## Who is this Sequence for?

The sequence can precede, accompany or follow work on algorithms for multiplying and dividing by fractions and decimals. Students need to know the meaning of fractions and decimals, but the complexity of the numbers involved can be adjusted by the teacher.

Students need to be able to use multiplication in situations of repeated addition and also to find a fraction of a quantity. In Year 6, that quantity is a whole number (ACMNA127). They need to understand how division applies in situations where a quantity is divided into a certain number of parts and the size of the part is required, (partition division e.g. 6 divided in 3 equal parts is 2) and also where a quantity is divided into parts of a certain size and the number of parts is required (quotition division e.g. how many 4s in 20?).

## Summary of learning goals

This short sequence aims to improve students’ understanding of the effect of multiplying and dividing by numbers that lie between 0 and 1. Some tasks involve fractions, others decimals and some both. Many students find it difficult to decide whether multiplication or division is required to solve a problem in a real context, and this sequence aims to improve students’ ability to do this by working on fundamental understandings of the meanings of these two operations.

## Rationale for this sequence

The aim here is to improve students’ understanding of the effect of multiplying and dividing by numbers that lie between 0 and 1, and to address two common misconceptions. A central feature of the middle years of mathematics is to move from working primarily with whole numbers to operating smoothly with all numbers. This does not only involve learning new algorithms for calculation. For multiplication and division, it requires extending students’ ideas of the meaning of the operations from the first understandings as repeated addition and partition/quotition, to include multiplicative comparisons and ideas of rates, ratios and proportional reasoning more generally. It also requires them to replace old ideas that may have served them well in the past. In particular, it is no longer true that they cannot divide a smaller number by a larger. It is also no longer true that the product of multiplication is larger than either of the factors, and that dividing makes a number smaller. This is the ‘multiplication makes bigger, division makes smaller’ (MMBDMS) misconception.

Having misconceptions such as these adversely affects students’ ability to solve problems in context and prevents them from easily checking the results of their own calculations. There is evidence of these two misconceptions all around the world and over decades, although the incidence and prevalence differ from place to place and from time to time. They are currently common in Australia, affecting a reasonably large group of students in the middle years. For example, there is an on-line SMART::test quiz on the MMBDMS misconception available from www.smartvic.com. Students need to identify the operation to find the cost of 3 kg of sausages at $11.80 per kg, 2.8 kg of fish at $9.20 and 0.7 kg of steak at $15.30. In a sample of students mainly from years 6 to 8, 422 answered the first two questions correctly (choosing multiplication). They recognise that multiplication is the correct operation to find a total cost from cost per kilogram and number of kilograms. However of these 422, 87 (26%) chose division for the third item. Most of these students know the cost of the steak is about $10. A major reason why they choose division instead of the correct multiplication is that they believe multiplying the numbers would give an answer greater than $15.30. The same misconception also operates when students fail to identify division – it is often because they believe that division would ‘make the number smaller’. Many students will naturally acquire this misconception when they learn whole number multiplication and division. The important teaching task is to address it as a key part of teaching about multiplication and division by fractions and decimals.

### reSolve Mathematics is Purposeful

Conceptual understanding is the main purpose of these lessons, and especially building up meaning and addressing misconceptions. This is done by (a) identifying the problem through diagnostic assessment, then (b) providing students with collaborative tasks designed to provoke thought and highlight the difficulties, then (c) resolving through whole class discussion. Follow up tasks consolidate this learning and develop fluency.

### reSolve Tasks are Challenging Yet Accessible

All students can participate in these games and group tasks. The rules of the games can be readily adjusted to begin at an appropriate place for students, and to increase in difficulty as their skills improve.

### reSolve Classrooms Have a Knowledge Building Culture

Collaborative tasks are used as an integral part of the lesson, so that students try out ideas as they work together to achieve a common goal. With the active leadership of the teacher, the whole class discussion phases draw out students correct and incorrect ideas, clarify issues and resolve the intellectual conflicts.