St Thomas of Canterbury, 17 January 2020

This week at the Curious Minds Club we continued our exploration of shapes, in two and three dimensions of space, by looking at (regular) pentagons. The children were able to identify that pentagons do not fit together without leaving a gap or an overlap.

pentagon problem

I showed them how to cut two irregular pentagons out of a regular hexagon. They made a set and explored how to get them to tessellate with all the edges meeting at a vertex. The solution involved flipping some of the tiles over.

I gave one group the puzzle called Pentamania to explore. This is a set of 54 so called folded pentagons. Two can be seen in the image above (one pink, one grey) when regular pentagons overlap. The children solved one of the three puzzles and enjoyed making a pattern with the pieces.

pentamania

A group of younger children enjoyed a quick game of Dotty Dinosaurs. They played the shape matching version.

dotty dinosaurs

We then made our first three dimensional object, a tetrahedron. I wanted to see how the children managed with the Polydron pieces I had, as they can be a little tricky for small hands to clip together. The children managed the activity well. One girl asked if she could take it home, so she must have liked it. I asked them what the object was called. One boy correctly said a triangle-based pyramid. They were all intrigued to learn its second name, the tetrahedron. I explained that tetra meant four in Ancient Greek, and this object has four faces.

. tetrahedron

We finished by getting out the pattern blocks from last week, and completing some of the pattern boards.

pattern boards

St Thomas of Canterbury, 10 January 2020

In this new term at the Curious Minds Club we started our exploration of shapes, in two and three dimensions of space.

I gave the children a collection of wooden triangles, squares and hexagons. I asked them to make a regular, edge to edge tessellation for each shape. It didn’t take long for every child to find the solutions:

regular tessellations

I explained that each tessellation has a vertex notation. I started with the square tessellation, explaining that its notation is 4,4,4,4 (every vertex is surrounded by a shape with four edges i.e. a square). I asked the children to work out the notation for the other two tessellations. With a little help they were able to find the answers: 3,3,3,3,3,3 and 6,6,6.

We then moved on to the semi-regular tessellations, of which there are eight. I used the same wooden pieces and some pieces that I had to cut out of card (octagons and dodecagons) as they are not available in wood. I gave each child a different vertex notation (e.g. 3,6,3,6 to make the pattern in the top left corner below) and asked them to put the pieces in the right order. When I had checked they had got it right (or offered a bit of help) I encouraged each child to take more pieces and extend the pattern out in each direction. I then rotated the activity between the children so they all got to try as many of the eight tessellations as possible.

semi regular tessellations

Here are some examples of completed tessellations:

3,4,6,4 tessellation by a Year 4 girl:

Y4 girl tessellation 3 4 6 4

3,3,4,3,4 tessellation by a Year 6 boy:

Y6 boy tessellation 3 3 4 3 4

3,12,12 tessellation by a Year 6 girl:

Y6 girl tessellation 3 12 12

For our final activity I gave each child a sheet showing all eight semi-regular tessellations and a piece of mirror card, and asked them to find the reflection symmetry for each tessellation (some have more than one). I asked them to find the ‘odd one out’. One boy was successful in identifying that 3,3,3,3,6 has no reflection symmetry. I explained that this is because it is chiral i.e. there are two different versions of it:

 

St Thomas of Canterbury, 13 December 2019

In our final week of this term at the Curious Minds Club we explored how to solve a maze. A maze is a type of topological puzzle. I showed the children a method of solving a maze by shading in all the dead ends: a dead end is chosen and shaded back to a point where a decision has to be made; at this point no further shading can be done so another dead end is chosen and the shading repeated. When no further dead ends can be found what remains is the solution to the maze. The children explored this method using some Christmas mazes I found online.

Last week we explored Borromean and Brunnian links. There was one activity we ran out of time for, so I brought it this week. Professor Tadashi Tokieda calls this a Borromean ribbon. His Numberphile You Tube video gives a great visual demonstration.

borromean ribbon

I tied some strips of ribbon into closed loops and showed the children how to set them up on one hand. The children soon got to grips with how to pull one ribbon down and come up with a surprising result.

The final activity was a make a Mobius paper chain to take home. I told the children we could have made regular Christmas paper chains, but they are boring. So much better to remember the session we did a few weeks ago on the Mobius loop. I had cut strips out of Christmas themed paper and reminded the children of the twist which produces the Mobius loop. The children then got to work, threading, twisting and taping. Here are some examples:

Year 5 boy:

Y5boyMobiusXmas

Year 6 boy:

Y6boyMobiusXmas

St Thomas of Canterbury, 6 December 2019

This week at the Curious Minds Club we explored Borromean and Brunnian links.  We started with the Borromean rings as they are the simplest Brunnian link. I showed the children how to turn coloured pipe cleaners into circles. They made a red one and a white one, then placed the white on top of the red one and threaded a blue one through, in an over-under-over-under sequence. I showed the children that as a ring of three this is a strong link, but if you remove any one ring the whole thing falls apart.

200px-BorromeanRings.svg

I asked the children to turn their flat ring into a three-dimensional structure, almost like a gryoscope.

220px-3d_borromean_rings_by_ronbennett2001

A drew a table to show that each ring is outside one ring and inside another ring.

Outside Inside
White Red Blue
Blue White Red
Red Blue White

I surprised the children by asking them how their Borromean ring is like the game ‘Rock, Paper, Scissors’. One girl was able to identify that each component has a different relationship with the other two components. I drew the table below and pointed out that white occupies the same positions in the table as rock; that blue corresponds to paper; and that red corresponds to scissors. I explained that the concept of ‘beating’ equates to being ‘outside’ and the concept of ‘losing to’ equates to being ‘inside’.

Beats Loses to
Rock Scissors Paper
Paper Rock Scissors
Scissors Paper Rock

By this point some of the children had discovered that they could turn their Borromean ring inside out. I encouraged all the children to practice turning their ring inside out and back again. I think this is one of the most fascinating aspects of the Borromean ring and is due to the fact that no two of the three rings are actually linked with each other.

We then moved on to making a Borromean ring of a Borromean ring! Yes, we went fractal! I asked the children to take three pipe cleaners of one colour, three of a second colour and three of a third colour. They made normal Borromean rings with their first and second colours and placed one on top of the other. They made another Borromean ring with their third colour to get the twists in the right place, undid the ends, threaded it through and retied the ends. When made three dimensional this is a very interesting object to look at.

Some children had raced a bit ahead, so I asked them to make triangles instead of circles and thread them into a ring. I showed them one I had made earlier. Two children thought the circles made a better three dimensional shape, but one boy argued in favour of his triangles. He had indeed succeeded in making something harmonious to look at.

The final task was a make a 4 component Brunnian link, with four different coloured pipe cleaners. This is trickier than the Borromean rings as the green link has to be made much smaller, but the threading is not too complicated. With a little assistance the children were able to complete theirs. I asked what would happen if one link was removed. Due to the effort involved in making it some of the children were reluctant to try this on theirs! I let them use my demo instead, and they were surprised to see the whole thing fall apart. I explained that as this happens with the Borromean rings we should expect it with the Brunnian link as they are part of the same class of links.

200px-Brunnian-4loops.svg

Here are some photos of the children’s work.

Year 2 girl:

 

 

Year 4 girl:

Y4girl Borromean

Year 5 boy:

Y5boy Borromean

Year 6 boy:

Y6boy Borromean

Year 6 girl:

Y6girl Borromean

St Thomas of Canterbury, 29 November 2019

This week at the Curious Minds Club we started by trying some more real world knots. I used the NetKnots website to learn these knots which I showed to the children: Clove Hitch, Transom Knot, Constrictor Knot, Highwayman’s Hitch, Barrel Hitch and Cat’s Paw.

Some children played a game called Knotted. The person who created this game put it online to download for free. I did this and printed it on thick card. I tried the rules but decided to adapt them to my version, in which the winner is the first player to lay a continuous path from their side of the board to the opposite side.

knotted2

All the children had a go at making a Celtic Knot pattern. I had created some patterns of my own which I asked the children to copy to get them familiar with the different pieces and how they fit together. I then challenged them to come up with their own design. Here are three examples:

Two strands interlinked (Year 2 girl):

Y2girl_rotate

Two separate strands (Year 6 boy):

Y6boy

One continuous strand (Year 6 girl):

Y6girl

St Thomas of Canterbury, 22 November 2019

A move away from mathematical knots this week at the Curious Minds Club, and a look at real world knots. We started with a look at the symmetry of shoelaces. I made some ‘shoes’ out of felt and threaded them with one blue lace and one yellow lace. The instructions asked the children to make the base knot with the lace on the left leading first, then make the actual shoe lace knot with the lace on the right leading; this resulted in a neat and tidy, balanced knot with two types of rotational symmetry. I then asked the children to take the other shoe and make the base knot with the lace on the right leading, then make the actual shoe lace knot as above (right lace leading). This lack of alternation resulted in an ugly knot with one part facing the toes and the other part facing the heel; this is the knot which is much more likely to come apart. Ian’s Shoelace Site has more information and really good diagrams.

I then showed the children how to make some actual knots, specifically some loops and some bends.  I used the NetKnots website to learn these knots. The loops we tried were: Slip, Bowline, Hanson and True Lover’s. The bends we tried were: Square (Reef), Flemish and Fishermen’s.

To end the session, one group of children played the Knot Game. Another group had a go at solving some of Adrian Fisher’s Celtic Knot Playing Card puzzles:

adrian fisher

St Thomas of Canterbury, 15 November 2019

Another week exploring Knot Theory. The first activity was to look at a page of knots and work out which ones were actually unknots. Some children could do this by looking at the knot, and other children made the knot and manipulated it. The second activity was to learn the Reidemeister moves then look at some pairs of knots and decide which Reidemeister move had been used to turn picture one into picture two. The last activity was to learn about crossing numbers, and to look at pictures of knots and see if they could be simplified in order to decide what their true crossing number was.

reidemeister

We ended the session by playing the Knot Game.

knot game