JewellMaths Club

The Maths Club is designed to inspire, stretch and challenge budding mathematicians and is open to all from Years 5 and 6. Whilst each session contains material to engage all pupils and provide extra practice of mathematical skills, the primary purpose is to encourage the higher skills of problem solving and genuine mathematical thinking. Many of the activities are investigations in which the pupils are encouraged to play first, and then develop a systematic approach, to notice and articulate patterns, to use their hypotheses to make predictions, to check their prediction and to explain why their hypothesis is correct (proof).

Success depends on using a variety of engaging activities within the same session, and having questions to pose to develop the investigation at a number of different levels, and the flexibility to move the investigation into a direction suggested by the children. At the end of the session, each child should have a record of their work in which they summarise their findings.

First half termEach half term is based around a theme with a progression of activities. The first seven weeks has been investigating rectangles, with number work based on multiplication and “tables practice”. Although the theme lasts for seven weeks, there is a good deal of variety within the theme so that each week is not the same. Each session is timed to run just over an hour with at least three things to do each week but the timings of the individual activities can be quite flexible if the children are engaged, or ready to move on.

Week One – rectangles and squares

Week Two Squares and square numbers

Week Three Area and Perimeter

Week Four Patterns on a tables square

Week Five Investigating perimeter of L and T shapes

Week Six Golden rectangle and drawing a spiral

Week Seven Cutting a hole

Week One – rectangles and squares

Aims: To introduce loop game for tables practice To review the concept of area and how to calculate the area of a rectangle To explore different rectangles with a given area To discuss which rectangles are essentially the same and consider whether a square is a

rectangle from the definition of a rectangle

Resources Loop game cards (90 card tables loop game)and a stopwatch nrich.maths.org activity “Torn shapes” 1cm squared paper, pencils and rulers

Outline Explain the loop game and how to play. Record the time to complete. Repeat if wanted (20

mins) Discuss what area means and how to find it for a rectangle (10 mins) Use nrich activity “Torn shapes” (20mins) Investigation (15mins) Game to finish (10mins)

InvestigationAsk the children to draw rectangles with area of 36cm2. Give no further guidance for the first few minutes. Be ready with the following questions

How many different ones have you got? Have you thought of any that won’t fit on the paper? Sketch it instead Record your answers in a sensible order (systematic thinking) Have you got them all? (Does the question make sense in this context?) Is the 4x9 rectangle the same as the 9x4? Can you think of one no-one else will have? (what assumptions have you made eg integer

values)o Can you have 8x… ? o Use fractions and decimals if you can

Is the 6x6 rectangle a rectangle at all? Does it fit the properties of a rectangle?o Does it have 4 sides?o Is the top parallel to the bottom?o Is the top equal in length to the bottom?o Are the sides parallel?o Are the sides equal?o Does it have 90⁰ angles?o Are the diagonals the same length?o THEN IT IS A RECTANGLE!

More open ended – what other shapes can you draw with an area of 36cm2? (No rules)

ReflectionThe first club is important in setting the tone of the club. The game establishes the etiquette for future games so it is best to take your time explaining the role of listening, of being prepared, or collaborating and helping others with their cards. There is a place for asking questions of the children to get them to think about the properties of multiplication, but is better left until the game is used again another week.

The nrich activity encourages problem solving and has a good progression of difficulty.

The investigation is a relatively simple one and begins with a very accessible activity for all children. The children may be used to closed questions and this is an open-ended one for which there are many correct answers. They will most likely assume that integer values for length are to be used and very quickly find 4x9 and 6x6. Some may need prompting to find 2x18 and 3x12, and may exclude 1x36 as it is too big to be drawn full size. It is interesting to observe whether the children adopt a systematic strategy to find all the factor pairs of 36. They need to be encouraged to record their answers in a systematic way.

The question of whether a square is a rectangle will come from the children themselves and should be used as a basis for discussion. The strategy of determining whether it is a rectangle is one that uses the profoundly mathematical approach of using the definitions to make a decision.

Use your judgement in the time available to decide whether to go into non-integer values as there may be enough discussion already.

It is always worth having open-ended questions like draw other shapes with the same area as it reinforces the concept of area as the number if squares inside the shape and leads to the method of splitting up compound shapes to calculate area. It is also accessible to the less able children who may have lost a little confidence while higher skills were being used.

Repeating the loop game at the end is a good note to finish, especially as the end of the session creates urgency in the game.

Week Two Squares and square numbers

Aims: To practice loop game and beat time from previous week To explore the square numbers and where they are to be found on a tables square To investigate the link between the nxn square and the (n-1)(n+1) rectangle To consider the question will the pattern always be true and how do you know? To use the pattern to help with recall of times tables

Resources Loop game cards (90 card tables loop game) 1cm squared paper, pencils and rulers Printed 10x10 tables square

Outline Play the times tables loop game – think how we can speed up from last week (15mins) Investigate the area of the square and the nearest rectangle (20mins) Colouring the square numbers and their immediate neighbours diagonally closest to the

square(10mins) Look at the differences between successive square numbers and the pattern of squares

which gives rise to it. (20mins)

InvestigationAsk the children to draw 6x6 square and write down its area inside it. Next to it draw the 5x7 rectangle and notice the answer (is one less). Repeat for squares of other sizes. Summarize what you notice predict area of a new rectangle before it is drawn. Try to explain why it works. Be ready with the following questions

Which is bigger in area, the 6x6 square or the rectangle? By how much? If I draw another square, how do I find the nearest rectangle? What if you start with 1x1 square? If I start with 15x15 square, the area is 225cm2. What rectangle would go with it and what

do you predict the area to be. Check by multiplying Do you think this will ALWAYS happen? How can you explain why it works? (Think about

removing a line of squares from the side of your square and placing them on top to make an extra layer. Notice you have one left over.)

Follow-up activities Colour the square numbers on the diagonal of the tables square.

o I had a child who wanted to look at the other diagonal – it is much more of a challenge to discern a pattern there)

Find the 6x6 and the notice location of the 5x7 and 7x5 diagonally touching it. Draw arrows linking them together and shade these in a different colour to make three diagonal stripes.

3536

35 Look at the sequence of square numbers and the differences. Write down the pattern as

clearly as you can. Use these to continue the pattern and check by multiplication. 1 4 9 16 25 (25+11)=6x6… +3 +5 +7 +9 +11

Think how an L shape containing an odd number of squares can be wrapped round each square to make the next one.

ReflectionThe repeat of the loop game further establishes the members of the club as “insiders” as they already know how to play. The aim is to beat the time from the previous week and reinforce the etiquette of the game.

The investigation is very easily accessible to all the children, and for a few of them, finding other squares and the correct “nearest rectangle” is a sufficient challenge. By about the third one, children instinctively want to generalise which is one of the skills the investigation hopes to encourage. It is worth channelling this into a written statement “I think that….” and drawing another one to check for further evidence. The absence of a counterexample will appear to provide proof to children some will be able to consider the possibility that more proof is required to be sure.

The relationship between n2 and (n-1)(n+1) is easy to prove algebraically but is more of a challenge to children who do not yet have the algebra skills to expand brackets. Removing the final column and replacing them with loose squares that can be stacked on top with one left over is quite graphic and can be done easily with a sketch on a whiteboard.

Spending time looking for patterns in a tables square can be helpful to children firstly in familiarising themselves with the answers but also in building ways of remembering tables. Children will sometimes feel that the answer to 8x8 is 60-something, the fact that it is one bigger than the other 60-something answer will help to distinguish between 64 and 63.

Finding the pattern that successive odd numbers are added to build the square numbers in easier for the children to see than articulate. It could be used to ask the “What if….”questions to develop other sequences.

It is often good to return to a game to finish on a high.

Week Three Area and Perimeter

Aims: To think about place value through a game To investigate which rectangle with area of 36cm2 has the smallest perimeter To encourage systematic thinking, to consider proof and thinking beyond that

which can be drawn

Resources Set of cards with the numbers 0 to 10 for each child (place value cards) with the option to

remove the decimal point cards) 1cm squared paper, pencils and rulers

Outline Place value game building three numbers to add together making smallest/largest total

(20mins) Investigation into rectangles with constant area to find smallest perimeter (30mins) To push the investigation into non-integer values for some of the children (within the 30

mins for previous activity) End with familiar tables game (15mins)

GameAsk the children to make each number without knowledge of the other numbers nor the target of highest total – first a number more than 500, the second a three figure number divisible by 5, the third to use all the other digits. Add the numbers and ask for totals, and the biggest is the winner. Play again, knowing what’s coming. Then allow the children to swap digits round on the table to get the largest total.

Repeat for smallest total with an even number, a number divisible by 3 and the rest.

InvestigationRefer back to the previous week’s study into rectangles with area 36cm2. For each rectangle calculate or measure the perimeter. Look for the smallest. After the children have explored, lead class discussion on how to collect the information in a systematic way. Tabulate the answers

Measurements Perimeter2x183x12etc

Be ready with the following questions

What is the smallest you have found? Have you got all the rectangles? Are you sure yours is the smallest possible? Is 2x18 rectangle the same shape as 18x2?

Move the discussion on to consider the longest perimeter. Ask

What is the longest you have found? Are there any longer ones? How could you make the answers bigger and bigger? Is there a limit to the answers you can draw or imagine?

Write a sentence to sum up what you have discovered about the shortest perimeter and another for the longest.

Possible extension – what happens to your answers if your shape does not have to be a rectangle?

ReflectionThe game is a good way to think about place value and divisibility rules. It is also an insight into how the absence of information early on changes who you want to use the digits. The largest total comes from using 9 in the Thousands, 8, 7 and 6 in the Hundreds, 5, 4 and 3 in the Tens, and 2, 1 and 0 in the units. The total is then 11223 which the children can obtain from several different sets of numbers. There is an interesting study into how many different permutations give this same answer.

The investigation needs the children to be clear about area and perimeter, and the units in which they are measured. They accept quite happily now that the square should be included and many are not surprised that it gives the minimum perimeter. It becomes clear when the table goes from 4x9 to 6x6 and on to 9x4 that the set of answers for perimeter will repeat and the symmetry of the pattern can be used as proof that the square is indeed the minimum. (Something for the adults is how calculus can be used to prove it formally)

The longest perimeter can be tackled at a range of levels from “the longest I have drawn is 2x18cm”, the longest possible with whole numbers is 1x36” to “there is no limit to the perimeter that is possible if the height of the rectangle can be decreased to 0.1, 0.01, 0.001cm ….) and one or two may start to talk about infinity!

The possibility of moving into other shapes gives an extension task that is useful if there is time to be used at the end of the session without taking the discussion beyond an appropriate level for any of the children.

Ending with the tables game brings all the children back on task if any have zoned out during the higher level discussion.

Week Four Patterns on a tables square

Aims: To be aware where odd numbers occur in the tables square To find answers on the tables square and be aware of the pattern they form To consider the special nature of square numbers in regards to factor pairs To consider prime numbers To think about checking strategies

Resources 10x10 tables square sheet (where are the odd numbers) 24x24 tables grid (tables square to 24)

Outline Reflect on the loop game played in previous weeks (3mins) Play loop game (15mins) Odd numbers investigation (20mins) Pattern spotting on big grid (20mins) Return to the loop game (10mins)

Loop game reflectionDistribute the cards and ask with the following questions

Which numbers come up many times (have lots of factor pairs) Which numbers occur only once? Why? Do you know which tables you are waiting for, having seen the answers you have? What slows the game down?

Play the game!

InvestigationPART ONE: Where are the odd numbers on a tables-square? Colour them and describe the pattern in words. Complete the sheet and share answers in a class discussion, at the end of which everyone should have written answers.

Why does odd x odd give odd when all other combinations give even?

Extension questions:

What fraction of the square is shaded? What percentage is this? What percentage of the square contains a number bigger than 50? (Should 50 be included?) Is there a number above which lie half the answers? (48% bigger than 24, 52% bigger than

23, so does 23.5 make sense as an answer?

PART TWO: What patterns can be made by shading a given number every time it occurs?

Begin with 24 and use the same colour every time it occurs.

How many times does it appear? Would there be any more if the table-square was bigger? Describe the pattern it makes Use the symmetry of the square to check you have not missed any

Repeat with the number 48 and reuse the questions

Does anything different happen when 144 is used?

Why is this the only one so far to give an odd number of times shaded? Write a hypothesis for square numbers. Choose a square number of your own and check your hypothesis. Can you explain why it works every time?

Are there numbers which occur only twice, no matter how big the tables-square is made?

EXTENSION: Extend the search for 24 to include non-integer values – Some children may build towards the reciprocal graph xy=24 and its properties.

ReflectionThis week’s activity has a different feel to the previous weeks’ and is designed to be more accessible and to develop facility with recall of tables.

Something for adults to think about is an algebraic proof that odd x odd is odd and all other combinations give even numbers (HINT: use 2n for an even number and 2n+1 for odd)

It was surprising how few children had ever noticed that pattern.

24 and 48 can both be found 8 times on the square given and there is a possible avenue here in predicting that all numbers are there 8 times (it’s not true!) In fact, 48 should really be found 10 times but 48x1 and 1x48 are not visible.

144 can be seen 7 times but has 13 appearances it total if the square was big enough for 1x144 to be seen. The fact that 12x12 is not different when the digits are reversed is the reason it appears an odd number of times and is the case for all square numbers.

The method of looking for primes on this square gives all the primes up to 2x29=58 which is not be seen on this square but is clearly not prime. 1 is considered to be a square number (odd number of appearances) and not prime, a prime being defined as a number with exactly two factors. 1 needs to be excluded so that the prime factorisation of each number is unique.

Week Five Investigating perimeter of L and T shapes

Aims: To explore shapes that can be made by removing corners from a square To investigate the perimeter of these shapes To encourage systematic thinking, pattern spotting, prediction and check Using a pattern to form a checking strategy Some children may think about general proof To think about strategy is a simple game

Resources 1cm squared paper Either sets of 20 counters, or small whiteboards and pens

Outline Investigation in three groups (could be grouped by ability) (45mins) Bachet’s game (20mins)

InvestigationBegin by drawing a 10x10 cm square. Remove 1, 2 or 4 corners to leave L, T of cross shape – no need to be symmetric, but is has to use all the height and width of the original square. For each shape find the perimeter. Work as a team to produce as many different ones as you can in the time, you should have someone in your team to lead the project.

Be ready with the following questions

Has your shape been drawn correctly following the instructions? Have you written drawn a selection of shapes? What do you notice about the area? What do you notice about the perimeter? Are you sure about your answers? Do you think this will always be the case for any shape in the set?

You could use the extension questions

Have you got all the possible shapes? How can you describe to the others in your team which shapes have already been drawn

and which are still needed? What notation could you use for it? How many different shapes are possible? Why does the pattern work?

Strategy gameThe game, sometimes known as Bachet’s game, is to be played in pairs. Each team needs 20 counters (or draw them on whiteboards). Take it in turns to remove 1, 2 or 3 counters and the winner is the one who takes the last counter. At the end of one game, move the children round so winners play winners, and the ones who haven’t won yet to play each other.

Be ready with the following questions

Is there a way to win every time? Do you want to go first, or let your opponent go first? At what point do you know you have lost? Keep playing at home and find a way to win. How does this compare with the game 21 which you might have played in school?

We can play again next week and find out the secret to winning!

ReflectionThe investigation encourages the children to work together and produce a set of shapes that meet the requirements. The children may begin with “Am I allowed to….” questions to establish exactly what they have to do. The children very quickly nominate who is to lead the team.

Where errors occur in the measuring, the pattern takes a while to emerge, but once they notice that the answer for perimeter is usually 40cm, the children will go back and check examples that seem to give other values. The children quickly realised there was no useful pattern in the areas and concentrated on perimeter.

Removing a corner from the square does not change the perimeter as the two edges that formed part of the perimeter of the original square have been replaced with the other two edges with the same lengths. No group would be able to produce an exhaustive list of shapes so it is insufficient to be sure that the perimeter is always 40cm based only on the evidence of the shapes they have.

The question of how to describe the shape proved an interesting one. I had envisaged they would describe the size and shape of the square(s) to be removed, which gives a handle on the problem of how many different shapes are possible – for the L shape, for example the rectangle could be 1, 2 …9 cm wide with any of 1, 2 …9 cm high, giving 81 possible rectangles and 81 possible L shapes. My group established a notation that gave the height of the top of the T shape and the width of the stem. They realised this did not give the exact shape as the stem could be drawn in different ways.

Bachet’s game is lost by the player who is faced with 4 counters on the table, as however many counters they take, their opponent takes the last one. The strategy is to leave four for your opponent, which can always be achieved if you leave 8 the previous time etc. With 20 counters to begin, allow your opponent to go first and make sure you take four minus the number of counters you opponent has already taken. If a child claims to know how to win and agrees to go second, they do not have the strategy!

I left this unfinished at the end of the session and returned to it at the start of the next one.

Week Six Golden rectangle and drawing a spiral

Aims: To articulate the strategy to win Bachet’s game and use it to improve strategy for

the game 21 To discover the special nature of the golden rectangle To use the golden rectangle to draw a spiral

Resources Whiteboards and pens Worksheet (golden rectangles) Calculators Scissors, rulers and compasses

Outline Feedback on Bachet’s game from last week and ask the ones who have grasped it how to

win every time (10mins) Play 21 (10mins) Investigate ratio of sides in golden rectangles (20mins) Use the printed sheet and compasses to draw the golden spiral(20mins) Look at the results of the Google search for images of the Golden rectangle and golden spiral

(5mins)

Game sectionBe ready with the following questions

Who thinks they can win every time? Is it better to go first or second? Does anyone want to come to the front and play against me/someone else who thinks they

know how to win?

21 is played as a whole class. Begin with everyone standing and decide in which order the children take a turn. Each child continues the counting, saying either one, two or three numbers that follow on from the previous child. The child who says 21 is out, the child who makes a mistake is out too and they must sit down. The game repeats until there are only two left, at which point the strategy for Bachet’s game comes into play. The winner is the last one standing.

Investigation and drawingAsk the children to cut out the rectangle, to write it’s measurements on the table and use a calculator to find the answer to the division (ratio of the sides). Discuss how to cut off an exact square either by measuring and drawing or by folding. Cut off the square to continue to write on and measure the new smaller rectangle. Fill in the next line of the table and repeat. The children should write what they discover, or what they ought to have discovered.

Be ready with the following questions

How could you get these answers from the numbers and not measurements? What do you notice about the answers? How should you round off your answers? Do you think this happens for other rectangles?

Use the second copy of the rectangle with the smaller rectangles drawn in for the construction. The instructions for drawing are on the worksheet.

ReflectionTo begin with, 21 is an exercise in keeping your friends in the game, or manoeuvring them to 21 so you beat them. When only two children remain, the winner should always be the one going first, making sure they stop counting on a multiple of four every time. There is a fruitful discussion of where it went wrong for the loser to be had.

The investigation into the golden rectangle was much more structured than previous investigations. The ratio of the sides proved easy to find, knowing how to cut the rectangle to produce the next rectangle proved a little tricky for some, and there was a lot of variation in the level of accuracy with which children could cut off the square. Once they got started, they quickly progressed, but errors in measurement tended to compound as the rectangles got smaller and smaller. The sheet is actually drawn using the Fibonacci sequence, so even children who were very precise eventually cut to the point where the rectangle was no longer similar to the original. However, they were happy to accept that all the answer were, or should have been close to 1.6.

For adults and GCSE students, there is an avenue of investigation using algebra and the ratio of the

sides of the first two rectangles to calculate

Many of my club are girls and they loved the construction of the spiral. At the size drawn on the sheet, the first quarter circle is very big for classroom compasses so some struggled at the start but got on better once the radius of the arc was reduced. Some finished the very small arcs by hand and they were delighted when they were allowed to colour it in! I felt this was an appropriate way to finish as it isn’t a lesson, it’s a club!

There are thousands of images to be found on a very simple Google search of golden rectangles in art and architecture, and spirals to be found in nature. I decided not to pursue the connection with Fibonacci at this stage but to return to it later in the year in the half term looking at sequences.

Week Seven Cutting a hole

Aims: To practise tables in the loop game To tackle an open-ended task of creating their own game To delight in the solution of the question “how do you cut a very large hole in a very

small piece of paper?”

Resources Set of cards (loop game tricky tables) Sheet on different colours of paper (Loop game template) Rectangles – one page makes 2 (Climb through the hole in the rectangle) Scissors and sticky tape (just in case!)

Outline Couple of times through the loop game (5mins) The children invent their own questions and use them to create a resource that can be used

in the club (30mins) Use the frame left when the cards are cut up to illustrate that the large hole is not large

enough to climb through and demonstrate the cutting needed to get through the hole (5mins)

Allow the children to have a go using the sheet (15mins) Finish with a loop game using (5mins)

Activities Play the loop game a couple of times to beat the record – explain there are no “twicers” in

this version of the game, so they have to be ready Ask the children to come up with 20 questions of their own – the sums must be ones they

can do in their head, can be on any topic or a mixture of topics and every answer must be different. They need only write the question with its answer as the sheet will mix them up onto the cards. Ask them to put the first question at the very bottom of the second column, its answer at the top of the first. The next question and then its answer can be written directly underneath each other and the question and answer will be separated when the cards are cut out

Demonstrate how to cut the rectangle very quickly to amaze the children! Then give out the sheets and ask them to cut out the rectangle, fold it as shown and only cut where there are printed lines – the end strips must not be cut along the fold and all the others do need to be cut. Take care not to cut all the way through to the other side, although if they do, they can usually get enough of a hole with the sections that are left.

Be ready with the following questions

How big do you think you could get a hole to be? Would it be better to fold the paper the other way and have fewer long zigzags or lots of

shorter ones?

Use a set of the children’s loop game cards to finish

ReflectionThe game is a slimmed down loop game in which all the children had just one card to play and there were no repetitions of answers. This has the advantage that the game is much more succinct (about 2 minutes) and disadvantages in that an inattentive moment can slow the whole game down, and the children are not engaged so closely once their card has been played. However, they were very happy to design their own game which is a good open-ended task. This took a substantial proportion of the time.

The cutting of the cards once made provided the starting point for the next activity as the frame is too small to get through although a couple of children were willing to try! I cut a small piece of paper so that I could get through and the children worked very enthusiastically on theirs. It was worth having sticky tape available to cope with mistaken cuts, and tears! A couple of the girls wanted to go home and make really large loops to decorate their rooms and this proved the start of a good discussion about which way to fold the paper and whether it was better to have lost of short sections or fewer long ones. Many asked for another copy of the sheet to take home. Point them to my website!