Understanding Numbers

Lesson 1: Place Value

The all-time best selling book is Guinness World Records. From October 1955 – June 2002 it sold, 94,767,083 copies This number can be represented in a place-value chart Hundred Millions Ten Millions Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones 9 4  7  6 7  0   8 3 

Place Value Information about a place value chart that you should know: From Right to Left, each group of 3 place values is called a period Within each period, the digits of a number are read as: hundreds, tens, ones Millions Period Thousands Period Units Period Hundred Millions Ten Millions (Ones) Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones

Place Value This place value chart shows the number 3 159 119 Here are some ways to represent this number: Standard Form: 3 159 119 Expanded Form: 3 000 000 + 100 000 + 50 000 + 9000 + 100 + 10 + 9 Number-Word Form: 3 million 159 thousand 119 Word Form: three million, one hundred fifty nine thousand one hundred nineteen Hundred Millions Ten Millions Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones 3 1 5 9

Lesson 2.1 Understanding Large Numbers Space after hundreds Lesson 2.1 Understanding Large Numbers A Few Notes: When writing a number with 5+ digits, leave a space between the periods: 3 159 119 Space after thousands Hundred Millions Ten Millions Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones 3 1 5 9

Lesson 2.1 Understanding Large Numbers A Few Notes: When we read / say large numbers we say the period name after each period except the units period: Three million One hundred fifty-nine thousand One hundred nineteen Hundred Millions Ten Millions Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones 3 1 5 9

Lesson 2:Multiples

Lesson 2: Multiples Explore (Partners) On Thursday morning the local radio station held a call-in contest. Every third caller to the station won a T- Shirt Every seventh caller won a baseball hat In 50 calls, which callers won a T-shirt? A baseball hat? Which numbers won both?

Lesson 2: Multiples In 50 calls, these callers won a T-shirt: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48 In 50 calls, these callers won a baseball hat: 7, 14, 21, 28, 35, 42, 49 In 50 calls, these callers won a won both: 21, 42

Lesson 2: Multiples The result of multiplying a number by a number Example: multiples of 5 are 10, 15, 20 5 x 2 = 10 5 x 3 = 15 5 x 4 = 20

Lesson 2: Multiples A method for finding multiples Use a chart. Start at the number and count by the number The multiples of 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Lesson 2: Multiples A method for finding multiples Use multiplication: 5 x 1 = 5 5 x 2 = 10 (factor x factor = multiple) 5 x 3 = 15 5 x 4 = 20 5 x 5 = 25

Lesson 2: Multiples KEY PHRASES 5, 10, 15, and 20 are all multiples of 5. 5, 10, 15, and 20 are divisible by 5. 5 divides 5, 10, 15, and 20.

Lesson 2: Multiples A method for finding multiples Use a pattern rule: Start at x. Add x each time In this example X = 4 4,(+4 =) 8, (+4 =) 12, (+4 =) 16,(+4 =) 20,(+4 =) The multiples of 4 are: 4, 8, 12, 16, 20, 24, …

Lesson 2: Multiples Here are the multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 Here are the multiples of 5: 5, 10, 15, 20, 25, 30 Which multiples are in both lists? 10, 20, 30 These are common multiples of 5 and 2 10 is the least common multiple of 5 and 2

Solve This… Wieners are sold in packages of 12. Hot dog buns are sold in packages of 8. Suppose you plan to sell about 75 hot dogs to raise money for charity. You do not want any wieners or buns left over. How many packages of each should you buy?

A Solution… Find the multiples of 8 8, 16, 24, 32, 40, 48, 60, 68. 72 12, 24, 36, 48, 60, 72, 84

A Solution… Identify the common multiples: 24, 48, 72 Since 72 is close to 75, you should buy 72 wieners and 72 buns. You skip counted by eight 9 times to reach 72, so buy 9 packages of buns. You skip counted by twelve 6 times to reach 72, so buy 6 packages of wieners.

Lesson 3: Factors

Factors Lesson 3: Factors Factors are the numbers we multiply together to get another number: Example: 5 x 3 = 15 5 is a factor 3 is a factor 15 is the product

Lesson 3: Factors A number can have many factors: Example: factors of 12 are 1, 2, 3, 4, 6 and 12 2 × 6 = 12 4 × 3 = 12 1 × 12 = 12

Lesson 3: Factors Factors of 9 Factors of 15 Common Factors A number that is a factor of each of the given numbers Ex. List the Factors of 9 and 15 9: 1, 3, 9 15: 1, 3, 5, 15 Factors of 9 Factors of 15 Common Factors 1 3 9 5, 15 Greatest Common Factor = 3

Lesson 4:Prime and Composite Numbers

2 x 8 = 16 Lesson 4: Prime Numbers 2 and 8 are factors of 16. What others can you name? 1, 4, 16 2 x 8 = 16 Factor Factor Product

Lesson 4: Prime Numbers Activity Cut out your squares Use the squares to make all the different rectangles you can make using each number of squares Complete the table to record your data

1 x 6 3 x 2 2 x 3 6 x 1

Lesson 4: Prime Numbers Activity (Part 2) use the information from the chart to complete the graph.

Lesson 4: Prime Numbers Activity (Part 3) What observations can you make by looking at patterns in the graph you completed?

Lesson 4: Prime Numbers Explore: Which numbers have only 2 factors? 2, 3, 5, 7, 11, 13, 17, 19 What do you notice about these numbers? All the numbers except 2 are odd numbers. What do you notice about the factors of these numbers? The factors of each number are 1 and the number itself. Which numbers have more than 2 factors? 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20

Lesson 4: Prime Numbers With 23 squares you can only make one rectangle 23 has two factors: 1, 23 A number with only two factors (1 and itself) is called a PRIME NUMBER 1 23

Lesson 4: Prime Numbers Simplify It… When a number can not be divided evenly by another number it must be a whole number greater than 1

Lesson 4: Composite Numbers 3 With 24 Squares you can make 4 different rectangles 6 8 4 24 has eight factors: 1, 2, 3, 4, 6, 8, 12, 24 1 24 12 A number with more than two factors is called a COMPOSITE NUMBER 2

Lesson 4: Composite Numbers Simplify It… When a number can be divided evenly by another number, it is a Composite Number

Brainpop: Prime Numbers http://www.brainpop.com/math/numbersa ndoperations/primenumbers/preview.wem l

Lesson 5: Prime Factorization

Lesson 5: Prime Factorization All Composite Numbers are made up of Prime Numbers that have been multiplied together It is like Prime Numbers are the building blocks of all numbers.

Lesson 5: Prime Factorizations Prime Factors: factors that are Prime Numbers Ex. 24 has eight factors: 1, 2, 3, 4, 6, 8, 12, 24 2, 3 are prime numbers therefore they are the prime factors of 24

Lesson 5: Prime Factorization What are the prime factors of 12 ? Method 1 Step 1 – Divide (It is best to start working from the smallest prime number) 12 ÷ 2 = 6 Yes, it divided evenly by 2. We have taken the first step! 12 = 2 x 6

Lesson 5: Prime Factorization What are the prime factors of 12 ? Method 1 Step 2 – Keep and Divide (keep all prime numbers and divide the rest) 12 = 2 x 6 6 is not a prime number, so we need to divide again 6 ÷ 2 = 3

Lesson 5: Prime Factorization What are the prime factors of 12 ? Method 1 Step 3 – Keep and Divide (keep all prime numbers and divide the rest) 12 = 2 x 2 x 3 2 and 3 are both prime numbers, so our factor phrase is: 12 = 2 × 2 × 3

Lesson 5: Prime Factorization Try These: What is the prime factorization of 48 ? What is the prime factorization of 17? What is the prime factorization of 147 ?

Lesson 2.5: Prime Factorization Method 2: The Factor Tree

Lesson 2.5: Prime Factorization Method 2: The Factor Tree

Lesson 5: Prime Factorization Try These: What is the prime factorization of 48 ? What is the prime factorization of 17? What is the prime factorization of 147 ?

Lesson 6: Order of Operations

Vocabulary Expression: A mathematical statement with numbers and operations A math question without an answer Example: 2×3 = 4x – 7 = 10 x 6 - 7 + 3 ÷ 5 = 𝜋 𝑟 2 =

Lesson 2.7: Order of Operations When you solve a problem that uses more than one operation, the answer depends on the order in which you perform the operations. Evaluate the expression: 3 + 6 x 4 If you add first, you get: 9 x 4 = 36 If you multiply first, you get: 3 + 24 = 27

Lesson 2.7: Order of Operations To avoid getting two answers, there is a rule that multiplication is done before addition. So, 3 + 6 x 4 = 3 + 24 27, which is the correct answer

Lesson 2.7: Order of Operations We use brackets if we want certain operations carried out first. To make sure everyone gets the same answer when evaluating an expression, we use this order of operations: Do the operations in brackets. Multiply or divide, in order, from left to right. Then add or subtract, in order, from left to right.

Lesson 2.7: Order of Operations Evaluate: 18 - 10 + 6 = 18 - 10 + 6 = 8 + 6 = 14

Lesson 2.7: Order of Operations Evaluate: 16 - 14 ÷ 2 = 16 - 14 ÷ 2 = 16 – 7 = 9

Lesson 2.7: Order of Operations Evaluate: 7 x (4 + 8) 7 x (4 + 8) = 7 x 12 = 84

Lesson 2.7: Order of Operations The order of operations is : Brackets Multiply / Divide (from left to right) Add / Subtract (from left to right)

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