Number Theory

Important Concepts / Examples
factors
Two or more numbers multiplied to get a product. / All the factors of 8 are 1, 2, 4, and 8 because 1 x 8 = 8 and 2 x 4 = 8.
multiple
The product of a given whole number and another whole number. / Some multiples of 8 are 8, 16, 24, 32, . . . .
Any number has an INFINITE number of multiples.
prime number
A number with exactly two different factors, 1 and the number itself. / Examples of prime numbers are 2, 3, 5, 7, and 11.
1 is not a prime number, because it only has 1 factor, itself.
composite number
A whole number with greater than two factors. / Examples of composite numbers are 4, 6, 8, and 9.
check one
# / List of factors / List of 5 multiples / prime / composite
12
30
49
100

PreAlgebra Concepts

Important Concepts / Examples
exponent
In a power, the number of times a base number is used as a factor / In 53, the exponent is 3. So, 53 is equal to 5 x 5 x 5 or 125.
order of operations
The rules which tell which operation to perform first when more than one operation is used.
  1. Simplify expressions inside parentheses.
  2. Find the value of all numbers with exponents.
  3. Multiply OR divide in order from left to right.
  4. Add OR subtract in order from left to right.
/ Find the value of the expressions.
3 x 62 + 4 = 3 x 36 + 4
= 108 + 4
= 112
60 ÷ (12 + 23) ∙ 9 = 60 ÷ (12 + 8) ∙ 9
= 60 ÷ 20 ∙ 9
= 3 ∙ 9
= 27
evaluating algebraic expressions
Substitute values for the variables and evaluate using the order of operations above. / Find the value of the expression if x = 3, y = 5, and z = 20.
2x + z ÷ y = 2 ∙ 3 + 20 ÷ 5
= 6 + 20 ÷ 5
= 6 + 4
= 10

Evaluate each algebraic expression for x= 12, y = 8, and z = 2. Write your work on the lines provided.

1.40 – y2 ÷ z 2.( x + y ) ∙ 32

______

______

______

3.( y + 1 )2 – x 4.5x – z3

______

______

______

Data Analysis

Important Concepts / Examples
mean
The sum of the values divided by the number of values in the set. / {1, 2, 3, 4}
1 + 2 + 3 + 6 = 12 and 12 ÷ 4 = 3 so the mean is 3
median
The number that marks the middle of an ordered set of data. / {4, 5, 6, 7, 8}
The median is 6 because it marks the middle value when the numbers are ordered from least to greatest.
mode
The mode is the value that occurs with greatest frequency in a set of data. / {1, 2, 2, 2, 3, 4}
The mode is 2 because it is the value that occurs most often.
range
The difference between the greatest and least values in a data set. / {1, 2, 3, 4, 5}
The range is 4 because 5 – 1 = 4.

Use the data table to answer each question.

Calories of Selected Vegetables (per serving)
15 35 50
31 5 25
85 2520

1.What is the mean of the calories? Round to the nearest tenth, if necessary.

2.Order the set of data from least to greatest. What is the median of the calories?

3.What is the mode of the calories? How do you know?

4.What is the range of the calories?

Fractions, Decimals, and Percents

Important Concepts / Examples
fraction
A number of the form where a and b are whole numbers. / means 1 part out of a total of two equal parts
decimal
A special form of a fraction based on the place value system. / = = 0.5
percent
Percent means “out of 100”. / = = 50%
improper fraction
An improper fraction has a value that is greater than, or equal to, one. / means (1 whole) and more
mixed number
A number that is written with both a whole number and a fraction. / = 1
equivalent fractions
Fractions that are equal in value but have different numerators and denominators. / = = = = . . .

Complete the missing values based on the given fraction, decimal, or percent.

fraction / decimal / percent
1. / 2. / 35%
/ 3. / 4.
5. / .09 / 6.
7. / 8. / 10%
/ 9. / 10.

Basic Integer Concepts

Important Concepts / Examples
integer
The set of numbers that includes all the whole numbers and their opposites, including 0. / {. . .-4, -3, -2, -1, 0, 1, 2, 3, 4. . .}
negative integers
All integers less than zero and found to the left of zero on the number line. /
-8 -7 -6 -5 -4 -3 -2 -1 0
positive integers
All integers greater than zero and found to the right of zero on the number line. /
0 1 2 3 4 5 6 7 8
opposites
Two numbers the same distance from zero but in opposite directions on the number line.
*The sum of two opposites is 0. / -4 and 4 -9 and 9 -36 and 36 -17 and 17
absolute value
The distance a number is from 0 on the number line.
Absolute value is never negative. / |-9| = 9 |8| = 8 |-13| = 13 |100| = 100

Identify numbers that meet the given requirements.

1.2 positive numbers that are not integers______

2.2 negative numbers that are not integers______

Graph the given numbers on the number line.

3.6, 0, -5, 1, -3

Insert <, >, or = to make a true statement.

4.-9 -85.-3 -66.-2 -4

Name the opposite of each integer given.

7.-12______8.9______

Identify the absolute value.

9.-10______10.8 + 7 x 6  14______

Ratio and Proportion

Important Concepts / Examples
ratio
A comparion of two numbers by division. Does NOT have to be a PART to WHOLE comparison. Can be written in 3 forms
, a to b, a:b /
ratio of triangles to circles , 3 to 7, 3:7
ratio of nonpolygons to polygons , 7 to 6, 7:6
proportion
An equation that shows that two ratios are equivalent. / Corresponding sides of similar shapes are in proportion. All measures shown are in units.

3 9 =
5
15

Use the numbers to write ratios for #1-3. Write each ratio in 3 forms.

-14 35 1 2 24 100 -4 -8 9 16

1.ratio of square numbers to nonsquare numbers______

2.ratio of negative integers to positive integers______

3.ratio of abundant numbers to total numbers______

Identify the missing value in the proportion.

4.= 5.= 6.=

Probability

Important Concept / Examples
probability
Probability is the likelihood that an event will occur. Probability can be expressed as a fraction, decimal, or percent.
P(event) = number of favorable outcomes
number of possible outcomes
The probability of 2 events occurring can be found by multiplying the probability of each independent event. / You are rolling a standard number cube {1, 2, 3, 4, 5, 6}. Find the probability of each event.
P(5) =
P(even number) = =
P(composite number) = =
P(4, 6) = x =
P(1, prime number) = x =

The letters T, H, E, O, R, E, T, I, C, A, and L are put in a bag. Find each probability.

1.P (A, I, or O) = ______

2.P (not O or T)=______

3.P (U)= ______

You choose 1 letter from the bag and then put in back before choosing a second letter. Find each probability.

4.P (T, vowel) =______

5.P (E, T) = ______

Operations with Fractions and Mixed Numbers

Addition Algorithm / Subtraction Algorithm
  1. Find a common denominator.
  2. Write equivalent fractions.
  3. Add numerators and keep denominator.
  4. Add whole numbers, if necessary.
  5. Simplify your answer.
/
  1. Find a common denominator.
  2. Write equivalent fractions.
  3. Borrow if necessary.
  4. Subtract numerators and keep denominator.
  5. Subtract whole numbers, if necessary.
  6. Simplify your answer.
OR
  1. Find a common denominator.
  2. Write equivalent fractions.
  3. Change to improper fractions so no borrowing is necessary.
  4. Subtract numerators.
  5. Simplify your answer.

Multiplication Algorithm / Division 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. /
  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.
  5. Simplify your answer.

Solve each problem. Use separate paper if necessary. NO CALCULATORS!

1.2 + 1 2. ÷ 33. x 1 3.3 - 1

5.10 - 1 6.2 x 37.10 ÷ 8.3 + 1

Operations with Decimals

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.
/
  1. Line up equal place values so you
are adding equal sized pieces.
  1. Put in zeros as place holders, if
necessary.
  1. Borrow and rename when
necessary.
  1. Subtract beginning with the
smallest place value.
  1. Bring down the decimal point into the difference.

Multiplication Algorithm / Division Algorithm
  1. Multiply as you would with whole numbers.
  2. Find the sum of the decimal places in both factors.
  3. Count the same number of decimal places going right to left in the product and place the decimal.
example: tenths x tenths = hundredths
1 place + 1 place = 2 places / 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.
  1. 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!

1.5.5( 9.9)2.0.6 + 5.83.3.4 – 0.972

4.25.041 – 8.3 5.1.009 + 12 + 0.876.936 ÷ 0.12

7.0.492 ÷ 48.12.2 ∙9.43.59 x 0.1

Mixed 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.

Solve each problem. Show work to support your answer. You may use a calculator on this section.

1.Laura has 121 carpet tiles. Each tile is a 1-foot by 1-foot square. Can she use all

of the carpet tiles to make a square pattern without cutting any of the tiles?

2.A florist has 28 white roses, 35 red roses, 14 pink roses, and 70 yellow roses to use in making floral arrangements. He wants to use all the flowers and place an equal number of each color rose in each arrangement. Find the greatest number of floral arrangements he can make.

3.According to a survey, 9 in 10 teens volunteer at least once a year. Of these, one-third help clean up their communities. What fraction of teen volunteers help by cleaning up their communities?

4.Andrew is going to plant three new types of vegetables in his garden. The garden store sells packages of tomatillo seeds for $1.67, chili pepper seeds for $0.89, and pumpkin seeds for $2.32. Andrew buys one of each package. How much change will he receive from a ten-dollar bill?

5.Find the perimeter and area of the figure below.

7 cm

5 cm

6.Claire and Chad want to design a rectangular pen for their new puppy. They want the pen to have an area of 48 square feet. Fencing costs $0.85 per foot. What would be the dimensions and the cost of the least expensive pen Claire and Chad could build, assuming the side lengths are whole numbers?

7.A triangle has side lengths measuring 3 and 7. The measurement of the longest side is missing. Ted says that one possibility for the unknown side length is 11. Is Ted correct? Why or why not?

8.The figure below shows the dimensions of Adam’s bedroom. Adam wants to put new carpet in the room. How much carpet will Adam need?

6

3 2 4

3

9.The Make A Difference Club raised 30% of their fundraising goal of $200. What percent do they have left to earn? How much money do they have left to earn?

10.Alexa wants to use ready-made 6-foot long fence sections for her yard. The yard is a rectangle with dimensions 30 feet by 36 feet. How many fence sections will she need to enclose her entire yard?