CONFIDENTIAL 1

Concept Page #

1. Divisibility 2-4

2. Prime Factorization 5-8

3. Powers & Exponents 9-13

4. Order of Operations 14-16

5. Variables & Expressions 17-19

6. Problem Solving 20-23

7. Practice Problems 24-30

Numbers & Algebraic Concepts

Numbers & Algebraic Concepts

CONFIDENTIAL 2

A whole number is DIVISIBLE by another number if the remainder is 0.

A whole number is EVEN if it is divisible by 2.

A whole number is ODD if it is not divisible by 2.

1

2

34

5

6

7

8 9

12

14

15

18

19

CONFIDENTIAL 3

DIVISIBILITY RULES

A whole number is divisible by:

2 if the ones digit is divisible by 2

3 if the sum of the digits is divisible by 3

5 if the ones digit is 0 or 5

10 if the ones digit is 0

CONFIDENTIAL 4

DIVISIBILITY RULES

A whole number is divisible by:

4 if the number formed by the last twodigits is divisible by 4

6 if the number is divisible by 2 AND 3

9 if the sum of the digits is divisible by 9

The rules for 4, 6, and 9 are related to the rules for 2 and 3.

CONFIDENTIAL 5

Remember that when two or more numbers are multiplied,each number is called a FACTOR of the product*.

1 x 6 = 6 and 2 x 3 = 6

1, 6, 2, and 3are the factors of 6

* Remember that "product" is the answer to a multiplication problem.

CONFIDENTIAL 6

COMPOSITE NUMBER - A number greater than 1 with more than two factors

How to identifyCOMPOSITE NUMBERS

andPRIME NUMBERS

Can you think of a number that we would classify as COMPOSITE?

What are its factors?

CONFIDENTIAL 7

PRIME NUMBERS

A Prime Number is a whole number that has exactly two factors.....1 and itself

CONFIDENTIAL 8

A factor tree can be used to find the PRIME FACTORIZATION of a number.

Write the number beingfactored at the top.

Choose any pair ofwhole number factors.

Continue to factor anynumber that is not prime.

Except for the order, theprime factors are the same.

54 54

3 18

3 2 9

3 2 33

x

xx

2 27

2 3 9

2 33 3 x x x

x

xx

x x x

THE PRIME FACTORIZATION OF 54 IS 2 x 3 x 3 x 3

CONFIDENTIAL 9

Numbers expressed using Exponents are called

Powers

Powers Words Expressions Value

25

32

103

2 to the fifth power

3 to the second power or 3 squared

10 to the third power or 10 cubed

2 x 2 x 2 x 2 x 2

3 x 3

10 x 10 x 10

32

9

1,000

CONFIDENTIAL 10

If we write 3 x 3 x 3 x 3 using an exponent,

the base is 3 AND

the exponent is 4

3 3 3 3 = 34 = 81 . . .

CONFIDENTIAL 11

We can also refer to writing 45 as a

PRODUCT OF THE SAME FACTOR(Remember that a "Product" is the answer to a multiplication problem.)

The Base is 4. The Exponent is 5. So 4 is a Factor 5 times.

45 = 4 x 4 x 4 x 4 x 4 = 1,024

CONFIDENTIAL 12

Exponents can be used to write the Prime Factorization

of a number.

Example: 24

2 12

2 62

32 2 2

x

x

x x

xx OR 23 x 3

(Start with the smallest prime factor)

CONFIDENTIAL 13

A Numerical Expression is a combination of numbers and operations.

Examples: 4 + 3 * 5

22 + 6 ÷ 2

(10 * 8) - 7

1 2 34 5

6

CONFIDENTIAL 14

Order of Operations

1. Parentheses

2. Exponents

3. Multiplication Division

4. Addition Subtraction

Simplifying the expressionsinside grouping symbolsexamples: (3+5) or (4*6)

Find the value of all powersexamples: 23 or 42

Perform multiplication or division in the order in which it occurs when reading the expression from left to right.Perform addition or subtraction in theorder in which it occurs when readingthe expression from left to right.

PE

MDAS

CONFIDENTIAL 15

We can remember the Order of Operations as PEMDAS

P E M D A Sarenthese

s

xponent

s ultiplication

ivision

ddition

ubtractio

n

"Please Excuse My Dear Aunt Sally"

CONFIDENTIAL 16

P E M D A Sarenthese

s

xponent

s

ultiplicatio

n

ivision

ddition

ubtractio

nwhichever comes first whichever comes first

"Please Excuse My Dear Aunt Sally"

20 ÷ 4 + 17 * (9 - 6) = Do the operations in Parentheses first.

20 ÷ 4 + 17 * 3 =

5 + 17 * 3 =

5 + 51 =

56 =

There are no Exponents.

Perform Multiplication or Division in the order in which they occur.The Division should be done first.Then perform the Multiplication.

Finally perform the Addition.

CONFIDENTIAL 17

3 + n is an "ALGEBRAIC EXPRESSION"

Numbers

Operations

Variables

Algebraic Expressions consist of Numbers, Operations, and Variables.

CONFIDENTIAL 18

The VARIABLES in an expression can be replaced with any number.

3 + x

If I substitute a 5 for the x ...........

I have 3 + 5 or 8

This is how we Evaluate (or find the value of) the Expression

CONFIDENTIAL 19

Let's Evaluate the Algebraic Expression 16 + b if b = 25

We replace b with the number 25

16 + b = 16 + 25

= 41

The Value of the Algebraic Expression when b = 25 is 41

CONFIDENTIAL 20

When solving math problems, it is often helpfulto have an organized problem-solving plan.

U

PS

nderstand

lan

olve

CONFIDENTIAL 21

UTo

nderstand the problem, we need to

-read the problem carefully-identify the facts that we know-identify what we need to know (WHAT IS THE QUESTION?)-determine if we have enough or too much information

(Many students find it helpful to highlight or underline theimportant facts in the problem.)

CONFIDENTIAL 22

Next, we must

Plan

-determine how the facts relate to each other-plan a strategy for solving the problem-estimate your answer

Key words play an important role in determining which operations to use.

Add

plussumtotalin all

Subtract Multiply Divide

minusdifferenceless

timesproductof

quotient

CONFIDENTIAL 23

And finally, we

S olve the problem

-use your plan to solve the problem-if your plan does not work, revise it or make a new plan-find the solution-make sure the answer makes sense and is close to your estimate

Keep in mind that numbers do NOT always appear in a problem inthe order in which they should be used to solve the problem.

CONFIDENTIAL 24

Practice problems

CONFIDENTIAL 25

1.) How can I tell if a number is divisible by

a.) 2

b.) 3

c.) 4

d.) 5

e.) 6

f.) 9

g.) 10

Example:

2.) Use Long Division.

41 1,886

Divisibility

CONFIDENTIAL 26

1.) What is a Prime Number? Can you give me some examples?

3.) Tell whether each number is Prime, Composite, or Neither.

a. 12b. 5c. 1d. 41

4.) Write the number 28 as the product of prime numbers.

5.) Use a factor tree to find the Prime Factorization of 60.

2.) What is a Composite Number? Can you give me some examples?

Prime Factorization

CONFIDENTIAL 27

1. Can you write this product using an exponent?

6 x 6 x 6 x 6

2. Can you find the value of this product?

4 x 4 x 4

3. Can you write this power as a product of the same factor?

36

4. Can you find the value of this power?

24

Let's see how well we know Powers and Exponents!

CONFIDENTIAL 28

1.) Can you give me an example of a "Numerical Expression"?2.) What do I mean by "Operations"?

3.) In what order do I perform the "Operations"?

4.) Find the value of each expression?

a.) 5 x 6 - (9 - 4) =

b.) 16 ÷ 2 + 8 x 3 =

c.) 43 - 24 + 8 =

Order of Operations

CONFIDENTIAL 29

1.) In the Algebraic Expression 14n + 5 - 6m

- what are the variables?

-what are the numbers?

-what are the operations?

Evaluate each expression if a = 4, b = 12, and c = 4.

2.) 7c ÷ 4 + 5a

3.) b2 ÷ ( 3 x c)

Variables & Expressions

CONFIDENTIAL 30

1.) In 1990, the population of Sacramento, CA was 370,000.

In 2000, the population was 407,000. How much did the

population increase?

2.) The Smith family wants to purchase a television set and

pay for it in four equal payments of $180. What is the

cost of the television set?

3.) Complete the pattern:

6, 11, 16, 21, ___, ___, ___? ? ?

Problem Solving