Year 6 Directed Number

Australian curriculum number (ACMNA124; ACMMG143)

Who is this Sequence for?

These lessons assume no previous experience of directed number. At the simplest level, students only need to be able to count up and back with positive whole numbers, including using a number line. In lessons 2, 3 and 4, there are opportunities to use decimal numbers, but whole number versions of the lessons are possible.  Lessons 1, 2 and 3 should be done in sequence. Lesson 4 follows these lessons at any time, perhaps as part of a unit on location. It is designed for students who can already locate places using a map grid reference.

Summary of learning goals

These lessons introduce negative numbers through a real world context. Students build a strong mental model that they can use to understand what negative numbers are and how they work. The intention is that they will be able to draw on this mental model as they work with negative numbers in later years. They learn how negative numbers are used in several everyday contexts, such as temperature. They use the positive and negative number lines to make the Cartesian plane with four quadrants and use it to describe location. 

Rationale for this sequence

For most Australian primary teachers, this is a new topic to teach. Students experience many difficulties in this topic in secondary school. They have difficulty making sense of the numbers themselves and of calculations: for example, students ponder how you can have -3 apples and how you can take 10 apples from 6 apples.

Therefore this sequence aims to provide a firm foundation for conceptual understanding of negative numbers (i.e. that they indicate directed quantities), and for understanding the meaning of a limited range of calculations with them. This is done by using an everyday situation (height above and below ‘ground level’) as a mental model from which students build the abstract numbers. The focus is on conceptual development and everyday applications (from height, to temperature and credit/debit). Students use a number line (physical or mental) to support any calculation that is required. Symbolic arithmetic is left to Year 7 whilst this sequence remains within direct sense-making. 

The sequence does extend the ACM specification to examine integers to include negative decimals and fractions, because it is easily supported by the real world model of height above and below ground, and in so doing, can address some commonly observed misconceptions. In Lesson 4, the sequence extends to work on the four quadrant Cartesian plane; the only other instance of negative numbers at this level of schooling.

The overarching plan of this sequence follows the principles of Realistic Mathematics Education (RME), which originated in the Netherlands and has been shown to be very successful in studies conducted in many countries around the world.

reSolve Mathematics is Purposeful

Conceptual understanding is the main aim of this sequence. Students start with a real world situation, from which they abstract directed numbers. They use the real world situation to reason about their properties. Later they see how this abstract system can be applied in other real world contexts such as temperature, credit/debit and for specifying location with Cartesian coordinates. 

Reasoning about directed numbers is intended to be done first by harnessing knowledge of everyday situations involving height, and later by using the number line to support calculations.  Reasoning with this mental model will address common misconceptions such as whether -0 is a separate number and the placement of negative decimals. There are similar opportunities in all lessons.

reSolve Tasks are Challenging Yet Accessible

Accessibility in this sequence arises from the use of the real world situation to develop a firm mental model. Transition from relying fully on the real world situation, to using the somewhat more abstract number line (physical, then later mental) occurs at a pace set by the student. The use of several real world situations throughout the sequence provides an opportunity to reinforce the new ideas, as well as extend them. 

There are many opportunities for teachers to adjust the level of challenge in this sequence, for example in deciding whether and when to use only integers or to include negative numbers with decimals, and in the nature of the patterns that students might discover.

reSolve Classrooms Have a Knowledge Building Culture

Several activities are in a game format, enabling discussion between students. Through this sequence, students build a deeper knowledge of directed number and also a bigger understanding of fractions and decimals. The teacher is central to the development of this knowledge building environment in promoting the positive attitudes of curiosity and independence.