News

July 2019 In this edition of the News you will find:   Engineering 'aha' Moments in Algebra   Mathematics Education is not an Enigma   Origami Sphinx Tiles   Get to Know a Cameo      ... Pythagoras 2      ... Triangles & Colours Please feel free to distribute this information.

Email News Services If you would like to subscribe please send an email request to: doug@blackdouglas.com.au: In the subject line write Subscribe Maths. In the body of message state which divisions you want to subscribe to, or simply write Subscribe All. Add your name and preferred email address. Update messages are sent from: Mathematics Centre monthly Calculating Changes each term Working with Mathematics Centre will help you find your way around our site Browsing Hint Use the View/Zoom options of your browser to 'move' the page closer or further away. Ctrl +, Ctrl - and Ctrl 0.

• Engineering 'aha' Moments in Algebra

  During this month we ran this workshop with Waverley Christian College secondary maths staff at their Wantirna campus. The session began with Task 147, Garden Beds as a whole class investigation. (See Link List below which includes the same task being used in this way in a Year 2 class).

  30 minutes into the session we had verbalised the problem, built a model on the floorboard, discussed and tried a few specific cases, recorded the data in our maths journal, been challenged to find the number of tiles needed to make...	...a complete tiled border around a linear garden of 100 plants and teachers were back at their tables, in pairs, with concrete materials, and the challenge of finding two ways to use the materials to explain how to find that answer.
  Can you see the explanation these teachers are working on?	The start is clear here. How were the other tiles used?

  By the end of the half day we had found at least six explanations and explored content in... Oral, written language and symbolic generalisations of each one Concept of a variable Substitution into equations Solution of equations Equivalent algebraic expressions Manipulation of algebraic expressions - collecting like terms, distributive law Number patterns Mental arithmetic Graphing ordered pairs Tables of values Gradient and intercepts Domain and range Differentiation of a function

  ...using teaching craft such as... story shell mathematical conversation making a model peer teaching keeping a journal visualisation / brain pictures kinaesthetic learning communication mathematics physical involvement interactive technology group negotiated decision making concept focus skill practice game context student ownership ...

  and ... considered the possibility of changing the shape of the garden bed; reviewed student output from a couple of classrooms; confirmed that throughout we were working like a mathematician; linked the session to a recent PD detailing the benefits of building relationships with the students; realised that related text book work would now go more quickly and successfully with kids; decided that this problem can pretty much teach the whole linear algebra curriculum and ... were ready to start all over again with a quadratic example.

  • A version of Garden Beds can be found as an exercise in any text book. For how many minutes would it be in front of the students' eyes. What might their hands be doing in those minutes. What would be the likely expected outcome from the textbook writers' viewpoint?
  Could this be a professional development workshop for your staff or cluster?

  It is available as a half day or full day. Both of these are derived from our 6 Day Program used by clusters or regions or systems of schools working together to support significant and sustainable curriculum shift.

  See Link List below for more on the 6 Day program and teacher review from one cluster's experience with the course. A similar range of professional development courses are available based on other content strands in both primary and secondary schools.

• Mathematics Education is not an Enigma

  This paper was recently added to our Web Papers link. (See Link List below.) Doug Williams prepared it as the Keynote Address at the Association of Teachers of Mathematics (ATM) Easter Conference 2012, Swansea University, Wales. The theme of the conference was 'Enigmas'. It was first published in ATM journal Mathematics Teaching in 2012. In her introduction, Margaret Jones, editor, writes:

  This is storytelling about the learning and teaching of mathematics at its best. The descriptions are of real classrooms with real teachers, and real learners.

  Also this month, Prime Number, the primary teachers journal of the Mathematical Association of Victoria has published two of Doug's articles.
  • Hotel Sevilla - a stimulus photo of the atrium of the Hotel Sevilla Center with an extensive set of learning challenges it could generate. (Contact doug@blackdouglas.com.au for a copy of the photograph to print or project.)
  • Win A Flat and Threading - a powerful place value game is used to illustrate teaching craft shown to engineer 'aha' moments in number.

  When you are looking for a good professional read, our Web Papers link includes several other papers and presentations with titles such as:

  • Aha! Teaching Maths is Simple
  • An Ocean of Possibilities
  • How Can Solving the World's Hardest Problem Inform Mathematics Teaching?
  • Maths Not At The Movies
  • Menu Maths & Other Models for Making Mathematicians

• Origami Sphinx Tiles

  Thanks to Matt Skoss who has found a Google Slides presentation showing how to fold a Sphinx tile. The link has been added to the Link List below and to the Sphinx Album. The Sphinx Album contains another approach to folding Sphinx tiles which was contributed by David Mitchell from Origami Heaven. David's approach begins with an A size page and doesn't use scissors. The slide show starts with a square of paper and does need scissors.

• Get to Know a Cameo

  Task 189, Pythagoras 2
  

  Pythagoras 2 is a classic demonstration of Pythagoras' Theorem. The triangle pieces only have to be translated and in doing so produce two versions of the same uncovered space. The first, one square. The second, the sum of two smaller squares. The geometric representation is there in squares and it can also be represented by the two sides of the classic Pythagorean equation c2 = a2 + b2. That is, Pythagoras Theorem is about shapes called squares before it is about numbers called squares which count the area of those shapes. The cameo provides an explanation of the task and an algebraic proof linking the geometry to the symbolic representation of the theorem. It also describes how a whole class lesson can be built from the task and suggests several extensions.

  In the eTask Package this task is in the 'more work' set because there is additional printing, laminating and cutting involved in its preparation.

  Task 221, Triangles & Colours
  

  This easy to start investigation only involves making triangles with coloured sticks the same length. At the level of, say, Year 2 it has plenty of mathematics related to sorting and classifying, counting and reasoning. At Year 12 it opens the door to mathematical induction. And there's heaps of pattern, generalisation and linear, quadratic and cubic algebra in between. Perhaps best of all it provides a wonderful opportunity to practise working like a mathematician. If only one colour is used, only one triangle can be made. If only two colours are used, there are four possible triangles. One all Colour A, one all Colour B, one 2A and 1B, one 1A and 2B. But what happens if we have three colours to use? The task card leads the students on to make all the possibilities for four colours, then, based on the data gathered so far, they are asked to predict for five colours.

  But why stop there? If I tell you any number of colours can you tell me the number of triangles that can be made?

  The cameo provides plenty of explanation of the patterns, all supported by colourful diagrams, and guides the exposition with the question "What might a mathematician do with this data?".

  In the eTask Package this task is in the 'easy to make' set because it only needs coloured pop sticks (or similar) which are available in craft shops.

Link List Garden Beds 6 Day 'aha' Algebra Course Cluster Review of 6 Day Course Working Mathematically Mathematics Education is not an Enigma Web Papers Origami Sphinx Tiles: Google Slides Sphinx Album Task 189, Pythagoras 2 Task 221, Triangles & Colours Task Library eTask Package Order Form Did you miss the Previous News? If so you missed information about: Real Mathematicians at Work Assessment Support Get to Know a Cameo ... Shape Algebra & Land of ET ...and more...

Did You Know? You can Search Mathematics Centre. Our Site Map acts as a Table of Contents to help you find what you need in Mathematics Centre. You can find photos of every task using the Task Library. You can find tasks by content using the Task Cameo Content Finder. You can create your own Task Library in a professional learning context with the eTask Package. Calculating Changes offers Content Finder & Year Level Finder tools. You can find tasks coded by Year Level and Curriculum Strand in the Task Centre Catalogue. (PDF file) You can search for lessons by Year Level, Curriculum Strand, Lesson Features & Keyword at Maths300. You can connect Tasks to their Maths300 companion lesson using Tasks & Maths300. (PDF file) Excel version provided by Pymble Ladies' College also links each Task to its Maths With Attitude content strand. All these links, and more, appear in Working with Mathematics Centre. (PDF file)