Gifted and Talented Mathematics for Key Stage 2 · Use a graph plotter like Geogebra to discover...

Illuminate – KS3 enrichment maths © www.themathszone.co.uk

Illuminate Enrichment maths for KS3

Sample

The Maths Zone 10 Staplehurst Road London SE13 5NB T 020 8318 6380 F 020 8318 6610 www.themathszone.co.uk [email protected]

http://www.themathszone.co.uk/

mailto:[email protected]


Complete Course Contents

Introduction 1

Aims, Structure and Management 3

Safe Surfing 6

Parents and Carers 7

Sessions

1 Imagine 8

2 Pythagoras and Proof 15

3 Proving Pythagoras 21

4 Constructing Roots 35

5 Winning Ways 43

6 Trisection 55

7 Gaussian Sums 63

8 Group Theory 68

9 Big Hands 83

10 Three Puzzles 91

11* Fractals 102

Soma Cards 110

* Extension Activity


Illuminate Sample Materials Welcome to the Illuminate enrichment course sample for Key Stage 3.

The materials are attractive and varied and all sessions are ready to go without too much preparation by the teacher.

Session plans are clear and concise and they are also supported by online expertise provided by experienced maths educators.

Also available online is a virtual learning environment that provides teacher resources, pupil worksheets, useful links to excellent websites and, most importantly, a forum where pupils can discuss mathematics with each other, their teacher and the online experts.

Each session comprises of a brief starter activity with which the pupils can engage immediately on arrival, the main activity and a take away puzzle for the students to complete at home.

The teacher is provided with the session plan, originals for the pupils' worksheets and take away puzzle and all other materials required for the successful delivery of the session as well as links to related websites.

It is important for session leaders to familiarise themselves with the activities and that they are confident with the content. This is especially true of the sessions where the pupils learn new games.

This teacher guide contains instructions of how to enrol your students on to the course, session plans, and originals of the worksheets and take away puzzles with their solutions.

Support is given weekly to both students and (if necessary) course leaders via the online forums. Pupils are encouraged to discuss their ideas and ask questions on their forum and the leaders can contribute to the course forum and ask for help on the teachers' forum.

Finally, to cater for the very talented pupils who may appear from time to time within your groups, there is an extended fractals project for them to attempt during the course. This is an open ended task which will be completed independently of the main sessions.


Course Aims The Illuminate maths course aims to enrich pupils' experience of mathematics in a number of ways.

To warm up the students, the first session establishes a link between algebra and geometry and the first starter activity and take away puzzle provide challenging number based problems that will extend pupils' number skills.

The main concept that is then addressed throughout most of the remaining sessions is that of proof; proof that certain theorems are always true, proof of why things work and proof that certain patterns continue infinitely. Proof is presented in an historical context as well, and pupils will learn about Euclid's elements and prove Pythagoras' theorem in a number of ways.

The second big idea of the course is Group theory.

Sets and groups are central to modern mathematics yet pupils do not meet them at all until they attend university. The session which looks at modular arithmetic and transformations may not be as challenging as those on proving Pythagoras' theorem but they give pupils an opportunity to explore a completely new area of mathematics.

Finally by solving puzzles and playing traditional games, pupils develop problem solving skills, analyse winning strategies and learn about mathematics from other cultures.

Planning Sessions To provide a manageable structure to the course, each session is provided with a starter, main activity and takeaway puzzle. However, due to the open ended nature of the activities and the challenging level of others it is envisaged that the timings given are only suggestions and that some activities could extend to two or even three sessions.

It is important that pupils are given time to explore, especially when investigating new areas of mathematics. It is also desirable that groups of pupils have enough time to present their ideas to the rest of the class and it may be necessary, for example, to allocate a whole session to pupils giving presentations on their Pythagoras proofs after all the work on proof has been completed.


Imagine Pupil Worksheet Imagine a cube with a square hole in the centre of every face. Each hole goes all the way through the cube. Calculate the surface area of the hollow cube if the cube has a side of 9cm and the hole has a side of 2cm What is the most efficient way of calculating the surface area? Calculate the surface area if the cube has side 3x and the hole has side x Calculate the surface area if the cube has side x and the hole has side y Use a graph plotter like Geogebra to discover the largest possible surface area of a holey cube if the original cube has a side of 9cm. How large would the hole have to be? Investigate this further for cubes of a different size. NB Should you find it difficult to start the investigation, then start with this alternative problem: Find the surface area of this cube with one hole through the centre. The length of the cube is 3cm and the hole has a 1cm square cross-section.


Imagine: Solutions For a cube of side y and hole size x: The surface area of the outer cube is 6 ( y² - x² ) Each face of the holes has a surface area of x(y-x)/2 The holes have a surface area (their sides) of 4x(y-x)/2 so their total surface area =6 × 4x(y-x)/2 = 12x(y-x) The centre "hole" has no surface area. Total surface area = 6 ( y² - x² ) + 12x(y-x) For the original cube Surface area = 6 ( 9² - 2² ) + 12×2×(9-2)

= 630cm² Optimisation When graphed the function f(x) =6(9² - x²) + 12x(9-x) looks like this: The point A suggests that the maximum surface area is obtained when x=3 and is equal to 648. This result is confirmed by differentiating (or by completing the square). The general result shows that the side of the hole has to be a third of the length of the side of the cube.


Session 3 You will need: Pythagoras cut-out sheets Proving Pythagoras Worksheets Copies of Take Away Puzzle 3 Starter 15 minutes Write this on the whiteboard: Subtracting three digit numbers without having to carry. 534 - 379 Take the second number away from 999 999 – 379 = 620 Add the first number with the new number 534 + 620 = 1154 Cross out the first digit and add it to the remaining 3 digit number. (154 + 1 = 155) The answer to 534 – 379 = 155 Practise doing this with other numbers to check that it always works Why does this work? Hint ………………………………. (Think about the 999 and the value of the digit 1 that you cross out and why you then add it on to the units.)


Pythagoras and Proof Part 2 Introduction 5 minutes

Introduce proving Pythagoras using the Powerpoint presentation "Proof II" Ensure that pupils understand why the dissection actually proves Pythagoras' theorem for all positive numbers. First Activity 10 Minutes

Place the cut versions of the first Pythagoras proof in the envelopes. Pupils work in groups to place an algebraic proof of Pythagoras' Theorem in the correct order. The teacher should draw attention to the comments that are an integral part of the proof and that using reasoning and logic as well as algebra and geometric drawings are an integral part of any proof. Second Activity 40 Minutes

Pupils use the four starting points to create more proofs for Pythagoras' theorem. Each group should choose a different activity so that they can present their proof to the rest of the class The diagram that involves the ratio of x:y should only be given to the strongest mathematicians - it is very hard. During the activity the teacher should again draw attention to comments and how the pupils progress from one stage of the proof to the next. Conclusion 15 minutes

Each group presents their proof to the rest of the class. Note that the timings are very approximate. The teacher may choose to extend the activity and ensure that all groups complete all proofs. The activity can also be structured as a "Jigsaw" activity. Each pupil is assigned a letter A B and C. Each group comprises of pupils A B and C. After each group has completed a different proof all the A pupils gather together to share their proofs and to explain to each other; similarly for the B and C pupils.


Proving Pythagoras 1 Group cut-out sheet Work in groups of two or three. Cut out along the dotted lines and rearrange the pieces into a logical order that proves Pythagoras' Theorem. Paste onto an A3 sheet and add comments of your own that explain exactly how it proves the Theorem.

y

x

z

y

y

y x

x

x

z


² 2 ² 2

²

( )² 2

( )²

2

² ²

x xy y xy

z

x y xy

x y

xy

x y

⇒ + + −

+ −

+

⇒ +

The shaded area is a square of area

The shaded area is therefore given by:

The area of the large square is given by:

The area of a triangle is equal to:

The area

4 22

² ² ²

²

xy xy

x y z

z

× =

⇒ + =

of four triangles is therefore

The shaded area is equal to the area of four triangles subtracted from the area of the large square.

The shaded area is a square of area


Proving Pythagoras 2b Pupil Worksheet There are at least 365 proofs to Pythagoras' Theorem; one for every day of the week. Some are similar, some are easy to understand and some are very difficult to follow. Proof A Explain how this proves Pythagoras' Theorem


Proof B What is the area of the small square? Add the four quadrilaterals and the small square to make the big square. Draw another right angled triangle of your own on squared paper. Draw in the three squares and try to reproduce the dissection (it's easier if you only use even numbers of squares for the two smaller sides. Find a rule for calculating the ratio of the two values x and y shown on the diagram.

x y


Proof C Start with a right angled triangle and duplicate it and rotate the images to create a square. What are the dimensions of the small square in the centre of the large square? What is the area of the triangle? What is the area of the large square? How can you link the solutions of these three questions to prove Pythagoras' theorem?

a

b c


Proof D Start with a right angled triangle and rotate it into the position shown.

Create a trapezium by inserting an extra straight line (see diagram).

Find the area of the trapezium in two different ways.

Use these two ways to create an equation that proves Pythagoras' theorem.

a

b

c

b

a c


Proof E How does this sequence of diagrams prove Pythagoras' theorem? Open the file Pythagoras to see an interactive version of this dissection/construction


Solutions Proof A

Start with two squares with lengths a and b The total area = a² + b² Construct two similar triangles with shortest sides a and b Rotate the two triangles through 90° to form a new square. The side of the new square is equal to the hypotenuse of the triangle with shortest sides a and b. Therefore a² + b² = c² where c is the hypotenuse of the right angled triangle constructed in step 2 Note that adding labels to points and lengths will clarify this proof and that all pupils should be encouraged to do so. The challenge in this proof is to write a description of the proof that is rigorous.


Proof B

The three shown angles are equal (this must be shown by the student). Therefore:

2 2

2

2

2 22

2

2 2

From the diagram:

and

From the diagram

p bq a

b bp q y

bb

b ay

ab bb y

a y ba by

a b b ax b y b

y a bx b a

=

= = −

=−

= −

= −+

=

+ −= − = − =

+=

−

This is a very difficult proof to discover - the following diagram could be given as a clue:

x y

p

q

b

a

b

a


Proof C

The proof using this construction is similar to the "cut-out" proof completed by the students in the introduction. The area of the large square is given by c² The small square by (a - b)² Each triangle by ab/2 Four triangles by 2ab Therefore, the large square is also given by (a - b)²+ 2ab c² = (a - b)²+ 2ab c² = a² - 2ab + b²+ 2ab c² = a² + b²

Proof D

The area of the trapezium is given by: (a+b)(a+b)/2 = (a+b)²/2 The area of the triangles are ab/2 and c²/2 So the total area of the trapezium is also given by ab+c²/2 (a+b)²/2 = ab+c²/2 (a+b)² = 2ab+c² a² + 2ab + b² = c² + 2ab a² + b² = c²


Proof E Construct squares on each side of the triangle Construct RS and PQ parallel to ZX and YZ respectively Shear the squares A and B to the point T along PQ and RS respectively to form the parallelograms A' and B' The triangles XYZ and ZTV are similar therefore the base of the parallelograms A' and B' are equal to the length YX (z) The areas of A' and B' are therefore equal to rectangles A'' and B'' respectively. Therefore Areas A +B = C and x² + y² = z²

The important step of this proof is to realise that XYZ and ZTV are similar. To make the proof completely rigorous then this similarity also needs to be proved.


Take Away Puzzle 3 Congruent Cuts

Cut this shape into two congruent pieces. Congruent means the same size and shape. The cut is not a straight line. Now cut the shape into three congruent pieces. And finally, into four congruent pieces. Cut these pieces into 2 congruent pieces each

How would you cut it into a 100 congruent pieces?


Take Away Puzzle 3 Solutions Congruent Cuts

Two congruent pieces. Three congruent pieces. Four congruent pieces.

How would you cut it into a 100 congruent pieces? Once the 4 piece problem is solved this becomes obvious.


Notes

Gifted and Talented Mathematics for Key Stage 2 · Use a graph plotter like Geogebra to discover...

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