beams_unit 1 negative numbers

8/9/2019 BEAMS_Unit 1 Negative Numbers

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Unit 1:Negative Numbers

UNIT 1

NEGATIVE NUMBERS

B a s i c E s s e n t i a l

A d d i t i o n a l M a t h e m a t i c s S k i l l s

Curriculum Development Division

Ministry of Education Malaysia


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TABLE OF CONTENTS

Module Overview 1

Part A: Addition and Subtraction of Integers Using Number Lines 2

1.0 Representing Integers on a Number Line 3

2.0 Addition and Subtraction of Positive Integers 3

3.0 Addition and Subtraction of Negative Integers 8

Part B: Addition and Subtraction of Integers Using the Sign Model 15

Part C: Further Practice on Addition and Subtraction of Integers 19

Part D: Addition and Subtraction of Integers Including the Use of Brackets 25

Part E: Multiplication of Integers 33

Part F: Multiplication of Integers Using the Accept-Reject Model 37

Part G: Division of Integers 40

Part H: Division of Integers Using the Accept-Reject Model 44

Part I: Combined Operations Involving Integers 49

Answers 52


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Basic Essential Additional Mathematics Skills (BEAMS) Module

Unit 1: Negative Numbers

1



MODULE OVERVIEW

1. Negative Numbers is the very basic topic which must be mastered by everypupil.

2. The concept of negative numbers is widely used in many AdditionalMathematics topics, for example:

(a) Functions (b) Quadratic Equations

(c) Quadratic Functions (d) Coordinate Geometry

(e) Differentiation (f) Trigonometry

Thus, pupils must master negative numbers in order to cope with topics inAdditional Mathematics.

3. The aim of this module is to reinforce pupils understanding on the concept ofnegative numbers.

4. This module is designed to enhance the pupils skills in using the concept of number line; using the arithmetic operations involving negative numbers; solving problems involving addition, subtraction, multiplication and

division of negative numbers; and applying the order of operations to solve problems.

5. It is hoped that this module will enhance pupils understanding on negativenumbers using the Sign Model and the Accept-Reject Model.

6. This module consists of nine parts and each part consists of learning objectiveswhich can be taught separately. Teachers may use any parts of the module as

and when it is required.


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TEACHING AND LEARNING STRATEGIES

The concept of negative numbers can be confusing and difficult for pupils to

grasp. Pupils face difficulty when dealing with operations involving positive and

negative integers.

Strategy:

Teacher should ensure that pupils understand the concept of positive and negative

integers using number lines. Pupils are also expected to be able to performcomputations involving addition and subtraction of integers with the use of the

number line.

PART A:

ADDITION AND SUBTRACTION

OF INTEGERS USING

NUMBER LINES

LEARNING OBJECTIVE

Upon completion of Part A, pupils will be able to perform computationsinvolving combined operations of addition and subtraction of integers using a

number lines.


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PART A:

ADDITION AND SUBTRACTION OF INTEGERS

USING NUMBER LINES

1.0 Representing Integers on a Number Line

Positive whole numbers, negative numbers and zero are all integers. Integers can be represented on a number line.

Note: i) 3 is the opposite of +3

ii) (2) becomes the opposite of negative 2, that is, positive 2.

2.0 Addition and Subtraction of Positive Integers

3 2 1 0 1 2 3 4

LESSON NOTES

Rules for Adding and Subtracting Positive Integers

When adding a positive integer, you move to the right on anumber line.

When subtracting a positive integer, you move to the lefton a number line.

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4

Positive integers

may have a plus sign

in front of them,

like +3, or no sign in

front, like 3.


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(i) 2 + 3

Alternative Method:

EXAMPLES

Adding a positive integer:

Start by drawing an arrow from 0 to 2, and then,

draw an arrow of 3 units to the right:

2 + 3 = 5

5 4 3 2 1 0 1 2 3 4 5 6

Start

with 2

Add a

positive 3


Start at 2 and move 3 units to the right:

2 + 3 = 5

Make sure you start from

the position of the first

integer.

5 4 3 2 1 0 1 2 3 4 5 6


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5



(ii) 2 + 5

Alternative Method:


Start by drawing an arrow from 0 to2, and then,draw an arrow of 5 units to the right:

2 + 5 = 3

5 4 3 2 1 0 1 2 3 4 5 6

Add a

positive 5


the position of the firstinteger.

5 4 3 2 1 0 1 2 3 4 5 6


Start at2 and move 5 units to the right:

2 + 5 = 3


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(iii) 25 =3

Alternative Method:

5 4 3 2 1 0 1 2 3 4 5 6

Subtracting a positive integer:

Start by drawing an arrow from 0 to 2, and then,

draw an arrow of 5 units to the left:

25 =3

Subtract a

positive 5


Start at 2 and move 5 units to the left:

25 =3

5 4 3 2 1 0 1 2 3 4 5 6



integer.


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(iv) 32 =5

Alternative Method:


Start by drawing an arrow from 0 to3, and

then, draw an arrow of 2 units to the left:

32 =5

5 4 3 2 1 0 1 2 3 4 5 6

Subtract a

positive 2

5 4 3 2 1 0 1 2 3 4 5 6


Start at3 and move 2 units to the left:

32 =5


the position of the firstinteger.


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3.0 Addition and Subtraction of Negative Integers

Consider the following operations:

41 = 3

42 = 2

43 = 1

44 = 0

45 =1

46 =2

Note that subtracting an integer gives the same result as adding its opposite. Adding orsubtracting a negative integer goes in the opposite direction to adding or subtracting a positive

integer.

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4

4 + (5) =1

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 44 + (6) =2

4 + (1) = 3

4 + (2) = 2

4 + (3) = 1

4 + (4) = 0


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Rules for Adding and Subtracting Negative Integers

When adding a negative integer, you move to the left on anumber line.

When subtracting a negative integer, you move to the righton a number line.

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4


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(i) 2 + (1) =3

Alternative Method:

5 4 3 2 1 0 1 2 3 4 5 6

Adding a negative integer:

Start at2 and move 1 unit to the left:

2 + (1) =3

EXAMPLES

5 4 3 2 1 0 1 2 3 4 5 6



then, draw an arrow of 1 unit to the left:

2 + (1) =3

Add a

negative 1



integer.

This operation of

2 + (1) =3

is the same as

21 =3.


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(ii) 1 + (3) =2

Alternative Method:

5 4 3 2 1 0 1 2 3 4 5 6


Start at 1 and move 3 units to the left:

1 + (3) =2

Add a

negative 3

5 4 3 2 1 0 1 2 3 4 5 6


Start by drawing an arrow from 0 to 1, then, draw an arrow of

3 units to the left:

1 + (3) =2



integer.

This operation of

1 + (3) =2

is the same as

13 =2


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(iii) 3(3) = 6

Alternative Method:

5 4 3 2 1 0 1 2 3 4 5 6

Subtracting a negative integer:

Start at 3 and move 3 units to the right:

3(3) = 6

5 4 3 2 1 0 1 2 3 4 5 6


Start by drawing an arrow from 0 to 3, and

then, draw an arrow of 3 units to the right:

3(3) = 6

Subtract a

negative 3

This operation of

3(3) = 6

is the same as

3 + 3 = 6



integer.


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(iv) 5(8) = 3

Alternative Method:

5 4 3 2 1 0 1 2 3 4 5 6


Start at5 and move 8 units to the right:

5(8) = 3

5 4 3 2 1 0 1 2 3 4 5 6

Subtract a

negative 8

This operation of

5(8) = 3

is the same as

5 + 8 = 3



then, draw an arrow of 8 units to the right:

5(8) = 3


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Solve the following.

1. 2 + 4

2. 3 + (6)

3. 2(4)

4. 35 + (2)

5. 5 + 8 + (5)

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

TEST YOURSELF A


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This part emphasises the first alternative method which include activities and

mathematical games that can help pupils understand further and master the

operations of positive and negative integers.

Strategy:

Teacher should ensure that pupils are able to perform computations involving

addition and subtraction of integers using the Sign Model.

PART B:


OF INTEGERS USING

THE SIGN MODEL

LEARNING OBJECTIVE

Upon completion of Part B, pupils will be able to perform computations

involving combined operations of addition and subtraction of integers usingthe Sign Model.


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PART B:


USING THE SIGN MODEL

In order to help pupils have a better understanding of positive and negative integers, we have

designed the Sign Model.

Example 1

What is the value of 35?

NUMBER SIGN

3 + + +

5

WORKINGS

i. Pair up the opposite signs.

ii. The number of the unpaired signs is

the answer.

Answer 2

+

+

+

LESSON NOTES

EXAMPLES

The Sign Model

This model uses the + and signs. A positive number is represented by + sign. A negative number is represented by sign.


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Example 2

What is the value of 53 ?

NUMBER SIGN

3 _ _ _

5

WORKINGS

There is no opposite sign to pair up, sojust count the number of signs.

_ _ _ _ _ _ _ _

Answer 8

Example 3

What is the value of 53 ?

NUMBER SIGN

3

+5 + + + + +

WORKINGS

i. Pair up the opposite signs.

ii. The number ofunpaired signs is the

answer.

Answer 2

_

+ + +

_

+

_

+


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1. 4 + 8 2. 84 3. 127

4. 55 5. 574 6. 7 + 43

7. 4 + 37 8. 62 + 8 9. 3 + 4 + 6

TEST YOURSELF B


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PART C:

FURTHER PRACTICE ON


OF INTEGERS


This part emphasises addition and subtraction of large positive and negative integers.

Strategy:

Teacher should ensure the pupils are able to perform computation involving addition

and subtraction of large integers.

LEARNING OBJECTIVE

Upon completion of Part C, pupils will be able to perform computationsinvolving addition and subtraction of large integers.


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PART C:

FURTHER PRACTICE ON ADDITION AND SUBTRACTION OF INTEGERS

In Part A and Part B, the method of counting off the answer on a number line and the Sign

Model were used to perform computations involving addition and subtraction ofsmallintegers.

However, these methods are not suitable if we are dealing with large integers. We can use the

following Table Model in order to perform computations involving addition and subtraction

of large integers.

LESSON NOTES

Steps for Adding and Subtracting

Integers

1. Draw a table that has a column for + and a columnfor.

2. Write down all the numbers accordingly in thecolumn.

3. If the operation involves numbers with the samesigns, simply add the numbers and then put the

respective sign in the answer. (Note that we

normally do not put positive sign in front of a

positive number)

4. If the operation involves numbers with differentsigns, always subtract the smaller number from

the larger number and then put the sign of the

larger number in the answer.


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Examples:

i) 34 + 37 =+

34

37

+71

ii) 6520 =+

65 20

+45

iii)

73 + 22 =

+

22 73

51

iv) 228338 =+

228 338

110

Subtract the smaller number from

the larger number and put the sign

of the larger number in the

answer.

We can just write the answer as

45 instead of +45.




answer.




answer.

Add the numbers and then put the

positive sign in the answer.

We can just write the answer as

71 instead of +71.


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v) 428316 =+

428316

744

vi) 863 127 + 225 =+

225 863

127

225 990

765

vii) 234 675 567 =+

234 675

567

234 1242

1008


negative sign in the answer.

Add the two numbers in the

column and bring down the number

in the + column.


the larger number in the third row

and put the sign of the larger

number in the answer.



in the + column.






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viii) 482 + 236 718 =+

236 482

718

236 1200

964

ix) 765 984 + 432 =

+

432 765

984

432 1749

1317

x) 1782 + 436 + 652 =+

436

652

1782

10881782

694



in the + column.







in the + column.





Add the two numbers in the +

column and bring down the numberin the column.






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1. 4789 2. 5448 3. 33125

4. 352556 5. 345437456 6. 237 + 564318

7. 431 + 366778 8. 652517 + 887 9. 233 + 408689

TEST YOURSELF C


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This part emphasises the second alternative method which include activities to

enhance pupils understanding and mastery of the addition and subtraction of

integers, including the use of brackets.

Strategy:

Teacher should ensure that pupils understand the concept of addition and subtraction

of integers, including the use of brackets, using the Accept-Reject Model.

PART D:


OF INTEGERS INCLUDING THE

USE OF BRACKETS

LEARNING OBJECTIVE

Upon completion of Part D, pupils will be able to perform computations

involving combined operations of addition and subtraction of integers, includingthe use of brackets, using the Accept-Reject Model.


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PART D:


INCLUDING THE USE OF BRACKETS

To Accept or To Reject? Answer

+ ( 5 ) Accept +5 +5

( 2 ) Reject +2 2

+ (4) Accept 4 4

(8) Reject 8 +8

LESSON NOTES

The Accept - Reject Model

+ sign means to accept.

sign means to reject.


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i) 5 + (1) =

Number To Accept or To Reject? Answer

5+ (1)

Accept 5Accept 1

+51

+ + + + +

5 + (1) = 4

We can also solve this question by using the Table Model as follows:

5 + (1) = 51

+

5 1

+4

EXAMPLES

This operation of

5 + (1) = 4

is the same as

51 = 4

Subtract the smaller number fromthe larger number and put the sign


answer.

We can just write the answer as 4

instead of +4.


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ii) 6 + (3) =


6+ (3)

Reject 6Accept3

63

6 + (3) = 9


6 + (3) =63 =

+

6

3

9

This operation of

6 + (3) =9

is the same as

63 =9




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iii) 7(4) =


7(4)

Reject 7Reject4

7+4

+ + + +

7(4) = 3


7(4) =7 + 4 =

+

4 7

3

This operation of

7(4) =3

is the same as

7 + 4 =3




answer.


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iv)5(3) =


5

(3)

Reject 5

Reject 3

5

3

5(3) = 8


5(3) =53 =

+

5

3

8

This operation of

5(3) =8

is the same as

53 =8




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v) 35 + (57) =3557 =

Using the Table Model:

+

35

57

92

vi) 123(62) =123 + 62 =

Using the Table Model:

+

62 123

61

This operation of

35 + (57)

is the same as

3557





of the larger number in the answer.

This operation of

123(62)

is the same as

123 + 62


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1. 4 + (8) 2. 8(4) 3. 12 + (7)

4. 5 + (5) 5. 5 (7) + (4) 6. 7 + (4)(3)

7. 4 + (3) (7) 8. 6(2) + (8) 9. 3 + (4) + (6)

10. 44 + (81) 11. 118(43) 12. 125 + (77)

13. 125 + (239) 14. 125 (347) + (234) 15. 237 + (465)(378)

16. 412 + (334) (712) 17. 612(245) + (876) 18. 319 + (412) + (606)

TEST YOURSELF D


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PART E:

MULTIPLICATION OF INTEGERS

Consider the following pattern:

3 3 = 9

623

313

003 The result is reduced by 3 in

3)1(3 every step.

6)2(3

9)3(3

93)3(

62)3(

31)3(

00)3( The result is increased by 3 in

3)1()3( every step.

6)2()3(

9)3()3(

Multiplication Rules of Integers

1. When multiplying two integers of the same signs, the answer is positive integer.

2. When multiplying two integers ofdifferent signs, the answer is negative integer.

3. When any integer is multiplied by zero, the answer is always zero.

positive positive = positive

(+) (+) = (+)

positive negative = negative

(+) () = ()

negative positive = negative

() (+) = ()

negative negative = positive

() () = (+)

LESSON NOTES


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1. When multiplying two integers of the same signs, the answer is positive integer.

(a) 4 3 = 12(b) 8 6 = 48

2. When multiplying two integers of the different signs, the answer is negative integer.

(a) 4 (3) =12

(b) 8 (6) =48

3. When any integer is multiplied by zero, the answer is always zero.

(a) (4) 0 = 0

(b) (8) 0 = 0

(c) 0 (5) = 0

(d) 0 (7) = 0

EXAMPLES


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1. 4 (8) 2. 8 (4) 3. 12 (7)

4. 5 (5) 5. 5 (7) (4) 6. 7 (4) (3)

7. 4 (3) (7) 8. (6) (2) (8) 9. (3) (4) (6)

TEST YOURSELF E


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PART F:


USING

THE ACCEPT-REJECT MODEL


This part emphasises the second alternative method which include activities to

enhance the pupils understanding and mastery of the multiplication of integers.

Strategy:

Teacher should ensure that pupils understand the multiplication rules of integers

using the Accept-Reject Model. Pupils can then perform computations involving

multiplication of integers.

LEARNING OBJECTIVE

Upon completion of Part F, pupils will be able to perform computationsinvolving multiplication of integers using the Accept-Reject Model.


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PART F:


USING THE ACCEPT-REJECT MODEL

The Accept-Reject Model

In order to help pupils have a better understanding of multiplication of integers, we havedesigned the Accept-Reject Model.

Notes: (+) (+) : The first sign in the operation will determine whether to acceptor to reject the second sign.

Multiplication Rules:

To Accept or to Reject Answer

(2) (3) Accept + 6

(2) (3) Reject 6

(2) (3) Accept 6

(2) (3) Reject + 6

Sign To Accept or To Reject Answer

( + ) ( + ) Accept +

() () Reject

( + ) () Accept

() ( + ) Reject +

LESSON NOTES

EXAMPLES


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1. 3 (5) = 2. 4 (8) = 3. 6 (5) =

4. 8 (6) = 5. (5) 7 = 6. (30) (4) =

7. 4 9 (6) = 8. (3) 5 (6) = 9. (2) (9) (6) =

10. 5 (3) (+4) = 11. 7 (2) (+3) = 12. 5 8 (2) =

TEST YOURSELF F


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This part emphasises the division rules of integers.

Strategy:

Teacher should ensure that pupils understand the division rules of integers to

perform computation involving division of integers.

PART G:

DIVISION OF INTEGERS

LEARNING OBJECTIVE

Upon completion of Part G, pupils will be able to perform computations

involving division of integers.


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PART G:


Consider the following pattern:

3 2 = 6, then 6 2 = 3 and 6 3 = 2

3 (2) =6, then (6) 3 =2 and (6) (2) = 3

(3) 2 =6, then (6) 2 =3 and (6) (3) = 2

(3) (2) = 6, then 6 (3) =2 and 6 (2) =3

Rules of Division

1. Division of two integers of the same signs results in a positive integer.

i.e. positive positive = positive

(+) (+) = (+)

negative negative = positive() () = (+)

2. Division of two integers ofdifferent signs results in a negative integer.i.e. positive negative = negative

(+) () = ()

negative positive = negative

() (+) = ()

3. Division of any number by zero is undefined.

LESSON NOTES

Undefined means thisoperation does not have a

meaningand is thus not

assigned aninterpretation!

Source:

http://www.sn0wb0ard.com
http://www.sn0wb0ard.com/out/meaning/dictionary.htmhttp://www.sn0wb0ard.com/out/meaning/dictionary.htmhttp://www.sn0wb0ard.com/out/interpretation/dictionary.htmhttp://www.sn0wb0ard.com/out/interpretation/dictionary.htmhttp://www.sn0wb0ard.com/out/interpretation/dictionary.htmhttp://www.sn0wb0ard.com/out/interpretation/dictionary.htmhttp://www.sn0wb0ard.com/out/meaning/dictionary.htm


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1. Division of two integers of the same signs results in a positive integer.

(a) (12) (3) = 4

(b) (8) (2) = 4

2. Division of two integers ofdifferent signs results in a negative integer.(a) (12) (3) =4

(b) (+8) (2) =4

3. Division ofzero by any number will always give zero as an answer.(a) 0 (5) = 0

(b) 0 (7) = 0

EXAMPLES


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1. (24) (8) 2. 8 (4) 3. (21) (7)

4. (5) (5) 5. 60 (5) (4) 6. 36 (4) (3)

7. 42 (3) (7) 8. (16) (2) (8) 9. (48) (4) (6)

TEST YOURSELF G


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PART H:


USING

THE ACCEPT-REJECT MODEL


This part emphasises the alternative method that include activities to help pupils

further understand and master division of integers.

Strategy:

Teacher should make sure that pupils understand the division rules of integers using

the Accept-Reject Model. Pupils can then perform division of integers, includingthe use of brackets.

LEARNING OBJECTIVE

Upon completion of Part H, pupils will be able to perform computations

involving division of integers using the Accept-Reject Model.


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PART H:

DIVISION OF INTEGERS USING THE ACCEPT-REJECT MODEL

In order to help pupils have a better understanding of division of integers, we have designedthe Accept-RejectModel.

Notes: (+) (+) : The first sign in the operation will determine whether to acceptor to reject the second sign.

: The sign of the numerator will determine whether to accept or

to reject the sign of the denominator.

Division Rules:


( + ) ( + ) Accept + +

( ) ( ) Reject +

( + ) () Accept

( ) ( + ) Reject +

)(

)(

LESSON NOTES


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To Accept or To Reject Answer

(6) (3) Accept + 2

(6) (3) Reject 2

(+6) (3) Accept 2

(6) (3) Reject + 2

Division [Fraction Form]:


)(

)(

Accept + +

)(

)(

Reject +

)(

)(

Accept

)()(

Reject +

EXAMPLES


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To Accept or To Reject Answer

)2(

)8(

Accept + 4

)2(

)8(

Reject 4

)2(

)8(

Accept 4

)2(

)8(

Reject + 4

EXAMPLES


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1. 18 (6)2.

2

12

3.8

24

4.5

25

5.

3

6

6. (35) 7

7. (32) (4) 8. (45) 9 (5)9. )6(

)30(

10.)5(

80

11. 12 (3) (2) 12. (6) (3)

TEST YOURSELF H


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This part emphasises the order of operations when solving combined operations

involving integers.

Strategy:

Teacher should make sure that pupils are able to understand the order of operations

or also known as the BODMAS rule. Pupils can then perform combined operations

involving integers.

PART I:

COMBINED OPERATIONS

INVOLVING INTEGERS

LEARNING OBJECTIVES

Upon completion ofPart I, pupils will be able to:

1. perform computations involving combined operations of addition,subtraction, multiplication and division of integers to solve problems; and

2. apply the order of operations to solve the given problems.


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PART I:

COMBINED OPERATIONS INVOLVING INTEGERS

1. 10(4) 3=10(12)

= 10 + 12

= 22

2. (4) (83 )= (4) (11 )= 44

3. (6) + (3 + 8 ) 5= (6 )+ (5) 5= (6 )+ 1

= 5

LESSON NOTES

EXAMPLES

A standard order of operations for calculations involving +, , , andbrackets:

Step 1: First, perform all calculations inside the brackets.

Step 2: Next, perform all multiplications and divisions,working from left to right.

Step 3: Lastly, perform all additions and subtractions, working

from left to right.

The above order of operations is also known as theBODMAS Ruleand can be summarized as:

Brackets

power of

DivisionMultiplication

Addition

Subtraction


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1. 12 + (8 2) 2. (35) 2 3. 4 (16 2) 2

4. (4) 2 + 6 3 5. (25) (35 7) 6. (20)(3 + 4) 2

7. (12) + (4 6) 3 8. 16 4 + (2) 9. (18 2) + 5(4)

TEST YOURSELF I


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TEST YOURSELF A:

1. 2

2. 3

3. 6

4. 4

5. 2

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

ANSWERS


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TEST YOURSELF B:

1) 4 2) 12 3) 54) 10 5) 6 6) 67) 0 8) 12 9) 7

TEST YOURSELF C:

1) 42 2) 102 3) 924) 908 5) 548 6) 97) 843 8) 282 9) 514

TEST YOURSELF D:

1) 12 2) 12 3) 194) 10 5) 8 6) 07) 8 8) 0 9) 110) 125 11) 161 12) 20213) 364 14) 238 15) 60616) 790 17) 19 18) 125

TEST YOURSELF E:

1) 32 2) 32 3) 844) 25 5) 140 6) 847) 84 8) 96 9) 72


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TEST YOURSELF F:

1) 15 2) 32 3) 304) 48 5) 35 6) 1207) 216 8) 90 9) 10810) 60 11) 42 12) 80

TEST YOURSELF G:

1) 3 2) 2 3) 3

4) 1 5) 3 6) 37) 2 8) 1 9) 2

TEST YOURSELF H:

1. 3 2. 6 3. 3

4. 5 5. 2 6. 5

7. 8 8. 1 9. 5

10. 16 11. 2 12. 2

TEST YOURSELF I:

1. 16 2. 16 3. 12

4 10 5 5 6 34

beams_unit 1 negative numbers

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