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27 Apr 2023
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Complete National 5 Maths Study Guide (Part 1): Expressions, Surds, and Indices Notes PDF
Sophie C
@sophiec_swmu
Completing the Square in National 5 Maths
This section of the Complete national 5 maths study guide surds free resource explains the process of completing the square, an important algebraic technique.
Definition: Completing the square is a method of factorizing an expression by using a square term and another term.
The guide outlines the steps to complete the square:
- Separate the first two terms from the last.
- Add ^2 inside the brackets.
- Subtract the same number outside the brackets.
- Write the expression in completed square form: y = a^2 + c
Example: x^2 + 10x + 27 = ^2 + 2
This technique is crucial for solving quadratic equations and understanding parabolas in Nat 5 Maths Questions.
Algebraic Operations in National 5 Maths
This section covers essential algebraic operations, including multiplying out brackets, factorizing, and working with trinomials.
For multiplying out brackets:
- Remove brackets by multiplying everything inside by everything outside.
- For double brackets, use the FOIL method .
Example: 3 = 3x + 6
Factorizing involves three steps:
- Take out a common factor.
- Look for a difference of two squares.
- Factorize any remaining expression.
Highlight: Trinomials are expressions with three terms and can be factorized using specific techniques.
For easy trinomials:
- Find numbers that multiply to give the last term and add to give the middle term.
- Place these numbers into brackets.
For harder trinomials:
- Multiply the first and last terms.
- Find factors of this new number that add to give the middle term.
- Replace the middle term with these two numbers.
These skills are essential for success in National 5 Maths Past Papers.
Algebraic Fractions in National 5 Maths
This section of the National 5 Maths Online course free resource focuses on simplifying and performing operations with algebraic fractions.
To simplify algebraic fractions:
- Follow the same process as simplifying regular fractions.
- Factorize when necessary.
For multiplying algebraic fractions:
- Simplify or cancel out terms.
- Multiply the remaining terms.
For adding and subtracting algebraic fractions:
- Find a common denominator by multiplying the denominators.
- Adjust the numerators accordingly.
- Add or subtract the fractions.
- Simplify the result if possible.
For dividing algebraic fractions:
- Change the division to multiplication.
- Flip the second fraction.
- Simplify and cancel out terms.
- Multiply the remaining terms.
These operations are crucial for solving complex equations in Nat 5 Maths Indices questions.
Volume and Significant Figures in National 5 Maths
This section of the Step by step algebraic operations national 5 maths notes pdf covers volume calculations and working with significant figures.
Volume formulas for various shapes are provided:
- Sphere: V = 4/3πr^3
- Pyramid: V = 1/3 × base area × height
- Cylinder: V = πr^2h
Highlight: For composite shapes, use multiple formulas and add the results.
The guide also explains significant figures:
- Any non-zero digit is significant.
- Zeros after the decimal point are significant.
- Zeros between two non-zero digits are significant.
Example: 803.0 has 4 significant figures.
These concepts are frequently tested in Nat 5 Maths Past Papers.
Circles in National 5 Maths
This section of the National 5 Maths textbook pdf focuses on calculations involving circles.
To find the length of an arc, use the formula: Arc length = × 2πr
Where θ is the angle at the center, and r is the radius.
Steps to solve circle problems:
- Choose the correct fractions.
- Substitute the given values.
- Cross multiply where needed.
These skills are essential for geometry questions in SQA nat 5 Maths exams.
Straight Lines in National 5 Maths
This final section covers the gradient and equation of straight lines, crucial concepts in National 5 Maths Higher.
To find the gradient of a straight line:
- Use the formula: m = /
- A positive gradient means the line slopes up, negative means down.
Example: Find the gradient of the line joining points A and B.
The equation of a straight line can be determined using the gradient and a point on the line.
These concepts are fundamental for graphing and analyzing linear functions in National 5 Maths topic List.
Surds and Indices in National 5 Maths
This section of the Complete national 5 maths study guide surds pdf focuses on surds and indices, two fundamental concepts in algebra.
Surds are irrational roots that result in decimal numbers when calculated. The guide explains key rules for working with surds, including simplification and operations.
Definition: Surds are irrational roots, meaning the root of the number will result in a decimal.
Example: √2 is a surd, while √4 = 2 is not a surd.
The guide covers important surd rules:
- No vaza: When a root is multiplied by itself, the result is the number without the root.
- Habox: When multiplying two numbers under the same root, they can be split into separate roots.
- Addition and subtraction of surds require the terms to be the same.
For indices, also known as powers, the guide presents five essential rules:
Vocabulary: Indices are another term for powers in mathematics.
- a^1 = a
- a^0 = 1
- To take a power of a power, multiply the indices.
- a^-n = 1/a^n
- For multiplication, add the powers; for division, subtract them.
These rules form the foundation for more complex algebraic operations in National 5 Maths.
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Complete National 5 Maths Study Guide (Part 1): Expressions, Surds, and Indices Notes PDF
Sophie C
@sophiec_swmu
The National 5 Mathsstudy guide covers essential topics including surds, indices, completing the square, algebraic operations, volume, circles, and straight lines. This comprehensive resource provides step-by-step explanations, examples, and practice questions to help students master key mathematical concepts and... Show more
Completing the Square in National 5 Maths
This section of the Complete national 5 maths study guide surds free resource explains the process of completing the square, an important algebraic technique.
Definition: Completing the square is a method of factorizing an expression by using a square term and another term.
The guide outlines the steps to complete the square:
- Separate the first two terms from the last.
- Add ^2 inside the brackets.
- Subtract the same number outside the brackets.
- Write the expression in completed square form: y = a^2 + c
Example: x^2 + 10x + 27 = ^2 + 2
This technique is crucial for solving quadratic equations and understanding parabolas in Nat 5 Maths Questions.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Algebraic Operations in National 5 Maths
This section covers essential algebraic operations, including multiplying out brackets, factorizing, and working with trinomials.
For multiplying out brackets:
- Remove brackets by multiplying everything inside by everything outside.
- For double brackets, use the FOIL method .
Example: 3 = 3x + 6
Factorizing involves three steps:
- Take out a common factor.
- Look for a difference of two squares.
- Factorize any remaining expression.
Highlight: Trinomials are expressions with three terms and can be factorized using specific techniques.
For easy trinomials:
- Find numbers that multiply to give the last term and add to give the middle term.
- Place these numbers into brackets.
For harder trinomials:
- Multiply the first and last terms.
- Find factors of this new number that add to give the middle term.
- Replace the middle term with these two numbers.
These skills are essential for success in National 5 Maths Past Papers.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Algebraic Fractions in National 5 Maths
This section of the National 5 Maths Online course free resource focuses on simplifying and performing operations with algebraic fractions.
To simplify algebraic fractions:
- Follow the same process as simplifying regular fractions.
- Factorize when necessary.
For multiplying algebraic fractions:
- Simplify or cancel out terms.
- Multiply the remaining terms.
For adding and subtracting algebraic fractions:
- Find a common denominator by multiplying the denominators.
- Adjust the numerators accordingly.
- Add or subtract the fractions.
- Simplify the result if possible.
For dividing algebraic fractions:
- Change the division to multiplication.
- Flip the second fraction.
- Simplify and cancel out terms.
- Multiply the remaining terms.
These operations are crucial for solving complex equations in Nat 5 Maths Indices questions.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Volume and Significant Figures in National 5 Maths
This section of the Step by step algebraic operations national 5 maths notes pdf covers volume calculations and working with significant figures.
Volume formulas for various shapes are provided:
- Sphere: V = 4/3πr^3
- Pyramid: V = 1/3 × base area × height
- Cylinder: V = πr^2h
Highlight: For composite shapes, use multiple formulas and add the results.
The guide also explains significant figures:
- Any non-zero digit is significant.
- Zeros after the decimal point are significant.
- Zeros between two non-zero digits are significant.
Example: 803.0 has 4 significant figures.
These concepts are frequently tested in Nat 5 Maths Past Papers.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Circles in National 5 Maths
This section of the National 5 Maths textbook pdf focuses on calculations involving circles.
To find the length of an arc, use the formula: Arc length = × 2πr
Where θ is the angle at the center, and r is the radius.
Steps to solve circle problems:
- Choose the correct fractions.
- Substitute the given values.
- Cross multiply where needed.
These skills are essential for geometry questions in SQA nat 5 Maths exams.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Straight Lines in National 5 Maths
This final section covers the gradient and equation of straight lines, crucial concepts in National 5 Maths Higher.
To find the gradient of a straight line:
- Use the formula: m = /
- A positive gradient means the line slopes up, negative means down.
Example: Find the gradient of the line joining points A and B.
The equation of a straight line can be determined using the gradient and a point on the line.
These concepts are fundamental for graphing and analyzing linear functions in National 5 Maths topic List.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Surds and Indices in National 5 Maths
This section of the Complete national 5 maths study guide surds pdf focuses on surds and indices, two fundamental concepts in algebra.
Surds are irrational roots that result in decimal numbers when calculated. The guide explains key rules for working with surds, including simplification and operations.
Definition: Surds are irrational roots, meaning the root of the number will result in a decimal.
Example: √2 is a surd, while √4 = 2 is not a surd.
The guide covers important surd rules:
- No vaza: When a root is multiplied by itself, the result is the number without the root.
- Habox: When multiplying two numbers under the same root, they can be split into separate roots.
- Addition and subtraction of surds require the terms to be the same.
For indices, also known as powers, the guide presents five essential rules:
Vocabulary: Indices are another term for powers in mathematics.
- a^1 = a
- a^0 = 1
- To take a power of a power, multiply the indices.
- a^-n = 1/a^n
- For multiplication, add the powers; for division, subtract them.
These rules form the foundation for more complex algebraic operations in National 5 Maths.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user