# Mathematical Worlds Teacher Guide

What are they? A series of ten maths investigations, produced on behalf of the FMSP Wales in Summer 2020, to help schools during lockdown.

What ages are they for? It varies across Y7-Y13 — see below.

What format are they? Typically:

• A video introducing the world
• A student handout of questions to explore
• A teacher handout of answers

The video is in English. The handouts are available in English and Welsh versions. The resources were created by maths researchers and others around the world, and the exact format varies according to who produced it.

Which should I choose if I only have time for one? Depends what you’re after. All years can get started on Number 8 Mad Vet World, but the ideas get increasingly profound. The questions in Number 2 Combinatorics are interesting and easy to engage with. The worlds Number 3 and Number 5 from Germany are engaging and clearly explained. Number 10 Difference is good for Y13 who like stories and puns, with some maths on the side. Worlds Number 6, Number 2 or Number 7 may help, among other things, to tackle stereotypes in mathematics. But this is just one view and may not match yours. Where can I find them? The video links are given below at the end of the resource descriptions.

All the handouts in English and Welsh are available on Dropbox

What are the details? In the list below given year is an estimate of the minimum school year in which students could tackle some of the questions in the investigation — but to fully understand the answers might require a higher level. And students vary a lot!

1. Factor - Find numbers with most factors; find palindromic numbers. The 5 minute video is not essential. From Australia. From Year 7

• Video
• Handouts
• 2. Combinatorics - Four questions to investigate: dates with distinct digits; numbers whose sum of digits is 8; paths for a robot on a chessboard; and a classic question about opening and closing doors. The 1 hour video goes through the first question and explores a second using Venn diagrams. From UK. From Year 7

• Video
• Handouts
• 3. Oracle - A series of questions investigating the probabilities of different outcomes when tying ropes together at random. Although the question can be understood at Y7, to make progress will require higher levels of maths, ultimately using recursive formulae. The 8 minute video clearly explains the situation and answers the first question. From Germany. From Year 10

• Video
• Handouts
• 4. Card tricks - This explores three card tricks, each shown in its own video (9, 12 and 4 minutes). The first involves repeatedly dealing into a number of piles, and surprisingly the order at the end is preserved; the second is a well-established trick where the identity of a card is determined from four other cards; the third is a “3-card reveal” based on repeatedly sifting the deck until just three cards are left. From Germany. From Year 10

• Video 1
• Video 2
• Video 3
• Handouts
• 5. Checkerboard - An investigation into removing counters from chessboards, under various rules. The 11 minute video clearly explains the situation and gives guidance. From Germany. From Year 10

• Videos
• Handouts
• 6. SDR - An investigation into Systems of Distinct Representatives. The abstract definitions of these are shown in the videos, but the presenter then explains them by example. The 6 minute video gives an overview before tackling the material in two areas: setting up parties; and playing an abstract game. From the USA. From Year 12

• Video
• Handouts
• 7. Topology - An introduction to topological ideas. These are explained in the 24 minute video. The 5 tasks then ask the students to identify shapes that are topologically distinct; identify shapes that are equivalent to simple closed curves; investigate connecting 3 houses to water, electricity and gas; investigate which graphs can be drawn without lifting your pen (Eulerian circuits); and finding shortest circuits for a postman. From the UK. From Year 7

• Video
• Handouts
• 8. Mad Vet - Considers the situation where a Mad Vet has a machine that can transform particular collections of animals into other collections. The 1 minute video is not necessary, but interesting to see the resource creator in Texas. From the USA. From Year 7

• Video
• Handouts
• 9. Möbius strips - This resource is an exploration of so-called Möbius strips. Good for students who like hands-on activities, and based around a great pre-existing YouTube video. From the UK. From Year 7

• Video
• Handouts

10. Difference - This is a story with embedded maths problems, exploring so-called Finite Calculus, and getting students to see its utility and parallels with ordinary Calculus. The 1 minute video is not necessary, just shows the presenter. From the UK. Year 13

• Video
• Handouts