• ### Where are they now?

Structured and supported spatial patterning or sharing games and activities as well as rhythmic songs or rhymes will help to build this concept.

• ### Where to next?

Makes and distributes small equal groups with support (e.g 2 paste bottles per table, 6 crayons per table)

• ### Purpose

Multiplicative thinking is a critically important ‘big idea’ as it underpins virtually all of the work in number and algebra in the middle years of schooling. (p28 Chapter 2: Working with the Big Ideas in Number and the Australian Curriculum: Mathematics.  Dianne Siemon, John Bleckly, Denise Neal.)

Building Multiplication and Division skills:

The skill of multiplying and dividing draws on all of the skills studied in the whole number, addition and subtraction sections.  A failure to progress will generally indicate one or more of these skills needs to be revisited.

Fluency with basic facts allows for ease of computation, especially mental computation, and therefore aids in the ability to reason numerically in every number-related area.  Although calculators and tedious counting are available for students who do not have command of the facts, reliance on these methods for simple number combinations is a serious handicap to mathematical growth (p167, elementary and middle school mathematics)

Student’s early multiplication and division knowledge is based fundamentally on the development of counting sequences and arithmetic strategies, along with skills of combining, partitioning and patterning (CMIT P30, LFN).  Early multiplication and division strategies focus on the structure and use of groups of things.  Rather than emphasizing individual items or number words, students develop increasingly sophisticated ideas of “composites”.  As they develop the concept of multiplication, students focus on groups of items and learn to treat the groups as items themselves (CMIT, LFN, P6).

Integral to solving multiplication and division problems is the ability to make and count equal groups.  Students also need to know that a set of objects has the same numerical quantity no matter how they are arranged (p24DENS1).  They need to be able to trust the count.

Specifically moving from step 1+2 to step 3:

Moving from key concept of:
Pre-counting – This is important because these concepts lay the foundation for children to later develop an understanding of the many ways that numbers are related to each other; for example five is two more than three, and one is less than six.

To the key concept of:

One-to-one counting – Counting is important because the meaning attached to counting is the key conceptual idea on which all other concepts are based.  (NZ Maths  – http:www.nzmaths.co.nz/number-early-learning-progression)

### References and links to other resources

Developing Efficient Number Strategies Book 1, Find-a-Group, p123 and Trains, p129.

Elementary and Middle School Mathematics: Teaching developmentally, John Van De Walle (7th edn)

http://www.nzmaths.co.nz/general-interaction-ideas-number

*Gives interaction ideas for pre-counting and one-to-one counting.

http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/continuum/Pages/learntocount.aspx

*Explains the principals involved in learning to count, ‘Even if young children can recite the number sequence (i.e say one, two , three, etc) we cannot assume that they can count small sets of objects.  Learning to count involves learning several important principals….’

Online

http://au.ixl.com/math

### Beehives – sharing

Focus: The ability to create and distribute equal groups is a precursor for multiplicative thinking. The beehives act as a unitising ‘container’ for the groups created, facilitating a count of groups as well as a count of the elements within each group.

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Beehives-sharing instructions

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Beehives (Blank)

### Assessment Rubric

Focus: To assess students’ ability to use perceptual counting and sharing to form groups of specified sizes. .

How: See Assessment rubric sheet

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Assessment rubric