Proof (new A level) - worksheet to teach & practise the whole topic + 3 homeworks/tests

Proof (new A level) - worksheet to teach & practise the whole topic + 3 homeworks/tests

This 10-page worksheet can be used to teach the whole topic of proof in the new A level specification for all exam boards. A great resource to help deliver this new topic - fully worked solutions included and a version with teaching notes added for some key points. It begins by reviewing all the required basic knowledge with some examples to work through, discusses particular errors in solutions/proofs, covers the use of ⇒, ⇐ and ⇔, interval and set notation, and then looks at the different methods of proving/disproving propositions. For each of the 4 methods (counter example, deduction, exhaustion and contradiction) there are a number of examples for you to work through as a class or get the students to attempt. I needed 4 hours of teaching time to get through this whole worksheet with my classes. There are also some suggested extension activities for students interested in doing some research that goes beyond the scope of the syllabus. Note that I designed this resource to work with a particular textbook accredited for the AQA course, so the references to exercises and page numbers may need to be amended for the exam board and textbook you are working from. 3 different homeworks/tests are also included, with fully-worked solutions provided.
langy74
Resolving forces - worksheet to teach and practise this skill (Mechanics 1)

Resolving forces - worksheet to teach and practise this skill (Mechanics 1)

I found it time-consuming tryingto teach my classes how to resolve forces by drawing diagrams on the board and asking them to copy them down - it seemed to take ages and they didn't get to work through that many examples themselves. So I created this worksheet with ready-made diagrams with all the forces and a blank copy of diagram for students to add on the resolved forces. I no longer dread teaching this skill and my classes get a lot more done in the lesson time. The worksheet starts with an introductory explanation and a worked example. There are then over 20 questions for students to attempt. Fully worked solutions are included.
langy74
All you need to teach the equation of a straight line!

All you need to teach the equation of a straight line!

I think this set of resources covers everything your classes need to learn and practice on straight line graphs (up to GCSE level). All the resources are suitable to be projected or printed for students to work on, saving a lot of time for drawing graphs and allowing them to annotate or work on diagrams. All resources come with solutions included. Here is a brief description of each resource: 1. Basic straight lines - lines of the form x=a, y=a and y=x or y=-x 2. Drawing straight lines - 10 questions using the equation of a line y=mx+c to complete a table of values and draw the graph. 3. Cover-up method - 12 questions to practise drawing lines of the form ax+by=c 4. Using the equation - test if a point lies on a line, determine y-coord given x-coord and vice versa (70 questions) 5. Finding the gradient - 18 questions to practise finding gradients, including where the scales on the axes are not the same 6. Matching y=mx+c to the graph - they find the gradient and y-intercept for each given graph and equation, learning the connection between the equation and properties of the graph 7. Equation to gradient and y-intercept - simple worksheet to practice writing down the gradient and coordinates of y-intercept from the equation, and vice versa (24 questions) 8. Finding the equation of a line - 24 questions to practise finding the equation of the line from its graph, including where the scales on the axes are not the same 9. Finding equation using point and gradient - 10 questions to practise doing this with a grid as an aid, then 26 questions without a grid 10. Pairs of lines - 4 graphs, each with a pair of parallel or perpendicular lines. By finding the equation of each line the students should start to see the rules for gradients of parallel and perpendicular lines 11. Parallel and perpendicular lines - almost 50 questions finding the equation of a line parallel / perp to a given line that passes through (0,b) or (a, b) 12. Using two points A and B - find midpoint M of AB, gradient of line through A and B, equation of line through A and B, equation of line perp. to AB through A, B or M. 10 questions to learn the methods with grids as an aid, then an exercise for each style of question (over 50 questions in total). 13. Multiple choice questions - quick assessment covering most of the topic 14. Straight lines revision - 60 questions to revise the whole topic 15. Homework - 19 questions on all aspects of the topic, fully works solutions included I have just worked through all these with my year 10 group and it took around 5 hours of lesson time to complete. A more able group may need less time but you have enough resources here to keep your classes busy for a number of lessons.
langy74
Solving linear inequalities worksheet + homework/test

Solving linear inequalities worksheet + homework/test

This worksheet covers how to solve single and double-sided inequalities and includes representing the solution on a number line as well as considering examples where integer solutions are required. The introduction covers what the solution to a linear inequality should look like and, by means of a few examples, explores the similarities and differences between solving equations and inequalities. The first exercise (52 Qs) then gives students practice solving inequalties of the form ax+b>c, x/a+b The second section focuses on double-sided inequalities such as 3 The final section is designed to help students consider the integer solutions to an inequality. In the examples students need to find the smallest possible integer value of n if n>p, the largest possible integer value of n if n Answers to all the exercises are provided, including the solutions on number lines. Also included is a homework/test with fully worked solutions.
langy74
First order differential equations worksheet

First order differential equations worksheet

This resource can be used to guide your students through the different techniques that may be used to solve some first order differential equations. It begins with a reminder about the solution of 'variable separable' equations, with a couple of examples to work through. By means of an example, the next section shows how the use of an integrating factor can help to solve 1st order linear diff.eqns. After the method is summarised there are a further 2 examples to work through with your class. The worksheet then mentions the use of a substitution to simplify a complex diff.eqn into either a linear or variable separable one. There are no examples of such equations, just a table for students to practise determining if the resulting simplified equation is linear or variable separable. The remainder of the resource introduces the important method of finding the general solution by adding the complementary function and the particular integral. It begins with the method for finding the complementary function from the auxiliary equation, and then goes on to explain the method for testing a suitable function f(x) for the particular integral (including the case where the function f(x) appears in the complementary function). There are several examples of this method to work through with your students, followed by an exercise with over 20 questions for students to complete themselves. Answers to the exercise are included.
langy74
Groups - teaching notes and examples (FP3)

Groups - teaching notes and examples (FP3)

I have used this resource a few times with my classes to cover the whole topic of groups. This 24-page worksheet covers all the required knowledge and skills for FP3. Each section starts with introductory notes or examples, followed by an exercise for students to attempt. The sections are: 1. Sets, binary operations, closed/commutative/closed operations, identity elements and inverses. 2. Groups - definition of a group, order of a group, group tables 3. Multiplicative groups and cancellation laws 4. Groups using modular arithmetic 5.Symmetries of shapes 6. The order of an element 7. Cyclic groups and generators 8. Subgroups 9. Lagrange's theorem 10. Isomorphic groups Note that answers to the examples/exercises are not included.
langy74
Second order differential equations worksheet

Second order differential equations worksheet

This 12 page resource covers the solution of 2nd order differential equations by finding the roots of its auxiliary equation, and its particular integral. The first section focuses on cases where the auxiliary equation has real roots (distinct or repeated). It begins by concentrating on finding only the complementary function - there are several examples to work through with your class and then an exercise with 14 questions for students to attempt. There are then a few examples that involve finding both the complementary function and the particular integral. The second section focuses on cases where the auxiliary equation has complex roots (a+/-bi or +/-bi). There are several examples to work through with your class and then an exercise with 18 questions for students to attempt. The exercise includes questions where students are required to consider the behaviour of the solution (bounded/unbounded oscillations) when x becomes large, as well as the function to which the solution approximates when x becomes large. Answers to both exercises are included.
langy74
Graph transformations and modulus function (new A level)

Graph transformations and modulus function (new A level)

These resources cover all the required knowledge and skills for the new A2 level specification for combined graph transformations and the modulus function. Please see the descriptions of each resource for more details.
langy74
Factor theorem worksheets

Factor theorem worksheets

These resources will give your class plenty of practice of using the factor theorem and the common questions that follow finding a factor of a cubic polynomial. The first resource focuses on showing that (ax+b) is a factor of f(x) and then using it to write f(x) as a product of a linear and quadratic factor. There is an example to work through as a group and then an exercise with 14 questions - answers are provided. The second resource has 2 sections. The first section focuses on factorising cubics fully, either as a product of a linear and quadratic factor, or as a product of 3 linear factors. The second section focuses on solving f(x)=0 and, in later questions, relates the solutions to the graph of f(x). In total there are 26 questions - answers are provided.
langy74
Modulus function (new A level) - notes, examples and exercises

Modulus function (new A level) - notes, examples and exercises

This resource is a great way to cover this whole topic using prepared notes and examples to explain it to your students. Projecting the notes/examples will save you a lot of work on the board and your students will save time by working on the provided spaces and axes when doing sketches. You could also email/print some or all of this for students who have missed lessons or need additional notes/practice/revision. The sections cover the following: 1. Sketching graphs of the form y=mod(f(x)) e.g. y=mod(x-2) 2. Sketching simple transformations of y=mod(f(x)) e.g. y=mod(x)+4 3. Solving equations involving the modulus function. This covers the different types of equations and explains when a sketch may/must be used. e.g. mod(x-4)=6 vs 2x+3=mod(x-1) 4. Solving inequalities involving the modulus function. This covers the different types of inequalities and explains when a sketch may/must be used. e.g. mod(x-4)=mod(2x+1) vs 3x-1=4-mod(x) There are almost 100 questions in total across the different exercises. Answers to all questions in the exercises are provided, including sketches.
langy74
Graph transformations and modulus function homework/test (new A level)

Graph transformations and modulus function homework/test (new A level)

This 2-page resource is a great way to assess your students after teaching them combined graph transformations and the modulus function for the new A level specification. The resource is designed for students to write on the sheet in the spaces or on the axes provided for questions that require sketches. Fully worked solutions are included.
langy74
Vector product (cross product) worksheet

Vector product (cross product) worksheet

This worksheet is a good way to give your class plenty of practice calculating and using the vector product. The first 2 questions just involve finding the vector product of two given vectors, both in column vector and in I,j,k form. The remaining questions introduce how the vector product can be used to answer particular questions such as converting vector eqn of plane to normal eqn, or finding the area of triangle in 3 dimensions. Fully worked solutions are provided to the questions.
langy74
Combined graph transformations (new A level)

Combined graph transformations (new A level)

This resource covers all the required knowledge and skills for the A2 topic of combined graph transformations. It begnis by reviewing the individual transformations and their effects on the graph or its equation. The first section focuses on finding the equation of the curve resulting from 2 transformations - there are some examples to complete with your class and then an exercise for them to do independently. The exercise does include some questions requiring a sketch of the original and the transformed curve. Within that exercise there are questions designed to help them realise when the order of the transformations is important. The second section focuses on examples where the transformations must be applied in the correct order. There are examples to complete and then an exercise for students to attempt themselves. The exercise includes questions where the resulting equation must be found, where the required transformations but be described, and some graph sketching. Answers to all the questions in the exercises are included.
langy74
Linear equations - examples, 200+ questions and a homework/test

Linear equations - examples, 200+ questions and a homework/test

These resources are a good way to quickly cover/revise the whole topic of linear equations. The first resource begins with a few notes on what forms linear equations can take and some of the steps or methods that may be required to solve them. There are some parts of the notes that need to be completed with your students, to practise the algebraic steps involved in solving linear equations. There are then several sections, each section focussing on a particular form of linear equation. There are a few examples to complete with your students as practice, then an exercise for students to complete on their own. There is also an exercise of mixed questions at the end. Answers to all the exercises are included. Section A - Solving x+a=b, x-a=b, a-x=b Section B - Solving ax=b Section C - Solving x/a=b and a/x=b Section D - Solving ax+b=c, ax-b=c, a-bx=c Section E - Solving x/a+b=c, x/a-b=c, a-x/b=c, a-b/x=c Section F - Solving (ax+b)/c=d, (ax-b)/c=d, (a-bx)/c=d Section G - Solving a(bx+c)=d, a(bx-c)=d, a(b-cx)=d Section H - Solving ax+b=cx+d, ax+b=c-dx Section I - Solving a(bx+c)=dx+e, a(bx+c)=d-ex Section J - Solving (ax+b)/c=dx+e, (ax-b)/c=dx+e, (a-bx)/c=d-ex Section K - Mixed exercise The second resource gives your students practice of solving linear equations using a graph. Worked solutions to this sheet are included. The final resource is a homework/test with 35 questions that cover the whole of the topic, including solving linear equations using a graph. Worked solutions are included.
langy74
Hypothesis testing (new A level) - notes, examples, exercises and a homework/test

Hypothesis testing (new A level) - notes, examples, exercises and a homework/test

This 17-page resource covers all the required knowledge and techniques for hypothesis testing in the new A level. It contains detailed notes, examples to work through with your class, and exercises of questions for students to attempt themselves (answers included). The topics covered are: 1. Sampling - different methods of sampling, biased and representative samples 2. Unbiased estimators - estimating the population mean and variance from a sample 3. Setting up a hypothesis test - null and alternative hypotheses 4. Making a conclusion - p-values, significance levels, 1-tail and 2-tail tests 5. Critical regions - finding the critical region for a hypothesis test 6. Significance levels and errors - probability of incorrectly rejecting null hypothesis, nominal vs actual significance level This projectable and printable resource will save you having to write out or create any notes/examples when teaching this topic. It also increases how much you can get through in lessons as students don't have to copy notes/questions and can work directly onto spaces provided for solutions. You could also email/print some or all of this for students who have missed lessons or need additional notes/practice/revision. Also included is a 2-page assessment that covers the whole topic. Fully worked solutions are included.
langy74
Applications of differentiation (new A level) - notes, examples, exercises and a homework/test

Applications of differentiation (new A level) - notes, examples, exercises and a homework/test

This 18-page resource covers all the uses/applications of differentiation as required for the new A level. In every section it contains examples to work through with your class followed by an exercise of questions for students to attempt themselves (answers included). The sections are: 1. Tangents and normals - finding the equations of tangents/normals to curves 2. Stationary points - finding them and determining their nature using first or second derivative 3. Smallest and largest values of a function - finding min&max value of f(x) in a set of values for x 4. Practical problems - using differentiation to find optimal solution to a problem in context This projectable and printable resource will save you having to write out or create any notes/examples when teaching this topic. It also increases how much you can get through in lessons as students don't have to copy notes/questions and can work directly onto spaces provided for solutions. You could also email/print some or all of this for students who have missed lessons or need additional notes/practice/revision. Also included is a 2-page assessment that can be used as a homework or test. Fully worked solutions to this assessment are provided.
langy74
All you need to teach the equation of a straight line!

All you need to teach the equation of a straight line!

I think this set of resources covers everything your classes need to learn and practice on straight line graphs (up to GCSE level). All the resources are suitable to be projected or printed for students to work on, saving a lot of time for drawing graphs and allowing them to annotate or work on diagrams. All resources come with solutions included. Here is a brief description of each resource: 1. Basic straight lines - lines of the form x=a, y=a and y=x or y=-x 2. Drawing straight lines - 10 questions using the equation of a line y=mx+c to complete a table of values and draw the graph. 3. Cover-up method - 12 questions to practise drawing lines of the form ax+by=c 4. Using the equation - test if a point lies on a line, determine y-coord given x-coord and vice versa (70 questions) 5. Finding the gradient - 18 questions to practise finding gradients, including where the scales on the axes are not the same 6. Matching y=mx+c to the graph - they find the gradient and y-intercept for each given graph and equation, learning the connection between the equation and properties of the graph 7. Equation to gradient and y-intercept - simple worksheet to practice writing down the gradient and coordinates of y-intercept from the equation, and vice versa (24 questions) 8. Finding the equation of a line - 24 questions to practise finding the equation of the line from its graph, including where the scales on the axes are not the same 9. Finding equation using point and gradient - 10 questions to practise doing this with a grid as an aid, then 26 questions without a grid 10. Pairs of lines - 4 graphs, each with a pair of parallel or perpendicular lines. By finding the equation of each line the students should start to see the rules for gradients of parallel and perpendicular lines 11. Parallel and perpendicular lines - almost 50 questions finding the equation of a line parallel / perp to a given line that passes through (0,b) or (a, b) 12. Using two points A and B - find midpoint M of AB, gradient of line through A and B, equation of line through A and B, equation of line perp. to AB through A, B or M. 10 questions to learn the methods with grids as an aid, then an exercise for each style of question (over 50 questions in total). 13. Multiple choice questions - quick assessment covering most of the topic 14. Straight lines revision - 60 questions to revise the whole topic 15. Homework - 19 questions on all aspects of the topic, fully works solutions included I have just worked through all these with my year 10 group and it took around 5 hours of lesson time to complete. A more able group may need less time but you have enough resources here to keep your classes busy for a number of lessons.
langy74
Probability (new A level) - teaching notes, examples and exercises & homework/test

Probability (new A level) - teaching notes, examples and exercises & homework/test

This 25-page resource covers all the required knowledge for probability in the AS part of the new A level. In every section it contains examples to work through with your class followed by an exercise of questions for students to attempt themselves (answers included). The sections are: 1. Sample space diagrams 2. Two-way tables 3. Tree diagrams 4. Venn diagrams and set notation 5. Independent, mutually exclusive and complementary events 6. Probability distributions 7. Arranging items (preliminary work for Binomial distribution) 8. Binomial distribution This projectable and printable resource will save you having to draw any tables/diagrams when teaching the topic and will make things easier for your students as they can just work directly on the provided tables and diagrams. Also included is a worksheet designed to specifically practise writing cumulative probability calculations in the required form for using a calculator. The 2 page assessment covers all aspects of the topic and fully worked solutions are provided. Lastly, I have included a spreadsheet that calculates and illustrates probabilities for any Binomial distribution with n up to 100. You may find this resource useful to show the shape of the distribution and, in later work, how the distribution approximates a Normal distribution in certain conditions.
langy74
Angles in quadrilaterals worksheet

Angles in quadrilaterals worksheet

This worksheet focuses on using the sum of angles in a quadrilateral to find missing angles. It assumes that students are already familiar with angles in triangles, on a straight line, vertically opposite angles, and angles in parallel lines. The first section covers different types of quadrilaterals and their properties. There is a short exercise where students practise choosing the correct type(s) of quadrilateral based on the information given. The second section begins with the result for the sum of angles in a quadrilateral. There are then some examples of finding angles - these are to be completed with your class. The exercise that follows is for students to attempt themselves. Answers to both exercises are included.
langy74
Introduction to differentiation (new A level) teaching notes, examples and exercises & homework/test

Introduction to differentiation (new A level) teaching notes, examples and exercises & homework/test

This 21-page resource introduces the method of differentiation as required for the new A level. In every section it contains examples to work through with your class followed by an exercise of questions for students to attempt themselves (answers included). The sections are: 1. Gradient function - sketching the graph of the derivative of a function 2. Estimating the gradient of a curve at a point, leading to differentiation from first principles 3. Differentiation of ax^n 4. Simplifying functions into the required form before differentiating 5. Using and interpreting derivatives 6. Increasing and decreasing functions 7. Second derivatives This projectable and printable resource will save you having to write out any notes/examples or draw any graphs when teaching the topic, and will make things easier for your students as they can just work directly on the given axes and spaces provided for solutions. Also included is a 2-page assessment that can be used as a homework or test. Fully worked solutions to this assessment are provided.
langy74
de Moivre's theorem and roots of polynomials - teaching notes and examples

de Moivre's theorem and roots of polynomials - teaching notes and examples

This worksheet can be used to introduce de Moivre's theorem to your class and show how it can be used to find multiple angle formulae (e.g. sin 4theta = ...) and how these formulae help us to relate trigonometric equations to polynomial equations. The introduction shows how we can arrive at 2 different results for (c + is)^n by using de Moivre's theorem and a binomial expansion. There are then 3 examples of using this technique to derive multiple angle formulae. The second section focuses on relating trigonometric equations to polynomial equations and how this allows us to find exact values of trigonometric functions or to express the roots of a polynomial in trigonometric form. There are 3 examples to illustrate this, the first one is deliberately straightforward to help students see the connection between the trigonometric work and the polynomial equation. The solutions version of the worksheet has fully-worked solutions to all the examples and the notes in the introduction section are also completed. Once you have worked through this worksheet with your students they should be able to attempt an exercise of questions on their own.
langy74
Statistical diagrams and calculations (new A level) teaching notes, examples, exercises & a homework

Statistical diagrams and calculations (new A level) teaching notes, examples, exercises & a homework

This 26-page resource covers all the required knowledge for diagrams and calculations to summarise or represent data in the new A level. In every section it contains examples to work through with your class followed by an exercise of questions for students to attempt themselves (answers included). The sections are: 1. Bar charts and pie charts - revision of interpreting these simple diagrams 2. Averages of a list of data 3. Range and interquartile range of a list of data 4. Histograms - drawing them, interpreting them and using them for probability 5. Cumulative frequency - using the diagram to find median, IQR, percentiles etc 6. Box-and-whisker plots - interpretation and use to compare 2 sets of data 7. Standard deviation - calculation from a list of data or summary statistics 8. Frequency tables - finding averages/measures of spread from (grouped) frequency tables 9. Scatter diagrams and correlation - interpretation of diagram, PMCC, use of line of best fit 10. Outliers - investigating presence of outliers in a list/table of data or a diagram This projectable and printable resource will save you having to draw any tables/diagrams when teaching the topic and will make things easier for your students as they can just work directly on the provided tables and axes, as well as drawing on the provided diagrams to help interpret them. Also included is a homework/test that covers the whole topic - fully worked solutions are provided.
langy74
Newton-Raphson method - teaching notes and exercises

Newton-Raphson method - teaching notes and exercises

This resource covers the use of the Newton-Raphson method for finding roots of equations. You can project/print the resource so that you save time teaching the required knowledge and your classes can focus their time on understanding the process and attempting questions. The resource is made up of four sections: 1. Applying the Newton-Raphson formula iteratively to find approximations to roots 2. Derivation of the iterative formula F(x) for different functions f(x)=0, and begin to consider F'(x). 3. Errors and convergence 4. Graphical representation of the Newton-Raphson method Each section contains an introduction with the required knowledge and explanations, followed by an exercise of questions. The answers to all exercises are included.
langy74
Calculations with fractions (all 4 operations)

Calculations with fractions (all 4 operations)

These are two 2-sided worksheets that cover all calculations with fractions. The adding/subtracting worksheet goes step-by-step through the process of making the denominators equal prior to the calculation. The first exercise (12 questions) involves adding/subtracting fractions and mixed numbers where the denominators match, the second exercise (34 questions) involves adding/subtracting fractions and mixed numbers where the denominators do not match. The multiplying/dividing worksheet begins with a reminder of the method, together with a few examples to work through as a group. There are then two exercises, each with 20 questions, first to practise multiplying and then to practise dividing fractions and mixed numbers. Fully worked solutions to all questions are provided.
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