## Who is this Sequence for?

These lessons are designed for students with a wide range of prior knowledge of multiplication. Some students may already be able to multiply a two digit number by a one digit number fluently, whereas for others this will be a new challenge. The focus on understanding multiplication and its properties through arrays is intended both to support learning of new multiplication skills and to deepen students’ appreciation of rules that they have learned.

## Summary of learning goals

This sequence of three lessons sets out to build students’ multiplicative agility – their efficiency, flexibility and accuracy in multiplication. The lessons aim to give students a range of strategies for multi-digit multiplication, highlighting strategies based on the distributive property and the associative property. The lessons develop the array as a tool to reason about multiplication, and they move in a careful developmental sequence from an array with all items perceived, to a grid array, and on to an open array and area model for multiplication.

## Rationale for this sequence

This sequence of lessons both strengthens prior learning about multiplication and prepares students to select and apply appropriate strategies for multi-digit multiplication. An overarching goal is to develop the array as a tool for reasoning about multiplication. Looking back, with this tool they will be able to deepen their understanding of earlier content. For example, they can explain the links between the number facts in the 6 times and the 3 times tables, and generalise these links. Looking forward, students can link partitioning principles to the multiplication of fractions and algebraic expressions.

Multiplicative thinking is a major theme of school mathematics with developments occurring from about Year 2 to about Year 9. It is well known that students often experience difficulty with the many transitions that need to be made.

An important feature of this sequence is the insight that it provides for teachers into students’ ideas about multiplication. This supports teachers to provide a differentiated experience to help each student develop a more complete understanding.

**reSolve Mathematics is Purposeful**

Conceptual Understanding: The carefully sequenced use of arrays to explain multiplication strategies is designed to build students’ understanding of multiplication. The problems in the sequence include both equal groups and rate situations.

Fluency: The fluency goals of this sequence lie in flexibility of calculation and having multiple methods available. The sequence provides an opportunity to use multiplication and division facts, and more importantly to select useful facts to make multiplications easier. For example, there is an opportunity to develop flexible mental computation skills.

Problem solving: In lesson 2, there is an opportunity for creativity when finding ingenious alternative ways to multiply numbers based on their factors. In lesson 3, there are interesting number patterns to find, some of which can be explained with the creative use of arrays.

Reasoning: Many of the activities require students to explain their reasoning about multiplication using an array.

**reSolve Tasks are Challenging Yet Accessible**

This sequence is accessible to all students. A range of work samples are provided for teachers so that they can identify the level of students’ understanding and assist students to move forward. For example, in lesson 1, there are work samples that help teachers to identify students who use counting strategies, through repeated addition and additive doubling, up to the flexible exploitation of the properties of multiplication. Similarly, the nature of the array model is carefully sequenced moving from an array with all items perceived, to the abstraction of a grid (which still represents all items) and on to the more abstract open arrays and area model for multiplication. Students can make these shifts when they are ready. The nature of the multiplication situation is also carefully sequenced, beginning with equal groups and moving to rates. Challenge is provided in the need to explain reasoning clearly, and in several opportunities for open investigation.

**reSolve Classrooms Have a Knowledge Building Culture**

In this sequence there are many opportunities for students to learn from each other. In discussions, students will see work samples presented by other students and hear their reasoning. This reasoning will be at a variety of levels of sophistication of multiplicative thinking. The teacher will actively orchestrate this sharing to highlight connections between solution strategies, explore the efficiency of some strategies over others and allow opportunities for students to ask questions.