In this Big Idea students explore the difference between general aspects of a mathematical situation; those that always occur, and those that are particular to a single instance. This introduces students to one of the essential concepts within mathematics and an essential component of the discipline; the search for genaralisations or ideas which remain the same while the specific details change. Generalising is the process through which mathematical rules are defined and discovered. When students are taught about generalisations they tend to see these as a method to be applied in a particular setting. Instead of being taught about generalisations, in this Big Idea students are taught to generalise and this will require students thinking mathematically, proposing possible generalisations based upon their observations and developing proofs.
Mathematics is a reasoning and creative activity employing abstraction and generalisation to identify, describe and apply patterns and relationships. (NESA Mathematics K-10)
Students will understand:
By the end of Stage 3, students ask questions and undertake investigations, selecting appropriate technological applications and problem-solving strategies to demonstrate fluency in mathematical techniques. They use mathematical terminology and some conventions, and they give valid reasons when comparing and selecting from possible solutions, making connections with existing knowledge and understanding.
This Big Idea builds on student's understanding of data displays, visual representations and diverse strategies to document their thinking. Student will look for patterns and develop explanations to suit. They will utilise their mathematical language to describe what they see and think is happening. They will develop and defend proofs.
Patterns & Algebra
Students explore making generalisations about numerical patterns presented graphically, to build an expansive understanding of what generalisations can be. (See page 223)
Questions for reflection:
We explore generalising strategies by playing the game Pennies and Paperclips and developing statements about how players can win.
The game Pennies and Paper clips has been developed to teach mathematical proof to college students with the 'Art of Mathematics' programme. Instructions are as follows.
Pennies and Paperclips is a two-player game played on a 4 by 4 checkerboard with a standard color pattern. One player, "Penny", gets two pennies as her pieces. The other player, "Clip", gets a pile of paperclips as his pieces. Penny places her two pennies on any two different squares on the board. Once the pennies are placed, Clip attempts to cover the remainder of the board with paperclips — with each paperclip being required to cover two adjacent squares. Paperclips are not allowed to overlap or to be placed diagonally. If the remainder of the board can be covered with paperclips then Clip is declared the winner. If the remainder of the board cannot be covered with paperclips then Penny is the winner.
Questions for reflection:
Encourage and support students in documenting their thinking to allow generalisations to be debated and built on.
Students investigate generalising from visual patterns by exploring networks of toothpick squares and the number of toothpicks it takes to construct them.
Questions for reflection:
