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Using Arrays to Explore Numbers

Arrays are useful models for multiplication which can be used in a variety of ways, ranging from highly structured lessons to games and open investigations. An array is formed by arranging a set of objects into rows and columns. Each column must contain the same number of objects as the other columns, and each row must have the same number as the other rows.

The following array, consisting of four columns and three rows, could be used to represent the number sentence 3 x 4 = 12.


Three rows, four columns

Building Number Facts

Arrays can be used for building multiplication facts in a meaningful way. Before drilling and memorising tables, children must understand how these facts are derived. For example, by progressively adding another column of three objects, children can build the three-times tables for themselves. This representation not only assists in understanding the process, but provides a visual image for children to draw upon as they begin to use and memorise the basic number facts.


Three rows, one columnThree rows, two columnsThree rows, three columnsThree rows, four columnsThree rows, five columns

Odds and Evens

The concepts of odd and even numbers are somewhat bewildering to many young children, so having a method to model numbers that clearly shows the difference between odds and evens is crucial. Even numbers can be made into two-row arrays, but odd numbers cannot - there being always one item left over. The making of two rows highlights the fact that even numbers are always divisible by two.

 
EightFiveTenEleven

Exploring factors

Another approach to discovering factors of numbers is to begin with a particular whole number and make as many different rectangular arrays as possible - see the Making Shapes problem. (Rectangles include squares.) For example, for the number 12, the possible rectangular arrays are:
Three rows, four columnsTwo rows, six columnsOne row, twelve columns
Thus, we find that the factors of 12 are: 3, 4, 2, 6, 1 and 12.

Exploring factors in this way will lead to the discovery that some numbers can be made into more than one array (that is; composite numbers), and some numbers can only be represented by one-row arrays (that is; prime numbers).


Three: One row, three columnsFive: One row, five columnsSix: Two rows, three columnsNine: Three rows, three columns

Another investigation of factors using arrays is the search for numbers that can form square arrays; namely, the square numbers .
 

Four is squareNine is squareEight cannot form a square

Grid paper can be useful for recording the findings of such investigations. Problems such as those below can provide consolidation of the discoveries made by young children.
 

 
This article provides further ideas on working with arrays.