## From Conjecture to Proof

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(grades 6-10) Ken Milton and Howard Reeves

### Introducing and developing ideas about proof

The study of proof in mathematics has historically been restricted to the geometry strand of the curriculum and studied at secondary school years. This book challenges the traditional treatment of proof. The authors show how school mathematics (not just geometry) is rich in opportunities for students to explore patterns, to seek and describe generalisations, and to make and prove conjectures. From Conjecture to Proof is about learning proof and acquiring skills of constructing proof. It is a book for teachers and teachers in training, which revisits the ideas of mathematical proof in a way that encourages students to learn by ‘doing’.

# The new student book for the Mathomat V2 template

### 120 pages in three sections

#### Section 1

40 illustrated investigations with the Mathomat V2 template. Put students 'into' a situation so they can make sense of the mathematical relationships and operations involved. This helps in forming and operating on the mental imagery that is central mathematical thinking.

#### Section 2

More about Mathomat. 14 pages of further investigations with the Mathomat V2 template. These less contextualised activities encourage students to build on the investigations from section1.

#### Section 3

Mathomat V2 diary. A place to deepen understanding of the Mathomat V2 activities through reflection and by learning to classify patterns that have been previously created; such as classifying the vertices formed in tessellation designs

### About the new investigations in the Mathomat V2 student book

#### The understanding angles series

In the new 'Understanding angles investigations' learners are encouraged to develop a concept of what an angle is, before using the Mathomat protractor for precision angle measurement. These activities reflect research studies* which argue that students develop a spatial structuring of angle by abstracting what is common between three distinct angle contexts; corner angles (where both arms of the angle are visible), slope angles (in which only one arm of the angle can be physically seen), and turning angles in which all features of the angle must be imagined dynamically. Our new Mathomat V2 instruction book gives students the opportunity to draw and measure the standard angle concept in these situations. This activity is scaffolded by asking students to visualise a drinking straw as they work. Students are asked to use the Mathomat protractor for precision measurement to confirm their initial estimates of angle size. A key skill in this precision measuring task is to be able to imagine, and to mentally place, the second arm of the angle after alignment of the physical Mathomat protractor.

#### Mathomat and symmetry

The Get transforming investigation in the Mathomat 4th edition student book is being extended in the Mathomat V2 student manual to give students a sense of the four transformations underlying earlier tessellation activities. These fully explain the transformation of motifs without changing their size or shape and are sometimes called the isometries of the plane. They are: rotationtranslationreflection and glide reflection.

The Mr Symmetrical activity from the Mathomat 4th edition student book is being extended in the Mathomat V2 student manual to give students experience with rotation and reflection as symmetry operations which transform a shape into itself. This includes classifying each of the Mathomat shapes according to its number of line and rotational symmetries.

#### New diary section of the Mathomat V2 student manual

In the diary section of the Mathomat V2 manual learners will be asked to 'size-up' the potential symmetry operations in Mathomat at a single glance through use of scientific classification of their symmetry properties.

### What's new in the Mathomat Version 2 template

The version 2 Mathomat will be available in June 2018, and is a more powerful product - being the same size as the current Mathomat but with many new functions and features built into its design. These include:

## Enhanced 2-D pattern drawing

The new large regular octagon, and the resized large regular pentagon combine to form a group of regular polygons with 15mm sides to compliment the very popular existing regular polygons with 10mm sides. Students now have more creative freedom in 2-D designs to suit their project style.

The Mathomat V2 manual is a rich source of creative drawing ideas; for classroom or individual student use.

Sometimes students want to fill the whole page with their drawing. At other times its best to leave space for the remainder of a presentation. The Mathomat V2 offers both creative options.

## Enhanced graphing

#### The Mathomat V2 template includes

• A larger sine/cosine curve
• Integration of unit circle, sine curve, linear radian scale and new trig scale to form a 'function machine' that can help students to learn trigonometry through visualisation of the unit circle method.

• Larger parabola

• A new normal distribution curve

#### Illustrated below:

The solution to a senior school trigonometry problem. Finding the hours of the day at which it is safe to cross the harbour bar of a fishing village by boat

The larger sine curve makes a striking improvement to hand drawn sketches. The integrated unit circle allows students to find angle values as distance travelled around its circumfrence before locating them on the x-axis of the graph of that function.Lesson 11 in Maths with Mathomat in the free resources section of this website provides a comprehensive plan for using Mathomat with the unit circle method of teaching trigonometry.

## Enhanced 3-D sketching in the Mathomat V2 template

• New kite for sketching square based pyramids

• New ellipse to assist with sketching conics

• A 1:2 scale for detailed engineering drawing
• More challenging fraction markings around circles for greater student interest

A new 1:2 scale ruler for detailed drawings, especially useful in engineering drawings

Ask students to redraw this section at a scale of 1:2 using their Mathomat V2 template

The many circles on Mathomat have a revised set of graduations to create interesting fractions such sevenths, and ninths, to challenge students to think flexibly about numbers