# MATHEMATICS MATERIALS FOR HIGH CLASS CLASS VI ELEMENTARY SCHOOL SEMESTER 1 AND 2

** PAPER**

**MATHEMATICS MATERIALS FOR HIGH CLASS CLASS VI ELEMENTARY SCHOOL**

**SEMESTER 1 AND 2**

**Supporting Lecturer : Reviva Safitri, S.Pd.I., M.Pd**

** **

** **

** **

** **

** **

** **

** **

** **

**Arranged by:**

**GROUP V**

** **

**NELSON SYAHRIL MANSYUR ZEBUA (21140122) **

**SITI RACMANIA HARAHAP (21140207) **

**SYAHRIL ROMADON RITONGA (21140215)**

**TRI ANNISAH HARAHAP (21140218) **

** **

** **

** **

** **

** **

**PRIMARY TEACHER EDUCATION**

**FACULTY OF SOCIAL SCIENCES AND LANGUAGE EDUCATION**

**SOUTH TAPANULI INSTITUTE OF EDUCATION**

**T.P 2022/2023**

**FOREWORD**

** **

** **Praise be to God, we pray for the presence of Allah SWT, for His mercy and grace we can finish this paper on time. The theme of this paper is " MATHEMATICS MATERIALS FOR HIGH CLASS VI SEMESTER 1 AND 2 "

** **On this occasion we would like to express our deepest gratitude to the lecturers of the HIGH CLASS MATHEMATICS LEARNING course who have given us assignments. We would also like to thank those who helped in the preparation of this paper ** **

We are far from perfect and this is a good step from the real study, because of the limitations of our time and abilities, we always welcome constructive criticism and suggestions. Hopefully this paper can be useful for us in particular and other interested parties in general.

Padang Sidimpuan, 05 October 2022

Compiler

**LIST OF CONTENTS**

** **

**FOREWORD**

**LIST OF CONTENTS**

**CHAPTER I INTRODUCTION**

** Background**

** Formulation of the problem**

** Writing purpose**

** **

**CHAPTER II DISCUSSION**

** **

**CHAPTER III CLOSING**

** Conclusion**

** Suggestion**

** **

**BIBLIOGRAPHY**

** **

** **

**CHAPTER I**

**INTRODUCTION**

** **

A. BACKGROUND

is one of the sciences whose function and application requires mathematics

for many problems in everyday life, one of which is in

development of mathematics and technology. In addition, mathematics is also a

disciplines that have unique or distinctive properties, the object of mathematics is

objects that are abstract and cannot be observed by the five senses.

Mathematics emphasizes activities in the world of ratios (reasoning), doesn't it

emphasizes the experimental results or the results of mathematical observations are formed

because of human thoughts, which are associated with ideas, processes, and reasoning

CHAPTER II

SEMESTER DISCUSSION 1

Mathematics is a basic subject taught since elementary school from grade 1 to grade 6.

Chapter 1 semester 1

A. INTEGER COUNTING OPERATIONS

1. DEFINITION OF INTEGER NUMBER

Integers are an expansion of whole numbers. The expanded set of integers consists of the set of natural numbers, namely {1,2,3,4…} which hereinafter are called positive integers, zero numbers, and the set of opposites of natural numbers, namely { - 1,-2, -3, -4, ...} which hereinafter is called the set of negative integers.

2. Operations to count whole numbers in addition, subtraction, multiplication, and division .

Addition and subtraction of integers. The way to add integers is as follows: If the two numbers have the same sign, then: The sign of the sum is the same as the sign of the two numbers and the result is the sum of the two numbers.

Example problem: The result of 15 + 15 = 30 The result of -14 + (-20) = - 34 If the two numbers have different signs, then: The sign of the sum is the same as the sign of the largest number in the sum, and the result is the difference between the numbers the largest to the smallest number in the sum.

Example problem: Result of 15 + 15 = 30 Result of -14 + (-20) = -34

If the two numbers have different signs, then: The sign of the result of the sum is the same as the sign of the largest number in the sum, and the result is the difference between the largest number and the smallest number in the sum.

reduced by – 45. It becomes as follows 120 – 45 = 75 According to the weather forecast, the temperature in the Thousand Islands is 300C, while the temperature in the Riau Islands is -100C, the temperature difference between the two villages is ... For this problem, please do it from to the left. That is, 85 – (-35) is changed to 85 + 35 = 120, it just needs to be reduced by – 45. It becomes like the following 120 – 45 = 75 To solve the above problem, it is necessary to first describe the concept of calculating it to be as follows: Difference in temperature = Temperature of the Thousand Islands – Riau Islands temperature Difference in temperature = 300C – (-100C) = 30 + 10 = 400C Multiplication and division of integers. Basically, multiplication of integers is almost the same as multiplication of whole numbers. But in multiplication of integers there are rules for multiplication of signs with a certain number, as follows: 34 x (-24) – (-4) = -816 – (-4) = -816 + 4 = - 812 (-75) : (-5) – (-13) = 5 – (-13) = 5 + 13 = 18 Addition of whole numbers. positive integer + positive integer the result is a positive integer. Example: 9 + 4 = 13 negative integer + negative integer the result is a negative integer. Example: - 12 + (- 6) = -18 negative integer + positive integer the result is a positive or negative integer negative integer + positive integer the result is a positive or negative integer Information on lessoncg.blogspot.com: for point 3 and the 4 steps for the solution are as follows Find the difference between the two numbers, which number is larger (positive or negative) and mark the sum with the same sign as the larger number. First example: 10 + (- 6) = ... where the difference between 10 and 6 is 4, then 10 is greater than 6, and 10 is positive, so the result is positive so, 10 + (- 6) = 4 Second Example 7 + (- 12) = ... where the difference between 12 and 7 is 5, so 12 is greater than 7, and 12 is negative, so the result is negative so, 7 + (- 12) = - 5 Third example -15 + 9 = ... where the difference between 15 and 9 is 6, so 15 is greater greater than 9, and 15 is negative, then the result is negative so, -15 + 9 = - 6 Fourth example -18 + 30 = ... where the difference between 30 and 18 is 12, then 30 is greater than 18, and 30 is the sign is positive, then the result is positive so, -18 + 30 = 12 Subtraction of integers. The operation of subtracting integers can be converted into an addition operation with the opposite number of the subtracting number. This means the opposite of a number, for example 5 versus -5; -12 versus 12; - 7 against 7; 9 against -9. Now consider the following example of subtracting integers: 9 – 4 = 9 + (-4) = 5 9 – 19 = 9 + (-19) = -10 - 12 – (- 6) = -12 + 6 = -8 10 – (- 6) = 10 + 6 = 16 -10 – 40 = - 10 + (-40) = - 50 Multiply integers. a positive integer x a positive integer is a positive integer. Example: 9 x 4 = 36 negative integer x negative integer the result is a positive integer. Example : - 12 x (- 6) = … 9 x 4 = 36 negative integer x negative integer the result is a positive integer. Example : - 12 x (- 6) = … 9 x 4 = 36 negative integer x negative integer the result is a positive integer. Example : - 12 x (- 6) = … 72 positive integers x negative integers result in negative integers. Example: 8 x (- 7) = - 56 negative integer x positive integer the result is a negative integer. Example : - 5 x 9 = - 45 Division of integers. positive integers : positive integers result in positive integers. Example: 72 : 8 = 9 negative integers: negative integers result in negative integers. Example: 120 : (- 10 ) = -12 negative integers : positive integers result in negative integers. Example : - 64 : 4 = - 16 negative integers : negative integers result in negative integers. Example : - 75 : -25 = 3.

3. The nature of the operation of round count

Properties of Counting Integer Operations and Examples

Properties of Counting Integer Operations

Properties of Integer Count Operations – An integer is a number that consists of positive numbers, zeros, and negative numbers. In performing operations on integers, there are several properties that we need to know. The following is a discussion of the properties of arithmetic operations on integers and examples of each.

Properties of Counting Integer Operations and Examples

Integers have properties in their calculations. The following are the properties that apply to the arithmetic operations of addition, subtraction, multiplication and division on integers.

1. Closed Nature

The closed property of integer operations only applies to addition and multiplication operations.

a. Closed Property of Addition

The closed property of addition is the addition operation on two integers, it will produce an integer.

a + b = c

Example:

2 + 3 = 5

(2 and 3 are integers, so 5 is also an integer)

4 + 6 = 10

(4 and 6 are integers, so 10 is also an integer)

b. Closed Properties of Multiplication

The closed property of multiplication is that the multiplication operation on two integers results in an integer.

a x b = c

Example:

2 x 3 = 6

(2 and 3 are integers, so 6 is also an integer)

4 x 5 = 20

(4 and 5 are integers, so 20 is also an integer)

2. Commutative nature

The commutative property is also known as the exchange property. This property applies to addition and multiplication of integers.

a. The Commutative Property of Addition

a + b = b + a

Example:

2 + 5 = 5 + 2 = 7

4 + 6 = 6 + 4 = 10

. The Commutative Property of Multiplication

a x b = b x a

Example:

2 x 5 = 5 x 2 = 10

3 x 4 = 4 x 3 = 12

3. Associative nature

The associative property is also known as the grouping property. This property applies to integer operations involving addition and multiplication.

a. The Associative Property of Addition

(a + b) + c = a + (b + c)

Example:

(2 + 3) + 4 = 2 + (3 + 4) = 9

(5 + 1) + 6 = 5 + (1 + 6) = 12

b. The Associative Property of Multiplication

(a x b) x c = a x (b x c)

Example:

(2 x 3) x 5 = 2 x (3 x 5) = 30

(4 x 5) x 6 = 4 x (5 x 6) = 120

4. Distributive Properties

The distributive property is the property of spreading. Distributive properties are grouped into two types, namely as follows:

a. The Distributive Property of Multiplication Over Addition

a x b) + (a x c) = a x (b + c)

Example:

(2 x 5) + (2 x 10) = 2 x (5 + 10) = 30

(3 x 4) + (3 x 5) = 3 x (4 + 5) = 27

b. The Distributive Property of Multiplication Against Subtraction

(a x b) – (a x c) = a x (b – c)

Example:

(5 x 3) – (5 x 2) = 5 x (3 – 2) = 5

(4 x 8) – (4 x 5) = 4 x (8 – 5) = 12

5. Nature of Identity

There are two groupings of identity properties in integer arithmetic operations, namely as follows:

a. Addition Identity Properties

The identity property of the integer addition operation is 0. If an integer is added to 0, the result is the number itself.

0 + a = a + 0

Example:

5 + 0 = 0 + 5 = 5

8 + 0 = 0 + 8 = 8

b. Properties of Multiply Identity

The identity property of the integer multiplication operation is 1. When an integer is multiplied by 1, the result is the number itself.

a x 1 = a

Example:

10 x 1 = 10

5 x 1 = 5

6. Additive Inverse Element

The inverse element of addition is the opposite number in the addition operation.

a + (-a) = 0

Example:

4 + (-4) = 0

7 + (-7) = 0

4. How to calculate mixed number operations

How to do mixed arithmetic operations - In mathematics lessons, we are familiar with various arithmetic operations, such as addition, subtraction, multiplication, and division. If you calculate just one arithmetic operation (for example: addition), you can definitely solve it easily. However, what if in one problem there are several calculations or mixed arithmetic operations? Which operation should be done first?

What is Mixed Arithmetic Operation?

Mixed arithmetic operations are arithmetic operations that involve more than one different calculation. For example addition operations with multiplication, subtraction with division, and so on. Then, is the way to do it directly starting from the left? Yes, there might be a point if the arithmetic operations are still at the same level. But if it's different levels, then such an answer is definitely wrong.

Please note that to carry out mixed arithmetic operations, there are several rules that must be followed. And we must follow these rules so as not to be wrong in calculating it. However, in reality there are still many elementary and junior high school students who do not know the rules. So there are still many mistakes in doing it.

For this reason, for those who are still confused about working on questions that involve several arithmetic operations in them, please refer to the following discussion on how to do mixed arithmetic operations correctly.

Rules for Performing Mixed Arithmetic Operations

To perform mixed arithmetic operations, we must understand the solution rules. The rules that apply in arithmetic operations with mixed mathematical numbers are as follows:

Addition and subtraction operations have the same level, so they are done sequentially starting from the left.

Multiplication and division operations have the same level, so they are done sequentially starting from the left.

Multiplication and division operations have a higher level than addition and subtraction operations, so if there are addition, subtraction, multiplication, and division operations at random, then do the multiplication and division operations first.

Operations that are arithmetic in parentheses, do it first.

How to do mixed arithmetic operations

Operations Count Addition and Subtraction

Mixed arithmetic operations that involve addition and subtraction have the same level, so to do them sequentially starting from the left.

Problems example

50 + 25 – 45 = …?

Answer:

The first step, count = 50 + 25 = 75

The second step, count = 75 – 45 = 30

So, 50 + 25 – 45 = 30

Operations to Count Multiplication and Division

Mixed arithmetic operations involving multiplication and division have the same degree, so to do them sequentially starting from the left.

Problems example

100 : 5 x 3 = …?

Answer:

The first step, calculate = 100 : 5 = 20

The second step, calculate = 20 x 3 = 60

So, 100 : 5 x 3 = 60

Mixed Count Operations (Addition, Subtraction, Multiplication and Division)

Multiplication and division operations have a higher level than addition and subtraction operations. So, if you encounter a problem which involves addition, subtraction, multiplication and division operations randomly, then do the multiplication and division operations first, then do the addition and subtraction. However, if there are parentheses, then do the calculations in the parentheses.

Problems example

25 + 3 x 10 – 50 = …?

Answer:

25 + (3 x 10) – 50 = …

The first step is to calculate the multiplication operation first = 3 x 10 = 30

Then the calculation operation becomes = 25 + (30) – 50 = …?

The second step, calculating the sum = 25 + 30 = 55

The third step, calculating the subtraction = 55 – 50 = 5

So, 25 + 3 x 10 – 50 = 5

Problems example

5 x [30 + 10] : 5 – 30 = …?

Answer:

5 x [30 + 10] : 5 – 30 = …

Because there are brackets, do what is inside the brackets first = 30 + 10 = 40

Then the calculation becomes = 5 x 40 : 5 – 30 = …?

The second step, first calculate the multiplication or division = 5 x 40 = 200

Then the result will be obtained = 200: 5 – 30 = …?

The third step, calculate the first division = 200: 5 = 40

Then the result will be obtained = 40 – 30 = …?

The fourth step, count = 40 – 30 = 10

So, 5 x [30 + 10] : 5 – 30 = 10

**Bab ll semester 1**

Volume measurement per time

** **

Units of Cubic Volume and Liters – Mathematics is an important lesson that needs to be learned. Not only at school, mathematical calculations are also often encountered in everyday life. One of them is calculating the volume.

Volume or content can be interpreted as a derivative of the principal quantity length. Units of volume are usually expressed with a cubic ending, such as cubic meters or cubic centimeters. Cubic is represented by a cube, for example m³ or cm³.

In addition to cubic, volume units are also often expressed in liters. The liter unit is usually used to calculate volume in three-dimensional spaces, such as cubes, blocks, cylinders, and so on.

Each volume unit, both cubic and liter, has a different conversion in its calculations. However, cubic units can also be converted to liters and vice versa. For more details, see the following discussion.

Cubic and Liter Volume Units and How to Convert Them

A. Cubic Volume Unit

A cubic is a unit of volume represented by a power of 3, eg m³, cm³, mm³. The international (SI) units for volume are m³ (MKS system) and cm³ (CGS system). To understand the conversion of volume units in cubic, please look at the following volume unit ladder image.

B. Unit Volume Liters

The liter is a unit of volume that is used to choose to calculate the amount of an object that occupies a cube-shaped space that has a side length of 10 cm. Thus, the value of 1 liter is equal to 1000 cm³. Liters are written with a lower case l. The following is a volume unit conversion ladder in liters.

In the picture above, it can be concluded that to change one unit to another, you can use the following steps:

Every time you go down one ladder, you multiply it by 1,000

Every time you go up one ladder, you divide by 1,000

Example:

5 km³ = 1 x 1.000 = 5.000 hm³

6000 mm³ = 6.000 : 1.000 = 6 cm³

The following are equivalent volume units in cubic units: 1 km³ = 1000 hm³

1 hm³ = 1000 dam³

1 dam³ = 1000 m³

1 m³ = 1000 dm³

1 dm³ = 1000 cm³ (cc)

1 cm³ = 1000 mm³

In the picture above, it can be concluded that to change one unit to another, you can use the following steps:

Every time you go down one ladder, you multiply it by 1,000

Every time you go up one ladder, you divide by 1,000

Example:

5 km³ = 1 x 1.000 = 5.000 hm³

6000 mm³ = 6.000 : 1.000 = 6 cm³

The following are equivalent volume units in cubic units: 1 km³ = 1000 hm³

1 hm³ = 1000 dam³

1 dam³ = 1000 m³

1 m³ = 1000 dm³

1 dm³ = 1000 cm³ (cc)

1 cm³ = 1000 mm³

In the picture above, it can be concluded that to change one unit to another, you can use the following steps:

Every time you go down one ladder, you multiply it by 1,000

Every time you go up one ladder, you divide by 1,000

Example:

5 km³ = 1 x 1.000 = 5.000 hm³

6000 mm³ = 6.000 : 1.000 = 6 cm³

The following are equivalent volume units in cubic units: 1 km³ = 1000 hm³

1 hm³ = 1000 dam³

1 dam³ = 1000 m³

1 m³ = 1000 dm³

1 dm³ = 1000 cm³ (cc)

1 cm³ = 1000 mm³

B. Unit Volume Liters

The liter is a unit of volume that is used to choose to calculate the amount of an object that occupies a cube-shaped space that has a side length of 10 cm. Thus, the value of 1 liter is equal to 1000 cm³. Liters are written with a lower case l. The following is a volume unit conversion ladder in liters.

Every time you go down one ladder, you multiply it by 10

Every time you go up one ladder, divide by 10

Example:

5 kl = 1 x 10 = 5.000 kl

60 ml = 60 : 10 = 6 cl

The following is the equivalent volume unit in liters:

1 kl = 10 kl

1 hl = 10 dal

1 from = 10l

1 l = 10 dl

1 dl = 10 cl

1 cl = 10 ml

C. How to Convert Cubic Units to Liters or Vice versa

To convert different volume units, for example cubic units to liters or liters to cubic, then on the other hand, first understand the equivalence between the two units. The following is the equivalent value between cubic units and liters.

1 liter = 1 dm³

1 ml = 1 cm³

To make it easier to understand, please refer to the following examples.

Problems example:

5 m³ = … dl

Completion:

The first step is to change the unit of m³ to dm³

From m³ to dm³ is 1 step down, so multiply by 1,000

5 m³ = 5 x 1.000 = 5.000 dm³

because 1 dm³ = 1 liter, then 5,000 dm³ = 5,000 liters

The second step is to convert from liter to dl

From liter to dl is down 1 ladder, then multiply by 10

5.000 liter = 5.000 x 10 = 50.000 dl

so, 5 m³ = 50,000 dl

**Chapter III semester 1**

Calculating area

Flat Shape Area Formulas and Example Problems - Flat shapes are flat planes that only have dimensions of length and width. So that only the area and perimeter can be calculated. On this occasion, we will discuss the area formula for flat shapes and examples of problems.

There are several types of flat shapes, including squares, rectangles, triangles, trapezoids, parallelograms, rhombuses, kites and circles. To calculate the area, the formula used is the formula for the area of a flat shape. Each plane shape has a different formula.

Well, for those who don't know how to calculate the area of a flat shape. Please refer to the following discussion.

Example of a Flat Shape Problem

Example Problem: How to Calculate the Area of a Square Shape

A square has a side size of 10 cm. Count how many squares it is!

Answer:

L = s x s

L = 10 x 10

L = 100 cm²

So, the area of the square is 100 cm².

Example Problem: How to Calculate the Area of a Rectangle Flat Shape

A rectangle has a length of 10 cm and a width of 5 cm. Calculate the area of the rectangle!

Answer:

L = p x l

L = 10 x 5

L = 50 cm²

So, the area of the rectangle is 50 cm².

Example Problem: How to Calculate the Area of a Triangle Flat Shape

A triangle has a base of 10 cm and a height of 5 cm. Calculate the area of the triangle!

Answer:

L = ½ x a x t

L = ½ x 10 x 5

L = 25 cm²

So, the area of the triangle is 25 cm².

Example Problem: How to Calculate the Area of a Trapezoid

A trapezoid has parallel sides 10 cm and 8 cm. If the height of the trapezoid is 5 cm, what is the area of the trapezoid?

Answer:

L = ½ x (a + b) x t

L = ½ x (10 + 8) x 5

L = ½ x 18 x 5

L = 45 cm²

So, the area of the trapezoid is 45 cm².

Example Problem: How to Calculate the Area of a Parallelogram

A parallelogram has a base side of 8 cm and a height of 5 cm. Count how wide the parallelogram is!

Answer:

L = a x t

L = 8 x 5

L = 40 cm²

So, the area of a parallelogram is 40 cm².

Example Problem: How to Calculate the Area of a Rhombus Flat Shape

A rhombus has the lengths of the diagonals 8 cm and 6 cm. Calculate the area of the rhombus!

Answer:

L = ½ x d1 x d2

L = ½ x 8 x 6

L = 24 cm²

So, the area of a rhombus is 24 cm².

Example Problem: How to Calculate the Area of a Kite Flat Shape

A kite has diagonals of 10 cm and 8 cm. Count how wide the kite is!

Answer:

L = ½ x d1 x d2

L = ½ x 10 x 8

L = 40 cm²

So, the area of the kite is 40 cm².

Example Problem: How to Calculate the Area of a Circle Flat Shape

A circle has a radius of 7 cm. Count how wide the circle is!

Answer:

L = Ï€ x r²

L = 22/7 x 7

L = 22/7 x 49

L = 154 cm²

So, the area of the circle is 154 cm².

**Chapter IV Semester 1**

Collect and process data

Data collection

Before obtaining data, we must carry out the data collection process. There are several ways that are usually done to obtain data, including through:

•Study

•Interview

•Polls / questionnaires

Direct counting

•Penyajian data

After obtaining the data, usually the data is presented in various forms. One example of data that can be presented is the math scores of students in a school. Here are some ways you can present data:

Using tables

The data can be described using tables, the following is an example of a table of data on the mathematics scores of SD students of Harapan Harapan Elementary School:

From the table above we can find out:

There were 5 students who scored 65

There were 9 students who scored 70

From the table above we can find out:

There were 5 students who scored 65

There were 9 students who scored 70

There were 14 students who scored 75

There were 10 students who scored 80

There were 5 students who scored 85

There were 7 students who scored 90

Using charts

There are various forms of diagrams, ranging from bar charts, pie charts, line charts, and line charts.

Bar Chart

Let's convert the data above into a bar chart:

**Pie chart**

** **

**to create a pie chart, we must find the percentage of the angles from the data obtained.**

**value 65 = 5/50 x 3600 = 360**

**value 70 = 9/50 x 3600 = 64.80**

**value 75 = 14/50 x 3600 = 100.80**

**value 80 = 10/50 x 3600 = 720**

**value 85 = 5/50 x 3600 = 360**

**value 90 = 7/50 x 3600 = 50.40**

**then the image diagram will be like this:**

Line Charts

Almost the same as a bar chart except that the shape is changed to a line.

Data processing

In processing a data there are several things that we have to look for using mathematical formulas, namely:

Mean

The mean is the average value of all the data obtained. The average value is obtained by adding up all values and then dividing by the amount of data.

Average = Total data

Lots of data

For example, from the data above, we can find the mean by adding up the scores and then dividing by the number of students, like this:

65+70+75+80+85+90 = 465 = 9.3

50 50

So the average score of class VI students for mathematics at SD Tunas Mekar is = 9.3

mode

The mode is the value that appears most frequently in the data. When viewed from the data on the mathematics scores of grade VI SD Tunas Mekar students, the score that appears most often is 70 because there are 14 students who scored 70.

Median

The median is the middle value. Obtained by sorting the existing values from smallest to largest. Consider the following examples:

Exercises

The daily test scores for grade IV SD Sumber Rejo are: 5,6,7,8,9,7,8,7,10,5 find the median of the data.

Answer:

Sort these values from the smallest: 5,5,6,7,7,7,8,8,9,10 -> there are 10 data

Take the value in the middle, if the number of data is even, take the two values in the middle, then divide by 2. As in the question above, because the number of data is even (10), then we take the two values in the middle, namely 7 and 7.

7+7 : 2 = 14 : 2 = 7

Then the median of the data is 7.

SEMESTER II DISCUSSION

Chapter 1

A. INTEGER NUMBER

→ Integer consists of natural numbers, zero, and opposite natural numbers Natural numbers are called positive integers Opposite natural numbers are called negative integers

→ An example of using negative integers is to write the position of an object below sea level or a temperature below 0°

→ How to read negative numbers : -32 is read as negative thirty two

How to write negative integers Negative 11 is written -11

Place integers on the number line

Numbers to the right of zero are positive integers, while to the left of zero are negative integers.

B. INTEGER COUNTING OPERATIONS

Addition of two integers

1. If both numbers have the same sign, then they can be added directly

as in whole numbers. The sign of the number on the result of the sum is the same

with the sign of the two numbers being added Example:

-275+ (-302)=-577

(the sign is the same negative, so it remains only to add and the result is negative) If the two numbers added have different signs, ignore the sign first, subtract the larger number by the larger number

2. small. The sign of the number in the sum is the same as the sign of the number

bigger Example:

-250 + 120=-130

(ignore the negative sign, the larger number is 250, subtract 120, then the sign of the sum is negative)

→ Subtraction of two integers

The subtraction of integers can be calculated by adding the opposite number of the subtraction

Example:

275-(-175) 275 +175 = 450 (note that the opposite of -175 is 175)

→ Multiplication and division of integers

The product of multiplication and division of integers can be determined in the same way as for whole numbers. If both numbers have the same sign then the result is positive. If the signs are different then the result is negative

Example:

25x4=100

equal sign

-30 x-5=150

40x-4=-160 difference sign

equal sign -20x3=-60 difference sign

C. PROPERTIES OF INTEGATE COMPUTATION OPERATIONS → Commutative (Exchange) Properties

a+b=b+a | axb = bxa

where a and b are integers

Example:

10 + 5 = 5 + 10 = 15 20 * 6 = 6 * 20 = 120

→ Associative Properties (Grouping)

(a+b)+c=a+ (b+c) (axb) xc=ax (bx c)

where a, b, and c are integers

Example:

(12 + 5) + 15 = 12 + (5 + 15) = 12 + 20 = 32 (2 x 3) x (-5)=2x (3 x (-5)) = 2 x (-15)=-30

> Distributive Properties (Spread)

ax (b+c)=(axb) + (a xc)

ax (bc)=(axb)-(axc) where a,b, and c are integers

Example:

10(- 5 + 3) = (10(- 5)) + (10 * 3) = - 50 + 30 = 0.2 5x(-20 -4) = (5 x (-20)) - (5 x (-) 4)) = -100 - (-20) = -120

D. COUNTING OPERATIONS OF MIXED INTEGER NUMBERS The rules for performing integer arithmetic operations are the same as whole number arithmetic operations, namely:

1. The arithmetic operations in parentheses are done first

2. Multiplication and division operations have a higher level than addition and subtraction, so multiplication and division are done first

3. The multiplication and division operations are of the same level, so they are done sequentially from the left

4. Addition and subtraction operations are at the same level, so they are done sequentially from the left

Example:

1. - 100 + (75 - (- 200)) = - 100 + 275 = 175

Worked first

2. 500+(-800): 4500+(-200) = 300

Worked first.

chapter 2

A. Definition of Circle

A circle is defined as a curved line whose ends meet at the same distance from the center. In Euclidean geometry, a circle is the set of all points in a plane within a given distance. which is called the radius, from a certain point, which is called the center. Whereas the inner circle Wahyudi (2013: 125) is a special simple closed curve. Each point on the circle has the same distance from a point called the center of the circle. The circle is the perfect form of all objects in the universe because the circle has no end and no beginning.

A circle for some people is defined as the set of all points (x, y) if the point (x, y) is the right angle of all possible right triangles formed from two points that are a certain distance away. Circles in mathematics are included in the category

flat shape whose area and circumference can be measured based on geometric mathematical formulas

.

2. The radius, also known as the radius of a circle, is the distance between the points on a circle and the center of a circle. Finger-jan notation is symbolized by letters. In the figure, AO and OB are the radii of the circle. The length of AO-BO=r

3. Diameter, a straight line that connects two points on a curved circle and passes through the center point. The length of the diameter of a circle is 2 times the length of the radius of the circle or can be written d-2r.

4. An arc, namely a circular arc that lies between two points on a circle. The notation for a circular arc is "~". Look at the picture, arc CD (CD) is one of the arcs of circle O. Arc CD is bounded by point C and point D on the circle.

5. A chord, which is a line segment that connects two points on a circle. A chord that passes through the center of the circle is also called a line

center or diameter. Thus, each centerline is a chord. However, not every bowstring is a centerline

C. Around the Circle

The circumference of the circle is the length of the curved line (a simple closed curve). We can measure the circumference of a circle by cutting the circle at a point, then straightening the curve of the circle and then we measure the length of the circle line with a ruler.

There are 2 ways to determine the circumference of a circle, namely:

By wrapping the rope / ribbon around the circle

Take an object with a circular surface, for example this object.

Second, by rolling the circle

Roll the object straight forward until the mark on the object is back on the table surface, make a mark on the table surface exactly aligned with the mark on the object (eg point B). Measure the distance traveled by the object (from point A to point B) with a ruler. The distance obtained is the circumference of the circle of the object.

In addition to the method above, the circumference of a circle can also be determined using the formula. However, this formula joins to a value, which is x (pronounced phi). To find out the value of a, then do the following activities:

1. Prepare materials such as paper compasses, mattress thread and ruler

2. By using a compass,

different diameters. Make 3 circles of length

3. Then, count each circle that has been made. You do this by squeezing the mattress threads in each circle earlier

4. Measure the length of the mattress thread earlier

5. Record the results in the table as follows:

From this research, the results of dividing the circumference divided by the diameter are always the same. This value is 3.141592...... hereinafter referred to as (pronounced phi). If rounded to the approximation is obtained

3,14. Therefore

22

22 7

= 3.14, then the value can also be expressed

with ♬ -

So, it can be written that:

d

K x d

Since Diameter (d)-2x radius, then:

Problems example.

1. Calculate the circumference of a circle with a radius of 7 cm!

Solution:

Dik r-7 cm

It

: K #

Answer.

= 2 x 22/7 x7

-44 cm

So, the circumference of the circle is 44 cm

2. A circle has a diameter of 35 cm. Determine Circumference!

Completion:

Dik: d= 14 cm

It: Can?

Answer:

Who

=xxd

2 x 14 cm

-44 cm?

So, the circumference of the circle is 44 cm³

D. Area of the circle

The area of a circle is the area contained within a circle. For

The formula for determining the area of a circle can be found by cutting the area of the circle to form wedges. Then the wedges are arranged in a crosswise manner so that they are close to a rectangular shape. The circular areas form wedges. Then the cut sections are arranged in a crosswise manner so that they are close to a rectangular shape. In the figure, a circle with radius r is cut across the center line to form 18 wedges. Then one of the wedges is divided in half to form a smaller wedge. Thus, there are 19 wedges. The wedges are arranged crosswise so that they are close to a rectangular shape. The smaller the pieces are, getting closer to the rectangle The rectangle that is formed has a length of half the circumference of the circle, and the width is equal to the radius of the circle. Thus, the area of the circle is equal to the area of the rectangle. So, the area of the circle is:

L

= pxl

2 Go around the xr circle

2 x 2 x r

=

LÏ€r

Problems example

1. Calculate the area of a circle with a radius of 20 cm!

Completion:

Height: r-20 cm

Say: Him?

Answer

= pi * r ^ 2

= 3.14 * 20 * 20

= 1256c * m ^ 2

So. The area of the circle is 1256 cm²

2. The area of a circle is 1386c * m^2 . Define the fingers!

Completion:

Dimensions: L=1386 cm³

This:r...?

Answer

L=pi*r^2

1386 = 27/(r ^ 2)

r ^ 2 = 7 22 *138

r ^ 2 = 7 * 63

r ^ 2 = 441

r ^ 2 = sqrt(441)

r = 21 So, the radius of the circle is 21 cm.

E. The properties of Circles

The properties of the circle are:

1. The circle is a simple closed curve

2. A circle has a diameter that is 2 times the radius

3. A circle has a center point

4. The radius of a circle is the distance from the center point to the edge of the circle.

5. It does not have a corner point or a 360 degree angle

6. Has infinite folding symmetry 7. Has infinite rotational symmetry

Chapter 3 Build Space

Understanding Build Space

A space shape is a mathematical shape that has a volume or volume. It can also be called the part of space that is bounded by the set of points that are found on the entire surface of the figure.

In each of these figures there is a formula for calculating the area as well as the content or volume. The various geometric shapes are prisms, blocks, cubes, pyramids, cylinders, cones and spheres. However, what we will discuss in this paper are prisms, blocks, cubes.

Various Build Space

In the following, we will provide various geometric shapes, starting from flat side shapes which include cubes, beams, prisms, and pyramids. To build a curved side space that includes cones, tubes, and spheres.

1. Cube

A cube is a three-dimensional shape bounded by six square sides.

The cube is also known by another name, namely a regular hexagon. A cube is actually a special form of a rectangular prism, because it is the same height as the base.

The nature of the cube

It has 6 square sides that are the same size

Has 12 ribs that are the same length

Has 8 corner points

It has 4 space diagonals

It has 12 diagonals

Formulas on the Cube

Volume: V= s x s x s = s3

Surface area: 6 sxs = 6 s2

The length of the plane's diagonal: s√2

Diagonal length of space: s√37

Diagonal area: s2√2

Information:

L= Surface area of the cube (cm2)

V= Volume cube (cm3)

S = length of the edge of the cube (cm)

2. Blocks

A beam is a geometric shape that has three pairs of rectangular sides. Where on each side facing has the same shape and size.

In contrast to a cube where all of its sides are congruent squares, and in a cuboid only the opposite sides are the same size.

And not entirely square, mostly rectangular.

Beam Properties

At least a cuboid has two pairs of rectangular sides.

Parallel ribs have the same length:

AB = CD = EF = GH, and AE = BF = CG = DH.

On each diagonal the fields on opposite sides are the same length, namely:

ABCD with EFGH, ABFE with DCGH, and BCFG with ADHE which have the same length.

Each space diagonal on the beam has the same length.

Each of the diagonal areas is a rectangle.

Formulas on Beams:

Volume: p.l.t

Surface Area: 2 (pl + pt + lt)

Diagonal Length of Field: √(p2+l2) or also √(p2+t2) or √(l2+t2)

Length of Space Diagonal: √(p2+l2+t2)

Information:

p: long

l : wide

t : height

3. Limas

A pyramid is a three-dimensional geometric figure bounded by an n-shaped base (can be a triangle, quadrangle, pentagon, etc.) and triangular upright sides that intersect at one vertex.

There are many types of pyramids which are categorized based on the shape of the base. Among others: triangular pyramids, rectangular pyramids, pentagonal pyramids, and others.

A pyramid with a circular base is known as a cone. Meanwhile, a pyramid with a square base is called a pyramid.

properties of pyramids:

Pyramids also have several properties or characteristics, including the following:

It has 5 sides, namely: 1 side is a quadrilateral in the form of a base and the other 4 sides are all triangular and are upright sides.

Has 8 ribs.

It has 5 corner points, including: 4 corners located at the base and 1 corner located at the top which is the apex.

The Formula on Limas

Limas Volume = 1/3 Base Area x Height

Surface Area = Total Base Area + Total Area of the Vertical Sides

4. Prism

A prism is a three-dimensional figure in which the base and lid are congruent and parallel in the form of n-angles.

The upright sides in a prism have several shapes, including: square, rectangle, or parallelogram. Judging from the uprights, prisms are divided into two kinds, namely: upright prisms and oblique prisms.

A upright prism is a prime in which the edges are perpendicular to the base as well as the lid. Meanwhile, an inclined prism is a prism in which the upright edges are not perpendicular to the base and also the lid.

If we look at the shape of the base, prisms are divided into several types, namely: triangular prisms, rectangular prisms, pentagonal prisms, and so on.

A prism whose base and lid are square is known as a block and a cube. Meanwhile, a prism that has a circular base and lid is referred to as a cylinder.

Prism properties

Building limas also has several properties or characteristics, among which are the following:

It has a base plane and also a top plane which are congruent triangles (the 2 bases are also sides of a triangular prism).

It has 5 sides (2 sides which are the top and bottom bases, the other 3 sides are upright sides which are all triangular).

Has 9 ribs.

Has 6 corner points.

Prism Formula

The formula for calculating area:

Area = (2 x area of base) + (area of entire vertical plane)

The formula for calculating circumference:

K = 3s (s + s + s)

The formula for calculating Volume:

Prism volume = area of triangle x height

or also can

Volume Prisma = 1/2 x a.s x t.s x t

5. Ball

A ball is one of the curved side shapes bounded by a curved plane. Or it can also be defined as a semicircular shape that is rotated around its center line.

Ball Properties

The ball has 1 side and 1 center point.

The ball has no ribs.

The ball has no corner points

It doesn't have a diagonal plane

It doesn't have a plane diagonal

The sides of the ball are referred to as the ball walls.

The distance from the wall to the center of the ball is known as the radius.

The wall to wall distance as well as past the center point is referred to as the diameter.

: