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# Poly Plug Puzzles A

 These notes begin with support for turning each menu item into a whole class investigation, then link to the companion Task Cameo, which in turn links to Maths300 (where appropriate). These investigations involve Computation, Pattern, Generalisation, Algebra and are suitable for Year Levels: 3 - 8 Discover Menu Maths here. Order Poly Plug here.

### 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)

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 the number of rings after 10 seasons?

### 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.

#### 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?

### 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.

#### 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?

### 4 Arm Shapes

#### Equipment

• One Poly Plug set per pair

#### Problem

A stepping stone (red plug) is placed in the middle of a park and a council worker proceeds to build equal length paths out from it leading to the North, South, East and West. She uses more red stepping stones and spaces them out equally. The finished shape is a cross (or plus sign).
If I tell you the length of one path, can you tell me the number of plugs needed to make the picture?

#### Procedure

1. Allow pairs to work on the problem.
2. 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.
3. Ask: Can you check this another way? and discuss suggestions.
4. Ask the working backwards question: If I tell you the total number of plugs, can you tell me the length of each path?
5. For a selection of path lengths, 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. Invite students to use their plugs and plug boards to make other patterns and set similar problems.

### Crossing The River 1

#### Equipment

• One Poly Plug set per pair

#### Problem

Eight adults and two children want to cross a river. Their canoe can hold:
• One child, or
• Two children
Everyone is able to row. What is the least number of moves for all to cross the river?

Task 173, Crossing The River 1

#### 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?

### 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?

#### Procedure

1. Act out the problem using the students.
2. 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.
3. 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.
4. 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?

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