Download Number Theory - Colts Neck Township Schools

‘Number Theory
Important Concepts
factor pair
 a pair of numbers whose
product equals a given
number
 dimensions of a rectangle
factors
A number that “fits evenly” into a
given number.
multiple
 what you say when you skip
count by a given number
 the product of a given whole
number and another whole
number
prime number
 a number with exactly one
factor pair
 has two different factors, 1
and the number itself.
composite number
A number that has three or more
factors and two or more factor
pairs
square number
A number you get when you multiply
two of the same numbers
Examples
The factor pairs of 18 are…
1 x 18
2x9
3x6
All the factors of 18 are 1, 2, 3, 6, 9, and 18
Some multiples of 4 are 4, 8, 12, 16, . . . .
Any number has an INFINITE number of multiples.
Examples of prime numbers are 2, 3, 5, 7, and 11.
1 is not a prime number, because it only has 1 factor,
itself.
Examples of composite numbers are 4, 6, 8, and 9.
16 = 4 x 4…..16 is the square number
36
Prime factorization
Process of writing a number a s a
product of prime numbers
2 x 18
9 x 2
3 x 3
Therefore, the prime factorization of 36 is…..
2 x 2 x 3 x 3
List all the factor pairs for 32._____________________________
List all the factors for 32. _________________________________
List the first four multiples for 32. ____, ____, ____, ____
Label each number prime or composite:
27_________
19_________
2_________
36_________
Circle the square numbers…
30,
25,
100,
40,
64,
21
Write the prime factorization for …
45
84
PreAlgebra Concepts
Important Concepts
order of operations
The rules which tell which operation
to perform first when more than one
operation is used.
PEMDAS
Parenthesis, Exponents,
Multiplication, Division, Addition,
Subtraction


Examples
Find the value of the expressions.
3+7x6÷3–4 =
3 + 42 ÷ 3 – 4 =
3 +
17 - 4 =
Multiply OR divide in order from left to right.
Add OR subtract in order from left to right.
evaluating algebraic expressions
Substitute values for the variables
and evaluate using the order of
operations above.
14 - 4 =
= 13
Find the value of the expression if x = 10, n = 5.
15 + n =
or
18x =
15 + 5
18 ( 10 )
= 20
= 180
Evaluate each algebraic expression for x= 15 and n = 2. Write your work on the lines
provided.
1. 8.6 + n
2. 3 + 5x
_________________
_________________
_________________
_________________
3. 4x - 2n
4. x
n
_________________
_________________
_________________
_________________
Data Analysis
Important Concepts
mean
The sum of the values
divided by the number of
values in the set.
median
The middle number or the
average of the middle
numbers in a set when the
numbers are arranged in
order from least to
greatest.
mode
The number(s) that occur(s)
most in a set of numbers.
range
The difference between the
greatest and least values in a
data set.
Examples
{1, 2, 3, 4}
1 + 2 + 3 + 6 = 12 and 12 ÷ 4 = 3 so the mean is 3
{4, 5, 6, 7, 8}
The median is 6 because it marks the middle value
when the numbers are ordered from least to
greatest.
{1, 2, 2, 2, 3, 4}
The mode is 2 because it is the value that occurs
most often.
{1, 2, 3, 4, 5}
The range is 4 because 5 – 1 = 4.
9, 11, 15, 11, 16, 6, 9
mean
_____
median ______
mode
______
range ______
If Mike’s bowling scores are 52, 38, 65, and 24, what would he need to score in
the fifth game to receive have an average of 44?
Fractions, Decimals, and Percents
Important Concepts
Examples
fraction
1
Describes one or more parts
means 1 part out of a total of two equal parts
2
of a whole that is divided
into equal parts.
decimal
1
5
=
= 0.5
A number that is less than 1
2
10
but greater than zero.
1
50
percent
=
= 50%
2
100
Percent means “out of 100”.
improper fraction
8
7
1
An improper fraction has a
means (1 whole) and more
7
7
7
numerator that is greater
than, or equal to, one.
mixed number
8
1
=1
A number that is both a
7
7
whole number and a fraction.
equivalent fractions
 fractions that are
equal in value but have
1
2
3
4
=
=
=
=...
different numerators
2
4
6
8
and denominators
 fractions that have
the same amount
Complete the missing values based on the given fraction, decimal, or percent.
fraction
1.
decimal
percent
1
83 %
3
2.
4
5
3.
5.
4.
.13
7.
6.
8.
3
4
9.
20%
10.
Probability
Important Concept
Examples
You are rolling a standard number cube {1, 2, 3,
probability
4, 5, 6}. Find the probability of each event.
Probability is the likelihood that an event
1
will occur. Probability can be expressed as
P(5) =
6
a fraction, decimal, or percent.
3 1
P(even number) =
=
6 2
P(event) = number of favorable outcomes
number of possible outcomes
P(composite number) =
2 1
=
6 3
All of the shapes above were placed in a bag. As I reached into the bag, what is
the probability that I will pull out:
 P (a polygon)?
 P (an equilateral triangle)?
 P (a quadrilateral)?
 P (an obtuse or a right triangle)?
Operations with Fractions and Mixed Numbers
Addition Algorithm
1.
2.
3.
4.
5.
Find a common denominator.
Write equivalent fractions.
Add numerators and keep denominator.
Add whole numbers, if necessary.
Simplify your answer.
1.
2.
3.
4.
5.
6.
Subtraction Algorithm
Find a common denominator.
Write equivalent fractions.
Borrow if necessary.
Subtract numerators and keep
denominator.
Subtract whole numbers, if
necessary.
Simplify your answer.
Solve each problem. Use separate paper if necessary. NO CALCULATORS!
1.
4.
2
2
5
+ 1
5
6
10
2
1
- 1
8
5
2.
4-
5.
6
5
8
3
4
+ 2
8
7
3.
6.
3
2
9
3
- 1
2
9
2
3
+ 1
2
3
Decimals
Place Value: The position of a digit in a number that is used to determine the value of the
digit.
Addition Algorithm
Subtraction Algorithm
1. Line up equal place values so you are
adding equal sized pieces.
2. Put in zeros as place holders, if
necessary.
3. Add beginning with the smallest place
value.
4. Bring down the decimal point into the
sum.
Multiplication Algorithm
1. Multiply as you would with whole
numbers.
2. Count the number of decimal places.
3. The total number of decimal places is
where you put the decimal in the
product
example:
2
1
3.42
x 5
17.10
(The total number of
decimal places was 2)
1. Line up equal place values so you
are adding equal sized pieces.
2. Put in zeros as place holders, if
necessary.
3. Borrow and rename when
necessary.
4. Subtract beginning with the
smallest place value.
5. Bring down the decimal point into
the difference.
Division Algorithm
1. If the divisor is a whole number,
bring the decimal straight up into
the quotient. Follow your division
algorithm for whole numbers,
adding zeros to the dividend as
necessary.
2. If the divisor is a decimal number,
multiply divisor and dividend by a
power of ten that will make the
divisor a whole number. Then
follow your division algorithm for
whole numbers, adding zeros to
the dividend as necessary.
Solve each problem. Use separate paper if necessary. NO CALCULATORS!
1.
123.5 x 0.25
2.
0.3 + 8.9
4.
264.051 – 2.3
5. 4.002 + 22 + 0.75
3.
5 – 0.671
6.
9.36 ÷ 12
Geometry
Important Concepts/Definitions
Point: An exact location in space.
Line: A straight path of points that goes on forever in two directions.
Line Segment: Part of a line that has two endpoints.
Midpoint: The point halfway between the endpoints of a line segment.
Ray: Part of a line. It has one endpoint and extends forever in only one direction.
Parallel Lines: never cross and stay the same distance apart
Intersecting Lines: pass through the same point
Perpendicular Lines: intersecting lines that form square corners (right, 90° angles)
Angle: formed by two rays intersecting at a common endpoint called a vertex
Acute Angle: an angle that measures less than 90°
Right angle: an angle that measures exactly 90°
Obtuse angle: an angle that measures more than 90° and less than 180°
Straight angle: an angle that measures exactly 180°
Triangle: a polygon with three sides
Equilateral triangles : all sides are the same length
Isosceles triangles: at least two sides are the same length
Scalene triangles: no sides are the same length
Equilateral triangles : all sides are the same length
Isosceles triangles: at least two sides are the same length
Scalene triangles: no sides are the same length
Right triangles: one angle measures 90°
Acute triangles: each angle measures less than 90°
Obtuse triangles: one angle measures more than 90° but less than 180°
Quadrilateral: a polygon with four sides
Parallelogram: both pairs of opposite sides parallel and equal in length
Trapezoid: only one pair of parallel sides
Rectangle: a parallelogram with four right angles
Rhombus: a parallelogram with all sides the same length
Square: a rectangle with all sides the same length
Problem Solving
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


Tips for Problem Solving
Read the question first so you know what you need to find.
Underline the important facts as you read the entire problem.
Look for key words as you read.
Plan how to solve the problem.
Find the solution and check your solution for reasonableness/accuracy.
Remember to label your solution.
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


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Answers should include…
Words that explain your strategy
Work that shows your calculations and steps
Picture(s) that visually show your understanding
Answer(s)
Label(s)
Solve each problem.
1. Complete the table with three ordered pairs that satisfy the equation.
Graph the equation.
Y = X – 1
X
1
2
3
4
Y
2. In Mr. Smith’s class, there are 8 students with brown hair, 5 students with
blond hair, and 12 students with black hair. One student is selected to answer a
problem on the board. What is the probability that the student selected has
black hair.
3. Consider the following pattern: 1, 5, 25, 125. If this pattern of numbers
continues, what are the next two numbers in the pattern? Explain.
4. Jim bought 5 pieces of wood. They measured 13.25 inches, 13.3 inches,
13.008 inches, 12.999 inches and 13.03 inches in length. List the pieces of wood
in order from shortest to longest.
Find the ordered pair for each point.
6. H (
,
)
7. J (
,
)
8. L (
,
)
9. K (
,
)
10. Which grid shows a triangle after a reflection across a line?