35
3. In a computer catalog, a computer monitor is listed as being 19 inches. This
distance is the diagonal distance across the screen. If the screen measures 10
inches in height, what is the width of the screen to the nearest hundredth of an
inch?
4. Mr. Sledjeski finds the largest slide in the world and is really excited to go down
it. The ladder goes straight up into the air 500 feet. The base of the ladder is 600
feet from the bottom of the slide. How long is the largest slide in the world to the
nearest hundredth of a foot?
5. A cable supports a 28 foot utility pole. The cable is fastened to the ground at a
point 21 feet from the base of the pole. Find the length of the cable.
Transformational Geometry:
Reflection: In mathematics, the reflection of an object is called its image. If the original
object (the pre-image) was labeled with letters, such as polygon ABCD, the image may
be labeled with the same letters followed by a prime symbol, A'B'C'D'. The line (where a
mirror may be placed) is called the line of reflection. The distance from a point to the
line of reflection is the same as the distance from the point's image to the line of
reflection. A reflection can be thought of as folding and "flipping" an object over the
line of reflection
Translation: A translation moves an object without changing its size or shape and
without turning it or flipping it. Think of polygon ABCDE as sliding two inches to the
right and one inch down. Its new position is labeled A'B'C'D'E'
There are several ways to indicate that a translation is to occur:
7 units to the left and 3 units down.
description:
(A verbal description of the translation is
given.)
mapping:
(This is read: "the x and y coordinates will
be translated into x-7 and y-3". Notice that
adding a negative value (subtraction),
moves the image left and/or down, while
adding a positive value moves the image
right and/or up.)
notation:
(The -7 tells you to subtract 7 from all of
your x-coordinates, while the -3 tells you to
subtract 3 from all of your y-coordinates.)
This may also be seen as
T-7,-3(x,y) = (x -7,y - 3).
Translate Point (5, 1) under the translation T5,6.
(5, 1)
(10, 11) (add 5 to the x, add 6 to the y)
Rotation: a counter clockwise turn on the coordinate plane. We use the following
formulas to rotate a figure:
Rotation of 90°:
Rotation of 180°:
EXAMPLES:
1) The point (-6,2) is reflected over the y-axis, the coordinates of the image point are:
c. (-6,-2)
a. (2,-6)
d. (6,2)
b. (-2.6)
2) If translation T4,5 maps (3,1) onto (x,6), what is the value of x?
a. 1
b. -3
c. 7
d. 8
3) Plot the image of quadrilateral PQRS under T3,-5 (10 points)
P’ (__,__)
Q’ (__,__)
R’ (__,__)
S’ (__,__)
4)
Draw the polygon with vertices A(1,3), B(5,6) and C(5,3)
Find the coordinates of the vertices under the specified rotation(s)
Draw the image(s) of each vertex under the transformation.
A (1,3)
B (5,6)
C (5,3)
Answer Key
R90
A’ (__,__)
B’ (__,__)
C’ (__,__)
R180
A’’ (__,__)
B’’ (__,__)
C’’ (__,__)
Order of Operations
1) (6 + 3) – (60 – 91)
2) -2
3) Diego is correct. The answer should be -56.
Evaluating Expressions
1) 96
2) 48
3) 100
Distributive Property
1) 2xy + 2wx
2) -2x – 3
3) 6x – 40y
4) 9ab – 2ab2 – 6a2b
Equations
1) No, She should have subtracted 8 from both sides first.
2) w = 8
3) p = 3
4) c = -1
5) y = 3.1
Inequalities
1) y > 7
2) y < -15
3) x < 1.75
4) y < -6
5) B
6) C
Proportions
1) a. yes b. no
2) a. y = 8 b. b = 18
3) a. 12 hours b. 720 minutes
Similar Figures
1) x= 3.6 y = 12.5
2) x = 27
Unit Price/ Best Buy
1) $.39
2) $.19
3) 5 cans
4) 24 oz
Percents
1) 15%
2) 56.25
3) 21
4) 16.7%
5) $66.04
6) $292.50
7) $43460000
Scientific Notation
1) 735
2) 49920
3) .0079
4) 4.62
1) 3.572 x 108
2) 4.52 x 10-4
3) 4.732 x 104
4) 9.87 x 10-2
Laws of Exponents
1) a7b8c9
2) x
y2
3) p7q10r3
4) 2y 1 5c4
b3
5) 10b9
a
Angles
1) x = 10
2) 2
3) x = 15
4) 75 degrees and 105 degrees
5) 65 degrees
Linear Equations
X
-2
-1
0
1
2
-2x + 5
-2(-2) + 5
-2(-1) + 5
-2(0) + 5
-2(1) + 5
-2(2) + 5
x
-3
2
3
-2
0
y
-18
7
12
-13
-3
Pythagorean Theorem
1) yes, yes, no
2) 48 ft
3) 16.16 in
4) 781.02 ft
5) 35 ft
Transformations
1) D
2) C
y
9
7
5
3
1