Number and algebra – Sharing counters in rows

• 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

Syllabus content descriptors

Develop efficient strategies for numerical calculation, recognise patterns, describe relationships and apply algebraic techniques and generalisation

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)

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

National Numeracy Learning Progression (NNLP))mapping to the NSW mathematics syllabus

When working towards the outcome MA1-6NA the sub-elements (and levels) of Quantifying numbers (QuN7), Additive strategies (AdS6), Multiplicative strategies (MuS2-MuS4), Number patterns and algebraic thinking (NPA4) and Interpreting fractions (InF1) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

NNLP links

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

• 14 counters if required
• sharing counters in rows image

Teacher instructions

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

• recognise patterns
• describe pattern relationships
• rhythmic or skip counting
• model division by sharing a collection of objects into a given number of rows
• describe a collection of objects ‘as rows of’
• describe the part left over when a collection cannot be distributed equally

Teachers read the question and instructions to the student and observe or record/photograph the student's response, looking for the strategy that the student is using (recreating with counters and then dealing the counters one at a time to make each row of 5, using the diagram to circle rows of 5 identifying the leftover counters, counting by 5’s, using a known fact 2 x 5 and similar)

Does the student recognise that the counters cannot be shared equally? Does the student express orally or through diagrams that some counters are left over? (Remainder)

Student instructions

• Share these 14 counters equally into rows of 5.
• How many equal rows are there?
• Show your answer using words, a drawing or counters.
• Can you describe what has happened to your counters? Why?

Where to next?

This is an example of grouping or quotitive division – How many groups are there?

For example – 'If I have 12 marbles and each child is to get four, how many children will get marbles?'

After students have divided a quantity into equal groups (e.g. they have divided 12 into groups of four), the process can be reversed by combining the groups, thus linking multiplication and division.

MA1-3WM - supports conclusions by explaining or demonstrating how answers were obtained

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

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