Download Number Theory - Colts Neck 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.
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
multiple
•
•
what you say when you
skip count by a given
number
the product of a given
whole number and another
whole number
Some multiples of 4 are 4, 8, 12, 16, . . . .
Any number has an INFINITE number of multiples.
prime number
•
•
a number with exactly one
Examples of prime numbers are 2, 3, 5, 7, and 11.
factor pair
1 is not a prime number, because it only has 1 factor, itself.
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
Examples of composite numbers are 4, 6, 8, and 9.
square number
A number you get when you
multiply two of the same
numbers
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
exponent
In a power, the number of times a
base number is used as a factor
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
In 53, the exponent is 3. So, 53 is equal to 5 x 5 x 5
or 125.
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= 1 5 and n = 2. Write your work on the lines
provided.
1. 8.6 + n
2. 3 + 5x
_________________
_________________
_________________
_________________
3. 4x - 2n
4. x
n
_________________
_________________
_________________
_________________
Fractions, Decimals, and Percents
Important Concepts
fraction
Examples
Describes one or more parts
of a whole that is divided
into equal parts.
1
means 1 part out of a total of two equal parts
2
A number that is less than 1
but greater than zero.
1
5
=
= 0.5
2
10
decimal
percent
1
50
=
= 50%
2
100
Percent means “out of 100”.
improper fraction
An improper fraction has a
numerator that is greater
than, or equal to, one.
8
7
1
means (1 whole) and more
7
7
7
mixed number
8
1
=1
7
7
A number that is both a
whole number and a fraction.
equivalent fractions
•
•
fractions that are
equal in value but have
different numerators
and denominators
fractions that have
the same amount
1
2
3
4
=
=
=
=...
2
4
6
8
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.
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.
Subtraction Algorithm
1.
2.
3.
4.
Find a common denominator.
Write equivalent fractions.
Borrow if necessary.
Subtract numerators and keep
denominator.
5. Subtract whole numbers, if
necessary.
6. Simplify your answer.
Solve each problem. Use separate paper if necessary. NO CALCULATORS!
1.
4.
2
5
2
+ 1
6
5
10
2
1
- 1
8
5
2.
4-
5.
6
5
8
3
4
+ 2
8
7
Multiplication Algorithm
1. Change all whole or mixed numbers to
improper fractions.
2. Multiply numerators.
3. Multiply denominators.
4. Simplify your answer.
Look to simplify before you multiply by
canceling a numerator with a denominator that
has a common factor.
3.
6.
3
2
9
3
- 1
2
9
2
3
+ 1
2
3
Division Algorithm
1. Change all whole or mixed
numbers to improper fractions.
2. Keep the dividend and change the
divisor to its reciprocal.
3. Then follow the rules for
multiplication.
OR
1. Change all whole or mixed
numbers to improper fractions.
2. Find a common denominator for
both fractions.
3. Write equivalent fractions.
4. Divide the numerators only.
Simplify your answer.
1.
2
2
5
x 1
5
6
2.
5
÷ 3
7
3.
7
9
x 1
1
2
3.
3
5.
2
1
÷ 1
6
5
6.
9
13
÷
14
24
Decimals
2
9
x 1
2
3
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
Polygon: a two-dimensional geometric figure formed of three or more straight sides
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
Classify each polygon. Be as specific as possible.
1.
2.
3.
_____________________
_____________________
_____________________
4.
5.
6.
_____________________
_____________________
_____________________
Area, Perimeter, and Volume
Important Concepts/Definitions
Perimeter: the distance around the outside of any polygon
Formula: add the lengths of all the sides = units
Area: the amount of surface a figure covers
Formula: length x width = square units
Volume : the number of cubic units needed to fill a solid figure
Formula: length x width x height = cubic units
Solve each problem. Use separate paper if necessary. NO CALCULATORS!
1.
2.
4.32 mm
3.
7
ft
12
3
2 in
7
5
ft
8
7.71 mm
`
5
4 in
6
Area_________________
Area _________________
Area_________________
Perimeter_____________
Perimeter _____________
Perimeter_____________
5.
6.
4.
1
1 ft
8
1
2 ft
4
3
2 ft
8
Volume________________
Length: 7.31 cm
3
4 ft
Length: 5
Width: 3.2 cm
3
Width: 8 ft
Height: 4.28 cm
5
Height: 2 ft
7
Volume________________
Volume______________
Problem Solving
•
•
•
•
•
•
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.
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. During the month of April, rain gauges were set up in various locations to
measure the amount of rainfall in inches. The line plot below shows the
findings.
April Rain
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
0
Inches
• What was the total amount of rain collected by all of the rain gauges?
• If all of the rain collected was poured equally among each of the rain
gauges, how many inches of rain would be in each gauge?
2. Consider the following pattern: 1, 5, 25, 125. If this pattern of numbers
continues, what are the next two numbers in the pattern? Explain.
3. 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.
4. Joe wants to pack a box with books. He needs a box that has a volume of
exactly 1,344 cubic inches. If the length of the box is 14 inches and the height
of the box is 8 inches, what must be the measure, in inches, of its width?
Find the ordered pair for each point.
6. H (
,
)
7. J (
,
)
8. L (
,
)
9. K (
,
)
Measurement Conversion
52 weeks = 1 year
365 days = 1 year
8 fluid ounces = 1 cup
2 cups = 1 pint
2 pints = 1 quart
4 quarts = 1 gallon
1000 milliliters (mL) = 1 liter (L)
12 inches = 1 foot
10 millimeters = 1 centimeter
3 feet = 1 yard
100 centimeters = 1 meter
5,280 feet = 1 mile
1,000 meters = 1 kilometer
16 ounces = 1 pound
2,000 pounds = ton
1,000 milligrams = 1 gram
1,000 grams = 1 kilogram
Solve each problem. Use separate paper if necessary.
1. A gallon contains 128 ounces. Paul wants to divide 3 gallons of apple cider equally among
the 2 dozen friends at his party. How many ounces of apple cider will each friend
receive?
2. If you only needed 1 cup of milk, what is your best choice at the grocery store – a
quart container, a pint container, or a ½ gallon container? Explain.