|Enrichment Powerpoint Presentation
This is a visually stunning, first class presentation on polyominoes with lots of amazing animation and lovely effects. We start by examining the ominoe, dominoe, triominoes and then get the students to try to draw/find all of tetrominoes. The amazing pentominoes and their properties are then introduced and discussed (after the class spends about half an hour trying to find them themselves). Lots of work follows on investigating which of the twelve have line /turn symmetry. We then check to see whether they have the same area/perimeter (they don’t) and then look at which ones can be folded to make open top boxes and the class are asked to shade the base of the ones that can. This extends to finding the 35 hexominoes and again we look at symmetry/area and perimeter and find the 11 of them that can form nets of a cube. We return to the pentominoes and realise that their combined area of 60 square units means that they could theoretically be made into a rectangle. The students are asked to investigate which rectangles could be made. Now comes the fun bit! Groups of students are given multi-link cubes and they build the 12 pentominoes and try to cover the possible rectangles (6 x 10, 5 x 12, 4 x 15 etc) that can be made from them without leaving any gaps. The children will find it astonishing that the 6 x 10 rectangle alone has over 2000 solutions. The worksheets for this are supplied. Once a solution is obtained using cubes they can transfer their solution using coloured crayons to the worksheet provided. These are great for classroom displays. The children will be begging you to take some cubes home. The last few slides contain an extract about the pentominoes from Arthur. C. Clarke's "Imperial Earth". This is designed to be printed off and perhaps given to students to read. There is a lot more besides the above.
If you haven’t got this presentation, then believe me there is a hole in your life! (just like one of the octominoes).
Below is a suggested outline of how to deliver and prepare the full activity that has previously been sent to registered customers in the form of a NEWSLETTER. We recommend that you print it for future reference. You will easily manage 5 hours work here with classes of all abilities. Everyone loves it.
This is an excellent and fun presentation that is cram-packed with good mathematics and has the added bonus of being suitable for all year groups and all levels of ability. A key feature is that it can help develop systematic thinking skills. There is easily a weeks' work here (and probably two weeks if you do the whole lot). You should regard this as mini maths project/investigation. Your students will find the jig-saw element addictive and some will probably want to take the twelve pentominoe pieces together with the template home for the holiday. It is recommended that you try to come up with your own solution first before inflicting it on you students. It is worth organising things well before-hand so that you can chill-out and largely relax while the kids do most of the work.
PREPARATION and EQUIPMENT
(a) Class set of A4 plastic wallets (or folders) with about ten 1 cm sheets of squared paper + label for name.
(b) Small A5 worksheets from back of presentation showing all 12 printed pentominoes. These are for use with open-top box, line symmetry, rotational symmetry and perimeter problems as described below. Issue 4 x A5's to each student only when you reach this point in the investigation. These may not be needed until the second lesson/hour depending on class ability.
(c) A large box with a good supply of 2 cm multi-link cubes. (not needed for the first lesson/hour). Ideally each student should have their own set so for a class of 30 students you will require 60 x 30 = 1800 cubes. Alternatively students work in pairs so 900. If the department is short on multi-link the pentominoes can be cut out from the worksheet provided after it has been photocopied onto some coloured card (slide 37).
The solution templates from the back of the presentation should be blown up to A3 so and copied back to back). The front side shows the 6 x 10 and 2 x 30 rectangles, the back shows the 4 x 15 and 3 x 20.
The students should build each pentominoe from the cubes (cardboard cut-outs) using the set shown on the template sheets. Encourage the students to lay them over each of the printed ones. This helps avoid constructing an incorrect one which would simply mess everything up.
OUTLINE of PROJECT
Everything that you need is built into the presentation so all you will need to do is rehearse it yourself.
1. A single click of slide 2 shows the 12 animated pentominoes fly in from all directions (with sound) to completely fill the blank 6 x 10 rectangle. You may notice that they look like letters of the alphabet. This slide is simply there to whet the student's appetite for what comes later on and you shouldn't dwell on it.
2. Use slide 3 to explain the rules to the students together with some terminology.
3. Slide 4 introduces mon/dom/tri-ominoes. After explaining that rotations and reflections are not allowed the students are then left to try and find all of the tetrominoes using the squared paper from their plastic wallets/folders. Allow about 10 - 15 minutes for this and give time for a good proportion of the class to get all five before displaying them. At this point you should emphasize that the best approach is to use systematic thinking by referring to the 5 x 1 tetrominoe and explaining that the others can be found by removing the end square and placing it in all possible positions to get some more. After this is exhausted take two end squares and move these about the remaining three until you have all 5. If there isn't an element of systematic thinking going on then it may be a struggle to find all 12 of the pentominoes.
4. Slide 5 displays the 5 x 1 pentominoe then they are on there own. It is probably best not to tell them how many there are. (although the more observant may remember from slide 2).
You will be bombarded with solutions after about 20 - 30 minutes (depending on the group) but is very likely that there will be duplicates so be on your guard.
When you are ready display them on the board. There will be a discussion "mine doesn't look exactly like yours" and you will have to explain about orientation. Again point out how systematic thinking yields all 12 solutions. Try to get the students to tick them off on their sheets after looking at the solutions on the board so that they can see what they have missed. You need to avoid a situation were you are having to check individual sheets. Although after you become familiar with them you can normally spot duplicates fairly easily. That should be about it for the first lesson. If there is time you may want to get them to draw them neatly in pencil on a fresh piece of squared paper and perhaps colour them in (lower groups like this).
5. Slide 6 displays the pentominoes looking like letters of the alphabet.
F, L, I. P, S, T, U, V, W, X, Y, Z. Hopefully the kids will spot this.
Which of the pentominoes can be folded to make open-top boxes? Students can be given the small A5 worksheets showing the printed pentominoes (give them 3 or 4 each but have a spare supply). They have to shade the base of the box. Display the SEVEN solutions on slide 7.
6. Slide 8. Which of the 12 pentominoes have line symmetry? Repeat as above and display the six solutions.
7. Slides 9 - 12. Which of the 12 pentominoes have rotational symmetry? Repeat as above then display the three solutions.
8. Slide 13. They all have the same area but do they all have the same perimeter? Repeat as above then display the odd one out (letter P).
9. Slide 14. What is the total area of all 12 pentominoes? How many rectangles are there that have an area of 60 square units and what are the dimensions of each? Display all the rectangle solutions on the board. What if the pentominoes could be placed on some of these rectangles so that they covered them completely (seems unlikely due to their awkward looking shapes) which ones could they cover?
10. Slide 15 is a re-run of slide 2 and shows that the 6 x 10 rectangle can in fact be covered completely by the set of 12 pentominoes. Ask the students how many ways they think that this could be achieved.
11. Slide 16 is a colourful animated slide that shows the pentominoes flying in to cover the 6 x 10, 5 x 12 and 4 x 15 rectangles. We show 3 of the 2339 solutions for the 6 x 10, 2 of the 1010 solutions for the 5 x 12 and 2 of the 368 solutions for the 4 x 15. Hard to believe!!!
12. Slide 17. Students to build the pentominoes out of cubes (or card) and start seeing if they can come up with there own solutions. Lower ability students will simply want to copy one from the board to start with which is fine. As time goes by some students will try to fool you into having a solution by reflecting or rotating one of the solutions on the board. They can try to cover any of the rectangles but try to encourage them to persevere with the 6 x 10 first for obvious reasons. The 3 by 20 is extremely difficult (see end slides Arthur C. Clarke "Imperial Earth") You may wish to print this out and give a copy to higher ability students that are perhaps showing particular interest).
13. Is it worth spending a whole lesson or two letting students try to come up with a solution? YES. When you get a solution use a print-out of slide 37 and ask the students to colour in the squares showing their solution. It makes a nice classroom display. By the way as an aside, the templates and 2 or 3 sets of pentominoe cubes go down very well on a department open evening. Lots of younger children like to have a go at them as admiring parents watch on.
14. Slide 19. Can you find the Hexominoes? You should regard this as optional as you need to be an addict to try and find all 35 of these but some kids ask if they can do it for homework. At this point you could simply show all 35 and move onto which ones can be folded into boxes. You end with the 11 nets of a cube as displayed on slide 20.
There are A5 print-outs of the 35 hexominoes for use with this and slides 21 to 23 look at their symmetry and perimeter in a similar way to the pentominoes earlier.
15. Slide 24 (for the more able students perhaps). What is the total area of all 35 hexominoes and what rectangles could you make with them. Which of these rectangles could you use in an attempt to fit all 35 pieces together with no gaps? Surprisingly perhaps, there are no solutions to any of them. A 14 x 15 rectangle is shown.
16. Slide 25 looks at the sequence of constructible polyominoes up to the octominoe and asks whether there is a rule for them.
Slide 26 is a printable investigation worksheet that you may wish to use with appropriate sets. When blowing the templates up to A3 you may not get an exact 2cm match so play around with the enlargement factor on the photocopier until you get it exact.
Enjoy what is a very nice activity.