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# News

## March 2024

In this edition of the News you will find:

Get to Know a Cameo
... Keith's Kubes
... Dice Footy

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• Number Slider

On the left is the Number Slider template provided to Calculating Changes members through the Threaded Activity with the same name (see Link List below). Two pages, a bit of tape and it's done. The image on the right shows how Poly Plug is used to create a hands on Number Slider. Non-members can easily design something similar that works for the counters or blocks they have available.

The ideas in this activity were provided by Alistair McIntosh, who developed the concept during the Mental Computation Project while at the University of Tasmania. In that project, the Number Slider is called a Place Value Board. A few introductory ideas are shown here to remind members of what they can access and to offer others a starting point.

What numbers are hiding under the plugs?
How do you know?
Can you check it another way?

 One yellow plug has replaced the two blue plugs above. How were the blue numbers combined to make the yellow number? One yellow plug has replaced the two blue plugs above. How were the blue numbers combined to make the yellow number?

Now the Number Slider can be used to show big numbers
What number is it showing at the moment?
What is the biggest number it can show like this?
How would you make the Slider show an even bigger number?

 What do you add to make this number into 66? Explain how you worked it out. Can you check this answer another way? What do you subtract to make this number into 37? Explain how you worked it out. Can you check this answer another way?

Members can access 29 more suggestions from Alistair which range from making one digit numbers to using the Number Slider to find pairs of numbers that sum to 1000.

Extension: Doubles 4 Points

 Back in the day, when I was teaching a regular class, two students designed a new game using 6-sided dice. Roll two dice. If you roll doubles you keep the total as points. If you don't roll doubles you get zero points. First to 100 or more wins. They also designed a new way to keep score. Each one had a Poly Plug board arranged as shown and two yellow/blue plugs. They demonstrated to the class how they could score every number from 1 - 99 on their own board. How might they have done it?

Over the next couple of days everyone wanted to try the game - of course. Then another pair asked if they could times the doubles to get points. Sure. Go for it. ... and then everyone wanted to ...

Things grew from there.

• Can we use three dice instead of two?
• Can we use those 10-sided dice?

One group decided the game finished too soon with 10-sided dice? Can we play up to 1000 or more? Sure. Go for it, if you can show me how to score it using Poly Plug boards. They did. ... and then everyone wanted to ...

Things went a bit crazy about then. Kids were soon showing me they could score up to 999,999. And why stop there?

It wasn't until much later that I realised I had missed opportunities for chance and data investigations to be included in this 'feel good' set of lessons.

• More on Princess Catharina's Trinket Chest

Have you watched the video yet? (See Link List below.) It was released last month. It's only two minutes but it captures key concepts necessary for understanding fractions.

Near the end Princess Catharina starts dreaming.
She studied maths at university, so perhaps it's second nature to daydream in fractions.

 What happens if the imagined central stack is only 1 drawer? Then that drawer would be ... of the whole trinket chest. And each drawer on the left would be ... of the whole trinket chest. And each drawer on the right would be ... of the whole trinket chest. And I could write a sentence about the fractions I see in the chest and include the word 'equals'. Actually I could write more than one.

Yes, Princess Catharina thinks and writes in pictures and words before she uses symbols.

• What happens if the central stack is 2 drawers? ...3 drawers? ...4 drawers? ...

• Get to Know a Cameo

 This task was contributed by Keith Windsor, mathematics teacher and consultant from UK. It's simply amazing that so much logical and spatial reasoning can flow from 4 cubes of one colour and two of another; skill development and practice too. Students each join their four cubes of the same colour to make a square. The problem is to join on the other two cubes in as many ways as possible so the new object still sits flat on the table. Working in pairs, each with their own cubes, encourages mathematical discussion and persistence. Recording becomes essential so that repeats are not included. But how can 3D shapes be recorded on 2D paper? Plan view is introduced on the card, but elevation view and isometric view can also be introduced.

The card also supports learning by quietly suggesting there are 6 solutions to this initial question. The Challenge on the card requires making all 6 objects by gathering more cubes from the class supply and then fitting them together to create a 6 x 6 square - in two different ways.

For the teacher the Cameo provides solutions, introduces the extension question:

• What happens if the cubes can also be placed on top or below the starting square?

and provides a link to the Picture Puzzle menu in which Keith's Kubes is one of the challenges. In this form the task is investigated using the pedagogy of one screen, two learners, concrete materials and a challenge. On this menu there are four other Picture Puzzle slide shows with similar content.

In the eTask Package this task is in the 'easy' set because making it only requires linking cubes which should be part of every school's resource stock.

 This task is a mathematical model of the great game of Australian Rules Football. Playing Aussie Rules takes a lot of skill and a bit of luck. Playing Dice Footy is all luck ... but it feels a lot like a real game when you score. The card explains the scoring rules then invites partners to play 5 rounds - producing 5 winning scores and 5 losing scores - and from that information calculate the average goals, behinds and points in a match. The card doesn't suggest why the average should be calculated, but students might wonder how their average compares with the averages in the current round of the real game.

(And if they don't wonder for themselves a question or two from the teacher will set them on that path.)

So the task encourages skill practice in a fun, game situation. But using dice also opens the door to chance and data investigations. The Challenge section of the card hints at making use of the highest and lowest possible scores to find a way to calculate the average theoretically. The Cameo goes further by introducing a method based on expected value.

Collecting and displaying data is a massive part of the real game. The Cameo, especially through its link to the Maths At Home version of Dice Footy, also offers possible ways to extend the task into this content area and provides a playing board and 'stats' sheet pro forma.

In the eTask Package this task is in the 'easy' set because, apart from the printing it only requires four dice, a pen for recording on the laminated playing board and a wiping cloth.

Keep smiling,
Doug.

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