Download ESL Summer Work - Solebury School

Summer Assignment:
Mathematics for New International Students
Number Theory:
Set: a collection of objects
 In math, we consider different sets of numbers (listed below).
Subset: a set within another set
 We will see examples below
Types of Number Sets: There are 8 different types of number sets (listed below).
1. Natural Numbers: 1, 2, 3, 4, 5, 6, …
2. Whole Numbers: 0, 1, 2, 3, 4, 5, 6, ...
 Natural numbers are a subset of whole numbers (the set of natural numbers is also in the
set of whole numbers)
3. Integers: …, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 5, 6, …
 Natural numbers and whole numbers are subsets of integers
 Integers can either be:
o Even: …, -4, -2, 0, 2, 4, 6, 8, …
o Odd: …, -3, -1, 1, 3, 5, 7, 9, …
4. Rational Numbers: any number that can be written as


is called a fraction: formed by dividing one integer by another integer
o “a” is called the numerator, and “b” is called the denominator
Natural numbers, whole numbers, and integers are subsets of rational numbers
5. Irrational Numbers: any number that cannot be written as


Examples: √ , π, e, …
The set of irrational numbers does not have a subset
6. Real Numbers: contains both rational and irrational numbers
 Natural numbers, whole number, integers, rational numbers, and irrational numbers are
subsets of real numbers
7. Imaginary Numbers: i = √
 Examples: 3i, ½i, -2i, i√ , …
 The set of imaginary numbers does not have a subset
8. Complex Numbers: of the form a + bi where a and b are real numbers
 Examples: 2 + 3i, √ + πi, …
Variable: a symbol that represents a number. Usually we use letters (x, y, n, t, …) for variables
 Examples:
o 3x + y  x and y are variables
o 2t2 – 3t + 1 = 0  t is a variable
Exercise 1: Complete the following problems. Use definitions and examples above to help you.
a. Express an even number using the variable x as an integer.
b. Write your own examples for each number set.
Algebraic Operations (6 Different Types):
1. Addition (+):
 Other names:
o Plus
o More than
o Sum
o Increase
o Gain
 Examples:
o 3+x
o 3 plus x
o 3 more than x
o The sum of 3 and x
o x increased by 3
2. Subtraction (-):
 Other names:
o Minus
o Take away
o Less than
o Less
o Difference
o Decrease
o Lose
 Examples:
o 3–x
o 3 minus x: 3 – x
o 3 take away x: 3 – x
o 3 less than x: x – 3
o The difference of x and 3 is x – 3
o x decreased by 3 is x – 3
3. Multiplication (×, ∗, ):
 Other names:
o Multiply
o Times
o Product
 Examples:
o 3 multiplied by x is 3 ∗ x = 3x
o 3 times x = 3x
o The product of 3 and x is 3x
4. Division (÷, /):
 Other names:
o Divide
o Over
o Quotient
 Examples:
3
o 3 divided by x is 3 ÷ x = 3/x =
𝑥
o 3 over x = 3/x
o The quotient of 3 and x is 3/x
5. Raising to a Power (or raising to an exponent):
 Exponent (or Power): the number of times a number is to be multiplied by itself.
 Base: the number that is raised to a power or exponent
 Examples of Powers:
o x2  has a power (or exponent) of 2 (means to multiply x by itself 2 times)
 Also called “x-squared,” “x to the second power”
 The base number is x.
o x3  has a power (or exponent) of 3
 Also called “x-cubed,” “x to the third power”
o x4  has a power (or exponent) of 4
 Also called “x to the fourth power”
o We can continue this for powers (exponents) of 5, 6, 7, 8, 9, …
o It also works for negative exponents:
1
 x-1 = 1  has a power (or exponent) of -1
𝑥

 Also called “x to the negative 1 power” or “1 over x to the first power”
We can continue this for powers (exponents) of -2, -3, -4, -5, …
6. Roots:
 Also called a radical
 Examples of Roots:
1
o
√𝑥 = 𝑥 2  has a power (or exponent) of ½
 Also called “square root of x” or “x to the ½ power”
o
3
o
o
o
1
√𝑥 = 𝑥 3  has a power (or exponent) of 1/3
 Also called “cubed root of x” or “x to the 1/3 power”
5
6
We can continue this for 4√𝑥 , √𝑥 , √𝑥 , 7√𝑥 , …
3
2
√𝑥 2 = 𝑥 3  has a power (or exponent) of 2/3
 Also called “x to the 2/3 power”
𝑚
We can continue this for √𝑥 𝑛 where m and n are integers (see page 1 for the definition)
Algebraic Expression: an expression that contains variables, operations, and numbers.
 Example:
o Write an algebraic expression for “the product of a number x and three times y”
 The product means we are going to multiply
 A number x means x is a variable
 Three times y means we are going to multiply 3 and y
 Algebraic Expression: x * 3 * y = x3y = 3yx
Exercise 2: Write an algebraic expression for the problems below. Use definitions and examples to help
you.
a. A number x increased by three times y
b. A number increased by itself
c. Three times the difference of x and 2
d. 9 less than 3
e. Half of y increased by three times the difference of x cubed and 7
f. Double the square root of 9
Equations: two algebraic expressions that could be joined by an equal sign (= is an equal sign)
 In word problems, the word “is” usually means “=”
 Examples:
o x+7=2
o 3=3
o 4x + 2y = x2
 Word Problem: The total length of rope, in feet, used to put up tents is 60 times the number of
tents. Write an equation and define variables.
o The total length = 60 times the number of tents
 We are not given the total length. So, we can use a variable L to represent the
total length. We also do not know the number of tents. So, we can use the
variable T to represent the number of tents.
 Variables:
 L = Total Length
 T = Number of Tents
 Equation:
 L = 60*T
Exercise 3: Define variables and write an equation to model each situation. Use definitions and examples
to help you.
a. The total cost is the number of cans times $0.70.
b. Maria receives $5 for each flower she sells. She already has $40 saved. The total amount
that Maria has in her bank account a month later is $100.
Order of Operations: We will use this many times. Here is a way to memorize the order in which you
perform an operation (multiplication, division, addition, subtraction, powers, etc.)
Parenthesis  ( )
Exponents
Multiplication OR Division
Addition OR Subtraction
We use order of operations (PEMDAS) when simplifying algebraic expressions. You always perform any
operations inside parentheses first. Then, simplify any exponents. Afterwards, you multiply or divide in
order from left to right. Finally, you subtract in order from left to right
Example Using PEMDAS:
 Simplify: (3*7)2 – 23 ÷ 4
o Step 1: are there any Parentheses?
 Yes  (3*7)2 – 23 ÷ 4. Evaluate inside parentheses first.
 212 – 23 ÷ 4
o Step 2: are there any Exponents?
 Yes  212 – 23 ÷ 4
 441 – 8 ÷ 4
o Step 3: do we have to Multiply or Divide?
 Yes, we have to divide  441 – 8 ÷ 4
 441 – 2
o Step 4: do we have to Add or Subtract?
 Yes, we have to subtract  441 – 2
 439 is our answer
Exericse 4: Evaluate each expression for c = 2 and d = 5. Explain each step.
a. 40 – d2 + cd * 3
b. c4 – d ÷ 2 * 6
Inequality: a symbol that compares the value of two expressions using an inequality symbol:
 < means less than
 > means greater than
 ≤ means less than OR equal to
 ≥ means greater than OR equal to
 = means equal to
 ≠ means NOT equal to
Example:
 -3/8 is less than ½. Using inequality symbols, -3/8 < ½
Graphing on the Coordinate Plane
Coordinate Plane: Two number lines that intersect at right angles.
x-axis: the horizontal axis on the coordinate plane (left to right).
y-axis: the vertical axis on the coordinate plane (up and down)
Origin: intersection of x-axis and y-axis (x and y are both equal to 0). Also divides the coordinate plane
into four parts called quadrants.
y-axis
Ordered Pair: identifies the location of a point.
 They are the coordinates on the graph.
 The first value in the ordered pair is the x-coordinate. The second
Value in the ordered pair is the y-coordinate.
Quadrant I
Quadrant II
 Example: (-2, 4) is an ordered pair
o -2 = x-coordinate. It tells you how far to move right
(positive) or left (negative)
Origin (0, 0)
x-axis
o 4 = y-coordinate. It tells you how far to move up
(positive) or down (negative)
Quadrant IV
Quadrant III
=
Slope =
2; 1
2; 1
=
 Example: The slope of the line is
Points Given: (-3, 3) and (-2, 0)
x1 = -3
x2 = -2
y1 = 3
y2 = 0
Slope =
3;0
;3; ;2
=
3
;3:2
=
3
;1
=
3
3
1
6
5
(-3, 3) 4
3
2
(-2, 0) 1
0
-6 -5 -4 -3 -2 -1-1 1 2 3 4 5 6
-2
-3
-4
-5
-6
Exercise 5: Graph the points on the coordinate plane and find the slope.
a. (3, 0) and (5, 0)
b. (-1.5, 2) and (-2, 1)
c. (-2, -3) and (-3, -2)
d. (4, -2.5) and (5, -1.5)
Linear equation: equation of a line (look at graph in slope example)
 The highest exponent of a variable is 1.
y-intercept: the y-coordinate of the point where a line crosses the y-axis
 Occurs when x = 0
 Example:
o In the linear equation 4x + 3y = 12, we can solve for the y-intercept by setting x = 0.
4(0) + 3y = 12
3y = 12
y = 4, so the y-intercept occurs at the point (0, 4)
x-intercept: the x-coordinate of the point where a line crosses the x-axis
 Occurs when y = 0
 Example:
o In the linear equation 4x + 3y = 12m we can solve for the x-intercept by setting y = 0.
4x + 3(0) = 12
4x = 12
x=3
The x-intercept occurs at the point (3, 0)
Slope-Intercept Form of a Linear Equation: y = mx + b
 m = slope
 b = y-intercept
Parallel Lines: lines in the same plane that never intersect.
 Two lines that are parallel have the same slope.
 Examples:
o y = 2x + 6 and 3y = 6x + 2 are parallel lines
 This is because we can rewrite the equation 3y = 6x + 2 in slope-intercept form
2
by dividing both sides of the equation by 3. Then, we have y = 2x + 3. The slope
of this line is 2, which is the same as the slope of the line y = 2x + 6
o Lines a and b are parallel (they will never cross each other)
Perpendicular Lines: lines that intersect to form right angles.
 Right Angle = 90°
 Two lines that are perpendicular have slopes that are negative reciprocals
 To know what a negative reciprocal is, first we need to know what a reciprocal is:
1
o Reciprocal: If we have a number a, then the reciprocal is
 We are switching the numbers in the numerator and denominator
 Negative reciprocal: the result of taking the reciprocal of a number and then changing the sign.
o Examples:
4
5
 The negative reciprocal of 5 is 4


The negative reciprocal of -6 is
1
6
Examples:
1
o y = 2x + 1 and y = 2x + 4 are perpendicular lines
 2 and – ½ are negative reciprocals, so the lines are perpendicular
o Lines e and c are perpendicular because they form a 90° angle (the square tells us that
the angle is 90°)
Exercise 6: Are the graphs of the lines in each pair parallel? Explain.
a. y = 4x + 12 and -4x + 3y = 21
b. y = -3x and 21x + 7y = 14
Functions
Function: for every input value, there is exactly one output value.
 The input value is also called the independent variable. The output value is also called the
dependent variable
 The possible values for the independent variable or input are called the domain of the function.
The possible values for the dependent variable or output are called the range of the function.

Example: Maria earns $7 per hour for baby-sitting after school and on Saturday. She works no
more than 16 hours a week
a. Identify the independent and dependent quantities.
Independent: The number of hours Maria spent baby-sitting
Dependent: the amount of money Maria earns
b. Write an equation to represent the situation.
H = number of hours Maria spent baby-sitting
M = amount of money Maria earns
M = $7H
c. Find the domain and range.
We are given that she works no more than 16 hours per week. This means that she
will earn no more than $7*16 = $112. We input (domain) 16 hours into our
equation (part b) to find the output (range) of $112.
If Maria works 0 hours, then she makes $7*0 = $0. So Maria makes $0.
Therefore, the range could go from $0 - $112. The domain could go from 0 hours –
16 hours.
Exercise 7: Identify the independent and dependent quantities for each situation, and find reasonable
domain and range values.
a. Tara’s car travels about 25 miles on one gallon of gas. She has between 10 and 12 gallons of
gas in the tank.
b. Sal and three friends plan to bowl one or two games each. Each game costs $2.50.
Ratios and Rates
Ratio: a comparison of two numbers, a and b, by division (a and b can be any real numbers).
 Other Names:
o The ratio of a to b
o a:b
o
 a and b must have the same unit (i.e. inches, centimeters, meters, gallons, etc.)
 Examples:
o 4 inches : 5 inches
o $2.00 : $3.00
Rate: a ratio where a and b do not have the same units (i.e. a is in centimeters and b is in meters)
 Examples:
o
o
0 72
16
 dollars and ounces are not the same unit of measurement
3372 km : 83.6 h  kilometers and hours are not the same unit of measurement
Proportion: an equation that states two ratios are equal.
 Examples:
4
2
o 6=3
o
5
9
= 6  In this case, you would solve for t. We can solve for t by cross-multiplying
5
9
= 6 Multiply t * 6 and set it equal to 9 * 5
= ∗
6t = 45
t = 7.5
Exercise 8: Solve each proportion. Use examples above to help you.
8
12
a.
= 30
b.
c.
1
2
=
:5
3
:12
5
=
9
9
Polynomials
Monomial: an expression that is a number, variable, or a product of a number and one or more
variables.
 Examples:
o 12
o y
o -5x2y
o 3
Binomial: an expression that contains two monomials.
 Examples:
o 2x2 + 1
o 6x + 5x2
Trinomial: an expression that contains three monomials.
 Examples:
o 3x4 + 5x2 – 7x
o 9 + 4x + 10x3
Polynomial: a monomial or the sum or difference of two or more monomials.
 Examples:
o 3x4 + 5x2 – 7x + 1
o 4x2 + 7
o 2x
Geometry
Point: can be thought of as a location.
 A point has no size. It is represented by a small dot and is named by a capital letter.
 Example: A is a point
A
Line: a series of points that extends in two opposite directions without end.
 Can name a line with two points on the line OR with a single lowercase letter.
 Example: ⃡ is a line; line t is the same line
t
B
A
Collinear Points: points that lie on the same line.
 Example: in the line above, points A and B are collinear because they are on the same line.
Plane: a flat surface that has no thickness.
 Contains many lines and extends without end in the directions of all its lines.
 Can name a plane by a single capital letter OR by at least three noncollinear points (points that
DO NOT lie on the same line)
 Example: P is a plane; Plane ABC is the same plane
B
P
A
C
Coplanar: points and lines in the same plane.
Segment: the part of a line consisting of two endpoints and all points between them.
 Example:
is a segment. (A and B are endpoints because they are points at the end of the
segment)
A
B
Ray: the part of a line consisting of one endpoint and all the points of the line on one side of the
endpoint.
 Example:
is a ray.
A
B
Angle ( ): formed by two rays with the same endpoint.
 The rays are called the sides of the angle.
 The endpoint is the vertex of the angle.
Side
Vertex
Side
Types of Angles:
 Acute Angles: angles that are less than 90°

Obtuse Angles: angles that are between 90° and 180°

Right Angles: angles that equal 90°
o We use a special symbol for right angles. The square in the corner of the angle tells us
that the angle is 90°

Straight Angles: angles that equal 180°

Adjacent (“next to”) Angles: two coplanar angles with a common side, a common vertex, and
no common interior (inside) points.
o Example: Angles 1 and 2 are adjacent because they share a side
1
2

Complementary Angles: two angles whose measures sum to 90°
o Example: Angles A and B are complementary because they add to 90°
B
50°
A

40°
Supplementary Angles: two angles whose measures sum to 180°
o Example: Angles 1 and 2 are supplementary AND they are adjacent
1
2
Example Using Angles
1. Two angles are supplementary angles and one is twice the other. Find those angles.
If two angles are supplementary angles, then the angles add to 180°
One of the angles is twice the other. This means that if one angle has a measure of x, then the
other angle has a measure of 2x (it is twice as large).
Therefore, since the two angles add to 180°, we can add:
x + 2x = 180°
Now, we solve for x:
3x = 180°
x = 60°
Therefore, one of the angles is 60°, so the other is 2*60° = 120°.
Exercise 9: In the picture to the right…
a. List all complementary angles.
b. List all supplementary angles.
c. List all acute angles.
d. List all obtuse angles.
e. List all right angles.
f. List all straight angles.
g. List all adjacent angles.
Types of Triangles (Shapes that have 3 Sides)
Acute Triangle: all angles are less than 90° inside the triangle.
Obtuse Triangle: one of the angles is more than 90°.
Right Triangle: one of the angles is a right angle (90°).
 Hypotenuse: the longest side of a right triangle; the side opposite
of the right angle in the right triangle.
Scalene Triangle: a triangle where none of the sides are equal.
4 32
5 1
Isosceles Triangle: a triangle where two of the sides are equal.
Equilateral Triangle: a triangle where all of the sides (and angles) are equal.
**NOTE: The angles in a triangle sum (add up) to 180°
Types of Quadrilaterals (Shapes that have 4 Sides)
Diagonal: a line that cuts through a quadrilateral from a corner to the opposite corner.
Trapezoid:
 Only one pair of opposite sides are parallel
Parallelogram:
 Opposite sides are parallel (do not intersect)
 Opposite angles are equal
Rectangle: a type of parallelogram
 Opposite sides are parallel
 Opposite sides are equal (have the same length)
 All angles are 90°
Rhombus: a type of parallelogram
 Opposite sides are parallel
 All sides are equal length
 Opposite angles are equal
Square: a type of parallelogram
 Opposite sides are parallel
 All sides are equal length
 All angles are 90°
**NOTE: The angles in a quadrilateral sum to 360°
Circles
Parts of a Circle:
Chord
Radius
Diameter
Circumference: the distance around a circle.
Diameter: a line in a circle that goes through the center and touches two points on the circle.
Chord: a line segment whose endpoints both lie on the circumference of the circle, but do not go
through the center of the circle.
 A chord is ALWAYS shorter than the diameter
Radius: a line in a circle that goes from the center of the circle to a point on the circle.
 The radius is ½ of the diameter
Area and Perimeter
Area: the amount of space an object occupies.
Perimeter: the distance around a shape.
Currency





A penny = 1¢ (1 “cent”)
A nickel = 5¢ (5 “cents”)
A dime = 10¢
A quarter = 25¢
A dollar = $1
Units of Measurement
Common Abbreviations:
 cm = centimeter
 gr = gram
 km = kilometer
 mi = mile
 US = United States
 oz = ounce
 yd = yard
 l = liter
 ft = foot
 in = inch
 lb = pound
 qt = quart
 gl = gallon
 kg = kilogram
 m = meter
Common Conversions:
 Length measures:
o 1 ft = 12 in
o 1 yd = 3 ft
o 1 mi = 5, 280 ft
o 1 mi = 1, 760 yd
 Liquid measures:
o 1 pint = 2 cups
o 1 quart = 2 pints
o 1 gallon = 4 quarts
o 1 gallon = 8 pints
 Weight measures:
o 1 lb = 16 oz
o 1 ton = 2000 lb
 Temperature:
5
o 1 degree Celsius (C) = 9 ∗
 F = Fahrenheit
3 )
General Mathematics Terms








Solve: to find an answer
o Most of the time, you will be asked to solve using a method (i.e. “Solve by Factoring”)
Evaluate: to compute the value of an expression.
o Example: Evaluate the expression x + 2 when x = 19.
x + 2 = (19) + 2 = 21
Substitute: To plug in a number for a variable.
o Example: In the example above, we substituted (or plugged in) 19 for x
Factor: expressing a quantity as a product of two or more quantities
o Example: Factor x2 – y2.
x2 – y2 = (x + y)(x – y)
Simplify: to convert a mathematical expression such as a fraction or equation to a simpler form
by removing common factors or regrouping elements.
Calculate: To perform mathematical operations.
o Example: Calculate 82.
82 = 64
Estimate: to use logic to make an educated guess that is close to the actual value.
o Another word that has the same meaning is approximate.
o Example: Estimate √ .
We know that √ is between √3 = and √ = . Therefore, the √ is between 6
and 7, most likely around 6.5.
Plot: to graph a point or line.
o Example: Plot the point (3, 4).
Final Test:
1. One number is 4 times larger than another. Their sum is 60. What are the two numbers?
2. The sum of two numbers is 33. Their product is 242. What are the numbers?
3. Write an algebraic expression for: nine times the difference of y and z, increased by 3z.
2
4. Simplify:
(25:22 ) ;5:4 2∗3
5 4:√4)
5. Order the numbers from least to greatest using inequality symbols: √3, 2.1,
6. Graph the following points and find their slope:
a. (-1, 0) and (-2, 0)
b. (3, 5) and (10, 8)
7. Find the x-intercepts and y-intercepts of the following equations:
a. 2x + 8y = 10
b. 0.2x = -0.6y – 14
8. Write the following equations in slope-intercept form:
a. 2x + 8y = 10
b. 0.2x = -0.6y – 14
9. Given the equations of two lines, how do you know they are parallel?
10. Given the equations of two lines, how do you know they are perpendicular?
11. Find the negative reciprocals of the following:
a. 0
4
b.
5
12. Find all possible range values for 3x – 7 = y when x = 1, 2, 3, and 4.
13. Solve the following proportions:
8
2
a.
=
:2
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
16
3
10 ;2353
,
3 2353
b.
=4
;2
State the type of polynomial:
a. 235x
b. 87x2 + 3x + 4 + 9x3
c. 5x + 9
Two angles are complementary. One angle has a measure of 3x + 5 and the other has a measure
of 2x. Find the value of x and find the measure of each angle.
In an isosceles triangle, two of the angles are equal to 4x and one angle has a measure of 80°.
Find the measure of all angles in the triangle.
If a chord is always shorter than the diameter, is it always shorter than the radius as well?
Explain.
An equilateral triangle has a 12 inch perimeter. What is the length of each side of the triangle?
What is the name of a triangle that has an angle greater than 90°?
What is the reciprocal of 2?
If you square the square root of x, what do you get?
A case of one dozen (12) glassware costs $244.80. Find the cost of a single glassware.
One inch equals 2.54 centimeters. A meter is 100 cm. How many inches are there in a meter?