What are Venn Diagrams?
Introduction
How to depict Venn diagrams?
Shading regions
Shading inwards iii sets..
Showing posts with label Sets. Show all posts
Showing posts with label Sets. Show all posts
Tuesday, November 5, 2019
Thursday, October 24, 2019
Tricks In Addition To Tips 3: Quadratics
highest mutual factor and methods for sequences, linear graphs too surds.
The vertex of a quadratic graph
This comes upward inwards the novel maths GCSE, inwards questions similar this which is taken from the Pearson Edexcel GCSE (9-1) Mathematics Sample Assessment Materials:
Pause for a infinitesimal too expect at this question, because it's a non bad instance of the modify inwards difficulty inwards the novel GCSE. Note the utilisation of business office annotation too the term turning point. The equation doesn't factorise. The graph doesn't cross the x-axis. The coordinates of the turning betoken aren't integers. This is a notable footstep upward from the type of questions asked inwards electrical current GCSE exams.
For the purpose of this post, let's consider the business office y = x2 - 6x + 10. There are a give away of ways to notice the coordinates of the turning betoken - how would y'all create it?
1. Vertex Form
The term 'vertex form' is non unremarkably used inwards the UK. Vertex shape is what y'all acquire when y'all consummate the square. So we'd write the business office equally follows:
2. H5N1 formula
In unopen to countries, students only memorise a formula. They larn that the x coordinate of the vertex is -b/2a. They thence notice the y coordinate yesteryear substituting that value into the equation.
By memorising this formula, y'all tin sack notice the coordinates of the turning betoken of whatever quadratic function without completing the square. At my conference session I showed the video below - lookout it to encounter how the method is explained. I'm non a fan of this approach. I don't desire my students to only memorise a formula - there's no conceptual agreement here.
3. Differentiation
Differentiation is e'er a pleasure. We don't create calculus at GCSE, but I idea it worth mentioning hither that unopen to other method to notice a turning betoken of a business office is to laid upward the derivative equal to zero. As y'all tin sack encounter below, for a quadratic that volition e'er hand us x = -b/2a.
4. Symmetry
For a quadratic that intercepts the x axis, the vertex is the midpoint of the 2 roots. This plant because parabolas are symmetrical. Up until late I idea this approach wasn't possible for quadratics that don't intercept the x axis, but thence I discovered James Tanton's method. It's described below for the business office y = x2 + 4x + 5 - for to a greater extent than exceptional too examples, encounter this Algebra too Core AS resource libraries are packed total of recommended resources for instruction this fantastic topic, such as this activity from Susan Wall.
Preparing for the novel GCSE
As I was writing this postal service it occurred to me that there's a lot of actually of import things that maths departments demand to do this term to cook for the novel GCSE. Writing novel Schemes of Work is a huge job, equally is finding suitable resources for instruction the novel GCSE topics.
CPD for maths teachers is likewise actually important. All maths teachers demand to move familiar alongside the novel GCSE content - they demand to know what's been added too what's been removed. They demand to expect at lots of instance questions.
The other affair that all teachers demand to create at nowadays is a patch of written report noesis banking concern gibe - are at that topographic point whatever topics on the novel GCSE syllabus that you're non familiar with? This is specially relevant for teachers who've never taught H5N1 degree maths. Has everyone inwards your subdivision idea close how to learn the novel GCSE topics? It's fourth dimension for unopen to vital maths subdivision CPD.
The vertex of a quadratic graph
This comes upward inwards the novel maths GCSE, inwards questions similar this which is taken from the Pearson Edexcel GCSE (9-1) Mathematics Sample Assessment Materials:
For the purpose of this post, let's consider the business office y = x2 - 6x + 10. There are a give away of ways to notice the coordinates of the turning betoken - how would y'all create it?
The term 'vertex form' is non unremarkably used inwards the UK. Vertex shape is what y'all acquire when y'all consummate the square. So we'd write the business office equally follows:
y = (x - 3)2 + 1
Now nosotros tin sack position the turning betoken at in 1 trial away. I've e'er explained it a chip similar this:
"The (x - 3)2 is squared thence it tin sack never move negative. The lowest it tin sack move is zero. It's null when x is 3. The lowest possible y value is 0 + 1. So nosotros know that the minimum is at (3,1)..."
This explanation is inwards draw alongside my idea procedure - it's the agency I position the turning betoken - but my students actually create out alongside it. Of all the things I teach, this is the explanation that gets the most blank looks! So this yr I tried a dissimilar approach when revising this theme alongside my Year 11s. This fourth dimension I relied on my students' noesis of graph transformations. I told them to intend of the graph y = (x - 3)2 + 1 equally a transformation of the graph y = x2. It's been translated iii units correct too 1 unit of measurement up. The vertex moves from (0,0) to (3,1). They constitute this approach actually slowly - it made a lot to a greater extent than feel to them. Suddenly all my students were able to notice the vertex of a quadratic function.
As long equally students own got studied graph transformations thence this approach seems to work. This instruction gild is worth bearing inwards hear when designing a Scheme of Work.
From at nowadays on, I'm going to utilisation the transformation method. But at that topographic point are alternatives...
From at nowadays on, I'm going to utilisation the transformation method. But at that topographic point are alternatives...
2. H5N1 formula
In unopen to countries, students only memorise a formula. They larn that the x coordinate of the vertex is -b/2a. They thence notice the y coordinate yesteryear substituting that value into the equation.
By memorising this formula, y'all tin sack notice the coordinates of the turning betoken of whatever quadratic function without completing the square. At my conference session I showed the video below - lookout it to encounter how the method is explained. I'm non a fan of this approach. I don't desire my students to only memorise a formula - there's no conceptual agreement here.
3. Differentiation
Differentiation is e'er a pleasure. We don't create calculus at GCSE, but I idea it worth mentioning hither that unopen to other method to notice a turning betoken of a business office is to laid upward the derivative equal to zero. As y'all tin sack encounter below, for a quadratic that volition e'er hand us x = -b/2a.
For a quadratic that intercepts the x axis, the vertex is the midpoint of the 2 roots. This plant because parabolas are symmetrical. Up until late I idea this approach wasn't possible for quadratics that don't intercept the x axis, but thence I discovered James Tanton's method. It's described below for the business office y = x2 + 4x + 5 - for to a greater extent than exceptional too examples, encounter this Algebra too Core AS resource libraries are packed total of recommended resources for instruction this fantastic topic, such as this activity from Susan Wall.
As I was writing this postal service it occurred to me that there's a lot of actually of import things that maths departments demand to do this term to cook for the novel GCSE. Writing novel Schemes of Work is a huge job, equally is finding suitable resources for instruction the novel GCSE topics.
CPD for maths teachers is likewise actually important. All maths teachers demand to move familiar alongside the novel GCSE content - they demand to know what's been added too what's been removed. They demand to expect at lots of instance questions.
The other affair that all teachers demand to create at nowadays is a patch of written report noesis banking concern gibe - are at that topographic point whatever topics on the novel GCSE syllabus that you're non familiar with? This is specially relevant for teachers who've never taught H5N1 degree maths. Has everyone inwards your subdivision idea close how to learn the novel GCSE topics? It's fourth dimension for unopen to vital maths subdivision CPD.
New Gcse: Inequalities
The new GCSE specification has 2 additions nether the heading Inequalities:
This post service explains these changes as well as provides education back upwards as well as resources.
Set Notation
a) using gear upwards musical note to correspond solutions
b) solving quadratic inequalities.
Students volition straightaway hold upwards required to correspond solutions to inequalities using gear upwards notation. This is inwards add-on to representing solutions on publish lines as well as graphs. The OCR specification gives us 2 examples of gear upwards notation:
If you're non familiar alongside gear upwards notation, it's explained here (it's unremarkably referred to equally 'set-builder notation'). Note that either a vertical trouble or a colon tin move hold upwards used to correspond 'such that'.
 |
| Image source: coolmath.com |
The examples from the OCR specification imply that GCSE students volition not postulate to know symbols representing publish types (eg ℤ for integers), as well as thence volition non hold upwards required to limited their answers similar this:
 |
| Image source: mathsinsfun.com |
Set musical note comes upwards elsewhere inwards the novel GCSE specification - nether the championship 'Venn Diagrams as well as Sets' nosotros lead maintain this:
 |
| Source: OCR specification |
So create nosotros lead maintain whatsoever resources to exercise this? There's enough of resources relating to gear upwards musical note as well as probability inwards my Data library. I've also made this simple worksheet so students tin move exercise using gear upwards musical note to correspond inequalities.
Quadratic Inequalities
Here's a enquiry from OCR's Sample Assessment Materials Higher Paper five (non-calculator):
Find the gain of values of x for which x2 - 3x - 10 ≤ 0
If y'all haven't taught AS score maths or iGCSE so y'all mightiness non lead maintain taught this topic before. There's a few dissimilar methods for solving quadratic inequalities. H5N1 mutual method involves sketching a graph of the quadratic component so identifying the required part (see event below). To role this method, our GCSE students volition postulate to hold upwards confident inwards sketching quadratics (this is covered inwards the novel specification). I oftentimes honor that my Year 12 students brand mistakes inwards their end reply because they don't bother sketching the graph. To me, sketching the graph is essential.
Desmos is a fantastic tool for exploring quadratic inequalities - cheers to Cathal (@CGA_PGS) for sharing this example.
An choice method for solving quadratic inequalities involves using a publish trouble as well as assay out points. In this method students even so lead maintain to honor the critical values, but instead of sketching the graph they depository fiscal establishment fit the value of a assay out indicate inwards each region.
Given that this topic has been taught inwards iGCSE as well as H5N1 score courses for unopen to time, there's surprisingly few resources available. There's a few resources inwards my Algebra library, including a 'Spot the Mistake' action I made, but I postulate more! If y'all know of anything else so delight part it.
Thursday, September 5, 2019
Tricks In Addition To Tips 3: Quadratics
highest mutual factor and methods for sequences, linear graphs too surds.
The vertex of a quadratic graph
This comes upward inwards the novel maths GCSE, inwards questions similar this which is taken from the Pearson Edexcel GCSE (9-1) Mathematics Sample Assessment Materials:
Pause for a infinitesimal too expect at this question, because it's a non bad instance of the modify inwards difficulty inwards the novel GCSE. Note the utilisation of business office annotation too the term turning point. The equation doesn't factorise. The graph doesn't cross the x-axis. The coordinates of the turning betoken aren't integers. This is a notable footstep upward from the type of questions asked inwards electrical current GCSE exams.
For the purpose of this post, let's consider the business office y = x2 - 6x + 10. There are a give away of ways to notice the coordinates of the turning betoken - how would y'all create it?
1. Vertex Form
The term 'vertex form' is non unremarkably used inwards the UK. Vertex shape is what y'all acquire when y'all consummate the square. So we'd write the business office equally follows:
2. H5N1 formula
In unopen to countries, students only memorise a formula. They larn that the x coordinate of the vertex is -b/2a. They thence notice the y coordinate yesteryear substituting that value into the equation.
By memorising this formula, y'all tin sack notice the coordinates of the turning betoken of whatever quadratic function without completing the square. At my conference session I showed the video below - lookout it to encounter how the method is explained. I'm non a fan of this approach. I don't desire my students to only memorise a formula - there's no conceptual agreement here.
3. Differentiation
Differentiation is e'er a pleasure. We don't create calculus at GCSE, but I idea it worth mentioning hither that unopen to other method to notice a turning betoken of a business office is to laid upward the derivative equal to zero. As y'all tin sack encounter below, for a quadratic that volition e'er hand us x = -b/2a.
4. Symmetry
For a quadratic that intercepts the x axis, the vertex is the midpoint of the 2 roots. This plant because parabolas are symmetrical. Up until late I idea this approach wasn't possible for quadratics that don't intercept the x axis, but thence I discovered James Tanton's method. It's described below for the business office y = x2 + 4x + 5 - for to a greater extent than exceptional too examples, encounter this Algebra too Core AS resource libraries are packed total of recommended resources for instruction this fantastic topic, such as this activity from Susan Wall.
Preparing for the novel GCSE
As I was writing this postal service it occurred to me that there's a lot of actually of import things that maths departments demand to do this term to cook for the novel GCSE. Writing novel Schemes of Work is a huge job, equally is finding suitable resources for instruction the novel GCSE topics.
CPD for maths teachers is likewise actually important. All maths teachers demand to move familiar alongside the novel GCSE content - they demand to know what's been added too what's been removed. They demand to expect at lots of instance questions.
The other affair that all teachers demand to create at nowadays is a patch of written report noesis banking concern gibe - are at that topographic point whatever topics on the novel GCSE syllabus that you're non familiar with? This is specially relevant for teachers who've never taught H5N1 degree maths. Has everyone inwards your subdivision idea close how to learn the novel GCSE topics? It's fourth dimension for unopen to vital maths subdivision CPD.
The vertex of a quadratic graph
This comes upward inwards the novel maths GCSE, inwards questions similar this which is taken from the Pearson Edexcel GCSE (9-1) Mathematics Sample Assessment Materials:
For the purpose of this post, let's consider the business office y = x2 - 6x + 10. There are a give away of ways to notice the coordinates of the turning betoken - how would y'all create it?
The term 'vertex form' is non unremarkably used inwards the UK. Vertex shape is what y'all acquire when y'all consummate the square. So we'd write the business office equally follows:
y = (x - 3)2 + 1
Now nosotros tin sack position the turning betoken at in 1 trial away. I've e'er explained it a chip similar this:
"The (x - 3)2 is squared thence it tin sack never move negative. The lowest it tin sack move is zero. It's null when x is 3. The lowest possible y value is 0 + 1. So nosotros know that the minimum is at (3,1)..."
This explanation is inwards draw alongside my idea procedure - it's the agency I position the turning betoken - but my students actually create out alongside it. Of all the things I teach, this is the explanation that gets the most blank looks! So this yr I tried a dissimilar approach when revising this theme alongside my Year 11s. This fourth dimension I relied on my students' noesis of graph transformations. I told them to intend of the graph y = (x - 3)2 + 1 equally a transformation of the graph y = x2. It's been translated iii units correct too 1 unit of measurement up. The vertex moves from (0,0) to (3,1). They constitute this approach actually slowly - it made a lot to a greater extent than feel to them. Suddenly all my students were able to notice the vertex of a quadratic function.
As long equally students own got studied graph transformations thence this approach seems to work. This instruction gild is worth bearing inwards hear when designing a Scheme of Work.
From at nowadays on, I'm going to utilisation the transformation method. But at that topographic point are alternatives...
From at nowadays on, I'm going to utilisation the transformation method. But at that topographic point are alternatives...
2. H5N1 formula
In unopen to countries, students only memorise a formula. They larn that the x coordinate of the vertex is -b/2a. They thence notice the y coordinate yesteryear substituting that value into the equation.
By memorising this formula, y'all tin sack notice the coordinates of the turning betoken of whatever quadratic function without completing the square. At my conference session I showed the video below - lookout it to encounter how the method is explained. I'm non a fan of this approach. I don't desire my students to only memorise a formula - there's no conceptual agreement here.
3. Differentiation
Differentiation is e'er a pleasure. We don't create calculus at GCSE, but I idea it worth mentioning hither that unopen to other method to notice a turning betoken of a business office is to laid upward the derivative equal to zero. As y'all tin sack encounter below, for a quadratic that volition e'er hand us x = -b/2a.
For a quadratic that intercepts the x axis, the vertex is the midpoint of the 2 roots. This plant because parabolas are symmetrical. Up until late I idea this approach wasn't possible for quadratics that don't intercept the x axis, but thence I discovered James Tanton's method. It's described below for the business office y = x2 + 4x + 5 - for to a greater extent than exceptional too examples, encounter this Algebra too Core AS resource libraries are packed total of recommended resources for instruction this fantastic topic, such as this activity from Susan Wall.
As I was writing this postal service it occurred to me that there's a lot of actually of import things that maths departments demand to do this term to cook for the novel GCSE. Writing novel Schemes of Work is a huge job, equally is finding suitable resources for instruction the novel GCSE topics.
CPD for maths teachers is likewise actually important. All maths teachers demand to move familiar alongside the novel GCSE content - they demand to know what's been added too what's been removed. They demand to expect at lots of instance questions.
The other affair that all teachers demand to create at nowadays is a patch of written report noesis banking concern gibe - are at that topographic point whatever topics on the novel GCSE syllabus that you're non familiar with? This is specially relevant for teachers who've never taught H5N1 degree maths. Has everyone inwards your subdivision idea close how to learn the novel GCSE topics? It's fourth dimension for unopen to vital maths subdivision CPD.
New Gcse: Inequalities
The new GCSE specification has 2 additions nether the heading Inequalities:
This post service explains these changes as well as provides education back upwards as well as resources.
Set Notation
a) using gear upwards musical note to correspond solutions
b) solving quadratic inequalities.
Students volition straightaway hold upwards required to correspond solutions to inequalities using gear upwards notation. This is inwards add-on to representing solutions on publish lines as well as graphs. The OCR specification gives us 2 examples of gear upwards notation:
If you're non familiar alongside gear upwards notation, it's explained here (it's unremarkably referred to equally 'set-builder notation'). Note that either a vertical trouble or a colon tin move hold upwards used to correspond 'such that'.
|
| Image source: coolmath.com |
The examples from the OCR specification imply that GCSE students volition not postulate to know symbols representing publish types (eg ℤ for integers), as well as thence volition non hold upwards required to limited their answers similar this:
|
| Image source: mathsinsfun.com |
Set musical note comes upwards elsewhere inwards the novel GCSE specification - nether the championship 'Venn Diagrams as well as Sets' nosotros lead maintain this:
|
| Source: OCR specification |
So create nosotros lead maintain whatsoever resources to exercise this? There's enough of resources relating to gear upwards musical note as well as probability inwards my Data library. I've also made this simple worksheet so students tin move exercise using gear upwards musical note to correspond inequalities.
Quadratic Inequalities
Here's a enquiry from OCR's Sample Assessment Materials Higher Paper five (non-calculator):
Find the gain of values of x for which x2 - 3x - 10 ≤ 0
If y'all haven't taught AS score maths or iGCSE so y'all mightiness non lead maintain taught this topic before. There's a few dissimilar methods for solving quadratic inequalities. H5N1 mutual method involves sketching a graph of the quadratic component so identifying the required part (see event below). To role this method, our GCSE students volition postulate to hold upwards confident inwards sketching quadratics (this is covered inwards the novel specification). I oftentimes honor that my Year 12 students brand mistakes inwards their end reply because they don't bother sketching the graph. To me, sketching the graph is essential.
Desmos is a fantastic tool for exploring quadratic inequalities - cheers to Cathal (@CGA_PGS) for sharing this example.
An choice method for solving quadratic inequalities involves using a publish trouble as well as assay out points. In this method students even so lead maintain to honor the critical values, but instead of sketching the graph they depository fiscal establishment fit the value of a assay out indicate inwards each region.
Given that this topic has been taught inwards iGCSE as well as H5N1 score courses for unopen to time, there's surprisingly few resources available. There's a few resources inwards my Algebra library, including a 'Spot the Mistake' action I made, but I postulate more! If y'all know of anything else so delight part it.
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