• ### Where are they now?

Can determine total number of elements in a collection of grouped items, but counts grouped items by 1s without any reference to group structure (i.e., uses make all/ count all strategy).

• ### Where to next?

Efficient counting using ‘easy’ composite units (2s, 5s, 10s).

Use group structure and stress or rhythmic counting for groups other than 2s, 5s or 10s.

• ### 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.

Focus in moving from step 4 to step 5:

Moving from key concept of

Making and counting equal groups (composite units)

To the key concept of:

Recognising and using the structure of equal groups to determine totals

### References and links to other resources

Developing Efficient Number Strategies Book 2, pp 92-10

### Counter Grab

Focus: The ability to see the structure of equal groups, and use this to calculate totals is an important step for understanding multiplication and for becoming proficient at solving multiplication problems. In this activity, students create groups from a random collection of counters and equate this arrangement with written total.

Beehives-sharing instructions