GCSE Mathematics for AQA Teacher's Resource Sample by CUPUKschools

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MATHEMATICS GCSE for AQA Teacher’s Resource Sample

Written from draft specification

Vector geometry

Vector geometry Student Book Foundation: 36 Vector geometry, p. 540 Higher: 38 Vector geometry, p. 602

What your students need to know • Basic arithmetic skills including addition, subtraction, multiplication and division (finding fractions of amounts) of both positive and negative numbers. • How to plot coordinates in all four quadrants, understanding that the x and y coordinates are distances in both horizontal and vertical directions from the origin. • Basic ratio including connections to proportion, e.g. 2 : 3 is _ 52 and _ 53 of the whole. • How to solve simple linear equations.

Additional useful prior knowledge • • • • •

What the laws of associativity, commutativity and distributivity are and how they apply to basic arithmetic operations. Be able to form and solve simultaneous linear equations. How to use Pythagoras’ theorem to find the length of a line segment in 2D. How to use the tangent function to find angles in right-angled triangles. (Higher only) What a mathematical proof is.

Learning outcomes Section 1 • Represent vectors as a diagram or a column vector. Section 2 • Add and subtract vectors. • Multiply vectors by a scalar. Section 3 • (Higher only) Use vectors to construct geometric arguments and proofs.

Vocabulary Vector, displacement, scalar, parallel vectors, magnitude, direction, opposite, commutative, associative, equal.

Common misconceptions and other issues • This can be used as a straightforward chapter for less able students, as a simple extension to their knowledge of how operations apply to scalars (i.e. numbers). • For higher ability students, who might be considering moving on to A/AS Level Mathematics, the following important concepts should be covered, since many students have misconceptions: – Confusing x- and y-values When working with column vectors, some students confuse the x- and y-axis and forget that the x-value controls movement left or right, and the y-value controls movement up or down. Ask students to recall how coordinates work. Useful phrases are ‘x is across’ and ‘y to the sky’.

GCSE Mathematics Teacher’s Resource

You could also give students a coordinate grid to practice rewriting coordinates as a positon vector from the origin, ___› 3    . Be careful that this does not reinforce or introduce the misconception that   or a =   for example A = (3, 5) so OA 5 all vectors must start from the origin (see next point).

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– Assuming all vectors start at the origin To avoid confusion between position vectors and displacement vectors, make sure students understand the following definitions:

position vectors describe the movement to a point on a grid and always start from the origin.

displacement vectors describe the movement between any two points on a grid, and therefore can start anywhere on the grid depending on the point(s) in question.

equal vectors have the same magnitude and direction; when written in column vector form, their x- and yvalues are the same.

Make it clear that a displacement vector can be equal to a position vector, but their starting points will be defined in a different way.

– Understanding that equal vectors are also parallel vectors The definition of parallel vectors is when one vector can be written as a (scalar) multiple of the other, i.e. x1 x 2   y1   = t y2   where t is the scalar to be found. This is often described to students when they know how to multiply a vector by a scalar, as one vector is a multiple of the other. It is also important that students understand that equal vectors are also parallel; this is the case when the scalar is 1.

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– Understanding what −a means geometrically in relation to a Ask students to draw three vectors and then to draw their opposite vector on the same grid. For example, the opposite of the vector drawn ‘2-right, 3-up’ would be ‘2-left, 3-down’. This should help students to identify that multiplying a vector by the scalar −1 results in moving in the opposite direction to the starting vector.

This can also be explained by describing a series of vectors in a geometric problem as a network with a series of paths, e.g.

B b

_ __› _ __› _ __› To find AB   you can travel back along a (i.e. AO   = −a) and then along b. This can be written as AB   = b − a (Higher ___› ___› ___›   + OB  .  only) If using only displacement vector notation, you can write AB   = AO

Be careful when using the idea of vector problems as networks because it can introduce new problems when working with midpoints and other values as proportional distances between two points, when students wish to stick to the grid only. For example, in addition to the information in the diagram above you are that ____› ____told › ___› 1 ___› OM     . By considering OM     = OA   + _  2  AB    coordinate M is the midpoint of the line segment AB and are asked to find ____› 1 1 _ _ you can simplify to give OM   =  2  a +  2  b. If students want to stick to the network and the drawn paths then this has little meaning to them whereas with a strong knowledge of equal and parallel vectors students can understand that it is the start and end points and the movement between them that are important rather than the routes taken.

– Understanding that addition is commutative in vector arithmetic. This links into the previous point about wanting to use only the drawn paths in the diagram above. Given the same vectors above, additional vectors can be drawn on the diagram to give:

C b

A a

B b

O 2

If students fail to_understand the importance of _parallel vectors in describing movement then they will fail to __ ___ ___ ___ ___ __

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understand that OC  = OA  + AC  = OB  + BC  , i.e. OC  = a + b = b + a. © Cambridge University Press 2014 Written from draft specification

Vector geometry

Hooks 1. A nice way to introduce the idea of a vector, and then ask students what they think it is, is through a clip of Despicable Me (movie), which you may be able to find online. It introduces the words ‘direction’, and ‘magnitude’ and produces a discussion point from which you can bring out the definition of a vector and how it differs from a scalar. 2. Another way to introduce vectors is to look at line segments on a grid joining two points, e.g. a map of an American city based on a grid/block network showing two points of interest. A discussion could be based on travelling ‘as the crow flies’ rather than by the grid network, and through this the discussion can move to the need for vectors to describe movement. Section 1: Vector notation and representation

Section 1 introduces the basics of vectors including notation and representation. There are opportunities for students to go beyond what a vector is and how they can draw it to also consider what it means for vectors to be parallel, equal and opposite. These ideas are drawn out through the first exercise, which asks students to compare their vectors to the diagrams of them.

Prompting questions Whilst introducing the need for (position) vectors, good prompts for promoting discussion may be:

• How can you describe any point on a grid/axis? Using coordinates to define the distance from the origin in terms of the horizontal and vertical distance. • What are you describing its position in terms of? Its distance from the origin. • What is the quickest way to travel from the origin to this point? Diagonally or ‘as the crow flies’. • What if you want to describe how to get from coordinate B to coordinate E? Why is this more challenging? Need to give starting point and x and y movement, and hence introduce need for (displacement) vectors. Whilst working through Exercise 36A (38A in Higher), good prompts for students might be:

• • • •

In question 2, what would the column vector be if the arrows were pointing in the opposite direction?

• • • • • •

What do we mean by opposite vectors?

How else could we describe this ‘opposite’ vector? Can you find any other pairs of vectors of interest in question 4? What is interesting about them? Could you draw on the right-angled triangle to show the movement represented by the vectors given when finding the new coordinate points? What do we mean by equal vectors? What do we mean by parallel vectors? Can you give three vectors and a coordinate point that together form a right-angled triangle? Can you do the same for an isosceles triangle? Can you do it for an equilateral triangle?

Starters, plenaries, enrichment and assessment ideas Starters • The ideas in the ‘Hooks’ section are nice ways to introduce this topic. Looking at a large coordinate grid (with or without the underlay of a city) would be a good way to replicate the example in hook 2. Providing multiple examples starting from both the origin and between two points immediately highlights the reason for vectors because coordinates are only useful in defining the distance from the origin.

GCSE Mathematics Teacher’s Resource

Starters or plenaries • In pairs, students play ‘Vector snakes and ladders’ – to help consolidate the use of column vectors. The board is designed so that the vectors placed on some squares on the board displace the counter in a helpful or not so helpful way. Filling in the column vectors could reverse this on a given ‘Snakes and Ladders’ board. • A card game of ‘follow me’ or dominoes. A series of points on a grid are projected at the front of the class. One side of the follow me/domino card gives the label of one of the points and the other side gives a column vector, e.g. using the diagram for question 4 in Exercise 36A (38A in Higher) you could have the following cards:

  2  ) (   −3

−8    4

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Students read out their column vector and calculate the point it arrives at using the grid on the board. If another student’s card has the letter of that point they then continue the game by reading out their column vector. In the example above, the next card would have the letter A. These cards can be made more challenging for use with Section 2 by giving a calculation to simplify first, e.g.

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−2      8     + 3      3 −12

• NRICH Vector Journeys task (http://nrich.maths.org/7453 ) pulls out a lot of the points that come through in Exercise 36A (38A in Higher) about opposite, equal and parallel vectors but there are many more options for exploration and introduction to addition of vectors (Section 2) in this simple, open task. • (Higher only) In addition to understanding what it means for vectors to be parallel, equal and opposite, it’s worth exploring the magnitude and direction of a vector using students’ knowledge of trigonometry and Pythagoras and their application to right-angled triangles, given the original definition of a vector. These ideas give ample opportunity for revision of other concepts. Section 2: Vector arithmetic

This section introduces students to the basics of vector arithmetic including addition and subtraction of vectors and multiplying by a scalar. Students are not introduced to other forms of vector multiplication until A Level and it’s probably worth explaining to students that this is one of many forms of multiplication with vectors.

Prompting questions Whilst working through Exercise 36B (38B in Higher), good prompts for students might be:

• How would these vectors look on the plane (grid)? You can choose any starting point and draw the movements of the vectors you need until you reach the end of your combination of vectors, then you can calculate the single vector you could travel. • What do you need for parallel vectors? To find a scalar multiplier. • What does it mean when a combination of two vectors makes a third? That the left-hand side combination for movements in the x-direction must be the same as the movement in the x-direction on the right-hand side of the equality sign. The same is true for the movements in the y-direction. • What do you need to set up in order to find the values x, y, z and t that create these vector equations? Linear and simultaneous linear equations.

Vector geometry

• If the ratio of AC : CB is 1 : 3, what must be the ratio of the movement in the x-direction of A to C and C to B? The same, 1 : 3. • If 20 is the movement between A and B in the x-direction, what must be the proportion of 20 for the movement in the x-direction for A to C and C to B? You can look at the ratio for the x-direction movement as 1 : 3, with your total of 20 you have an equivalent ratio for the x-movement as 5 : 15 for A to C and C to B. • What fraction of the line segment AB is AC if the ratio of AC : CB is 1 : 3? One quarter. • How can you write down a vector equation in terms of the points you could visit when moving from C to A? C to A = backwards along A to C. • (Higher only) What different types of triangles do you know about and how do you tell them apart? Equilateral, isosceles, scalene (magnitudes of vectors) and right-angled (in 2D use gradient of the line segments).

Starters, plenaries, enrichment and assessment ideas Starters • As mentioned in Section 1, the NRICH Vector Journeys task (http://nrich.maths.org/7453 ) offers opportunities for students to discover addition of vectors through exploration. It also gives students concrete examples for drawing on right-angled triangles and forming a justification for why the x-values and y-values are summed when adding vectors. Starters or plenaries • Short problems from NRICH, that can be used as starters and plenaries, are given by this link: http://nrich.maths.org/9357. Enrichment activities • The NRICH Vector journeys task can be extended into a second problem, Vector Walk (http://nrich.maths.org/6572 ), which explores all possible combinations of vector addition to arrive at a new coordinate point. By allowing multiple uses of the two vectors, students can also discover vector multiplication by a scalar as repeated addition. (Higher only) The possible extension of finding any two vectors to get you to any point on the grid revisits some ideas x1 x 2 of number properties (see Chapter X) as given two vectors   y1    and y2    the highest common factor (HCF or greatest common divisor) of x1 and x2 must be 1, the HCF of y1 and y2 must be 1, the HCF of x1 and y1 must be 1 and the HCF of x2 and y2 must be 1 for this to be possible (i.e. the only pairs of numbers allowed to share a common factor greater than 1 are y1 and x2, x1 and y2).

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• Once students are able to add vectors they can explore those that sum to zero through a closed path (polygon). The NRICH task Spotting the Loophole (http://nrich.maths.org/5812 ) offers opportunities to practise basic arithmetic on vectors with a meaningful purpose (aim to the task) and further consolidates students’ knowledge through the use of diagrams). Section 3: Using vectors in geometric proofs (HiGHER ONLY)

Section 3 is part of the higher syllabus and explores the use of vectors to prove geometric results and applies vector arithmetic to solving problems including finding midpoints. This is the section where we often see the most misconceptions in our students, as they are able to apply procedure and manipulate vectors but struggle to reconcile them with their geometric meaning to solve problems. In addition, students’ understanding of ‘mathematical proof’ can limit their ability to successfully complete the problems in this section. The most success is generally found with students’ when they are encouraged to sketch the vectors they are working with (note: this is a SKETCH and not an accurate representation of the vectors they are working on).

GCSE Mathematics Teacher’s Resource

Prompting questions Whilst working through Exercise 38C, good prompts for students might be:

• What are the properties of a rectangle, specifically a square? One of the properties of a rectangle is that opposite sides are parallel, and in the case of the square all sides are of equal length. • Which line segments form opposite sides and thus which vectors are parallel? Line segments NO and MP are parallel and MN and PO are parallel. • If going from N to O is the same distance and direction as going from M to P, what does that mean about the vectors? They are equal. • If two vectors are parallel, of the same magnitude and going in the same direction what can we say? They are equal. • •

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If M is the midpoint of line segment BC (q), can we write down a vector MC   in terms of the vector q? ____

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MC   = _ 21  q Can we write down a list of points we can move to in the diagram that go from A to C? One possible path is A to B, B to M and M to C.

• What is the definition of a parallelogram and how might that help here? Two pairs of parallel sides, so sides BC and AD are parallel. • What do you need to show two vectors are parallel? If two vectors are parallel, one is a scalar multiple of the other. • What does congruence mean? Links to Chapter X. • Which angles are the same? Equal angles IHF = HFG, HIF = IFE, etc. • Which line segments are the same? EJ = FI, JI = IH, etc. • What other vectors can we label with m and n in the diagram? _ _› _ __› _ _› _ __› _ __›   = m and JI =   EF   = FG   = n. Since they are congruent and the pattern tessellates, we can label JE   = HG • How far are midpoints along a line segment? Halfway. • What does ‘similar’ mean in a mathematical sense? Links to Chapter X. • How can we use question 7 to help us answer question 8? We can use our answers to parts a, c and d. • In question 10, what do we know about the point U on the line segment WV and how does that help? It is halfway along and so the vector from W to V is the twice the size of the vector from U to V. • How did you answer question 5 in Exercise 38B? You thought about the proportion of the line you need to travel for the ratio you were given. • If the ratio is of RT : TU : US is 1 : 3 : 2 what fraction of the line segment is between RT, TU and US? RT is  _61 of the line segment RS and so TU is _ 63 and US is  _62  . • What diagram could you draw for question 12? Assume parallel sides for the banks.

Vector geometry

• How can you represent the flow of the river as a vector? East to West flow is in the negative x-direction. • How can you do the same thing for James? You assume James will try to swim 1.5 m per second straight to the other side of the bank. So 1.5 in the positive y-direction. • What will the resulting vector be for the movement?   −3   . The vector will look like    1.5 • What lengths and angles could we label? The vector for the movement will have a direction and this will help compute the distances by using trigonometry and the width of the river.

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Starters, plenaries, enrichment and assessment ideas Plenaries • Students could form a poster summarising the vectors material as a class. Each student contributes a sticky note to an area of the poster based on one of the topics or highlighted points in the chapter. The teacher then helps compile the poster, either on a white board or the beginnings of a wall display, and the students comment on the points that have been drawn out or any that have been missed. Enrichment activities • An investigation can be formed from question 12 in Exercise 38C and linked into the trigonometry and kinematics chapters. What happens to the distance he moves off course if James chooses to swim at 1.5 m every second at an angle less than 90 degrees to the bank? Is there an angle made with the positive x-direction that James could swim at 1.5 m every second and reach the opposite point of the bank to where he set off? topic links

Previous learning This topic provides a good opportunity to return to work on straight line graphs (Chapter X), Pythagoras (Chapter X) and trigonometry (Chapter X). There are opportunities to consider the connections between both a line segment, the vector that describes it, its gradient, its associated right-angled triangle and therefore magnitude and associated angles. These are all extension tasks that help consolidate how all these concepts are connected and prepare students for future study at KS5.

Future learning

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x Column vectors are also used in transformations (see Chapter X) to denote a translation   y    in the 2D plane.

Gateway to A Level This is a relatively straightforward topic at GCSE that will be built upon at KS5. Having strong foundations in this concept will be necessary for students to extend their knowledge in A Level modules. In addition to column vectors and displacement vector notation, students will also learn to write and operate on vectors in their component form and extend in to 3D. They will also work mainly with position vectors and learn to write the equation of a straight line using a position vector, displacement vector and scalar. Some exam boards also convert between Cartesian and vector forms of straight lines. For some exam boards, vectors form a large part of the mechanics content in which students look at acceleration in a given direction and form vector equations of straight lines where the scalar represents time. In A Level Further Mathematics, students will build on their knowledge again to learn additional vector operations, work with vector equations of planes as well as lines, and extend their linear algebra knowledge to a new concept: matrices.

Want to explore further? As well as Student Book sample chapters included here, you can browse samples of the additional GCSE Mathematics components, such as the Problem-solving Book and Teacherâ&#x20AC;&#x2122;s Resource, online at

www.cambridge.org/ukschools

For more information or to speak to your local sales consultant, please contact us:

www.cambridge.org/ukschools

ukschools@cambridge.org

01223 325 588

CUPUKschools

GCSE_Edexcel_Mathematics_SC.indd 2

20/08/2014 11:45

Blended digital and print resources specifically created for the new AQA GCSE Mathematics specification, available from early 2015.

Brighter Thinking

For more information or to speak to your local sales consultant, please contact us:

www.cambridge.org/ukschools ukschools@cambridge.org 01223 325 588 CUPUKschools

MATHEMATICS GCSE for AQA Teacher’s Resource Sample

Written from draft specification