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# News

## October 2022

In this edition of the News you will find:

Get to Know a Cameo
... Rectangle Nightmare
... What's In The Bag?

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• New eTask Schools

Welcome to

• Holy Cross College (WA)
• Thomas Hassall Anglican College (NSW)
• Frankfurt International School (Germany)

all which have begun building their own Task Library using the eTask Pack since our previous edition of eNews. We wish you success in setting up and making use of this resource to help your students learn to work like a mathematician. We hope to hear from your classrooms in the future.

• A Different Angle on Angles

 You will be amazed at what children can learn about angles without measuring in degrees. This part of the geometry curriculum is so much more than the skill of counting to 180. A Rotagram is a clear plastic circle set into a clear plastic square. The circle rotates against the square. One arm of an angle is marked (and fixed) on the square. The other is marked on the circle and consequently rotates. Focus is on the concept of an angle and the way it is measured, rather than on the skills related to using a protractor to find a number to represent the size of the measurement. In the photo an angle has been drawn on the paper and the Rotagram circle has been turned and fitted on to the drawing to match the size of the angle. An action that recreates the drawn angle and measures it at the same time. This captured angle with its measurement are now portable and can be compared to other angles elsewhere.
Last century, Geoff Giles designed and produced these simple, powerful learning tools and backed them up with work booklets and guide sheets arranged in units of work. The units are designed as self-directed and can be used by partners, small groups or the whole class. They have been extensively tested in classrooms. The three units are titled Equal Angles, Rotation and Directions and each includes a Target Test.

This century, Mathematics Centre had Rotagrams manufactured in Australia when they were no longer available elsewhere. We also obtained permission to have Geoff's Rotagram print resources freely available from our site. See Link List below for Rotagrams, Work Booklets and Worksheets. Rotagrams are also used in Task 214, Angle Estimation.

• Trial, Record & Improve

This threaded activity from Calculating Changes (see Link List below) models the way many real world equations are solved by mathematicians. Neat first and second degree equations that can be solved by a recipe of doing something to both sides are important. However, Trial, Record and Improve, which on the Working Mathematically page (see Link List below), is the 'Guess, check and improve' strategy, deserves equal air time.

• When it is used children's number sense is exposed, explored and enhanced in a way that just can't happen with a formatted solution.

The key to the approach is the freedom to use a calculator at will, in the same way as mathematicians use computers to try an intelligent guess at a solution, then build the next intelligent guess on that outcome.

 7 + + 3 - 5 = 7 For example, even children in Year 2, perhaps younger, can be encouraged to take a guess when they know they can use their calculator to check their hypothesis. Trial ... I guess 8 and try it with the calculator Record ... 13 too big Improve ... I'll try 6

Trial, Record & Improve (see Link List below) includes several examples of Year 2 and 3 children's work using this process to solve:

+ = 55

Also, as a challenge for you personally to explore, the approach offers:

1·8 ÷ + 6 = 6·36

and an important cautionary note about using calculators.

• Get to Know a Cameo

Task 84, Rectangle Nightmare
 Rectangle Nightmare is a geometric paradox which begins as two jigsaw challenges. Here's 5 shapes and a frame to place them in. Make them fit. Here's the same 5 shapes and the same frame and an extra unit rectangle. Make the 6 pieces fit into the frame. It sounds a bit crazy because 5 pieces take up all the space inside the frame, so how can there be room for an extra shape. All the more astounding when you find it can be done. Actually it can be done in more than one way. What happened to conservation of area?
For some - middle primary students for example - just exploring these 'jigsaw' challenges will be enough.

For others, the depth of the task's iceberg is revealed by its creator in a paper specially prepared for this cameo. From the 'jigsaw' Geoff Giles leads us into non-proportional and proportional scaling, ratio and gradient and Fibonacci Numbers. We have placed the paper in the From The Classroom section of the cameo, because Geoff writes to teach us. The explanation is indeed from his personal classroom. With its emphasis on what you see being an insufficient measure of proof, this task may become an essential part of your middle secondary geometry curriculum.

In the eTask Package this task is in the 'special' set because it requires printing, laminating and careful cutting of a special set of shapes based on a unit rectangle and similar preparation of the matching frame. A template is supplied.

Task 198, What's In The Bag?
 Using a classroom experiment totally consistent with the way mathematicians work in situations, such as polling to predict election outcomes, the students have an opportunity to design their own method for interpreting data from a population sample. In this case its data about the colour of cubes in a bag. Ten mixed coloured cubes are hidden in a bag. Three times a sample is taken, colours are recorded and the cubes are returned to the bag. Based on the three samples students are asked to predict the number of each colour in the bag.

Although students will want to know (and can easily find out) if their prediction is correct, the main focus is the reasoning that led to the prediction. To help you support students to extend their thoughts about how to predict, the cameo supplies teachers with four possible methods. Your students are very likely to design something different. In this inexact investigation, who is to say which method is better?

In the eTask Package this task is in the 'easy' set because it only needs an opaque bag and small coloured cubes.

Keep smiling,
Doug.

• Did you miss the Previous News?
If so you missed information about:
1. New eTask Schools
2. Tasks, 3 Lives & Poly Plug
3. Target Range
4. Get to Know a Cameo
... Number Game, Cover Up

### Did You Know?

Previous News

Continue exploring our history back to July 1992 through the Sense of History link.