Poly Plug & Tasks

Poly Plug can be used to create whole class investigations which explore the iceberg of many tasks from the Mathematics Task Centre Project. Doing so allows teachers to model how to work like a mathematician. Tasks which can be converted in this way are listed below. In several activities students can record their explorations with these sheets: Poly Plug Frame ...1 full size board Poly Plug Paper ...12 small size boards If a task has a Task Cameo, it has been linked below. The Task Cameo will also tell you if the task has a companion Maths300 lesson. Information about ordering Poly Plug can be found here.

Converting Task 62, 4 & 20 Blackbirds, to a whole class investigation. Kingsley Park Primary School, Victoria

Click the photo to enlarge it. Evidence shows that all students can learn to work like mathematicians in: mixed ability inclusive multiple intelligence classrooms. Through professional development teachers become aware of Principles which guide the selection and use of tasks. One of these is that a task has three lives. The three lives are: as a small group investigation, usually two students as a whole class investigation as a personal investigation guided by a teacher-prepared sheet Teachers also learn that success happens for students in a Working Mathematically Curriculum which balances: the invitation to work like a mathematician, as when choosing a task modelling how a mathematician works, as with whole class investigations toolbox lessons that develop and practise the skills of a mathematician - something which seemed to be the total purpose of traditional curriculum All three aspects are important. Finding the balance is the essence of building a curriculum through which students, at all levels, learn that they can do mathematics because they can work like a mathematician.

Task List Task 2, Cars In A Garage Task 5, Make A Snake Task 6, Counter Escape Task 9, Row Points Task 11, Lining Up Task 13, The Frog Pond Task 19, Cookie Count Task 21, Tactical Task 22, Make Four Game Task 23, Two Colours Game Task 27, Can Stack Task 28, Plate Triangles Task 34, Dice Differences Task 38, The Mushroom Hunt Task 45, Eric The Sheep Task 49, Take A Chance Task 51, Staircase Task 52, Which Floor Task 53, Have A Hexagon Task 59, 13 Away Task 60, Back To Back Building Task 62, 4 & 20 Blackbirds Task 82, Snail Trail Task 94, Crossing The Desert Task 95, Reflections Task 106, Crossing The River 2 Task 111, Square Numbers Task 114, Where Is The Rectangle? Task 117, 12 Counters Task 120, Nim Task 123, Bob's Buttons Task 124, Human Moves Monster Task 130, Protons & Anti-Protons Task 131, Walk The Plank Task 132, Red To Blue Task 139, Squound Task 140, Time For Tiling Task 145, Land Of ET Task 154, 4 Arm Shapes Task 173, Crossing The River 1 Task 182, Jumping Kangaroos Task 201, Rectangle Fractions Task 211, Soft Drink Crates Task 212, Monkeys & Bananas Task 235, Tables For 25 Task 236, Star Numbers There are more Poly Plug activities to be found in the Free Tour section of Calculating Changes Click this logo.

Task 2: Cars In A Garage Equipment One red and two yellow/blue plugs per person Problem This task relates to Professor Morris Puzzles 27 & 28. The problem is that three cars of different colours park in three adjacent garages. How many ways can this be done? How do you know when you have found them all?

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Procedure

1. Use masking tape or an alternative to mark out three garages large enough to take a child.
2. Use large pictures of three different coloured cars. It is probably less distracting to use the same car coloured in three ways, and this can easily be produced with computer clip art and a colour printer. Choose three students to 'become' the cars.
3. Explore some of the ways they could 'park' and ask the class how these could be recorded.
4. Ask for predictions of how many ways there would be altogether. Discuss.
5. Ask the students to quickly draw three garages in a line. Provide the three plugs above. Students now have a red 'car', a blue 'car' and a yellow 'car'.
   Note: It is preferable for each student (or pair of students) to have three (and more) small model cars to use, but this may not be practical, thus the Poly Plug alternative.
6. Students explore and record the possibilities.
7. Class discussion to record an agreed result.
8. What do you think would be the number of ways if there were four cars and four garages?
9. Develop a class wall story report on the problem as a model and ask students to complete their own version of it.

Task 5: Make A Snake Equipment Yellow/Blue Poly Plug sets for each student Problem When Mungo the Maths Snake is born her body has only one yellow ring (plug). Birth (Season 0)

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Each season Mungo grows longer, and each season each yellow ring is replaced by a yellow/blue/yellow combination.

Season 1

This replacement (or substitution) pattern continues every season after that. Blue rings are not changed. They are just 'pushed along' as each yellow ring grows.

Seasons 0, 1 and 2 look like this:

Investigate Mungo's ring patterns as she grows. Can you predict blue and yellow numbers? Can you predict how many rings after 10 seasons?

Task 6: Counter Escape Equipment Three plugs per student - one uses yellow, the other uses blue One dice per pair Three 'cells' labelled A, B & C in which to place the plugs - easily sketched on paper. Problem The plugs are placed in the cells in any way, but the objective is for each player to place their three so that they are first to have them all removed under the following rules: Roll 1: Remove one of your plugs from Cell A. Roll 2 or 3: Remove one of your plugs from Cell B. Roll 4, 5 or 6: Remove one of your plugs from Cell C.

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Procedure

1. Pairs try the problem a few times to get a sense of what happens.
2. Class discussion to develop placement strategies.
3. Testing of strategies. One way is for each pair to play 25 games.
4. How much data is enough? How are the data sets from two strategies to be compared?
5. Discussion of the strategies most likely to be best. Do we have enough time to collect sufficient data to compare these likely strategies? How could we organise/arrange to collect the needed data? Following through on these ideas if possible.
6. What happens if we change the dice rules?

Task 9: Row Points Submitted by Charles Lovitt. I like this activity because it is an 'easy to start' game suitable for Years 2 - 10 which provides a context for: practising arithmetic applying the process of Working Mathematically Equipment One Poly Plug set for each student, or pair of students. Row Points Recording Paper.

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Procedure

Note
The NRICH web site has a neat, software-based, introduction to this task which they call Spot Thirteen.

Hold up what you have made. Hey they're nearly all different. Show your picture to three others near you.

Scoring

• 5 in a row - 5 points
• 4 in a row - 4 points
• 3 in a row - 3 points

I usually have to make the obvious comment that 'in a row' means side by side without gaps - and yes, diagonally does count - and no if it counts as a five it isn't also a four and a three.

Example This picture scores as shown: Horizontal 5 (Row 1) + 3 (Row 2) + 3 (Row 3) = 11 Vertical 5 (Col 1) + 3 (Col 2) + 3 (Col 3) = 11 Diagonally - falling left 3 (Diag 3) + 4 (Diag 4) = 7 Diagonally - falling right 3 (Leading diagonal) = 3 Total = 32 This generates lots of discussion along the lines of What did you get? and I record a few scores on the board with the names of the students. After there are half a dozen or so on the board I write What is the highest possible score? under the heading Challenge. That sets off another burst of investigation. It is going to take me too long to check everyone's so checking has to be your responsibility. Before a score goes on the board it must be checked by one other person. Don't forget to check that there are only 13 plugs turned over. As the investigation continues I add more focus questions: What is the lowest possible score? Are there any scores between the lowest and the highest which can't be made? What scores do symmetric designs get? There is no need to attempt to complete Row Points in one session - in fact you can't. I use it on a daily basis for a week or so for mental arithmetic practice. It depends on the students' interest level. We also keep a display board of results and there is always blank Poly Plug Paper available for the students to record their designs. The best score we have found so far is 40 points. Can anyone beat that?

David Pahl, Wangaratta Primary School, introduced a recording system like this. You can sketch up a similar one or use the one supplied above. David's class also used a neat strategy to try to find the highest score. They worked their way up using first 3 plugs, then 4, then 5... What is the maximum score with 3 plugs? What is the maximum score with 4 plugs? What is the maximum score with 5 plugs? and so on to 13. (You can get 6 points with 5 plugs!) These students also think the top score is 40. Can anyone do better?

Variations

1. For younger children the game can be played on a 4 x 4 board, with scoring for only 4 in a row or 3 in a row, by using the red board and taking out 4 rows of 4 plugs. The yellow plugs are then placed in the red playing board. Only 9 plugs are used. Seven holes will therefore be blank.

Task 11: Lining Up Equipment One Poly Plug set per pair Problem The class is lining up in one line. You are told how many you are from each end. Work out how many there are in the line.

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Procedure

1. Explore the problem using the children and examples such as 6th, 10th or 14th from each end.
2. Students continue exploring using a yellow or blue plug to represent themselves and red plugs for 'the others'. The objective is to find at least two ways to explain how to work out the number of students in the line.
3. Can you work out the number in the line if you are a different number from each end?
4. Can you work out the number in the line if you are told the fraction of the line that is represented from one end to you?

Task 13: The Frog Pond Equipment One Poly Plug set per pair One pond each - easily drawn on paper One dice per pair marked 1IN, 2IN, 3IN, 1OUT, 2OUT, 3OUT - stickers on wooden cubes can work. An alternative is to use a dice code, ie: 1, 2, 3 mean 1IN, 2IN, 3IN and 4, 5, 6 mean 1OUT, 2OUT, 3OUT

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Problem

Procedure

1. All pairs begin with say 5 plugs in the 'ponds' to represent the frogs.
2. Take turns to roll the dice and put frogs in and out accordingly. The winner in each pair is the first to have an empty pond.
3. Students need to keep a record of their number of rolls.
4. Collect class data on the number of rolls to finish the game.
5. Pose the problem: Would the game be more fun if you started with a different number of frogs in the pond?
6. Pairs investigate alternatives with a view to preparing a television commercial lauding the advantages of their chosen length of game.

Task 19: Cookie Count Equipment One red Poly Plug board each One paper plate per pair Problem Given the number of cookies on a plate and the number of people present to share them, how many cookies will each person receive?

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Procedure

1. With the class gathered in a circle, or around a central table, present them with a plate of red Poly Plug cookies. Something like 48 is a good number and easy to extract from two boards.
2. There are 48 cookies on this plate. There are three people at the party. (Choose three to act the part.) How many cookies would each person get if they were shared equally?
3. Ask the class to show/explain at least two ways to work this out. Record the answer and all the methods.
4. Put all the cookies back on the plate.
5. Before the cookies could be shared, one more person came to the party. Now there are four people. How many cookies would they each get?
6. Apply the various methods to work out the answer.
7. Put all the cookies back on the plate.
8. Before the cookies could be shared, one more person came to the party. Now there are five people. How many cookies would they each get?
9. This example introduces the situation of 'left-overs'. Discuss what could be done with these. Depending on the age and experience of the students, you might want to just deal with groups and left-overs. However, it is likely at all levels, that students will suggest interesting ways of sharing these few which will introduce fractions.

For this task, each time it is used the instructions are something like:

Make a big pile of cookies on your plate. Count how many there are. Decide how many are coming to your party. Share your cookies and record how many each person will get. Then change your number of people, share again and record. Now do it again.

Task 21: Tactical Equipment One Poly Plug set per pair Sheets of Poly Plug Paper for the teacher which have several blank boards on each sheet. Problem This task is a game and the problem is to decide the best strategy to win.

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Procedure

1. Prepare a red board so that it shows a 4 x 4 set of empty spaces. Fill the spaces with either yellow or blue plugs.
2. Place it in the middle of a student circle or gather the students at a central demonstration table.
3. Invite two students to play the game according to the following rules.
   • Players take turns to remove plugs.
   • Plugs can be removed from either one row or one column each time.
   • On each turn players can remove any number of plugs from a chosen row or column, provided the plugs are next to each other.
   • The person who has to take the last counter loses.
4. Invite pairs to play the game several times.
5. Hold a class discussion about winning strategies. Record suggestions.
6. Over time invite students to play the game and continue to refine the strategies.
7. As the students play, sometimes record partially played games on your Poly Plug Paper. Make a display, or offer them to students in other ways with the challenge:
   
   If it was your turn to move now,
   what would you do and why?

Task 22: Make Four Game (Discontinued)

Equipment

• One Poly Plug set per pair
• Sheets of Poly Plug Paper for the teacher which have several blank boards on each sheet.

Problem

Procedure

1. Prepare a red board so that it shows a 4 x 4 set of empty spaces. You will also need 4 yellow plugs and 4 blue plugs.
2. Place the board in the middle of a student circle or gather the students at a central demonstration table.
3. Invite two students to play the game according to the following rules. One player uses yellow and one uses blue.
   • Players take turns to put plugs into the board until all 8 plugs are placed in the sixteen gaps.
   • When all plugs are placed, take turns to move them one space at time parallel to the sides of the board.
   • The aim is to make four of your colour as a row, column or square.
4. Invite pairs to play the game several times.
5. Hold a class discussion about winning strategies. Record suggestions.
6. Over time invite students to play the game and continue to refine the strategies.
7. As the students play, sometimes record partially played games on your Poly Plug Paper. Make a display, or offer them to students in other ways with the challenge:

   If it was your turn to move now, what would you do and why?

Task 23: Two Colours Game Equipment One Poly Plug set per pair Problem This task is a game and the problem is to explore strategies which force the other player to lose.

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Procedure

1. Prepare a red board so that it shows a 5 x 5 set of empty spaces. You will also need all the yellow/blue plugs.
2. Place the board in the middle of a student circle or gather the students at a central demonstration table.
3. Invite two students to play the game according to the following rules. One player uses yellow and one uses blue.
   • Players take turns to put plugs into the board one at a time.
   • Plugs next to each other horizontally or vertically must be different colours.
   • The first player who cannot place a plug is the loser.
4. Invite pairs to play the game several times.
5. Hold a class discussion about strategies that could force the opponent to lose. Suggest that it is better to consider you are playing against the 'world's best' because then any winning strategy would work on any player, and would not depend on the other player making a mistake. Record suggestions.
6. Over time invite students to play the game and continue to refine the strategies.
7. Make a class display of the developing insights about the game. Perhaps organise a premiership play off between interested class members.

Task 27: Can Stack Equipment One Poly Plug set per pair Problem Given a starting placement of cans stacked in a supermarket aisle predict the number of cans in the next layer and the number of cans needed altogether to make the stack.

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Procedure

1. You may need to allow students to freely stack the yellow/blue counters first before focusing on the problem - a free play time.
2. Introduce the story shell of stacking cans in a supermarket display. All the cans have to be up the same way, so encourage using blue on the bottom. Suggest a starting stack such as:

   Top View

   Side View

3. Discuss how the next layer would be formed. There may be more than one way.
4. For an agreed continuation of the stacking pattern, invite students to work out the number of cans in the next layer and the total number of cans needed to make the whole stack so far.
5. Develop the problem further by asking students to work out these two numbers for the tenth layer. Discuss and record the various methods the students use to solve this problem.
6. Some may be willing to tackle the general question, ie:

   If I tell you any number of layers, can you tell me how many cans in the layer and how many cans altogether to make a stack with that number of layers?

Task 28: Plate Triangles Equipment Two Poly Plug sets for each group of 4 One set of counters numbered 1-36 for each group of 4

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Problem Arrange the plugs to make an equilateral triangle. How many plugs are added to make each new row? If I tell you any number of rows can you tell me how many plugs you need in total? If I tell you the number of plugs I have can you tell me the size of the largest equilateral triangle I could make? Later, place the numbered counters on the plugs, in order from the top, and explore number patterns.

Procedure

1. Begin the lesson as suggested in the Task Cameo using paper plates on the floor to make a large model.
2. As the triangle builds, explore the connection between the Natural Numbers and the Triangle Numbers.
3. Ask the students (in groups of 4) to reconstruct the triangle on their tables using the plugs.
4. Suggest that the real challenge is to find the total number of plugs for any size triangle. Allow time to explore this problem then share the various approaches, successful or not.
5. Suggest that Greek mathematicians saw mathematics as pictures rather than numerals, hence sets of numbers like Square and Triangle. Remind students that Greek mathematicians taught us that it is easy to count numbers arranged in equal rows which make a rectangle - Rectangle Numbers. Challenge them to see if they can picture a solution to the triangle counting problem using the plugs. Two possible approaches are shown below.

Making A Parallelogram This will work for any size triangle. So twice the total is the longest side times the longest side plus 1. If N = the longest side, then 2T = N x (N + 1) T = [N x (N + 1)]/2

Making A Rhombus This will work for any size triangle. So twice the total is the square of the longest side plus the longest side. If N = the longest side, then 2T = N2 + N T = [N2 + N]/2

One More

This approach was suggested by Julie Riga, Maryborough Education Centre. Look at the triangle one size less and reflect it across the diagonal. The two triangles will form a parallelogram which is (N - 1) rows of N. This will work for any size triangle. So the total is half this parallelogram plus the diagonal. If N = the longest side, then T = (N - 1)N/2 + N

Task 34: Dice Differences Equipment Six plugs and two dice per pair A rectangle marked into six sections which are numbered 0 to 5 Problem To find the best way to place the plugs so that all six are removed in the least number of rolls given that: To begin, any number of plugs may be placed in any section. A player may remove one plug from a section on each turn, if they have a plug in the section which is the difference of the two dice numbers.

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Procedure

1. Divide the whiteboard into sections marked 0 to 5. Write these numbers high enough to be seen above the students heads. Make the sections as wide as one child.
2. Select six students, explain the rules, and ask them to decide where to place themselves. Record the placement strategy as an ordered sextuplet. For example [1, 2, 1, 2, 0, 0] means:
   • 1 student in section 1
   • 2 students in section 2
   • 1 student in section 3
   • 2 students in section 4
   • 0 students in section 5
   • 0 students in section 6
3. Play the game using one larger set of dice that most of the class can see. It will help the students to analyse the probabilities later if the two dice are different colours.
4. As you play, call out the two numbers and ask the class to call back the difference. Also choose a student to be the recorder. On an easel (the whiteboard is being used remember) they record both the differences and a tally of the number of rolls.
5. Discuss what this experiment tells us about the placement strategy. "Not much." should be the response.
6. Ask each pair to play the game with the plugs, their own dice and this first placement strategy. Collect class data about the number of rolls to remove all six.
7. Discuss again what this data demonstrates about this placement strategy. Does this information mean that this is the best placement strategy?
8. Invite students to choose their own strategies and collect sufficient data to be confident about the average number of rolls for that strategy.
9. Over time, collect and display the students' data and keep reconsidering the initial question until the class agrees that the best placement strategy has been found.
10. What happens if we change the number of plugs?

Task 38: The Mushroom Hunt Equipment Two Poly Plug sets per four students Six paper plates, or margarine containers per four students Problem Six people go mushrooming. They have a basket each. When they return home each person counts the mushrooms in their basket. They realise that these six numbers are very special. If they total the six numbers the answer is 63. AND If they can combine the numbers in different ways they can make every number from 1 up to 63.

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What were the numbers of mushrooms in the baskets when the six people came home?

Procedure

1. Introduce the problem and using a sample set of equipment try a case that doesn't satisfy both conditions. Two sets of Poly Plug provides 100 plugs, so, as part of the demonstration ask:

   What is the largest number of plugs that I would have to press out of the boards?

2. Ask the students to explore the problem and keep a record of the arrangements they try and (if possible) why they chose each possible arrangement.
3. When the solution has been found, ask the solvers to try to explain how they arrived at it. Their notes should assist this explanation.
4. Suppose there was one less basket, but the structure of the problem was the same. What would be the new total number and what would be the basket numbers?
5. Suppose there was one more basket, but the structure of the problem was the same. What would be the new total number and what would be the basket numbers?

Task 45: Eric The Sheep Equipment One Poly Plug set for each pair of students One set of counters numbered from 1 - 25 Problem This well known problem is about a line of sheep waiting to be shorn. Eric is last in the line and too impatient to wait his turn, so every time the shearer takes a sheep from the head of the line, Eric jumps forward 2 sheep. The problem is:

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Given that you know how many sheep are in the line in front of Eric at the start, how many will be shorn before Eric gets to the head of the line?

Procedure

• The first step in a class approach to this problem is a kinaesthetic one. Use the children as sheep and try a few numbers of sheep in front of Eric.
• The second step is a consolidating illustration or two using the yellow/blue plugs in a place where the whole class can gather around. (Eric is the blue sheep and the others are yellow). It is important that every student understands the structure of the problem before proceeding.
• The third step is to make use of a previously prepared set of plastic counters numbered from 1 to 25 (say). Hand one counter at random to each pair and tell them the number on it is the number of sheep in front of Eric. Ask them to use yellow/blue Poly Plugs to solve the problem for that number.

Experimenting with the number of sheep in front of Eric
Thorne Grammar, UK, Year 8

• When the partners have solved that problem they bring you the counter and the actual 'sheep' shorn before Eric.

While they are busy, prepare a line of red boards with plugs in. As the children return with their counters and shorn sheep build a graphical representation of the class results as below by popping out plugs and placing the yellow shorn sheep in their pens.

When all results are in, gather the class and consider the graph. There will be errors, but there will be enough of the pattern for the students to suggest those data points which need to be checked. Once checked and corrected the pattern can be used to predict the number of sheep shorn before Eric reaches the head of the line for 50 or 100 or 1000 sheep in front of him to start with.

Then, there are all the associated what if investigations:

• What if the jumping rule changed?
• What if the number of shearers changed?

I have taught this lesson in this way from Year 1 to Year 9 and in every class the power of the Poly Plug graph, combined with the class discussion and revision of it, has given every student the confidence to investigate the 'what ifs', and hypothesise about the generalisation. In fact, in every class from Year 1 to Year 9 there has been at least one person who could explain what the "bunches of three numbers" in the graph has to do with the structure of the problem. I also find this approach a fruitful way to highlight the steps of Working Mathematically, ie: working in context to: gather data organise data seek and see patterns communicate and record make and test hypotheses Collecting Class Data Conway Elementary School South Carolina, USA, Year 4

Task 49: Take A Chance Equipment One Poly Plug set for each student One deck of cards - a large size deck works well Problem In this game the deck is shuffled and the top two cards are turned up as the 'ends'. Players risk 1, 2 or 3 counters (or pass) on whether the next card will fall between the ends. Players start with 10 red plugs out of their board.

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If they win they get back the plugs they risked and take the same number again from the board. If they lose they put the risked red plugs back in the board. The game ends when one pair has 20 plugs out of the board and wins, or one pair puts all their plugs back in the board and loses. The challenge is to decide the best strategy for risking counters on each play. NB: All cards are returned to the deck and it is reshuffled after each play. Procedure Teacher shuffles and places the ends. Groups decide what to risk and place their risked plugs in the centre of the table. Teacher turns the next card. (Class groans or cheers appropriately.) Pairs take plugs from, or put plugs into, their board. Repeat 1 - 4 until one pair wins or loses.

At this point each group must leave its current plug supply in the middle of the table so that class data can be collected as in the photograph. In this class there were 10 pairs. Each began the game with 10 red plugs. That's a class stake of 100 plugs. If the class had done well, they would have more than 100 plugs when the game finished.

This class has not done so well, but this begs the question 'Have we been making the best risk decisions?' Exploring the chances of a win for each possible 'ends' situation should lead the class to making better decisions. After each pause in the lesson to discuss strategy, play the game again to see if the class is doing better.

To explore a situation, consider, for example, 3 and 9 as the ends. This is a situation with 5 cards between (4, 5, 6, 7, 8) which could produce a win. But there are four of each in the pack, so 20 possible winning cards of the 50 cards left in the deck. A 2 out of 5 chance of winning. Should we risk any counters? Investigate all the possible 'betweens' in a similar way. Perhaps the investigations can be shared among the pairs of students.

Task 51: Staircase Equipment One Poly Plug set per pair Problem Calculate the number of plugs needed to build a staircase of any height.

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Procedure Ask students to build a staircase as shown as far as is possible with one red board. The challenge for today is: If I tell you any number for the highest step in the staircase, can you tell me how many plugs there would be altogether? This is not an easy task. Encourage visual ways of working. For example, in the photo above, the central column of plugs is three high the columns to its right are, respectively, one and two plugs higher the columns to its left are, respectively, one and two plugs lower shifting the excess plugs from the right to the left would make five columns, each three plugs high, so a total of 15.

Explore What happens if ...? questions such as: there are an even number of columns in the photo? the staircase rises two for each column it moves across? the staircases rises to the highest then comes back down in the same pattern, creating a double staircase?

Task 52: Which Floor Equipment Two Poly Plug sets per pair or group of 4 Problem We know the journey someone took in an elevator - how many floors up and how many floors down. We know the floor they came out of the elevator. The challenge is to work out the floor (or floors) where they began.

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Procedure This is a great activity to begin as a physical involvement situation outside, or in an appropriate inside space. The picture on the right is the table top version of the problem, but it can readily be adapted to a whole-body scale using a set of number cards from 0 - 9. A problem to begin the investigation is: Andrew gets into an elevator. He goes down 2 floors up 4 floor down 3 floors He gets out at the 2nd floor. Which floor did he enter the lift? Students will tackle this initial problem either by guess and check, or, perhaps with a little encouragement, by working backwards. One of the main thrusts of the problem is indeed to introduce, or refresh, working backwards as a problem solving strategy.

After appropriate practice with this level of problem - physically, on the table top with Poly Plug, and creating their own challenges for each other - the challenge level can be increased with questions like:

If Andrew does the same journey, but doesn't get out at Level 2, what other floors could he enter the lift?

• How many solutions are there?
• How do you know when you have found them all?

The next level of development comes from realising what can be changed in the problem and asking What happens if...?. The journey can be changed. The number of steps in the journey can be changed. The exit floor can be changed. The height of the building can be changed. ...and... What happens if there are twin elevators and Andrew and Angela enter at different floors, make different journeys but both step out at the same floor?

Task 53: Have A Hexagon Equipment Hexagon Board & Blank Board (see Task Cameo) Red Poly Plug board per pair Two dice per pair (preferably numeral rather than spot dice) Problem Roll two dice and multiply the numbers together. Place a red plug beside the product on the appropriate hexagon. Continue until one hexagon has a plug beside each of its sections. Which hexagon do you predict will be finished first? Which hexagon is actually more likely to be finished first? Rearrange the numbers on the hexagons so that they are equally likely (or nearly equally likely) to be finished first.

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Procedure

• Play the game about 20 times. This should be enough to show that the middle hexagon is more likely to be finished first.
• Investigate the possible ways each product on the hexagons can be made from the dice numbers.
• Make and test a hypothesis about how to rearrange the numbers so that the hexagons have a more equal chance of being finished first.

Task 59: 13 Away Equipment One Poly Plug board per pair - it doesn't matter which colour One calculator per pair Problem This task is a game with these rules. Each pair put 13 plugs in a pile. Players take turns to remove 1, 2 or 3 plugs from the pile. The loser is the person who takes the last plug. The problem is to find a winning strategy.

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Procedure

1. Introduce the game and invite students to play it a few times.
2. Discuss anything students have noticed about the game and suggest that you want their assistance to find a way to be sure of winning.
   • Do you really have to play the last move to know who wins?
   • If you can 'see' the result of the last move without doing it, can you see the result of the move one before the last without doing it?
3. Encourage further investigation, but explain that it is better to consider you are playing against the 'world's best' because then any winning strategy would work on any player, and would not depend on the other player making a mistake. Record suggestions.
4. Encourage testing various hypotheses about the game by playing it on a calculator. Begin with 13 on the screen and subtract 1, 2 or 3 at each turn. The person who has to make the calculator show zero is the loser.
5. Write individual reports on the game and its solution. Alternatively, consider a class 'wall story' report, or the possibility of making a power point presentation 'to explain to the parents at parents' night'.
6. How would you play the game if the person who took the last plug was the winner?
7. What if we change the game, eg: 21 plugs to start and being allowed to remove 1, 2, 3 or 4?

Task 60: Back To Back Building

This task is a 3d activity using cubes that connect in three dimensions. One student secretly makes a shape, the students sit back to back, and the second student has to rebuild the shape from the first student's instructions only. This can't be exactly reproduced with Poly Plug, but the same principle is involved in the activity Copy Cats stored in Calculating Changes. Also in Calculating Changes you will find that children from West Ulverstone Primary School invented the same activity for themselves and called it Poly Plug Pictures.

Task 62: 4 & 20 Blackbirds Equipment One Poly Plug red board per pair Blackbird Sheet - you need to print 12 sheets to get 24 birds Royal Garden Sheet - one sheet per pair Problem When the pie was open, the birds began to sing ... and the queen was delighted. Every day after that she would look out of her window into the courtyard garden where there were eight feeding platforms arranged like this:

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And every morning she would see two things:

• a total of 24 birds, and
• 9 birds along each side of three feeding platforms - left/right/top/bottom.

Procedure

• Arrange 8 tables or chairs like the feeding platforms.
• Hand out 24 blackbirds and ask the class to demonstrate a solution to the problem.
• It is likely that they will soon enough come across a solution with 3 birds on each platform.
• Well that's exactly how it looked to the queen, but the next morning the arrangement was different but the 24/9 was still the same.
• Suggest that each pair could use red birds (plugs) and a Royal Garden to look for another arrangement.
• Record the arrangements on the whiteboard as they develop.
• How many solutions are there?
• How do we know when we have found them all?
• Hint: Focus on a particular corner (say top left) and try every possible number in this corner.
• Could the same number of birds be arranged with a different same number along each side, eg: 24/8 or 24/5?
• What happens if the number of birds was different? Could they still be arranged 9 along each side?
• What happens if the king had built a triangle courtyard with 6 feeding platforms in total, one at each corner and one in the middle of each side? Could the blackbirds still be arranged 24/9?

Task 82: Snail Trail Equipment Three Poly Plug sets per group of 4 Problem A snail starts at the bottom of an 11 metre well and climbs 5 metres in one hour. Then it rests for one hour during which it slips back 3 metres. If this climbing resting pattern continues, how long will it take the snail to climb out of the well?

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Procedure This is a great activity to begin as a physical involvement situation outside, or in an appropriate inside space. The picture on the right is the table top version of the problem, but it can readily be adapted to a whole-body involvement by stepping out and marking the asphalt with chalk. A opening problem is as above: 11 metre well crawl up 5m in one hour slide back 3m while resting for one hour climbing and resting pattern continues When will the snail get out of the well? After exploring this problem in teams outside, return to the classroom so each team can set up the Poly Plug version on their table. Check the outside work with this model, then discuss the variables in the problem. What can be changed? height of the well up distance sliding back distance length of the time unit (eg: half hour units instead of one hour) whether the snail crawls up or down Students select one combination of alternatives and explore their own problem in groups. Facilitate a class discussion of the range of the results. Is there a way to tell if a chosen set of variables won't work?

Task 94: Crossing The Desert Equipment One Poly Plug set per pair Problem Two travellers set out on a 9 day journey across a desert to deliver a message and return with the answer. They can each only carry enough food for 12 days. How can the message be delivered and the answer returned?

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Procedure See the Task Cameo for a full explanation and a printable playing board. One red board supplies the 24 pieces of food that can be carried by the two travellers and they can then be made into piles moved across the desert by two students acting as the travellers. The students will soon realise that one must be the messenger and one is along to carry (and bury food). But on which days and how many portions? How many solutions are there? How do you know when you have found them all?

Task 95: Reflections Equipment One Poly Plug set per pair Problem If I give you a reflection line (mirror) and a part of a picture, can you complete the picture so it is symmetric? Is there more than one way? Procedure To model the equivalent of this task, the plugs of a Poly Plug could be thought of as the nails of a 5 x 5 geoboard. But there is no need to join the 'nails' with rubber bands to see a symmetric pattern.

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Start with an empty 5 x 5 red board and mark in a line of symmetry:

         or         

Player A plugs in blue (perhaps either side of the line of symmetry) and when finished, Player B has to plug blues into symmetric places.

Encourage students to record their creations on Poly Plug Paper. What happens if Player A inverts a plug in the line of symmetry? What happens if players use two board or four boards? These ideas of symmetry are reinforced and extended by a collection of activities on Exploring Symmetry with Poly Plug created by teachers from Mayfield, Rocherlea and Kingston Primary Schools for Calculating Changes.

Task 106: Crossing The River 2 Equipment One Poly Plug set per group of 4 Problem Three adults and three children have to cross a river in a row boat. Their situation is: The boat can only take two people. All the adults can row but only one of the children can row. At no stage is there allowed to be more children than adults.

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Procedure

Each pair can easily set up the problem on their table using blue plugs for the adults, yellow for the child rower and red for the other children. This is not a straightforward problem, so expect it to take some time, a few false solutions that don't obey Rule 3 and considerable discussion. Use the problem to highlight the work of a mathematician, in particular, the strategies of 'make a model' and 'try every possible case'.

Once all are confident with the solution, shift the emphasis to that part of a mathematician's work that involves recording, explaining to others and publishing. Students might record in their journal, instruct someone else in the solution while keeping their hands behind their back, or publish a poster, comic strip, PowerPoint or video for others to understand the solution. The video could perhaps be presented as a news report.

One solution and extensions to the lesson are described in the Task Cameo.

Task 111: Square Numbers Equipment One Poly Plug set for each student Problem Take all the plugs out of a red board. Start with one yellow plug in the corner. Build up a pattern of L-Shapes like this: Wherever there is a visual pattern, there is always a number pattern. What number patterns can you discover?

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Procedure

1. Introduce the problem and invite students to explore it.
2. Discuss what has been discovered and make a class recording.
3. For any of the discoveries ask: Can you check it another way?
4. Encourage students to work together to explore how the pattern grows to 10 Ls.
5. If I tell you any number of Ls what can you tell me about ... the yellow plugs? ... the blue plugs? ... the total of the plugs?

Task 114: Where Is The Rectangle? Equipment One yellow/blue board and Poly Plug Paper for each pair. Problem To identify the position of the four (or sometimes two) corner plugs of the hidden rectangle in the least number of guesses. Procedure Start with a completely blue board. Turn over any number of blue plugs to make equal rows in a board and show it to the students, eg:

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...or... ...or...

2. Point to the plug in the bottom left corner.
   If we say this plug is at zero this direction (indicate horizontal) and zero this direction (indicate vertical), what is the position of this plug (point to one of the yellow ones)?
3. Explore the positions of other plugs using this ordered pair system.
4. Introduce students to the two person game. The aim is to find the corner plugs, given the plugs are in equal rows.
   (You might introduce this by having one student hide to make the equal rows in the plug board. You work with an overhead transparency of the Poly Plug Paper to deduce the corners.)
   The rules of the game are:
   • Player A secretly makes equal rows of plugs and tells Player B the total number of plugs turned over.
   • Player B writes this total above the first picture on their Poly Plug Paper and guesses an ordered pair.
   • Player A tells whether this plug is on the edge, inside or outside the rectangle of equal rows.
   • Player B records this answer and continues guessing until the four (or in some cases two) corners can be identified by their co-ordinates.
   Note: Player A must not turn their board around once the equal rows have been made.

When students become proficient with the 5 x 5 board, use two boards to make 10 x 5 or 5 x 10, or four boards to make 10 x 10.

Note
The NRICH web site has a neat, software-based complement to this task which they call Poly Plug Rectangle. It offers several levels of challenge in the Teachers' Notes and video from classrooms.

Task 117: 12 Counters Equipment Twelve plugs per pair One board per pair divided into 12 sections numbered 1 to 12 - easily made by folding an A4 paper Two dice per pair Problem To find the best way to place the plugs so that all twelve are removed in the least number of rolls given that: To begin, any number of plugs may be placed in any section. A player may remove one plug from a section on each turn, if they have a plug in the section which is the sum of the two dice numbers.

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Procedure

1. Use a large board (say A3 size) showing 12 sections as two rows of six. Place the board on the floor with the students in a circle, or use a central table.
2. Start with the 12 plugs more or less randomly placed, but do include 12 and don't include 1. Explain the game (without commenting on your plug placement) by asking a few students to roll two dice and work out the sum. Remove the appropriate plug each time.
3. This game is our investigation for today. The challenge is to decide the best way to place the plugs so that they are all removed from the board in the least number of moves. ... Constance, could you suggest a way of placing the plugs to start us off.
4. Record the suggestion then, ask each pair to make a board and play the game with the plugs, their own dice and the placement strategy suggested by Constance. Collect class data about the number of rolls to remove all twelve.
5. Discuss what this data demonstrates about this placement strategy. Does this information mean that this is the best placement strategy?
6. Invite students to choose their own strategies and collect sufficient data to be confident about the average number of rolls for that strategy.
7. Over time, collect and display the students' data and keep reconsidering the initial question until the class agrees that the best placement strategy has been found.

Task 120: Nim Equipment One Poly Plug board per pair - it doesn't matter which colour Problem This task is a game with these rules. Arrange plugs like this to begin the game. Players take turns to remove one or more plugs from one row. This includes being allowed to remove the whole row. The loser is the person who takes the last plug. The problem is to find a winning strategies.

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Procedure

1. Introduce the game and invite students to play it a few times.
2. Discuss anything students have noticed about the game and suggest that you want their assistance to find a way to be sure of winning.
   Have there been any times when you knew you were going to win? How did you know?
3. Encourage further investigation with a focus on knowing when you are going to win (or lose).
4. It should start to become clear that you can win if you leave your opponent with an odd number of moves to the end, for example: The investigation now becomes a search for other positions to leave your opponent which will cause them to lose.
5. Make a classroom display of the winning strategies that students discover over time.

Task 123: Bob's Buttons Equipment One Poly Plug set per pair Problem The task tells us that Bob has a certain number of buttons to share. If he shares them between four people there are two left over. If he shares them between five people, there is one left over. The problem is to find the number of buttons. Procedure Investigate the original problem. There is more than one answer. Investigate the pattern in the answers. How does it relate to the original problem? Explore What happens if? questions, eg: What happens if we change the number of friends? What happens if we change the number of left overs?

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Task 124: Human Moves Monster Equipment One Poly Plug set per pair or per person Problem Nine monsters are placed in a given arrangement on a 5x5 grid. These are the blue plugs in the red board. The human is shown with a yellow plug. The human begins in any unoccupied gap and moves as a knight in chess, that is, one cell left/right or up/down combined with one move diagonally to the next gap after that. The diagonal move cannot make a 'sharp' angle. The human can jump over monsters as part of the move and it captures a monster when the end of any move is on a monster. The aim is to capture all the monsters in the least number of moves.

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Procedure

1. Set up the board like this: for the first challenge and this: for the second challenge.
   
2. In each case, use the human to capture all the monsters in the least number of moves.
3. Explore What happens if? questions, eg:
   • What happens if we change the arrangement of monsters again?
   • What happens if we change the capture rule?
   • What happens if we change the number of monsters?

Task 130: Protons & Anti-Protons I particularly like the model of integer arithmetic illustrated with this activity. It is true to pedagogical and mathematical principles: Pedagogical - proceed from the known to the unknown. Mathematical - the importance of the identity element in the number system. Equipment One yellow/blue board for each pair

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Problem

The problem is to discover how these particles behave in collections.

Procedure

First explore the many ways of making a collection with a particular value (say 3P where the blue side means proton and the yellow means anti-proton):

etc.

It is important to consolidate here and discuss the zero effect of a proton/anti-proton pair. It is then a natural step to use the materials to investigate situations such as:

It is the last of these which requires the fullest understanding of the idea that a proton/anti-proton pair is worth zero. In order to have two anti-protons to take away, they must first be added in as part of two zero collections:

Now 2A can be taken away and the result will be 5P.

Again, after consolidation, the concept of protons and anti-protons having opposite charge can be reintroduced. By precise analogy the four exercises listed above become:

The students have now essentially derived the rules for adding and subtracting integers from their own experiments. Further consolidation often means that rules are not necessary because the students 'picture the operations in their heads'. Teachers also frequently comment that it is surprising with this approach how those normally classified as poor students in the chalk/talk/text approach become quite confident with this approach; and, sometimes, those normally classified as good students in the chalk/talk/text approach become insecure.

Task 131: Walk The Plank Equipment One Poly Plug set per pair Two cube dice in two colours, eg: yellow and white. Problem The captain of a pirate ship is making you walk the plank. But he is a bit of a sportsman and is giving you a chance to be saved. It all depends on the rolls of the dice. The plank is 15 units long (15 red plugs). You start on the middle unit (the edge of the boat) using a yellow/blue plug. One dice tells you which direction to face - right is towards the boat and left is towards the sharks. The other dice tells you whether you will walk forwards or backwards and the number of steps to take. Yellow Dice: Even roll - face the sharks. Odd roll - face the boat White Dice: If you roll 1, 2, 3 walk forward 1 or 2 or 3 steps. If you roll 4, 5, 6 walk backward 1, 2, 3 steps. Sometimes you reach the safety of the boat. Sometimes you are eaten by the sharks. Either way the game ends. What do you think is the best plank length for a good game?

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Procedure

1. Play the game enough times to have a good idea of how many turns it takes to end.
2. Choose other plank lengths to explore and repeat the experiment.
3. Use the results of at least three different plank lengths to make a case for the best game.

Task 132: Red To Blue Equipment One yellow/blue board per pair Problem The task becomes Yellow To Blue when using Poly Plug. Starting with 4 yellow plugs the objective is to turn them over to be all blue in the least number of moves. The turning rule is: Turn over all except one on any move.

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Procedure

1. Pairs investigate the original problem.
2. When they are confident with that, ask: What happens if we change the number of plugs?
3. What happens if we change the turning over rule?

Task 139: Squound Equipment One red board per pair One piece of A4 paper Problem The playing board is displays a square and a circle that overlap. Given a total number of plugs and the numbers in the square and circle, find out how many are in the Squound, which is the name given to the overlap region.

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Procedure

1. Ask each pair to orient their A4 sheet to landscape and draw an approximate square as big as possible on the left end. Then overlap it with a circle as large as possible on the right end. Sketch what you want on the board or have an example prepared.
2. Explain which region is the Squound.
3. Ask each pair to select 12 red plugs and explain that for now, that is all they need.
4. So you will always know that we are dealing with 12 plugs. Now I am going to give you some clues and for each one the objective is to tell me the number of plugs in the Squound
5. Set a number of exercises such as 10 in the square and 9 in the circle, or 8 in the square and 4 in the circle. These can be given orally, or written on the board, or prepared as an investigation guide.
6. Explain that the first challenge is to discover and explain a method for always being able to find the Squound number.
7. Go on to explore other challenges as described in the Task Cameo, such as:
   • What happens if we chose a different number of plugs, say 3, and ask how many squound problems can be created?
   • What happens if we explore the number of squound problems that can be created for 0, 1, 2, 3, 4, 5, ... counters?

Task 140: Time For Tiling Equipment One Poly Plug set per pair Graph paper Problem Douglas is planning to build an outdoor play area for his grandchildren. It will have circular stepping stones arranged in equal rows with a diagonal pattern of contrasting stones. The spaces will be filled with white sand or pebbles. He thinks his grandchildren will make up lots of games to play in an area like this.       or             or       ... The problem is to help Douglas know how many blue and how many yellow stones to buy for any size square area.

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Procedure

1. Gather the students at a central table and explain the story shell as one pair of students helps to build an example (say 5 x 5).
2. Pose the problem and ask students to work as pairs or as small groups to explore some different sizes.
3. Gather the data on the whiteboard (students can write their own results and ask students to record it in their journals in an organised way.
4. Look for patterns in the table of results and use them to predict blue and yellow for other size squares.
5. The question, Can we check this another way?, can be used to encourage seeking a visual understanding to match the number pattern, as described in the cameo.
6. Follow on with the other ideas in the iceberg information in the cameo.
7. At the end of the investigation by asking students to reflect on how the class has worked like a mathematician. Use the Working Mathematically Process as a guide.

Extension

• Write a letter to Douglas (or prepare a poster or slide show) to advise which size play area he should build and explain how you know the number of blue and yellow tiles he should buy. Suggest at least three games his grandchildren might play on the stepping stones in this area.

Task 145: Land Of ET

Equipment

• One yellow/blue board per pair

Problem

• BBB next to each other is removed (or included)
• YY next to each other is removed (or included)
• BYB next to each other is replaced by Y
• Y can be replaced by BYB

Procedure

1. Make up words of any length and apply the Royal Rules. Record the new words you make each time.
2. Including the 'nothing word', how many words are there in the new language of ET?
3. When you have found all the new words, choose any two and add them together. What happens?
4. Explore and record all the possible ways of adding any new word to any other new word. What happens?

Task 154: 4 Arm Shapes Equipment One Poly Plug set per pair Problem There is monument (one yellow/blue plug) in the centre of a park. There are equal length paths leading to the monument from the North, South, East and West. If I tell you the length of one path, can you tell me the number of plugs needed to make the picture? Procedure Allow pairs to work on the problem. Discuss and record how students work it out. This is an example of visual algebra. The recording can be in words and the words can then be converted to a mathematical sentence, or equation. Ask: Can you check this another way? and discuss suggestions. Ask the working backwards question: If I tell you the total number of plugs, can you tell me the length of each path? For a selection of path lengths, build up a table of values on the whiteboard. It is important that these are not in order. Each row of the table of values is an ordered pair and ordered pairs can be graphed. Plot the ordered pairs. What happens? How does this picture relate to the problem? Invite students to use their plugs and plug boards to make other patterns and set similar problems.

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Task 173: Crossing The River 1

Equipment

• One Poly Plug set per pair

Problem

• One adult, or
• One child, or
• Two children

Procedure

1. Act out the problem using the students.
2. Once solved, ask each pair to demonstrate the solution with their plugs. Use yellow/blue plugs for adults and red plugs for children. The objective is to find the smallest number of moves. It may take a little time before there is consensus on this.
3. What would happen if we changed the number of adults? Explore examples. A pattern will develop. If I tell you any number of adults, can you tell me the number of crossings?
4. Ask the backwards question: If I tell you the number of crossings, can you tell me the number of adults?
5. For a selection of adult numbers, build up a table of values on the whiteboard. It is important that these are not in order.
6. Each row of the table of values is an ordered pair and ordered pairs can be graphed. Plot the ordered pairs. What happens? How does this picture relate to the problem?
7. What happens if we change the number of children?
8. Can we create a new problem by changing the rules about who can safely use the canoe?

Task 182: Jumping Kangaroos Equipment One Poly Plug set per pair Problem Three blue kangaroos and three yellow kangaroos are facing each other on the stepping stones (red plugs) of a narrow mountain trail. There is one vacant stepping stone between them, ie: a trail of 7 stepping stones altogether. Kangaroos are able to hop onto a vacant stepping stone if it is just in front of them, or they are able to jump over one on-coming kangaroo to land on a vacant stepping stone. Are they able to swap sides?

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Procedure Act out the problem using the students. Once solved, ask each pair to demonstrate the solution with their plugs. The objective is to find the smallest number of moves. It may take a little time before there is consensus on this. Remember, the objective is to find the least number of moves, so if any kangaroo has done a backward move, the total can't be the least number since that kangaroo would still have to go forward again to get back to its initial position, ie: two additional moves. What would happen if we changed the number of kangaroos on each side? Explore examples - it is a good idea to use the mathematician's strategy of trying a simpler example. Display the results for 1, 2 and 3 kangaroos on each side? Can these be used to predict the result for four kangaroos? Can we check the prediction another way?

5. Once the predicted number of moves is decided and checked, can the students actually do the moves with their plugs?
6. If I tell you any number of kangaroos on each side, can you tell me the number of moves?
7. Ask the backwards question: If I tell you the number of moves, can you tell me the number of kangaroos on each side?
8. What happens if there are unequal numbers of kangaroos on each side?

Task 201: Rectangle Fractions

Equipment

• One Poly Plug set per pair
• Poly Plug Paper for recording

Problem

The challenge is to become more and more efficient at making any whole rectangle by 'unplugging' the red board and exploring the fractions it illustrates by plugging in yellow and blue plugs. For example:

    
Plugging one third yellow and plugging two fifths blue.

Procedure

1. Use a demonstration Poly Plug rectangle as above to talk about the physical/spatial parts - rows, columns and cells - of the whole.
2. Change the whole and go through the same discussion again.
3. Ask students to record the main points of the discussion in word and picture on Poly Plug Paper.

   Rectangle Fractions is a rich task. It can be visited frequently, and although the structure of the problem remains constant, the challenge in it changes with each different whole rectangle. Consequently, once the students understand the structure, it can be threaded into the curriculum for a few minutes a day, two or three times a week over several weeks. Threading is a powerful curriculum model which allows genuine time for the students to construct their own learning. The students do not become bored with the activity and, in fact, find security in the familiar aspects of the challenge.

   In particular, where fractions are concerned, the longer students have had an exposure to fractions through symbolic manipulation with little reference to materials, the more there is a tendency to see the area as meaningless, and the longer this threading may need to continue. The emphasis throughout is that the spatial/geometric representation of rows, columns and cells is the solution to any question.

4. Ask students to work backwards: What rectangle would you choose that could show halves and thirds? ... eighths and halves? ... Answers can be described in words or drawn.
5. When students appear confident with spatial representation, introduce the idea of showing two different fractions on the same whole. For example, showing one third and two fifths on the same whole requires thinking like this:

   Showing one third means plugging one row...   
   

   Showing two fifths as well means plugging two columns...   
   But where will the two blue plugs go when the yellow ones are already in their spaces?

   Simple. Shift them somewhere else that is empty, then the
   blue plugs still show an amount that is the same as two fifths...   

6. In this way students can be helped to find a visual representation for the addition of fractions. In this case, it is clear that:

   1 third + 2 fifths = 11 fifteenths

7. Eventually encourage students to picture in their mind, rather than actually make, the whole rectangle that would be needed to work out the addition of fractions given symbolically.

Task 211: Soft Drink Crates Equipment Two Poly Plug sets per pair 6x4 Crate suitable for holding plugs as an alternative to the red boards (PDF version). Recording Paper with multiple copies of 6x4 crate. Problem A person was packing cans into a crate with spaces like this: When 10 cans (yellow/blue plugs) were packed, it was noticed that: EVERY row and column had an EVEN number of cans How were the cans arranged? Procedure Introduce the problem using a centrally placed Poly Plug arrangement as above. Pack a few cans and examine whether rows and columns have an even or odd number of cans. Emphasise that zero is an even number. Students investigate the problem in pairs. Is there more than one answer? What happens if we change the number of cans to, say, 18? Explore all numbers of cans. Which can be placed so that all rows and columns are even?

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Task 212: Monkeys & Bananas Equipment One Poly Plug set per pair Problem A little twist of the story is necessary to investigate this problem with Poly Plug because they don't look much like bananas. Kids & Cup Cakes works well and a story shell something like:

One morning Peter, Paul & Mary helped bake some cup cakes. They were so tired when they finished that they went to have a sleep.

• During nap time Peter woke up, snuck out to the kitchen, took one third of the cakes and hid them in a safe place. But he had to eat one of the cakes first before he could take exactly one third. Then he went back to bed.
• Later Paul woke up, snuck into the kitchen, ate one cake and took one one third of the cakes that were left and hid them in a safe place. Then he went back to bed.
• Finally Mary woke up, went to the kitchen, ate one cake and took one third of what was left and hid them before she went back to bed.

After nap time the three children were able to equally share all the cakes left in the kitchen.

• How many cakes might have been in the kitchen before nap time?
• How many did each child get at the sharing after their sleep?

Procedure

1. Tell the story to introduce the problem. It is a complex story so it helps to have it backed up with a printed version for each group.
2. Students investigate the problem in pairs with a view to contributing to a class 'journal' that develops on the board as things are tried out. Since many students use a 'Guess, Check & Improve' strategy, it is useful to keep a table of numbers that do/don't work for the original cake collection.
3. Encourage identification of strategies that are being, or could be, used based on the Working Mathematically process. There are several ways to solve the problem, working backwards for example, but eventually the class will discover that 25 works for the original number of cakes and leads to a share of two cakes each after the nap.
4. But is 25 the only possible solution? Share the work of finding more. Agree to either start with a number for the original pile or a number for the final share so all groups are 'working in tune'. This work does not have to be done all at once. Each group could take one number each day and the results could be recorded in the class 'journal'.
5. Another approach, depending on the age, is to use a spreadsheet to do the testing. The problem involves a step by step set of mathematical actions that fit brilliantly with this tool. In fact, the problem might be used to introduce students to the spreadsheet tool. One way or another they will hopefully find the next solution, which is an original cake collection of 106, with a share number at the end of 10. The next solution after that is (187, 18) and if you get this far, a pattern begins to show up.
6. What happens if the problem involved only two children who hid one half of the cakes after eating one each time? Or four children each of whom takes one fourth after eating 1? ...?

Task 235: Tables For 25 Equipment One Poly Plug set per pair or group of 4 Pieces of A4 or A5 paper to use as tables Problem There are 25 students in the class today and 14 are boys. The teacher wants them to sit in groups of 5 with at least two boys in each group. How many ways could the students arrange themselves?

Procedure Set the context through a story shell about a colleague who had 25 students in the class and wanted them to sit in groups the same size. What sizes could he choose? Students soon see that 5 is the only realistic possibility. Well, on this day, there were 14 boys in the class, you can use your Poly Plug blue side up to show boys. Mr. Edwards said he wanted 5 students at a table and at least 2 boys in each group. How could the students arrange themselves?

3. It soon becomes apparent that there is more than one solution, so the next step is to ask the mathematician's questions:
   • How many solutions are there?
   • How do we know when we have found them all?
4. To work towards answering those questions, students have to realise that 'at least two boys' means 10 must be used around the 5 tables. There are only four left to 'play with'. The girls, in this case, don't matter.
5. One way to solve the problem now is to examine every possible way to arrange these extra four boys around the tables. Remember the tables can't be distinguished, so order doesn't matter. Following this through, the possible arrangements for boys are: (5, 3, 2, 2, 2), (4, 4, 2, 2, 2), (4, 3, 3, 2, 2), (3, 3, 3, 3, 2).
6. What happens if...?
   • Mr. Edwards says there should be at least one girl at each table instead of at least two boys?
   • We change the number of students in the class? What equal size groups are possible with the new number?
   • We change the ratio of boys to girls?
   • We change the seating restriction?
7. Design and investigate your own Tables For... situation.

Task 236: Star Numbers Equipment One Poly Plug set each or one per pair Problem If I show you how to construct one Star Number, say Size 4, can you tell me how many plugs I need to construct a Size 10 Star Number?

Once this starting point question has been answered 100 + 4 (1 + 2 + 3 ... + 9) = ... there are many others: How many plugs to make Star Number 20? 50? 100? If I tell you I have 1000 plugs, what it the largest Star Number I can make? Make pairs of numbers (Star Number Size, Number of Plugs). If you graph these pairs what do you expect to see? Graph the points to check your prediction. If I tell you any size Star Number can you tell me how many plugs I will need to make it? If I tell you any number of plugs can you tell me how to work out the largest Star Number I can make from them? Suppose I want to make a Christmas decoration that starts with Star Number 1 and shows all the Star Numbers up to 10. How many plugs will I need?

Procedure With student help, construct a Size 4 Star Number on a central table and pose the question. Students might choose to work it out by building the Size 10, which will take quite a deal of co-operation between groups, or they might decide to collect more data about Star Numbers, and organise it to see if a pattern develops.

And for those who like a considerable algebraic challenge, derive a formula to find the number of plugs needed to create all the Star Numbers from 1 to n.