Number and algebra – multiplication and division

• Uses a range of mental strategies and concrete materials for division MA1-6NA
• Describes mathematical situations and methods using everyday and some mathematical language, actions, materials, diagrams and symbols MA1-1WM

Content

• model division by sharing a collection of objects into groups of a given size, and by arranging it into rows or columns of a given size in an array, for example, determine the number of columns in an array when 20 objects are arranged into rows of four

Linked syllabus outcome

MA1-5NA uses a range of strategies and informal recording methods for addition and subtraction involving one- and two-digit numbers (repeated addition)

National Numeracy Learning Progression Mapping to the NSW mathematics syllabus

When working towards the outcome MA1-6NA the sub-elements (and levels) of Additive strategies (AdS6), Multiplicative strategies (MuS4-MuS5), Number patterns and algebraic thinking (NPA5) and Interpreting fractions (InF1) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

Materials

• Blank paper and a pencil
• Students can be provided with 24, 15, 12 counters or teddy bear manipulatives if required.

Teacher instructions

The purpose of this task is to gauge students’ understanding of multiplication and division concepts such as:

• recognise patterns
• describe pattern relationships
• model division by sharing a collection of objects into groups
• rhythmic or skip counting
• the use of equal groups of objects

Teachers can provide students with 24, 15 or 12 counters or teddy bear manipulatives if they have any difficulty drawing their solution. Teachers read the question and instructions to the student and observe the students’ response, looking for the strategy that the student is using.

When students have drawn the cupboard or arranged the manipulatives, ask students to label or tell you how many rows there are and how many in each row.

Extension questions

How many shelves could be in the cupboard?

Is there only one possible solution?

What other solutions can you make?

(Possible answers for 24 include 1, 2, 3, 4, 6, 8, 12 shelves)

Investigate how students share amounts:

• by dealing each item by ones but unable to coordinate the number of equal groups and the number of items in each group to calculate the product.
• uses rhythmic or skip counting.
• does not use the visible support materials but counts using fingers as perceptual markers to represent each group
• uses composite units in repeated subtraction starting at 24, 15 or 12 and counting back using the unit a specified number of times.
• applies known multiples and strategies for division to mentally calculate e.g. 24 ÷ 4 = 6 (without using any diagrams or concrete materials, relies on known facts).

Student instructions

24, 15 or 12 (teacher discretion) teddy bears are on shelves in the cupboard. There are the same number of teddy bears on each shelf. What might the cupboard look like? Draw your answer or make your answer with the counters/teddy bears. Is there only one possible answer? How do you know?

Where to next?

Represent division as grouping into equal sets and solve simple problems using these representations (ACMNA032)

Model division by sharing a collection of objects equally into a given number of groups, and by sharing equally into a given number of rows or columns in an array.

Describe the part left over when a collection cannot be shared equally into a given number of groups/rows/columns (Communicating, Problem Solving, Reasoning)

Model division by sharing a collection of objects into groups of a given size, and by arranging it into rows or columns of a given size in an array, e.g. determine the number of columns in an array when 20 objects are arranged into rows of four.

Describe the part left over when a collection cannot be distributed equally using the given group/row/column size, e.g. when 14 objects are arranged into rows of five, there are two rows of five and four objects left over (Communicating, Problem Solving, Reasoning)