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estimation and

number theory

UNIT 2

MATH PACKET

Estimation Factors Multiples

(2)

RECALL PRIOR KNOWLEDGE

In 7,465 -

the digit 7 is in the thousands place the digit 4 is in the hundreds place the digit 6 is in the tens place and the digit 5 is in the ones place. In 7,465 -

the digit 7 stands for 7,000 the digit 4 stands for 400 the digit 6 stands for 60 and the digit 5 stands for 5.

Round 745 to the nearest 10.

Look at the digit to the right of the tens digit, which is the ones digit. Round up if the ones digit is 5 or greater.

Round down if the ones digit is less than 5.

745 rounded to the nearest 10 is 750.

Using place value to find the value of each digit.

(3)

Round 2,839 to the nearest 100.

Look at the digit to the right of the hundreds digit, which is the tens digit. Round up if the tens digit is 5 or greater.

Round down if the tens digit is less than 5.

2,839 rounded to the nearest 100 is 2,800.

Estimate 7,615 + 2,487. The leading digit of 7,615 is 7. The leading digit of 2,487 is 2.

The estimated sum is 9,000.

Estimate 6,434 - 1,592.

The leading digit in 6,434 is 6. The leading digit in 1,592 is 1.

The estimated difference is 5,000.

Rounding numbers to the nearest 100

Estimating sums using front-end estimation

(4)

3 x 4 = 12

12 is the product of 3 and 4.

The product can be divided exactly by 3 and 4.

Quick check

Complete.

The digit 6 is in the place. The digit 6 stands for . . The digit 2 is in the place. The digit 9 stands for . . The digit 4 stands for .

Round to the nearest 10.

819 is about 724 is about

Round to the nearest 100.

4,236 is about 5,982 is about

Multiplying two numbers to find the product

(1) (2) (3) (4) (5)

(9) (7) (6)

(5)

Estimate each sum or difference using front-end estimation.

+ =

+ =

Find the correct numbers.

42 = 3 x


42 is the product of 3 and .

The product can be divided exactly by 3 and . (10)

(11)

(6)

Lesson 2.1 - estimation

Eva’s Market sold 196 jars of grape jelly in September. In October, the market sold 389 jars. How many jars of grape jelly did they sell over the two months?

196 + 389 = 585

The market sold 585 jars of grape jelly altogether.

Estimate to check that the answer is reasonable. Round each number to the nearest hundred.

Add: 200 + 400 = 600

The estimated sum rounded to the nearest 100 is 600.

Since both numbers are rounded up, the estimate is greater than the actual sum.

The answer 585 is reasonable.

The same method of estimation can be used to check the reasonableness of differences.

Use rounding to check the reasonableness of sums & differences.

NUMBER Rounded to the Nearest 100

196 200

(7)

guided learning

Find the difference. Then use rounding to check that your answer is reasonable.

Round each number to the nearest hundred.

Find 786 — 453. 786 — 453 =

Estimate to check that your answer is reasonable. Round each number to the nearesthundred.

Subtract: — =

The estimated difference rounded to the nearest 100 is . Is the answer reasonable?

Explain:

Find 472 — 184. 472 — 184 =

Estimate to check that your answer is reasonable. Round each number to the nearest hundred.

472 — 184

The estimated difference is . Is your answer reasonable?

Explain: (1)

NUMBER Rounded to the Nearest 100

786

453

(2)

=

7 8 6 — 4 5 3

(8)

Find 7,840 — 3,622.

7,840 — 3,622 = 4,218

The answer is 4,218

Estimate to check that the answer is reasonable.

The estimated difference is 4,000. The answer 4,218 is reasonable.

The same method of estimation can be used to check the reasonableness of sums.

guided learning

Find the sum. Then use front-end estimation to check that your answer is

reasonable.

Find 6,572 + 8,144. 6,572 + 8,144 =

Estimate to check that your answer is reasonable.

6,572 + 8,144

+ =

The estimated sum is . Is your answer reasonable? Explain:

Use front-end estimation to check the reasonableness of sums & differences.

(3)

(9)

Find each sum or difference. Then use rounding to check that your answers are

reasonable. Round each number to the nearest hundred.

326 x 2 = 652 Estimate to check that the answer is reasonable.

Round 326 to the nearest hundred and multiply by 2.

The estimated product rounded to the nearest 100 is 600. The answer 652 is reasonable. (1)

5, 4 2 6 + 3, 2 1 0

8, 4 7 5 — 3, 3 5 6

9, 6 3 3 + 4, 2 5 0

1 6, 8 6 2 — 1 2, 5 5 1

+

+ =

=

+

+ =

=

(4) (3) (2)

(10)

guided learning

Find the product. Then use rounding to check that your answer is reasonable.

Round the 3-digit number to the nearest hundred.

Find 242 x 4. 242 x 4 =

Estimate to check that your answer is reasonable.

The estimated product rounded to the nearest 100 is . Is your answer reasonable? Explain.

Find 134 x 5. 134 x 5 = 670 The answer is 670.

Estimate to check that the answer is reasonable.

Multiply the value of the digits in the greatestplace of each number.

The estimated product is 500. The answer 670 is reasonable.

2 4 2

x 4

(1)

NUMBER Rounded to the Nearest 100 x 4

242 x 4 =

(11)

guided learning

Find the product. Then use front-end estimation to check that your answer is

reasonable. Find 471 x 2.

471 x 2 =

Estimate to check that your answer is reasonable.

471 x 2

The estimated product is . Is your answer reasonable? Explain:

Novak bought a box of 72 building blocks. He shared the blocks equally with his 2 friends. How many blocks did each of them get?

72 ÷ 3 = 24

Each of them got 24 blocks.

Use related multiplication facts to check that your answer is reasonable. Since division is the opposite of multiplication, find a multiple of 3 that is close to 7.

3 x 2 = 6 3 x 20 = 60

3 x 3 = 9 3 x 30 = 90

72 is closer to 60 than to 90. So, 72 ÷ 3 is about 60 ÷ 3.

60 ÷ 3 = 20, so the estimated quotient is 20.

4 7 1

x 2

(1)

2

x =

(12)

Find each quotient. Then use related multiplication facts to check that your

answers are reasonable.

Find 92 ÷ 2 . 92 ÷ 2 =

Use related multiplication facts to check that your answer is reasonable. Since division is the opposite of multiplication, find a multiple of 2 that is close to 9.

2 x 40 = 2 x 50 =

92 is closer to than to . So, 92 ÷ 2 is about ÷ 2.

÷ 2 = The estimated quotient is . Is your answer reasonable? Explain:

Find 76 ÷ 4. 76 ÷ 4 =

Use related multiplication facts to check that your answer is reasonable. Since division is the opposite of multiplication, find a multiple of 4 that is close to 7.

4 x = 4 x = 76 is closer to than to .

So, 76 ÷ 4 is about ÷ 4.

÷ 4 = The estimated quotient is . Is your answer reasonable? Explain:

(1)

(13)

Find 85 ÷ 5. 85 ÷ 5 =

Use related multiplication facts to check that your answer is reasonable. Since division is the opposite of multiplication, find a multiple of 5 that is close to 8.

5 x = 5 x = 85 is closer to than to .

So, 85 ÷ 5 is about ÷ 5.

÷ 5 = The estimated quotient is . Is your answer reasonable? Explain:

Find each product or quotient. Then use one of the methods above to check that

your answers are reasonable.

123 x 7 = Estimate:

Is your answer reasonable? Explain:

54 ÷ 3 = Estimate:

Is your answer reasonable? Explain:

(3)

(5)

(4) 1 2 3

(14)

let’s explore

Use two methods to estimate.

EXAMPLE:

Estimate 73 ÷ 8. Method 1

73 is close to 72.

So, 73 ÷ 8 72 ÷ 8 = 9. Method 2

8 x 8 = 64 8 x 9 = 72 8 x 10 = 80

8 x 9 = 72 is closest to 73, so the estimated quotient is 9.

EXAMPLE:

Estimate 9 x 26. Method 1

9 is close to 10.

So, 9 x 26 10 x 26 = 260. Method 2

9 x 26 9 x 25 = 225 Method 3

9 x 26 10 x 25 = 250

Use any method to estimate the following quotient and product.

658 ÷ 8 = 6 x 52 = (1)

(2)

(15)

The Senior Citizen Committee has raised $1,000 for the Senior Center. Is $1,000 enough to buy the items shown?

Decide if you need an exact answer or an estimate.

Estimate the total cost by rounding to the nearest hundred dollars.

$600 + $700 = $1,300

The committee does not have enough money to buy both items.

An estimate is all that is needed to decide if the committee has enough money to buy both items.

About how much more money does the committee need to buy both items? Because the question asks about how much more money is needed, an estimate is needed.

$1,300 - $1,000 = $300

The committee needs about $300 more to buy both items.

4 people are invited to Marcia’s home. Each guest gets one carton of cranberry juice, which costs 45¢. How much money does Marcia need to buy the juice?

Because the question asks how much money Marcia needs to buy the juice, an exact answer is needed.

4 x 45 = 180 cents = $1.80

She needs $1.80 to buy the juice.

(16)

guided learning

Solve. Decide whether to find an estimate or an exact answer.

The table shows the number of dogs that were adopted in a state within a year.

How many dogs were adopted altogether that year?

Mr. Sousa has $250. He wants to spend $63 on a sweatshirt, $45 on running shoes, and $120 on sports gear. Does he have enough money?

A family has 7 people. Each person is supposed to drink

h

quart of water every day.

About how much water is needed for the whole family each day?

Kerry bought 15 roasted chickens. Each chicken cost $6. Find the total amount Kerry spent on the chickens.

Joel bought 3 apples at 26 cents each and 4 oranges at 32 cents each. About how much money did he spend on the apples and oranges?

(1)

(2)

(3)

(17)

SHOW ALL WORK IN YOUR MATH NOTEBOOK.

Find each sum or difference. Then use rounding or front-end estimation to check

that your answers are reasonable. Round each number to the nearest hundred.

536 + 289 + 109 = 320 + 478 + 215 = 8,530 — 1,286 = 7,271 + 1,335 = 26,235 — 1,451 = 15,422 + 13,130 = 18,726 + 29,343 = 31,540 — 24,622 =

Find each product. Then use rounding or front-end estimation to check your

answers. Round the 3-digit number to the nearest hundred. 232 x 4 = 148 x 5 =

212 x 3 = 498 x 2 =

Find each quotient. Then use related multiplication facts to check your answers.

42 ÷ 3 = 56 ÷ 2 = 80 ÷ 5 = 88 ÷ 4 =

Solve. Decide whether to find an estimate or an exact answer.

A rectangular lawn is 11 meters long and 4 meters wide. About how many square meters of grass sod is needed to cover the entire lawn? (HINT: To find the number of square meters, multiply the length times the width.)

A rectangular courtyard is 12 feet long and 8 feet wide. A tile is 2 feet long and 2 feet wide. How many tiles are needed to pave the courtyard?

(18)

Lesson 2.2 - FACTORS

Lesson Objectives:

• Find the common factors and greatest common factor of two whole numbers. • Identify prime numbers and composite numbers.

1 x 6 = 6

Can 6 be divided exactly by 1? Yes. So, 1 is a factor of 6.

Can 6 be divided exactly by 6? Yes. So, 6 is a factor of 6.

6 is the product of 1 and 6. 1 and 6 are factors of 6.

Break down whole numbers into factors.

VOCABULARY

factor - common factor - greatest common factor

(19)

GET READY- FINDING FACTORS

GET READY

Mrs. Pompei is arranging computers in her classroom. There are 24 computers. How many ways can she arrange the computers so that the number of computers in each row is the same?

Think of number pairs that result in a product of 24. 1 x 24 = 24

!!!!!!!!!!!!!!!!!!!!!!!!

2 x 12 = 24

!!!!!!!!!!!! !!!!!!!!!!!!

3 x 8 = 24

!!!!!!!! !!!!!!!! !!!!!!!!

4 x 6 = 24

!!!!!! !!!!!! !!!!!! !!!!!!

Factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. So, the computers can be arranged in eight ways.

FACTOR PAIRS are two numbers being multiplied to give you a product. Example: Factor pairs of 6 are 1 x 6 & 2 x 3.

NOTE: Every number will have 1 and itself as a factor pair.

THINK: There are 4 arrays: 24 x 1 8 x 3 12 x 2 6 x 4 Two or more numbers that are multiplied together to form a product are called factors. To find the different arrangements

(20)

GET READY- FINDING FACTORS

MAKE A T-CHART TO FIND THE FACTORS

Find the factors of 32.

• 32 = 1 x 32 • 32 = 2 x 16 • 32 = 4 x 8

The factors of 32 are , , , , , and .

Find the factors of 24.

• 24 = x

• 24 = x

• 24 = x

• 24 = x

Find the factors of 18. • 18 = x

• 18 = x

• 18 = x

32

1 32

16 8 2

4

24

(21)

Find the factors of each number.

12 28

(1) (2)

56 100

(22)

Guided learning

Find the common factors of 9 and 36.

The factors of 9 are , , and .

The factors of 36 are , , , , , , , , and .

Answer each question with yes or no.

Is 5 a factor of 20? Is 5 a factor of 35?

Is 5 a common factor of 20 & 35?

Is 2 a factor of 24? Is 2 a factor of 27?

Is 2 a common factor of 24 & 27?

Is 4 a factor of 32? Is 4 a factor of 12?

Is 4 a common factor of 32 & 12?

(1) 20 35

(2) (3)

(6) (4) (5)

24 27

32 12

(23)

Greatest common factors

Guided learning

Find the greatest common factor of 16 & 48.

The factors of 16 are ____ , ____ , ____ , ____ , and ____.

The factors of 48 are ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ and ____. The common factors of 16 & 48 are ____ , ____ , ____ , ____ , and ____.

Therefore, the greatest common factor of 16 & 48 is ____ .

Find the greatest common factor of 18 & 72.

(Make T-charts and list the factors in your notebook.)

Find the greatest common factor of two whole numbers.

(24)

practice

(2) (1)

(3)

(25)

prime & composite numbers

guided practice

Find all the factors. Then decide whether the numbers are prime or composite.

(Make T-charts to name the factors.)

21 (prime - composite)


33 (prime - composite)

59 (prime - composite)

77 (prime - composite) (1)

(26)

Answer each question.

6 - 14 - 15 - 20 - 23 - 25 - 28 - 32 - 33 - 35 - 39

Which of these numbers have 2 as a factor? Which of these numbers have 5 as a factor?

Find a possible number which has only these factors.

Factors of a Number

Complete.

Find all the prime numbers from 1 to 10.

Find all the prime numbers for each set of numbers. Complete the table.

Prime Numbers

Which set has more prime numbers?

Solve.

Julia’s birthday is on 3/7/1998. Using the digits from her birthday, form three 2-digit prime numbers, and three 2-digit composite numbers.

(1) (2)

Factors Number

1, 2, 3, and 6

1, 2, 5, and 10

(3) (4)

(5) (6)

1 - 16 16 - 32 32 - 48

(7)

(27)

Think about multiplication and division. Then find the missing numbers.

28 = 1 x 28 = 2 x 28 ÷ 1 = 28 ÷ 2 = 28 = 4 x

28 ÷ =

The factors of 28 are , , , , , and .

Find all the factors of 42.

42 = x 42 = x 42 = x 42 = x

The factors of 42 are , , , , , , , and .

Complete.

The factors of 8 are , , , and .

The factors of 24 , , , , , , , and . The common factors of 8 and 24 are , , , and .

The greatest common factor of 8 and 24 is .

The factors of 52 are , , , , ,and . The common factors of 24 and 52 are , , and .

(28)

These are the steps to find the factors of 12.

1 Think of all the factors you can multiply to achieve the product of 12.

12 = 1 x 12 12 = 2 x 6 12 = 3 x 4

2 The factors are 1, 2, 3, 4, 6, and 12.

Write the steps for finding the common factors of 12 and 15.

Step 1:

_______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ Step 2:

_______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ Step 3:

_______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________

READING AND WRITING MATH

(29)

Lesson 2.3 - multiples

Lesson Objectives:

• Find multiples of whole numbers.

• Find common multiples and the least common multiple of 2 or more numbers.

VOCABULARY

(30)

guided learning

Complete. Write Yes or No.

Is 24 a multiple of 8? Is 42 a multiple of 5?

REMEMBER: A multiple of a number is simply the product of that number and any whole

number. An easy way to remember to name multiples is to skip count. Is 24 a multiple of 8? Start skip counting by eights, starting with the number 8. Will you eventually get to 24? Do the same with problem #2. Start skip counting by fives. Will you eventually get to 42?

Find the first five multiples of each number.

2 10 6 8

Complete.

What is the fourth multiple of 7? What is the fifth multiple of 7? What is the twelfth multiple of 7?

(1) (2)

(3)

(6) (5)

(4)

(31)

PRACTICE

Skip counting a number produces multiples of that number.

Skip count by 2s.

2 , 4 , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , 24

Skip count by 3s.

3 , 6 , ____ , ____ , ____ , ____ , 21 , ____ , ____ , ____ , ____ , ____

Skip count by 4s.

4 , 8 , ____ , ____ , ____ , ____ , ____ , ____ , 36 , ____ , ____ , ____

Skip count by 5s.

(32)

Skip count by 6s.

6 , 12 , ____ , ____ , ____ , ____ , ____ , ____ , 54 , ____ , ____ , ____

Skip count by 7s.

7 , 14 , ____ , ____ , ____ , ____ , 49 , ____ , ____ , ____ , ____ , ____

Skip count by 8s.

8 , 16 , ____ , ____ , ____ , ____ , ____ , ____ , ____ , 80 , ____ , ____

Skip count by 9s.

(33)

Skip count by 10s.

10 , 20 , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , 120

Skip count by 11s.

11 , 22 , ____ , ____ , ____ , ____ , ____ , ____ , ____ , 110 , ____ , ____

11 x 4 = _____ 11 x 7 = _____ 11 x 1 = _____ 11 x 5 = _____ 11 x 3 = _____

11 x 11 = _____ 11 x 6 = _____ 11 x 9 = _____ 11 x 8 = _____ 11 x 12 =_____

Skip count by 12s.

12 , 24 , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , 144

(34)
(35)

guided learning

List the first twelve multiples of 4 and 6. Then find the common multiples of 4

and 6 from the first twelve multiples.

The first twelve multiples of 4 are , , , , , ,

, , , , , .

The first twelve multiples of 6 are , , , , , ,

, , , , , . From the list of twelve multiples, the common multiples of 4 and 6 are

, , , and .

The least common multiple of 4 and 6 is .

Complete.

List the first twelve multiples of 5 and 8. From the list, circle a common multiple of 5 & 8.

5 8

Find a common multiple of each pair of numbers.

3 and 4 5 and 4 2 and 7

Solve.

15 and 30 are common multiples of 5 and X. X is a 1-digit number. X is not 1. What is the value of X?

(1)

(2)

(3)

(4)

(36)

least common multiples

Guided Practice: List the first twelve multiples of 3 and 4. Then find the common multiples

of 3 and 4 from the first twelve multiples.

The first twelve multiples of 3 are ____ , ____ , ____ , ____ , ____ , ____ , ____ ,

____ , ____ , ____ , ____ and ____ .

The first twelve multiples of 4 are ____ , ____ , ____ , ____ , ____ , ____ , ____ ,

____ , ____ , ____ , ____ and ____ .

From the list of the twelve multiples, the common multiples of 3 and 4 are

____ , ____ and ____ .

The common multiple that is less than all the others is _____ or the Least Common

Multiple. (LCM)

(1)

(2)

(3)

(37)

let’s practice

Find the first five multiples of each number.

9 , , , , 7 , , , ,

Find the numbers in the box that are multiple of each given number.

4 9

Complete.

List the first twelve multiples of 6 & 8. From the list, find the common multiples of 6 & 8.

6 8

Common multiples:

Solve.

Here is a list of numbers. Circle the multiples of 5. Two of the multiples of 5 are common multiples of 5 and X.

The value of X is:

From the list, what are the common multiples of 5 and X? and

Solve.

12 is the least common multiple of 4 and X. Find three possible values of X.

(1) (2)

6 - 16 - 18 - 27 - 36 - 42 - 63

(4) (3)

(5)

4 - 5 - 7 - 10 - 15 - 24 - 27 - 35 - 42 - 45 - 49

(6) (7)

(38)

Critical Thinking Skills

put on your thinking cap

Mrs. Lim wrote a number on a card without showing it to her students.

She asked her students what the number was. She gave them only three hints.

• The number can be divided exactly by 3.

• When I add 3 to the number, it can be divided exactly by 5. • The number is less than 32 but greater than 23.

What is the number?

➋Jonathan bought an item that cost less than $100. He could pay for the item

(39)

Critical Thinking Skills

put on your thinking cap

A farmer has a rectangular field. The length

of the field is 5 times its width. The length and width are whole numbers. The perimeter of the filed is 50 meters when rounded to the nearest ten meters. What are the width and length of the field?

Width Length Perimeter Perimeter Rounded to the Nearest 10m

(40)

CHAPTER WRAP UP: estimation & number theory

STUDY GUIDE: You have learned…

(41)

number theory

factors

(42)

CHAPTER review

VOCABULARY: Choose the correct word.

You can to check that an answer is reasonable. A number that has only 2 different factors is a . 6 is a of 36, and 36 is a of 6. The of two or more numbers is less than all other common multiples.

CONCEPTS AND SKILLS: Find each sum or difference. Then use rounding to check

the your answers are reasonable. Round each number to its greatest place.

74 + 53 = 216 — 39 =

Estimation: Estimation: 568 + 329 = 707 — 183 = Estimation: Estimation:

WORK AREA:

(1) (2)

least common multiple - estimate - multiple - prime number - factor

(3) (4)

(5) (6)

(43)

Find each sum or difference. Then use front-end estimation to check that your

answers are reasonable.

23 + 64 = 31 — 16 = Estimation: Estimation:

8,142 + 3,154 = 7,214 — 3,645 = Estimation: Estimation:

Complete. Then estimate to check that your answers are reasonable.

8,012 + 1,569 = 568 — 127 = 3,516 — 1,657 = 59 x 6 = 72 ÷ 3 = 78 x 5 = 42 x 3 = 95 ÷ 5 =

WORK AREA:

(13)

(17)

(14) (15)

(9) (10)

(12) (11)

(16) (18)

(44)

Find the factors of each number.

16 36

Find the common factors of 16 and 36 . The greatest common factor of 16 and 36 is .

Find the first eight multiples of the following numbers.

4 5

Find a common multiple of 4 and 5 . The least common multiple of 4 and 5 is .

Find the factors of each number. Then list the prime numbers & composite

numbers.

Prime Numbers: Composite Numbers:

(21)

16 36

(22)

(24) (23)

(25) (26) (27) (28)

(29) (30)

(45)

PROBLEM SOLVE:

Jared has 98 oranges. He packs them into 4 crates. How many oranges does he have left over?

There are 147 erasers, 215 pencils, and 327 pens in a stationery store. ABOUT how many erasers, pencils, and pens are there altogether?

A tourist agency is expecting 83 visitors in a week. Each of the agency’s cars can carry 4 passengers. How many cars will be needed for all the visitors?

Ms. Clarkson has $315 to spend on kitchen appliances. She has to choose between two of three options; a microwave for $220, a coffee machine for $83, and a waffle maker for $98. Should she buy the microwave and the coffee machine, or the microwave and the waffle maker?

(1)

(46)

glossary review

Factor - A number that divides a whole number evenly. Also a number that is multiplied by another number.

Example: 4 x 5 = 20 (4 & 5 are factors)

Factor Pairs -

A

ny two numbers being multiplied to give you a certain number.

Example: Factor Pairs of 16 are 1 x 16, 2 x 8, and 4 x 4

Greatest Common Factor - The highest number that divides exactly into two or more numbers.

Example: Factors of 8 are 1, 2, 4, 8 & factors of 12 are 1, 2, 3, 4, 6, 12 Which factors are common? 1, 2, 4

Therefore, 4 is the greatest common factor

Prime Number - Numbers that have only two factors.

Example: The only two factors of 11 are 1 & 11 The only two factors of 3 are 1 & 3

Composite Number - A number that has more than two factors.

Example: The number 10 is composite because 1 x 10 = 10 & 2 x 5 = 10

Multiple - A multiple of a number is the product of that number and any whole number.

References

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