# Maths Cafe

## Singapore Maths Problem

This problem went viral across the world on 14 April 2015. It is from a test given to 14 year old students competing in the Singapore and Asian Schools Maths Olympiad, aimed at the top 40% of students.

We had a fantastic response from our community regarding the correct answer. It’s an interesting maths challenge that relies on logical thinking.

How do you approach this problem? What strategies do you use to solve it? And imagine being in a test situation too. That is an added distraction for your brain.

Do what’s the answer???

To start, Cheryl gives 10 possible dates for her birthday. Albert is then told the month and Bernard the day of Cheryl’s birthday.The challenge is to work out the actual date of Cheryl’s birthday.

Albert and Bernard at first don’t know the actual date from the information they are given. So it can’t be May 19 or June 18. Otherwise Bernard would definitely know if he had 18 or 19 as his day.

And for Albert to say that he definitely knows Bernard doesn’t know, this means it can’t be any date in May or June (when these two numbers 18 and 19 appear) either.

So that leaves July or August.

But then Bernard says he does know when the date of Chery’s birthday.

It can’t be July 14 or August 14 as then Bernard would not know which month is the correct one.

So that leaves July 16, August 15 or August 17.

And then Albert says he knows now too. That means there is no double up.

It can’t be August 15 or 17 as then Albert wouldn’t know which day to select.

It must be July 16.

Phew!

## Crescent Moon

Watching the moon every month is a great way to think about both time and 3D objects. Twenty-eight days to observe and record, to think about and spot patterns. That’s what Galileo did way back in 1609 when he created a telescope that could magnify 20 x. He observed the moon each day and worked out the surface of the moon has valleys, shadows and peaks like on Earth. It was not perfectly spherical after all. And like Copernicus, Galileo believed the Sun was the centre of our system, not the Earth. Up until then Aristotle’s idea maintained the Earth was the centre of the Universe. Galileo sparked the birth of modern astronomy.

Your students can create their own daily record of the shape of the moon. Compare it with our March calendar which shows Galileo’s own recorded observations.

## Monument Valley

If you like mathematical games, Monument Valley for iphones or ipads, is pretty spectacular – one of the most beautiful games we have ever seen. If you like MC Escher, and we love his work, the creators have been influenced by his impossible puzzles. At the conclusion of each segment of the game, the little girl returns a mathematical shape that belongs to a tower. Very addictive.

## Giant Bubbles

What a wonderful way to celebrate the brilliance of everything! Create giant bubbles and watch them float away, burst, contort. Just try some detergent and water and a large metal ring on a stick. You can bend metal coat hangers, for example. What is the largest bubble you can create? The longest in length? The longest time it stays afloat?

Here are some great bubble video clips to use as a class stimulous:

Bubbles in slow motion – https://www.youtube.com/watch?v=q4BByh4zrWs

Bubbles on a beach – https://www.facebook.com/video.php?v=10153082881424742

Giant bubble net – https://www.youtube.com/watch?v=Q9kBL1cYd-8

## IMPROVE your problem-solving skills using these questions

One reason we learn mathematics is so that we can solve problems. The latest educational research shows that the IMPROVE method of metacognitive questioning, developed in Israel by Mevarech and Kramarski, is very effective. Here are 4 IMPROVE questions to help your Stage 2 and 3 students think about their own thinking. These self-directed questions can be used as a scaffold. They’ll be especially useful when tackling complex, unfamiliar and non-routine problems. Problems like these should be available for all students to solve, not just your low blockage students. Most textbooks are not suitable as they include only routine problems using a known algorism. As Andreas Schleicher (Director for Education and Skills, OECD) says, ” … good mathematics education can … foster the innovative capacities of the entire student population. including creative skills, critical thinking, communication, team work and self-confidence”.

You can read more here.

Or view Andreas Schleicher’s TED talk Use Data to build better Schools

## MV Road to Mandalay

There are plenty of websites that summarise key facts about ships and boats across our planet – a wonderful source of mathematical discovery. For example, http://www.wildearth-travel.com/ships/.

Bev Dunbar was recently on a river cruise down the Irrawaddy (Ayeyarwady) River in Myanmar when she spotted this magnificent river boat – MV Road to Mandalay. It’s a luxury boat that was originally brought from Germany in the 1990s but it was completely refitted after the devastating Cyclone Nagis in 2008. It is 101.6 metres long, with an 11.7 beam (width), a 1.45 m draft (underwater depth) and 900 tonnes gross tonnage. There are 4 decks which include 43 cabins that can take 82 passengers down the river in style from Mandalay to Bagan and back again. There are 108 crew members. And there are plenty of maths discussions just regarding the size of the cabins on this vessel.

## Amazing maps of Australia

Imagining the size of things is a mathematical skill. It’s called spatial visualisation. Maps are one way humans have invented to imagine what the earth looks like broken up into smaller chunks we call countries. Mercator was a Flemish geographer who imagined the earth as a cylinder in 1569. Most maps we see today are based on this cylinder, opened out to create a rectangle. We can imagine the shape of Australia on this cylinder/rectangle, but what does this shape look like in comparison with other countries.

Your students will enjoy seeing these amazing maps of Australia imposed one on top of another. They will help them visualise the huge shape of Australia better than seeing it alone.

http://www.buzzfeed.com/simoncrerar/mind-boggling-australian-maps#.ntO0lMxbO2

## Anthony Fisher Maze Maker

Anthony Fisher is a world famous designer of mazes – all shapes large and small. Explore his maze maker website for an amazing (ha, ha …) insight into the wide variety available throughout the world today. Maze exploration can be part of your study of Space: Position. Solving a maze puzzle uses spatial visualisation skills, an essential aspect of coping with our modern world.

You can see Anthony’s work at : http://mazemaker.co.uk or http://www.mirrormaze.com

## David Bailey’s spectacular Escher-type tessellations

David Bailey has created the most amazing set of Escher-type tessellations, many of which will appeal to your primary students. There is a vast range to explore from geometric designs, birds and animals. A tessellation is when you put tiles together on a flat surface so that there are no gaps or overlaps. Decorative tessellating patterns have been found in many ancient cultures, such as Ancient Rome, Egypt and throughout many monuments in the Middle East. A “tessella” is a Latin word for a small square. The Dutch graphic artist MC Escher created magnificent tessellations using irregular interlocking shapes. Escher had travelled extensively through Italy and also Spain where he was particularly inspired by the tilings he saw at The Alhambra.

Why not encourage your students to explore some of David Bailey’s work too? He has created a fantastic website packed full of information about tessellations with his own drawings plus real-life photographs. http://www.tess-elation.co.uk/new-hom

## How to use our Place Value Number Photographs

Maths Matters Resources is dedicated to helping you create real-life links in your maths classroom.

What are you trying to do?

- Use a real life example of number for a Place Value mental warm-up with your students

What do you need?

- Any of our Maths Matters Resources Place Value to 100 Photographs, such as house and letterbox numbers (e.g. on your electronic whiteboard).
- Any of our Place Value from 100 Photographs.
- Up to the whole class

What do you do?

- Talk about all the places you see real-life examples of numbers.
- Show the photograph e.g. 52. Ask the students to face a partner. On a given signal, each team has 1 minute to think of as many different things they know using that number as an example.

- “It is a 2-digit number, it’s larger than 50 but less than 100, it’s an even number, it’s a multiple of 2, 3 fewer than 55, half of 104, double is 26, 10 groups of 5 and 2 more, 52 is the age of my aunty, you could buy a pair of shoes for $52, 10 less than 65, that’s 10 pasta sticks plus 2 extra pieces of pasta, if I put 52 buttons into groups of 5 I would have 2 left over …”.

- At the end of the minute, discuss some of the discoveries with the whole class. Are there any misunderstandings?

Variations?

- Collect your own photographs of real life numbers for future mental warm-ups. e.g. How many different ways can you model this number? Beansticks, pasta sticks, Base 10 materials, groups of 10s and 1s pelicans, shells, alpacas (see our Place Value to 100 Photographs …)

- Use the photograph on an activity card or electronic whiteboard as part of a place value discussion with your students. e.g. Count by 1s, 2s, 5s and 10s to and from 52, try to add multiples of 52 – what’s the highest number you can reach?

## How to use the Maths Photographs

Any of our vast collection of Maths Matters Resources Photographs can be used as a suitable stimulus for your maths sessions. First identify the appropriate maths sub-strand. Next find a photograph that best suits your needs and the students’ interests.

For example, how will you use this picture of the 9 robots as part of your daily mental warm-ups? Ask everyone to brainstorm different maths ideas based on this photograph. Challenge them to be creative, to think of as many maths possibilities as they can. We need to develop maths confidence in our students so that they can think for themselves, without us always providing the mathematical links for them.

Stage 1 responses might be:

“one more makes 10”, “2 fewer will be 7”, “double these are 18”, “that’s 3 rows of 3”, “3 and 3 and 3 makes 9”, “there are 9 x 2 legs – that’s 18 legs”, “nine robots with 2 fingers on each hand makes 9 x 4 – that’s 4 less than 10 x 4 – that’s 36 – they have 36 fingers altogether”, “if I put them into groups of 4 there will be 1 left over”.

Stage 2 responses might be:

“4 groups like this would be 36 robots in total”, “if each robot costs $7 that would be 9 x $7 – that’s $63 to buy the lot”, “ if each robot has a mass of 80 grams that makes 9 x 80 – that’s a total mass of 720 grams or 0.72 kg”, “if you need 3 small balls to make each robot arm that’s 6 balls to make each robot, or 9 x 6 = 54 small green balls altogether”, “ if each robot displaces 50 mL of water, 9 robots have the same volume as 9 x 50 = 450 mL of water”.

Stage 3 responses might be:

“If it takes 3 robots 3 days to do a job, it will take only 1 day for 9 robots to do the same job because they can work 3 times as fast”, “if the area taken up by these 9 robots is about 20 x 20 cm, you will need a box about 25 x 25 cm and probably about 5 cm high to pack them in for sale in a shop”, “ if robot eyes come in packs of 100, you can make 50 robots”, “if they are battery-powered and each battery lasts for about 75 hours, you could get 9 x 75 hours of work out of them – 9 x 70 is 630 and 9 x 5 is 45 so that’s about 675 hours altogether”.

Congratulate students for specific maths suggestions using the photograph as an initial stimulus, such as the most mathematical links, the clearest example of a maths calculation, the quality of their discussion. The more specific they can be the better. Your students will soon look forward to inspiring you with their quick responses, depth of thinking and breadth of vision.

And of course you can use each photograph as part of your maths session activities – part of a smartboard discussion, a worksheet, an activity card, a group challenge question.

## Brazil World Cup Soccer Ball 2014

The traditional soccer ball is the 10th Archimedean solid – a truncated icosahedron. But times are changing! I don’t think there is an actual mathematical term for the latest ball, the Brazuca. Here are some interesting facts collected by Maths Matters Resources to help you integrate the World Cup soccer ball into your maths sessions.

Did you know that the special Adidas Brazuca, the Brazil 2014 World Cup soccer ball, was tested for over 2½ years? This name is slang for “Brazil” and was selected by a vote of over 1 million Brazil football fans. It has the smallest number of panels (6 identical pieces) ever, making it a very efficient design. Small dimples give it an irregular surface to create better aerodynamics. It was tried out by over 600 of the world’s top footballers. The colours symbolize the wish bracelets worn in Brazil. The actual ball is made in Pakistan and they have orders to produce over 42 million balls.

Imagine how you can integrate these facts into your maths sessions. For example, how much money will be spent if each ball costs at least $100? How many official games were in the World Cup tournament? How many times might one ball be kicked during one game? What’s the longest distance a ball can be kicked?

What’s your estimate for the surface area of this ball in square metres? Square centimetres? Do all soccer balls have the same surface area? How heavy is a soccer ball? What’s your estimate of the mass in kg?

What about other World Cup soccer ball facts? You can challenge your students to work in teams to come up with their own 10 interesting facts about soccer balls.

## Design a clock

Layla Mehdi Pour, a young designer from Iran, created this adjustable “Sophie” clock. Another designer from Britain designed a colour clock. Rnd Lab for Progetti designed what at first glance looks like a random clock. These can be the basis for a whole unit on Time. What sort of clock can your students invent? Normally we think of an analogue clock as round. What other shapes can they be? Do all clocks need a number? Why? Why not? Do they all need hands? Why? Why not? You could select a group of students to research clocks online and share one interesting clock each day of your Time Unit.

## Sharpen your imagination

Using real-life examples in your maths sessions will sharpen your students’ imaginations. Just like in this beautiful poster from Tang Yau Hoong. Encourage them to see maths in everything. A good start is to provide daily mental warm-ups using our maths photographs and maths graphics for mathematical brainstorms. Try not to rely on commercial textbooks and worksheets – these can stifle your students’ imaginations. Can they work with a partner to explain what they think and why? Maths language needs to pour out of them as they argue, explain, describe, compare and extend their thinking.

## Architecture and geometry

Of course architects create huge 3D geometric spaces all over the world. A wonderful source of investigation for primary students.

The 2014 Pritzker Prize for Architecture went to Japanese architect, Shiguru Ban. He created the temporary Cathedral for Christchurch, New Zealand, after the 2011 earthquake – made from recycled equal length cardboard tubes, wood and glass. These provided quick and efficient construction and the cathedral is large enough to hold 700 people.

Paper tubes are fairly inexpensive and Shiguru has used them for disaster relief structures in Ahmedabad India, Rwanda and Kobe, Japan. Imagine what your students could create using cardboard tubes, just like this prize-winning architect.

## Data investigations with large numbers

In 2013 The Rolling Stones earned $26 215 121.71. A pretty large number. Your Stage 3 students can discover amazing large number facts hidden behind their favourite pop groups. Long ago, my favourite group was the Rolling Stones. I was fascinated to see that in 2013 they still earned so much money. What are the 10 favourite pop groups in your class this year? How will they find out? Who will collect this data? How much did each of these groups earn last year? How do you know? How will you find out? Who will do this? Can they estimate first the order for the top 10 earners? Once you have your ten $ amounts you can put them into ascending or descending order, find the $ differences between the 1st and the 2nd highest earners. You can also create a column graph – what will the scale look like? Why? What will be the title of your graph?

Challenge your students to create further mathematical questions e.g. Who is the highest earning pop group of all time? Who is the highest earning individual performer this year? Ever? Who is the highest earning musician this year? Ever? Do pop stars earn more than other musicians?

## How to use the People Photographs

Any of our People Photographs can be used as a suitable stimulus for your maths sessions. Look through the collection and find the photograph that best suits your needs and the students’ interests. For example, show the picture of the ballet dancer. Ask everyone to brainstorm 10 different maths ideas based on this photograph. Challenge them to be creative, to think of as many maths possibilities as they can. We need to develop maths confidence in our students so that they can think for themselves, without us always providing the mathematical links for them.

How many right angles can you spot? Acute angles? Obtuse angles? Reflex angles? How long can a dancer stay on their toes? What is the longest time on record? Look at her hair – how long might it really be? Why has it been tied up so tightly? Who are the 10 most famous ballet dancers in the world today? Why are they so famous? Who is/was the most famous ballet dancer ever? How do you know this? How many students in your class are learning ballet? What percentage is this? What would it look like as a column graph? What is the ratio of boy to girl ballet dancers in your class? What is the ratio of ballet to non-ballet dancers in your class? How many times can a dancer spin before they lose their balance? What’s the longest time a dancer can stand on just one foot? How much does a pair of ballet shoes cost? What is the most popular form of dance? How will you find out?

Once you have a suitable collection of questions, these can then form the basis of further class investigations, or problems to solve. Teams can select the question they most want to investigate. Again challenge your students to find as much mathematics hidden in their topic as they can. You can always use the Maths Sub-strand Checklist proforma as a prompt for their thinking. You can find this in the first column (General Info).

Congratulate students for specific maths achievements using the photograph as an initial stimulus, such as the most mathematical links, the clearest example of maths calculations, the quality of their presentation details.The more specific they can be the better. You don’t need lots of worksheets and textbooks to stimulate your students in maths. You need something that excites their interest, that intrigues them, that encourages them to ask more questions.

Try to make time at least once each week to use a photograph as a quick mental warm-up. Your students will soon look forward to inspiring you with their quick responses, depth of thinking and breadth of vision.

## 3D and 2D Reflections

Reflections are part of the Geometry strand in Primary Mathematics. Being able to visualise what something looks like along an axis of symmetry is more difficult than you think. Spatial visualisation skills are used frequently in our daily life and we need to practise reflecting or flipping images in a variety of ways. Flips can be in any direction. These zebras are flipped across a horizontal line formed by the waterhole. A vertical flip would create an entirely different picture. How can you encourage your students to flip images both mentally and by using pieces of silver cardboard or a safety mirror?

## Looking down from above

Looking down is an essential skill in Position/Location, a sub-strand of Geometry. Your ability to imagine things from a different perspective helps you solve problems. Close you eyes and think of a 3D object. Now try to see it from above, from the left side, from below, from the right side, from behind. What will it look like in these different positions? What looks the same? What looks different? Very few people get to see the famous Uluru from above. The rock is now not instantly recognisable as an icon of Australia. Most people can only recognise Uluru from its side-on position seen from ground level. How can you help your students develop this skill through a daily Mental warm-up?

## World population statistics

Lego mini figures were created in 1978. By 2019 there will be more LEGO mini figures than humans in our world. That’s more than 8 billion plastic figures. Amazing.

How can world populations inspire your students? Do you regularly look at world population counters? You can easily set up one on your class computer and challenge your students to suggest maths facts, problems and related or inspired thoughts daily.

For example, Worldometers gives access to a wide range of population statistics for daily analysis.