This document provides information about rational numbers. It defines rational numbers as numbers that can be expressed as fractions p/q where p and q are integers and q is not zero. Examples of rational numbers in different forms are given, including fractions, terminating decimals, and repeating decimals. The key properties of rational numbers are that they can be located on the real number line and the number line is used to visually demonstrate the location of sample rational numbers. The document asks students to determine if given numbers are rational and if so, locate them on the number line. It also asks students to convert rational numbers between fraction/mixed number and decimal forms.
Powerpoint on K-12 Mathematics Grade 7 Q1 (Fundamental Operations of Integer...Franz Jeremiah Ü Ibay
This document provides information on addition, subtraction, multiplication, and division of integers. It begins by explaining that when adding or multiplying integers with the same sign, you keep the same sign, and with different signs, the result is negative. Examples are provided to illustrate addition, subtraction, multiplication, and division of integers. The document then discusses properties of integer operations like closure, commutativity, associativity, distributivity, identity, and inverses. Activities are included for students to practice integer operations.
The document discusses the real number system. It defines rational and irrational numbers, and provides examples of each. Rational numbers can be written as fractions, while irrational numbers can only be written as non-terminating and non-repeating decimals. The document also covers operations like addition, subtraction, multiplication, and division on integers, using rules like keeping or changing signs depending on whether the signs are the same or different.
This document provides instruction on multiplying integers. It begins with the rules for multiplying integers:
1) Positive x Positive = Positive
2) Negative x Negative = Positive
3) Negative x Positive = Negative
4) Any Number x 0 = Zero
Examples are provided to illustrate each rule. The document emphasizes that if the signs are the same, the answer is positive, and if the signs are different, the answer is negative. Students then practice multiplying integers in a group activity before evaluating additional examples.
The document is a mathematics lecture on integers. It discusses the four integer operations of addition, subtraction, multiplication, and division. It provides examples of how to perform each operation on integers and the rules for determining if the result is positive or negative. Addition and subtraction are explained using rules about combining positive and negative integers. Multiplication and division are covered together, as their rules are the same - the result is positive if the signs are the same and negative if the signs are different.
Pre-Calculus Quarter 4 Exam
1
Name: _________________________
Score: ______ / ______
1. Find the indicated sum. Show your work.
2. Locate the foci of the ellipse. Show your work.
𝑥2
36
+
𝑦2
11
= 1
Pre-Calculus Quarter 4 Exam
2
3. Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4. Graph the function. Then use your graph to find the indicated limit. You do not have to
provide the graph
f(x) = 5x - 3, f(x)
5. Use Gaussian elimination to find the complete solution to the system of equations, or state
that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
Pre-Calculus Quarter 4 Exam
3
6. Solve the system of equations using matrices. Use Gaussian elimination with back-
substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per
minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336
calories in her workout. Write an inequality that describes the situation. Let x represent the
number of minutes running and y the number of minutes swimming. Because x and y must be
positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8. A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true. Show your work.
Sn: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
𝑛(6𝑛2−3𝑛−1)
2
Pre-Calculus Quarter 4 Exam
4
9. A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying
Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10. Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and
70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast
blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea
and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade
tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on
each pound of the afternoon blend, how many pounds of each blend should she make to
maximize profits? What is the maximum profit?
11 Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86
and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a
$35 profit on each one. You expect to sell at least 100 laser printers this month and you need to
make at least $3850 profit on them. How many of what type of p
This chapter introduces integers and their operations. Students will learn to use negative numbers, draw integers on a number line, compare integers, and order integers in sequences. Key terms include integers, positive integers, negative integers, and number line. The chapter discusses representing temperatures below zero as negative numbers, finding opposites on the number line, and using properties like commutativity and associativity to simplify integer calculations mentally.
STRAND 1 NUMBERS.pptx CBC FOR GRADE 8 STUDENTSkimdan468
This document provides information about integers, fractions, decimals, squares, and square roots for an 8th grade mathematics strand. It includes specific learning outcomes, definitions of key terms, examples of operations and conversions, and a quiz. The learner is expected to identify integers, represent and operate on them using a number line, and understand their real-world applications. Fractions, decimals, squares, and square roots are similarly defined and examples are provided of related operations and conversions. A quiz at the end assesses these concepts.
The document provides examples and explanations for adding, subtracting, and evaluating expressions with integers on a number line and using absolute value. It includes step-by-step work with integers like finding the sum of -7 + -7, evaluating expressions like 13 + r for r = -15, and an example word problem about the number of dogs in a shelter.
Powerpoint on K-12 Mathematics Grade 7 Q1 (Fundamental Operations of Integer...Franz Jeremiah Ü Ibay
This document provides information on addition, subtraction, multiplication, and division of integers. It begins by explaining that when adding or multiplying integers with the same sign, you keep the same sign, and with different signs, the result is negative. Examples are provided to illustrate addition, subtraction, multiplication, and division of integers. The document then discusses properties of integer operations like closure, commutativity, associativity, distributivity, identity, and inverses. Activities are included for students to practice integer operations.
The document discusses the real number system. It defines rational and irrational numbers, and provides examples of each. Rational numbers can be written as fractions, while irrational numbers can only be written as non-terminating and non-repeating decimals. The document also covers operations like addition, subtraction, multiplication, and division on integers, using rules like keeping or changing signs depending on whether the signs are the same or different.
This document provides instruction on multiplying integers. It begins with the rules for multiplying integers:
1) Positive x Positive = Positive
2) Negative x Negative = Positive
3) Negative x Positive = Negative
4) Any Number x 0 = Zero
Examples are provided to illustrate each rule. The document emphasizes that if the signs are the same, the answer is positive, and if the signs are different, the answer is negative. Students then practice multiplying integers in a group activity before evaluating additional examples.
The document is a mathematics lecture on integers. It discusses the four integer operations of addition, subtraction, multiplication, and division. It provides examples of how to perform each operation on integers and the rules for determining if the result is positive or negative. Addition and subtraction are explained using rules about combining positive and negative integers. Multiplication and division are covered together, as their rules are the same - the result is positive if the signs are the same and negative if the signs are different.
Pre-Calculus Quarter 4 Exam
1
Name: _________________________
Score: ______ / ______
1. Find the indicated sum. Show your work.
2. Locate the foci of the ellipse. Show your work.
𝑥2
36
+
𝑦2
11
= 1
Pre-Calculus Quarter 4 Exam
2
3. Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4. Graph the function. Then use your graph to find the indicated limit. You do not have to
provide the graph
f(x) = 5x - 3, f(x)
5. Use Gaussian elimination to find the complete solution to the system of equations, or state
that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
Pre-Calculus Quarter 4 Exam
3
6. Solve the system of equations using matrices. Use Gaussian elimination with back-
substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per
minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336
calories in her workout. Write an inequality that describes the situation. Let x represent the
number of minutes running and y the number of minutes swimming. Because x and y must be
positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8. A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true. Show your work.
Sn: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
𝑛(6𝑛2−3𝑛−1)
2
Pre-Calculus Quarter 4 Exam
4
9. A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying
Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10. Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and
70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast
blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea
and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade
tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on
each pound of the afternoon blend, how many pounds of each blend should she make to
maximize profits? What is the maximum profit?
11 Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86
and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a
$35 profit on each one. You expect to sell at least 100 laser printers this month and you need to
make at least $3850 profit on them. How many of what type of p
This chapter introduces integers and their operations. Students will learn to use negative numbers, draw integers on a number line, compare integers, and order integers in sequences. Key terms include integers, positive integers, negative integers, and number line. The chapter discusses representing temperatures below zero as negative numbers, finding opposites on the number line, and using properties like commutativity and associativity to simplify integer calculations mentally.
STRAND 1 NUMBERS.pptx CBC FOR GRADE 8 STUDENTSkimdan468
This document provides information about integers, fractions, decimals, squares, and square roots for an 8th grade mathematics strand. It includes specific learning outcomes, definitions of key terms, examples of operations and conversions, and a quiz. The learner is expected to identify integers, represent and operate on them using a number line, and understand their real-world applications. Fractions, decimals, squares, and square roots are similarly defined and examples are provided of related operations and conversions. A quiz at the end assesses these concepts.
The document provides examples and explanations for adding, subtracting, and evaluating expressions with integers on a number line and using absolute value. It includes step-by-step work with integers like finding the sum of -7 + -7, evaluating expressions like 13 + r for r = -15, and an example word problem about the number of dogs in a shelter.
Okay, let's break this down step-by-step:
* The population is dropping at a rate of 255 people per year
* We want to know how long it will take for the change in population to be 2,040 people
* So we set up an equation: Rate x Time = Change
* Rate is -255 people/year
* Change is -2,040 people
* So the equation is: -255x = -2,040
* Solve for x: x = 2,040/-255 = 8 years
Therefore, it will take 8 years for the change in population to be 2,040 people.
This document provides an overview of several topics in number and operations that may be covered on the SAT, including properties of integers, operations with integers, rational numbers and fractions, number lines, squares and square roots, scientific notation, elementary number theory, ratios, proportions, percents, sequences, and arithmetic and geometric sequences. Sample problems are included to illustrate key concepts.
This document discusses essential concepts of algebra including:
- Different types of numbers such as integers, rational numbers, irrational numbers, and complex numbers.
- Properties and rules for operations involving integers, fractions, order of operations, and factoring.
- How to perform operations like addition, subtraction, multiplication, and division on fractions and mixed numbers.
- Key terms like numerator, denominator, common factor, and lowest common multiple that are important for working with fractions.
Integers include whole numbers and their negative counterparts. They are ordered on a number line with positive integers to the right of zero and negative integers to the left. The absolute value of an integer is its distance from zero, regardless of sign. Addition and subtraction of integers follows rules where numbers with the same sign are added, and different signs are subtracted and take the sign of the greater number.
The document discusses various topics related to numbers including:
1) Perfect numbers which are numbers whose factors sum to the number.
2) Classification of numbers as natural, whole, integers, rational, and irrational.
3) Rules for divisibility including by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.
4) Formulas for finding cubes of two-digit numbers and number of zeros in expressions.
The document discusses the order of operations (PEMDAS) and provides examples of how to evaluate expressions and solve equations. It explains that parentheses, exponents, multiplication/division from left to right, and addition/subtraction from left to right have precedence based on the PEMDAS acronym. Integers, absolute value, adding, subtracting, multiplying, and dividing integers are also covered along with writing algebraic expressions and solving different types of equations.
The document discusses the order of operations (PEMDAS) and provides examples of how to evaluate expressions and solve equations. It explains that parentheses, exponents, multiplication/division from left to right, and addition/subtraction from left to right have priority in calculations. Algebraic expressions and equations are introduced along with rules for manipulating integers and solving different types of equations.
This document provides an introduction to negative numbers for students in Year 8 maths. It includes examples of where negative numbers are used, such as temperatures below zero and bank account balances in debt. Students are introduced to concepts like adding and subtracting negative numbers using a number line. Rules for multiplying positive and negative numbers are explained, such as a positive times a negative equals a negative, and a negative times a negative equals a positive. Students are provided practice problems to solve involving addition, subtraction, and multiplication of negative numbers.
1. This document discusses positive and negative numbers, integer numbers, and operations involving integers such as addition, subtraction, multiplication, division, powers, and square roots.
2. Key points include defining positive and negative integers and representing them on a number line, and establishing rules for adding, subtracting, multiplying, and dividing integers. Properties of integer powers and the square roots of positive integers are also covered.
3. Vocabulary words introduced include deposit, withdraw, forward, backward, integer, number line, opposite, sign, absolute value, and properties related to order of operations.
- An integer is a positive or negative whole number including 0, such as -3, -2, -1, 0, 1, 2, 3.
- The four basic integer operations are addition, subtraction, multiplication, and division.
- For addition and subtraction of integers, if the signs are the same, add the numbers and the answer takes the sign of the larger number. If the signs are different, subtract the numbers and the answer takes the sign of the larger number.
- For multiplication and division of integers, if the signs are the same, the answer is positive. If the signs are different, the answer is negative.
This document discusses multiplying integers. It begins with examples of using a number line and counters to model integer multiplication. Students are instructed to complete examples multiplying 3(2), 3(-2), and -3(-2) using counters and a number line. The document concludes by stating the rules for multiplying integers: the product of two integers with the same sign is positive, and the product of two integers with different signs is negative.
Integers, roots, powers, order of operationsmathn3rd
The document provides information about integers and order of operations. It defines integers as positive and negative numbers including zero. It explains how to compare integers using less than, greater than, and equal signs. It also defines absolute value as the distance from zero on a number line and provides examples. Rules for multiplying, dividing, adding, and subtracting integers are outlined. Finally, it discusses order of operations and provides examples of solving expressions using PEMDAS.
The document discusses integers and integer operations. It defines integers as positive or negative whole numbers including 0. It describes the four basic integer operations as addition, subtraction, multiplication, and division. It provides rules for performing each operation with integers, such as adding when signs are the same and subtracting when signs are different. It gives examples of performing each operation and provides summaries of the rules for multiplying and dividing integers.
The document provides review sheets for a basic mathematics course covering key concepts in whole numbers, fractions, decimals, and mixed numbers. It lists over 60 review questions addressing skills like operations, word problems, rounding, order of operations, exponents, prime factorization, and conversions between fractions and decimals. The purpose is to help students refresh their math skills and determine the appropriate level course to begin study.
A Summary of Concepts Needed to be Successful in Mathematics
The following sheets list the key concepts that are taught in the specified math course. The sheets
present concepts in the order they are taught and give examples of their use.
WHY THESE SHEETS ARE USEFUL –
• To help refresh your memory on old math skills you may have forgotten.
• To prepare for math placement test.
• To help you decide which math course is best for you.
This document defines integers and the four basic integer operations - addition, subtraction, multiplication, and division. It provides rules for performing each operation on integers, such as the product of two integers with the same sign is positive and the product of two integers with different signs is negative. Examples are included to demonstrate applying the rules to solve integer operation problems.
The document is about integers and their properties. It begins with an introduction to integers and then discusses various topics related to integers like natural numbers, whole numbers, addition, subtraction, and their properties. It provides examples of adding, subtracting, and multiplying integers. It also contains practice questions related to integers with their step-by-step solutions.
The document contains information about rational numbers including integers, fractions, and decimals. It provides examples of adding and subtracting rational numbers on a number line. Key points include:
- Rational numbers include integers, fractions, and decimals.
- Zero is a whole number but not a positive integer.
- Examples are given of comparing rational numbers and performing addition and subtraction on a number line.
- Properties of addition like commutativity and inverses are illustrated.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Okay, let's break this down step-by-step:
* The population is dropping at a rate of 255 people per year
* We want to know how long it will take for the change in population to be 2,040 people
* So we set up an equation: Rate x Time = Change
* Rate is -255 people/year
* Change is -2,040 people
* So the equation is: -255x = -2,040
* Solve for x: x = 2,040/-255 = 8 years
Therefore, it will take 8 years for the change in population to be 2,040 people.
This document provides an overview of several topics in number and operations that may be covered on the SAT, including properties of integers, operations with integers, rational numbers and fractions, number lines, squares and square roots, scientific notation, elementary number theory, ratios, proportions, percents, sequences, and arithmetic and geometric sequences. Sample problems are included to illustrate key concepts.
This document discusses essential concepts of algebra including:
- Different types of numbers such as integers, rational numbers, irrational numbers, and complex numbers.
- Properties and rules for operations involving integers, fractions, order of operations, and factoring.
- How to perform operations like addition, subtraction, multiplication, and division on fractions and mixed numbers.
- Key terms like numerator, denominator, common factor, and lowest common multiple that are important for working with fractions.
Integers include whole numbers and their negative counterparts. They are ordered on a number line with positive integers to the right of zero and negative integers to the left. The absolute value of an integer is its distance from zero, regardless of sign. Addition and subtraction of integers follows rules where numbers with the same sign are added, and different signs are subtracted and take the sign of the greater number.
The document discusses various topics related to numbers including:
1) Perfect numbers which are numbers whose factors sum to the number.
2) Classification of numbers as natural, whole, integers, rational, and irrational.
3) Rules for divisibility including by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.
4) Formulas for finding cubes of two-digit numbers and number of zeros in expressions.
The document discusses the order of operations (PEMDAS) and provides examples of how to evaluate expressions and solve equations. It explains that parentheses, exponents, multiplication/division from left to right, and addition/subtraction from left to right have precedence based on the PEMDAS acronym. Integers, absolute value, adding, subtracting, multiplying, and dividing integers are also covered along with writing algebraic expressions and solving different types of equations.
The document discusses the order of operations (PEMDAS) and provides examples of how to evaluate expressions and solve equations. It explains that parentheses, exponents, multiplication/division from left to right, and addition/subtraction from left to right have priority in calculations. Algebraic expressions and equations are introduced along with rules for manipulating integers and solving different types of equations.
This document provides an introduction to negative numbers for students in Year 8 maths. It includes examples of where negative numbers are used, such as temperatures below zero and bank account balances in debt. Students are introduced to concepts like adding and subtracting negative numbers using a number line. Rules for multiplying positive and negative numbers are explained, such as a positive times a negative equals a negative, and a negative times a negative equals a positive. Students are provided practice problems to solve involving addition, subtraction, and multiplication of negative numbers.
1. This document discusses positive and negative numbers, integer numbers, and operations involving integers such as addition, subtraction, multiplication, division, powers, and square roots.
2. Key points include defining positive and negative integers and representing them on a number line, and establishing rules for adding, subtracting, multiplying, and dividing integers. Properties of integer powers and the square roots of positive integers are also covered.
3. Vocabulary words introduced include deposit, withdraw, forward, backward, integer, number line, opposite, sign, absolute value, and properties related to order of operations.
- An integer is a positive or negative whole number including 0, such as -3, -2, -1, 0, 1, 2, 3.
- The four basic integer operations are addition, subtraction, multiplication, and division.
- For addition and subtraction of integers, if the signs are the same, add the numbers and the answer takes the sign of the larger number. If the signs are different, subtract the numbers and the answer takes the sign of the larger number.
- For multiplication and division of integers, if the signs are the same, the answer is positive. If the signs are different, the answer is negative.
This document discusses multiplying integers. It begins with examples of using a number line and counters to model integer multiplication. Students are instructed to complete examples multiplying 3(2), 3(-2), and -3(-2) using counters and a number line. The document concludes by stating the rules for multiplying integers: the product of two integers with the same sign is positive, and the product of two integers with different signs is negative.
Integers, roots, powers, order of operationsmathn3rd
The document provides information about integers and order of operations. It defines integers as positive and negative numbers including zero. It explains how to compare integers using less than, greater than, and equal signs. It also defines absolute value as the distance from zero on a number line and provides examples. Rules for multiplying, dividing, adding, and subtracting integers are outlined. Finally, it discusses order of operations and provides examples of solving expressions using PEMDAS.
The document discusses integers and integer operations. It defines integers as positive or negative whole numbers including 0. It describes the four basic integer operations as addition, subtraction, multiplication, and division. It provides rules for performing each operation with integers, such as adding when signs are the same and subtracting when signs are different. It gives examples of performing each operation and provides summaries of the rules for multiplying and dividing integers.
The document provides review sheets for a basic mathematics course covering key concepts in whole numbers, fractions, decimals, and mixed numbers. It lists over 60 review questions addressing skills like operations, word problems, rounding, order of operations, exponents, prime factorization, and conversions between fractions and decimals. The purpose is to help students refresh their math skills and determine the appropriate level course to begin study.
A Summary of Concepts Needed to be Successful in Mathematics
The following sheets list the key concepts that are taught in the specified math course. The sheets
present concepts in the order they are taught and give examples of their use.
WHY THESE SHEETS ARE USEFUL –
• To help refresh your memory on old math skills you may have forgotten.
• To prepare for math placement test.
• To help you decide which math course is best for you.
This document defines integers and the four basic integer operations - addition, subtraction, multiplication, and division. It provides rules for performing each operation on integers, such as the product of two integers with the same sign is positive and the product of two integers with different signs is negative. Examples are included to demonstrate applying the rules to solve integer operation problems.
The document is about integers and their properties. It begins with an introduction to integers and then discusses various topics related to integers like natural numbers, whole numbers, addition, subtraction, and their properties. It provides examples of adding, subtracting, and multiplying integers. It also contains practice questions related to integers with their step-by-step solutions.
The document contains information about rational numbers including integers, fractions, and decimals. It provides examples of adding and subtracting rational numbers on a number line. Key points include:
- Rational numbers include integers, fractions, and decimals.
- Zero is a whole number but not a positive integer.
- Examples are given of comparing rational numbers and performing addition and subtraction on a number line.
- Properties of addition like commutativity and inverses are illustrated.
Similar to powerpointfull-140924104315-phpapp02 (1).pdf (20)
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
4. • When integers have different
sign, find the difference
between two numbers.
• The sum will have the sign of
the integer with a larger
absolute value.
ADDITION OF DIFFERENT
SIGNED INTEGERS
7. Activity: Using the number line,
find the sum of the following:
1. 5 & 4
2. 6 & 11
3. 23 & -25
4. -17 & 21
5. -13 & -3
ADDITION USING NUMBER
LINE
8. • This is another device that can
be used to represent integers.
• The tile represents
integer 1, the tile represents
integer -1.
ADDITION USING SIGNED TILES
+
-
11. Activity: Find the sum of the following signed
tiles (Column A) on its corresponding value
(Column B):
____ 1. + a. 7
____ 2. + b .-4
____ 3. + c. 3
____ 4. + d. 1
____ 5. + e. 0
+ + +
- - + + +
+ + + + +
+ +
- - - -
+ + + +
- - - -
12. 1. Mrs. Reyes charged P3752 worth of
groceries on her credit card. Find her
balance after she made a payment of
P2530.
2. In a game, Team Azkals lost 5 yards in
one play but gained 7 yards in the next
play. What was the actual yardage gain of
the team?
SEATWORK
13. 3. A vendor gained P50.00 on the first day;
lost P28.00 on the second day, and
gained P49.00 on the third day. How
much profit did the vendor gain in 3 days?
4. Ronnie had PhP2280 in his checking
account at the beginning of the month. He
wrote checks for PhP450, P1200, and
PhP900. He then made a deposit of
PhP1000. If at any time during the month
the account is overdrawn, a PhP300
service charge is deducted. What was
Ronnie’s balance at the end of the
month?
14. Using the number line, find the
sum of the following:
1. 6 & 3
2. -40 & 11
3. 1 & --1
4. -15 & 8
5. -9 and -8
ASSIGNMENT
15. Using the signed tiles, find the sum
of the following:
1. 5 & 3
2. -3 & -3
3. 1 & 4
4. -1 & -6
5. --5 & -1
21. Using the number line, find the
difference of the following:
1. 8 & 18
2. 6 & 3
3. 1 & --1
4. 16 & -7
5. -8 & -10
GROUPACTIVITY
22. Using the signed tiles, find the
difference of the following:
1. 6 & 2
2. -3 & -3
3. 3 &1
4. -5 & 3
5. 6 & -6
23. 1. Maan deposited P53400.00 in her
account and withdrew P19650.00 after a
week. How much of her money was left in
the bank?
2. Two trains start at the same station at the
same time. Train A travels 92km/h, while
train B travels 82km/h. If the two trains
travel in opposite directions, how far apart
will they be after an hour? If the two trains
travel in the same direction, how far apart
will they be in two hours?
SEATWORK
24. 3. During the Christmas season, the student
gov’t association was able to solicit 2,356
grocery items and was able to distribute
2,198 to one barangay. If this group
decided to distribute 1,201 grocery items
to the next barangay, how many more
grocery items did they need to solicit?
25. Read the rules in multiplying
integer and we will have a graded
recitation.
ASSIGNMENT
27. • When integers have the same
sign, simply multiply the
absolute value of the integers.
• The product of same signed
integers is always positive.
MULTIPLICATION OF SAME
SIGNED INTEGERS
28. • When integers have different
signs, simply multiply the
absolute value of the integers.
• The product of different signed
integers is always negative.
MULTIPLICATION OF
DIFFERENT SIGNED INTEGERS
29. 1. 3 cars with 4 passengers each,
how many passengers in all?
4 x 3 = 4 + 4 + 4 = 12
EXAMPLES
30. 2. 4 cars with 3 passengers each,
how many passengers in all?
3 x 4 = 4 x 3 =3 + 3 + 3 + 3 = 12
3. When a boy loses P6 for 3
consecutive days, what is his total
loss?
(-6) + (-6) + (-6) = (-6) (3) = -18
31. How can a person fairly divide 10
apples among 8 children so that
each child has the same share?
To solve the dilemma, match the
letter in column II with the number
that corresponds to the numbers in
column I.
ACTIVITY
(MATH DILEMMA)
32. Column I
____1. (6) (-12)
____2. (-13) (-13)
____3. (19)(-17)
____4. (-15)(29)
____5. (165)(0)
____6. (-18)(-15)
____7. (-15)(-20)
____8. (-5)(-5)(-5)
____9. (-2)(-2)(-2)(-2)
____10. (4)(6)(8)
Column II
C. 270
P. -72
E. 300
K. -323
A. -435
M. 0
L. 16
J. -125
U. 169
I. 192
33. 1. Jof has twenty P5 coins in her coin
purse. If her niece took 5 of the
coins, how much has been taken
away?
2. Mark can type 45 words per minute,
how many words can Mark type in
30 minutes?
SEATWORK
34. What was the original name for the
butterfly?
To find the answer, find the
quotient of each of the following
and write the letter of the letter of
the problems in the box
corresponding to the quotient.
ASSIGNMENT
35. R −𝟑𝟓𝟐 ÷ 𝟐𝟐
T 𝟏𝟐𝟖 ÷ 𝟏𝟔
U −𝟏𝟐𝟎 ÷ 𝟖
L −𝟒𝟒𝟒 ÷ −𝟏𝟐
Y 𝟏𝟒𝟒 ÷ (−𝟑)
B 𝟏𝟎𝟖 ÷ 𝟗
E 𝟏𝟔𝟖 ÷ 𝟔
T −𝟏𝟒𝟕 ÷ 𝟕
F −𝟑𝟏𝟓 ÷ (−𝟑𝟓)
9 37 -15 -8 -21 28 -16 12 -48
37. • When integers have the same
sign, simply divide the absolute
value of the integers.
• The quotient of same signed
integers is always positive.
• If possible, express the quotient
in lowest term.
DIVISION OF SAME SIGNED
INTEGERS
39. • When integers have different
signs, simply divide the absolute
value of the integers.
• The quotient of different signed
integers is always negative.
• If possible, express the quotient
in lowest term.
DIVISION OF DIFFERENT
SIGNED INTEGERS
44. • When two integers is multiplied
or added, the result is also
belongs to Z.
a, b ∈ Z, then a + b ∈ Z, a∙b ∈ Z
CLOSURE PROPERTY
45. Z= {…-3, -2, -1, 0, 1, 2, 3 …}
It is closed to:
• Addition
• Multiplication
• Subtraction
EXAMPLE
46. • Any order of two integers that
are either added or multiplied
does not change the value of
sum or product.
COMMUTATIVE PROPERTY
For addition
a + b = b + a
For multiplication
ab = ba
50. • When two numbers have been
added or subtracted and then
multiplied by a factor, the result will
be the same when each number is
multiplied by the factor and the
products and then added or
subtracted.
a(b + c) = ab + ac
DISTRIBUTIVE PROPERTY
52. Additive Identity
• The sum of any number and 0 is the
given number.
• Zero (0) is the additive identity.
a + 0 = a
Multiplicative Identity
• The product of any number and 1 is the
given number.
• One (1) is the multiplicative identity.
a ∙1 = a
IDENTITY PROPERTY
54. Additive Inverse
• The sum of any number and its additive
inverse is zero.
• -a is the additive inverse of the number a.
a + (-a) = 0
Multiplicative Inverse
• The product of any number and its
multiplicative inverse is one.
•
1
a
is the multiplicative inverse of the
number a.
a ∙
1
a
= 1
INVERSE PROPERTY
56. Complete the Table: Which
property of real number justifies
each statement?
ACTIVITY
57. Given Property
1. 0 + (-3) = -3
2. 2(3 - 5) = 2(3) - 2(5)
3. (- 6) + (-7) = (-7) + (-6)
4. 1 x (-9) = -9
5. -4 x (−
1
4
)= 1
6. 2 x (3 x 7) = (2 x 3) x 7
7. 10 + (-10) = 0
8. 2(5) = 5(2)
9. 1 x −
1
4
= −
1
4
10. (-3)(4 + 9) = (-3)(4) + (-3)(9)
58. Fill in the blanks and determine
what properties were used to solve
the equations.
1. 5 x ( ____ + 2) = 0
2. -4 + 4 = _____
3. -6 + 0 = _____
4. (-14 + 14) + 7 = _____
5. 7 x (_____ + 7) = 49
ASSIGNMENT
64. • Rational numbers can be located on
the real number line.
• A number line is a visual
representation of the numbers from
negative infinity to positive infinity,
which means it extends indefinitely
in two directions.
65. • It consists of negative numbers on
its left, zero in the middle, and
positive numbers on its right.
66. EXAMPLES OF RATIONAL
NUMBERS IN THE NUMBER
LINE
Example 1: Locate 1/4 on the number
line.
a. Since 0 < 1/4 < 1, plot 0 and 1 on
the number line.
67. b. Divide the segment into 4 equal
parts.
c. The 1st mark from 0 is the point
1/4.
68. Example 2: Locate 1.75 on the number
line.
a. The number 1.75 can be written
as 7/4, and 1 < 7/4 < 2. Divide the
segment from 0 to 2 into 8 equal
parts.
70. Determine whether the following
numbers are rational numbers or not.
_____1. -3 _____4. √36
_____2. π _____5. ∛6
_____3.
3
5
_____6. 2.65
71. If the number is rational, locate them
on the real number line by plotting:
72. Name one rational number x that
satisfies the descriptions below:
a.
1
4
< x <
1
3
b. 3 < x < π
c. -
1
8
< x < -
1
9
d.
1
10
< x <
1
2
e. -10 < x < -9
ASSIGNMENT
79. NON - DECIMAL
FRACTIONS
• A non-decimal fraction is a
fraction whose denominator is
cannot be expressed as a
power of 10, which results to a
non-terminating but repeating
decimals.
84. SOLUTIONS
1. Let r = 0.2222…
10r = 2.2222…
Note: Since there is only one repeated
digit, multiply the first equation
by 10.
Subtract the first equation from the
second equation:
9r = 2.0
r =
2
9
85. 1. Let r = -1.353535…
100r = -135.353535…
Note: Since there is two repeated digit,
multiply the first equation by 100.
Subtract the first equation from the
second equation:
99r = -134
r = -
134
99
= −1
35
99
87. Find the sum or difference of the
following.
1.
3
5
+
1
5
= _____
2.
1
8
+
5
8
= _____
3.
10
11
−
3
11
= _____
4. 3
6
7
−1
2
7
=_____
88. TO ADD OR SUBTRACT
FRACTION WITH THE SAME
DENOMINATOR
If a, b and c ∈ Z, and b ≠ 0, then
a
b
+
c
b
=
a + c
b
and
a
b
−
c
b
=
a − c
b
If possible, reduce the answer to
lowest term.
89. TO ADD OR SUBTRACT
FRACTION WITH DIFFERENT
DENOMINATOR
With different denominators,
a
b
and
c
d
, b
≠ 0 and d ≠ 0, if the fractions to be
added or subtracted are dissimilar
• Rename the fractions to make them
similar whose denominator is the
least common multiple of b and d.
90. • Add or subtract the numerators
of the resulting fractions.
• Write the result as a fraction
whose numerator is the sum or
difference of the numerators
and whose denominator is the
least common multiple of b and
d.
• If possible, reduce the result in
lowest term.
93. Give the number asked for.
1. What is three more than three and
one-fourth?
2. Subtract from 15
1
2
the sum of
2
2
3
and 4
2
5
. What is the result?
3. Increase the sum of 6
3
14
and 2
2
7
by
3
1
2
. What is the result?
94. Solve each problem.
1. Michelle and Corazon are
comparing their heights. If
Michelle’s height is 120
3
4
cm. and
Corazon’s height is 96
1
3
cm. What is
the difference in their heights?
2. Angel bought 6
3
4
meters of silk,
3
1
2
meters of satin and 8
1
2
meters of
velvet. How many meters of cloth
did she buy?
ASSIGNMENT
96. There are 2 ways of adding or
subtracting decimals.
1. Express the decimal numbers
in fractions then add or
subtract as described earlier.
2. Arrange the decimal numbers
in a column such that the
decimal points are aligned,
then add or subtract as with
whole numbers.
97. 1. Express the decimal numbers
in fractions then add or
subtract as described earlier.
Example:
Add: 2.3 + 7.21
=2
3
10
+7
21
100
=2
30
100
+7
21
100
= 2+7 +(
30+21
100
)
=9 +
51
100
=9
51
100
=9.51
99. 2. Arrange the decimal numbers in a
column such that the decimal
points are aligned, then add or
subtract as with whole numbers.
Example:
Add: 2.3 + 7.21 Subtract: 9.6 – 3.25
2.3 9.6
+7.21 - 3.25
9.51 6.35
101. Solve each problem.
1. Helen had P7500 for shopping
money. When she got home, she
had P132.75 in her pocket. How
much did she spend for shopping?
2. Ken contributed P69.25, while John
and Hanna gave P56.25 each for
their gift to Teacher Daisy. How
much were they able to gather
altogether?
ASSIGNMENT
102. 3. Ryan said, “I’m thinking of a number
N. If I subtract 10.34 from N, the
difference is 1.34.” What was
Ryan’s number?
4. Agnes said, “I’m thinking of a
number N. If I increase my number
by 56.2, the sum is 14.62.” What
was Agnes number?
5. Kim ran the 100-meter race in
135.46 seconds. Tyron ran faster by
15.7 seconds. What was Tyron’s
time for the 100-meter dash?
104. MULTIPLICATION OF
RATIONAL NUMBERS IN
FRACTION FORM
• To multiply rational numbers in
fraction form, simply multiply
the numerators and multiply the
denominators.
a
b
∙
c
d
=
ac
bd
, where b ≠ 0 and d ≠ 0
105. DIVISION OF RATIONAL
NUMBERS IN FRACTION
FORM
• To divide rational numbers in
fraction form, take the
reciprocal of the divisor(second
fraction) and multiply it by the
first fraction.
a
b
÷
c
d
=
a
b
∙
d
c
=
ad
bc
, where b, c
and d ≠ 0
109. SEATWORK
1. Julie spent 3
1
2
hours doing her
assignment. Ken did his
assignment for 1
2
3
times as
many hours as Julie did. How
many hours did Ken spend
doing his assignment?
2. How many thirds are there in
six-fifths?
110. 3. Hanna donated
2
5
of her
monthly allowance to the Iligan
survivors. If her monthly
allowance is P3500, how much
did she donate?
4. The enrolment for this school
year is 2340. If
1
6
are
sophomores and are seniors,
how many are freshmen and
juniors?
111. MULTIPLICATION OF
RATIONAL NUMBERS IN
DECIMAL FORM
1. Arrange the numbers in a vertical column.
2. Multiply the numbers, as if you are
multiplying whole numbers.
3. Starting from the rightmost end of the
product, move the decimal point to the left
the same number of places as the sum of
the decimal places in the multiplicand and
the multiplier.
112. DIVISION OF RATIONAL
NUMBERS IN DECIMAL
FORM
1. If the divisor is a whole number, divide the
dividend by the divisor applying the rules of
a whole number. The position of the
decimal point is the same as that in the
dividend.
2. If the divisor is not a whole number, make
the divisor a whole number by moving the
decimal point in the divisor to the rightmost
end, making the number seem like a whole
number.
113. ACTIVITY
Perform the indicated operation:
1. 3.5 ÷ 2
2. 3.415 ÷ 2.5
3. 78 x 0.4
4. 3.24 ÷ 0.5
5. 9.6 x 13
6. 27.3 x 2.5
7. 9.7 x 4.1
8. 1.248 ÷ 0.024
9. 53.61 x 1.02
10.1948.324 ÷ 5.96