1. CONFIDENTIAL1 ConceptPage # 1.Divisibility2-4 2.Prime Factorization5-8 3.Powers & Exponents9-13 4.Order of Operations14-16 5.Variables & Expressions17-19 6.Problem Solving 20-23 7.Practice Problems24-30 Numbers & Algebraic Concepts

2. CONFIDENTIAL2 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 3 4 5 6 7 8 9 12 14 15 18 19

3. CONFIDENTIAL3 DIVISIBILITY RULES A whole number is divisible by: 2if the ones digit is divisible by 2 3if the sum of the digits is divisible by 3 5if the ones digit is 0 or 5 10if the ones digit is 0

4. CONFIDENTIAL4 DIVISIBILITY RULES A whole number is divisible by: 4if the number formed by the last two digits is divisible by 4 6if the number is divisible by 2 AND 3 9if 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.

5. CONFIDENTIAL5 Remember that when two or more numbers are multiplied, each number is called a FACTOR of the product*. 1 x 6 = 6and2 x 3 = 6 1, 6, 2, and 3 are the factors of 6 * Remember that "product" is the answer to a multiplication problem.

6. CONFIDENTIAL6 COMPOSITE NUMBER - A number greater than 1 with more than two factors How to identify COMPOSITE NUMBERS and PRIME NUMBERS Can you think of a number that we would classify as COMPOSITE? What are its factors?

7. CONFIDENTIAL7 PRIME NUMBERS A Prime Number is a whole number that has exactly two factors.....1 and itself

8. CONFIDENTIAL8 A factor tree can be used to find the PRIME FACTORIZATION of a number. Write the number being factored at the top. Choose any pair of whole number factors. Continue to factor any number that is not prime. Except for the order, the prime factors are the same. 54 3 18 3 2 9 3 2 3 3 x x x 2 27 2 3 9 2 3 3 3 x x x x x x x x x THE PRIME FACTORIZATION OF 54 IS 2 x 3 x 3 x 3

9. CONFIDENTIAL9 Numbers expressed using Exponents are called Powers WordsExpressionsValue 2 5 3 2 10 3 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

10. CONFIDENTIAL10 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 = 3 4 = 81...

11. CONFIDENTIAL11 We can also refer to writing 4 5 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. 4 5 = 4 x 4 x 4 x 4 x 4 = 1,024

12. CONFIDENTIAL12 Exponents can be used to write the Prime Factorization of a number. Example: 24 2 12 2 62 3 2 2 2 x x xx x x OR 2 3 x 3 (Start with the smallest prime factor)

13. CONFIDENTIAL13 A Numerical Expression is a combination of numbers and operations. Examples:4 + 3 * 5 2 2 + 6 ÷ 2 (10 * 8) - 7 1 2 3 4 5 6

14. CONFIDENTIAL14 Order of Operations 1. Parentheses 2. Exponents 3. Multiplication Division 4. Addition Subtraction Simplifying the expressions inside grouping symbols examples: (3+5) or (4*6) Find the value of all powers examples: 2 3 or 4 2 Perform multiplication or division in the order in which it occurs when reading the expression from left to right. Perform addition or subtraction in the order in which it occurs when reading the expression from left to right. P E M D A S

15. CONFIDENTIAL15 We can remember the Order of Operations as PEMDAS P E M D A S arentheses xponents ultiplication ivision ddition ubtraction "Please Excuse My Dear Aunt Sally"

16. CONFIDENTIAL16 P E M D A S arentheses xponents ultiplicatio n ivision ddition ubtraction 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.

17. CONFIDENTIAL17 3 + n is an "ALGEBRAIC EXPRESSION" Numbers Operations Variables Algebraic Expressions consist of Numbers, Operations, and Variables.

18. CONFIDENTIAL18 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

19. CONFIDENTIAL19 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

20. CONFIDENTIAL20 When solving math problems, it is often helpful to have an organized problem-solving plan. U P S nderstand lan olve

21. CONFIDENTIAL21 U To 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 the important facts in the problem.)

22. CONFIDENTIAL22 Next, we must P lan -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 plus sum total in all Subtract Multiply Divide minus difference less times product of quotient

23. CONFIDENTIAL23 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 in the order in which they should be used to solve the problem.

24. CONFIDENTIAL24 Practice problems

25. CONFIDENTIAL25 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

26. CONFIDENTIAL26 1.) What is a Prime Number? Can you give me some examples? 3.) Tell whether each number is Prime, Composite, or Neither. a.12 b.5 c.1 d. 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

27. CONFIDENTIAL27 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? 3 6 4. Can you find the value of this power? 2 4 Let's see how well we know Powers and Exponents!

28. CONFIDENTIAL28 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.)4 3 - 24 + 8 = Order of Operations

29. CONFIDENTIAL29 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.)b 2 ÷ ( 3 x c) Variables & Expressions

30. CONFIDENTIAL30 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