FLATLAND: A Romance of Many Dimensions
Edwin A. Abbott was born on December 20 1838. Abbott was an English school headmaster and theologian who in 1884 wrote a very unusual book, Flatland: A Romance of many dimensions, under the pseudonym of “A square”. It predated Einstein’s ideas about the 4th dimension and is a social satire which tells the story of a world where there are only two dimensions, no third dimension that would give us height. Men are polygons, women are line segments and a circle is the perfect shape (an infinite polygon). A film was made about it in 2007.
The funny thing is that when Duffy and I first met way back in the late 1980s we discovered that we both had a copy of this unusual book. I know of only one other person who had a copy, my friend Bill Barry, a famous mathematics educator from Sydney in the 1960s through to the 1980s.
Perhaps thinking about the world as only 2D can help our primary students understand 3D better. Imagine as a girl you are a straight line. When you move you can move forwards or backwards and if you bump into anyone you would sort of bounce off them. However if you are a straight line and you move sideways, when you bump into someone you might hurt them as you would have a sharp point on either end.
Boys are polygons which mean they have at least 3 straight sides. So depending on how they move they might bounce off their 3 straight sides or hurt someone with their 3 sharp corners. And boys might also be a square or an oblong, or a pentagon and so on. They become more dangerous as the number of sides increase.
If you are a circle, no matter how you move you will bounce off other people in your area as long as their pointy ends or corners don’t stick into you. You can sort of move anywhere you like without worrying about who you might bump into but still trying to avoid those sharp points. But when you think about it, you would appear to be a straight line when viewed by a non-circle. Your centre would be closer to them and perhaps the outer parts of your “line” would be a little shadowy. It would be quite a subtle distinction. I think inhabitants would learn to tell the difference between encountering a circle or a non-circle and act accordingly.
So Flatland would be quite a scary place to live. It certainly gives you heaps to think about and discuss together with your students, especially with Stage 3.
MEASUREMENT – TIME: Exploring the Calendar
Calendar skills are a part of TIME (Measurement sub-strand) in Primary Mathematics across the world. Today is 2 November 2019. It is the 306th day of this year (and it would be the 307th day if this was a Leap Year). Today’s date can be written in many ways – in Australia we write it as 2.11.19. Calendar discussions should be a daily part of your class mathematics routines, whether you teach Kinder ( 5 year olds) or Year 6 Students (11 year olds). Encourage your students to create a set of 4-5 quick questions each day. e.g. What day is it today? It is Saturday and it is the first of 5 Saturdays in this month. How many days are there in this month? There are 30 days in November so there are 28 more days to go to the end of the month. It is the 306th day of 2019 and there are still 59 more days until the end of 2019. What season is it now? November is in the last month of Spring. There are 3 months in Spring – September, October and November. The next season will be Summer. Is today someone in your family’s birthday? Or someone in your class? Your school? Did something memorable happen on this day in the past? In Mexico it is celebrated as part of the Day of the Dead.
You can find more TIME Mental Maths Warm-ups here.
You can find suggested TIME Classroom Activities here.
You can find useful TIME Photographs here.
You can find useful TIME Graphics here.
Measurement comparisons matter
We recently found on our Facebook page that one post outperformed any other post in the 6 years since we created Maths Matters Resources. It relates to Length Measurements and is a graphic designed for Northern Territory Tourism.
So why is this height graphic so amazing? It very clearly shows us well-known tourist icons that help us visualise the actual height measurement of Uluru in the Northern Territory.
The ability to accurately measure height and length seems to be important to every culture on earth. You could fill a page with all the different words associated with length e.g. circumference, distance, kilometre, shorter than, furlong, as long as, narrow, centimetre.
The funny thing is, in length measurement, everything is relative. There is no higher than or not as high as, longer than or shorter than without a second length for comparison. A snake is longer than a crocodile only in some, not all circumstances. You are shorter than the height of most doors, but not all doors. You need at least two things to measure before you can make your judgment.
You can now easily estimate height using the icons on this graphic. The Eiffel Tower is almost as tall as Uluru so those of you who have visited Paris will have a pretty good sense of what that looks like. Similarly, if you have been to Egypt, the height of Uluru is about as high as two Pyramids of Cheops.
You can also think about length. Uluru appears to be a bit longer than 5 Sydney Harbour Bridges. If you are a Sydney-sider, that helps you imagine the actual Uluru rock far better than just saying it is 348 m high and 3.6 km long. The graphic doesn’t show the actual width measurement in metres of the bridge but you can imagine the image repeating more than 5 times to measure the same width as Uluru. The arch span is actually 504 m, plus the width of each pylon probably means the graphic image of Sydney Harbour Bridge represents a width of about 650 m.
The graphic also doesn’t mention that Uluru is about 1.9 km deep, has a circumference of 9.4 km and an area which covers about 7 square kilometres. Or that it takes about 3 hours to walk right around the circumference. Measurements galore.
The most important concept we can take from all this is that as humans we need help to visualise large measurements.
What are you doing to help your students do this? How high is your main school building for example? Perhaps you can find the tallest student in your class and make a paper strip to match this length. Everyone can estimate how many of these strips you would need to match the height of the building. You could then paste different colour copies of this length to make an even longer strip that has clearly marked sections. Is there somewhere you can go to drop this strip from a height to the ground? Using this measurement you can then estimate the height of your building e.g. 10 Jessicas. Check what you find against everyone’s estimates. Were they in the right ballpark? This paper strip can then be used to measure the width and circumference as well.
Pi Day – 14 March
Happy Pi Day. In some countries the calendar dates are written with the month first. So the 14th day of March is written as 3/14. Just for fun, this day is now celebrated all around the world as PI day as the first few digits of pi are 3.14.
There are now so many sites that help you celebrate this event with your students. Pi is the mathematical tern for the relationship between the diameter and the circumference of a circle. The diameter fits around the circumference more than 3 times. This relationship is the same no matter what size circle you create. It is a number that never ends. But it is also a number where absolutely no pattern has ever been discovered. Humans find this astounding. They can’t believe that a number can exist without a pattern attached. They keep searching for one.
This relationship known in the times of the ancient Babylonians and Egyptians. The symbol π was first used to denote the circumference-to-diameter ratio in 1706 by Welsh mathematician William Jones. But it didn’t catch on until Swiss mathematician Leonhard Euler adopted its use in the 1730s.
In 2016, a Swiss scientist, Peter Trueb, used a computer with 24 hard drives and a program called y-cruncher to calculate pi to more than 22 trillion digits — the current world record for the enumeration of pi. If you read one digit every second, it would take you just under 700 000 years to recite all those digits.
March 14 is also Albert Einstein’s birthday. And physicist Stephen Hawking, considered by some to be Einstein’s intellectual successor, died on March 14 2018.
What can you do to celebrate PI DAY with your primary students?
Geometric adventures with a CymaScope
A CymaScope is a scientific devise that makes sound visible.
By adding water to the membrane surface geometrical patterns made by different sounds become visible. The word “Cyma” is based on the Greek word for wave.
Beautiful patterns are created by musical instruments and also human voices.
These images might inspire your Stage 3 students to explore their own pattern-making using a pencil and compass. These ones are made by notes on a piano.
IM Pei Pyramid at The Louvre Paris
I don’t know if you have been to The Louvre but if you have you would know about the beautiful pyramid entrance structure built by the world famous Chinese American architect IM Pei. There is one large pyramid and three small ones in the main courtyard. It opened in 1989 to much controversy – an ultra modern structure next to the traditional French palace structure.
Made entirely of clear glass and metal poles, it is 21.6 m high with a 34 m square base. The surface area is 1000 square metres. The volume is about 9050 cubic metres. There are 603 rhombus shapes and 70 triangular shapes.
The Great Pyramid of Egypt, the Cheops Pyramid, is 138.8 m high now but was originally thought to be 146.5 m high – 280 Egyptian cubits. Each base side was 230.4 m long but with erosion the sides are now 230.24 m long. The total volume is 2 583 283 cubic metres.
Sbahle Zwane – 10 year old maths whizz
Sbahle Zwane is a 10 year old boy in Johannesburg South Africa who has been wooing everyone with his maths thinking – solving problems in his head without the use of a calculator. His mother says that “he only wants to talk about numbers.” She realised when he was a preschooler that he had an ability to work with numbers beyond his age expectations. Sbahle claims he sees all the numbers inside his head. He is using mental maths strategies to calculate. Do you have students in your class who show extraordinary maths skills beyond their age expectation? If so, how are you providing for their thinking?
You can see him calculating here. When he grows up he would like to be a pilot.
Happy 2019 – a free subscription for everyone
We now offer a free subscription for everyone all over the world. We wish you all a brilliant lifetime of fun creating effective maths sessions for your students using all our resources. We will continue to provide you with the highest quality maths activities, photographs and graphics, all carefully sorted into both age and curriculum substrands.
Real Life Maths Examples
Area is one of the most difficult Measurement sub-strands for your students to understand. We use spatial visualisation skills to estimate areas large or small. Areas include surface areas which can be curved and squiggly and difficult to think about. Area calculations also require skills iusing decimal operations (+ – x ÷). And when we talk about Area with our students, we need to use real life examples so that our students deepen their concepts of how area works in the world around them. It can’t be taught as just conversion facts and figures (e.g. How many square metres in a 2 x 12 rectangle?)
This old photo shows a man painting the Eiffel Tower. The total area of metal struts and surfaces that need to be painted regularly is about 250000 square meters. One litre of paint covers about 16 square metres. If the maintenance team need to give the Eiffel Tower 2 coats, how many litres do they need? How many tonnes of paint is that? If bulk paint works out at $5 per litre, how much does just the paint cost? The whole upkeep must be so expensive.
Photographs are a great way to springboard your lesson into real life.
2018 Year 3 NAPLAN NUMERACY NSW Results Analysis
In an effective NAPLAN Numeracy test, we should expect 100% of any Core Stage 1 questions to be answered correctly by 80% or more Year 3 students. This result would show that the majority of Year 3 students demonstrated an understanding of Core Stage 1 content. In this NAPLAN test only 4 out of 15 Core Stage 1 questions were answered correctly by 80% or more Year 3 students. Does this shock you?
That means that our Year 3 students do not demonstrate a solid understanding of basic Stage 1 mathematics content. Call it basic, core whatever, these questions are the ones that should be easy to handle, easy to demonstrate for 4 out of 5 students. That still allows for a small group of 20% or 1 in 5 students to not demonstrate an understanding. We think this is a very reasonable assumption.
Question 14 is a good example.
Students had to select two digits that could be rearranged to make the largest number – 75. Only 62% of Year 3 students did this correctly. As this was a free response question, perhaps the others wrote the largest number that could be made from all 3 digits – 754. If so, this would indicate a strong understanding of place value, just a poor interpretation of the actual instructions. As a general comment, 40% of all 10 free response questions in this year’s paper scored ≤ 50%.
This question was repeated in the Year 5 paper. Still only 76% of Year 5 students answered correctly. And in all 11 Year 3 questions that were repeated in Year 5, only one scored ≥ 80%.
Question 31 was about Area.
Students had to analyse 4 shapes and find two shapes with an identical area. This requires adding square units and understanding that two half units add to make one square unit. The number of square units in each shape was 9 or less. Yes this task requires students to persevere but the task itself was simple. Only 30% of Year 3 students could work out the correct answer. This makes me want to tear my hair out. We need our students to be able to tackle more that just a one step problem. We need them to persevere on a task and not give up too easily. We need them to think logically, eliminate ones that don’t work. What are we doing to help them succeed at problem solving in general?
Question 31 was about time – how to read a calendar.
It was straight forward and did not involve having to imagine the month before or the month after. Why on earth could only 34% of our Year 3 students work this out correctly?
The low results indicate Year 3 students are inexperienced at reading a calendar. The text tells them they are looking for the 3rd Saturday. This should be obvious. It is a pity we can’t access deeper data to show the most common errors. Did most students select 3 October? Or 18 October?
If I were you, I would interview a selection of my Year 3 students to identify what it was that they misunderstood. And you can practice these ideas with your students using our Time Activities – Calendar 1-step S1 Mental Warmups
Year 3 NSW State Results 2018 NAPLAN Numeracy:
67% were Stage 1 questions (24 out of 36 questions)
42% were Core Stage 1 questions (15 out of 36 questions)
73% of these Core Stage 1 questions scored ≤ 80% correct (11 out of 15 questions)
13% of these Core Stage 1 questions scored ≤ 50% correct (2 out of 15 questions)
See the complete analysis in Whole School Planning – Mathematics Improvement Plans – Terms 3/4
Position Numbers and the Calendar
Position terms can be quite subtle for many of our students. For example nine and ninth. One word refers to a finite quantity of 9 objects, the other to the position of an object in a line that starts at 1 and continues to at least 9 places. There are 8 objects before or in front of it and perhaps there are objects behind it too. It is a relationship to other objects not a quantity in itself.
The calendar is packed with position language. Days, weeks and months fit together in a continuous cycle. Nine can be the 9th day of a particular month, the ninth month of a particular year or the ninth year of a particular group of years.
And did you know that in the ninth month 26 September is the only day of the year that is written the same as its position number – 26/9 – 269th day – but not in a leap year (the next leap year is 2020). Amazing.
Victor Vasarely Museum Aix-en-Provence France
Have you visited the Vasarely Museum in Aix-en-Provence France? Victor Vasarely was a famous Hungarian mathematical artist who created amazing, gigantic Op Art tapestries and artworks from the 1940s to the 1980s. I love his black and white linework zebra. Op Art links beautifully to 2D line discussions and the creative power of mathematical thinking. Lots of inspiring images for our Stage 2 and 3 students.
Chance – Crazy Coin Contests
Our website contains a huge variety of maths activities, graphics and photographs. It is easy to get overwhelmed, especially if you are a new subscriber.
When you are preparing your ideas for a particular maths substrand, take a look at what we offer to see if there is something that matches your content needs. For example, Chance and Data. For Stage 2 students, Grades 3/4, there are some wonderful activities for whole class events.
I particularly like Crazy Coin Contests, Y34 ACMSP094. There are 4 activities for your students to work on in teams of 2:
In Millimetre Match you toss a coin with your partner. One person is Heads, the other is Tails. Whoever wins draws a 10 mm line on their “head”. You are trying to be the first to draw 10 lines, but only by experiencing the power of a chance outcome. Plus you are linking to length and drawing 10 mm lines. Millimetre Mike is also available for you to design your own activity in Graphics – People Graphics. Have fun.
Squillons and squillions …
“Squillion” is a made-up word to describe a very large number … but so is “googol”. Milton Sirotta was the 9 year old nephew of US mathematician Edward Kasner. Kasner needed a word for 10 to the 100th in decimal notation. That’s a 1 followed by one hundred 0s, or 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. So Milton suggested “googol” and it stuck. When Google created their company, they thought they were naming it after the mathematical term but they misspelt it.
A google is so large that it is larger than all the atoms in the universe.
Australia’s population reaches 25 million
Today is 7 August 2018 and at 11:00 pm this evening Australia’s population will officially reach 25 million.
The Australian Bureau of Statistics estimates:
- one birth every 1 minute and 42 seconds
- one death every 3 minutes and 16 seconds
- one person arriving to live in Australia every 1 minute and 1 second
- one Australian resident leaving Australia to live overseas every 1 minute and 51 seconds, leading to
- an overall total population increase of one person every 1 minute and 23 seconds.
- 24 million in January 2016
- 23 million in February 2013
- 22 million in May 2010
- 21 million in December 2007
- 20 million in October 2004
- 15 million in October 1981
- 12.5 million in 1970 (half of 25 million)
- 10 million in 1959
- 5 million in 1918.
Australia is expected to reach 26 million in the next 3 years.
Imagine all the amazing discussions you can have with your Stage 2 and 3 students thinking about this data and then researching similar data from different countries around the world. Remember there is also a World Population Clock too. This tells your students the estimated population of the whole world at any one point in time.
More maths jokes
Don’t forget to encourage your students to collect, share and discuss examples of maths jokes. There are so many online sites now that include ones suitable for primary students. Here are some of my latest favourite jokes.
Why is this one so clever? What do you need to know to understand this joke? What are the key aspects of a russian doll that you need to understand before this joke makes any sense?
What is this white shape? What does it represent? What are the key aspects of pi you need to know to understand why this joke works?
A brand new geometric shape – the scutoid
How astounding. In 2018, scientists have found a completely new geometric shape existing in our body. It tessellates. It is compact. It is quite beautiful. It’s called a scutoid. That’s pronounced SCOO TOY D.
The scutoid shape was named after this top-down view of a beetle’s scutellum.
The shape let’s things pack together in the most efficient way. We;ll find out more once scientists conduct further research.
Causes of death in Country Songs
I know this is probably too gruesome for your students to see .. but isnt this a fascinating graph. I wonder what sort of data your students could collect about their favourite songs. What criteria could they look out for? How would they record their data? Will they demonstrate their findings using a column graph, a pie chart or some other sort of graph? Could they talk about their discoveries at the next school assembly?
Maths and knitting
Knitting involves the creation of mathematical patterns and design using ‘wool” and thick needles. You create a type of knot, so it could also be called “knotting”.
As a child, I grew up in the era of “french” knitting, where you hammered small nails along the top edge of an empty cotton reel and used your fingers to create a long knitted ‘rope”.
And now people use very thick wool and just their fingers to create similar ropes. The longest finger knitting is 19 369.5 m long, achieved by Ida Sofie Myking Veseth from Norway on 4 March 2016. Ida lost her sight due to a degenerative disease, which meant that she wanted to find new hobbies.
You might like to try some finger knitting with your students. e.g. Follow these instructions on youtube to create a beanie,
Problem solving is the key to mathematical success
Your students need to be able to apply their maths knowledge to a wide variety of problems. And most problems we meet in real life are quite complicated. They have more than one step to their solution. Yet in school we regularly give students problems that require only one step to answer. Challenge your students to accept this challenge. If they give up at the slightest hint of being unable to solve a problem immediately then they won’t be prepared to accept similar challenges as they grow into adulthood.
My latest book, Blake’s Guide to Maths Problem Solving has over 120 problems with detailed suggested solution strategies to help students get started in their thinking. And it spotlights 7 famous problem solvers to inspire your students to take on a problem solving mentality.
It is available in most newsagents and educational bookstores. Great for use with students in Years 4 – 6.