Multiplication Using Tiles © Math As A Second Language All Rights Reserved next #3 Taking the Fear out of Math

Multiplication

Using Tiles

Multiplication

Using Tiles

© Math As A Second Language All Rights Reserved

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#3

Taking the Fearout of Math

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However, unlike the Romans who would represent the numbers one through nine by writing…


The Egyptians had a numeral system that was similar to

Roman numerals.

Tiles and Multiplication

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the Egyptians recognized that by arranging the tally marks in geometricpatterns, it was easier to recognize the number. Thus, they might representthe whole numbers from one to nine by writing them as…

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Using patterns to represent numbers led to many interesting connectionsbetween number theory, arithmetic,

and geometry.

To make the results more visual

for young learners it is helpful to replace the tally marks by

equally-sized square tiles.

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In this way the tally mark representations of the numbers one through nine would become…

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The above representations make it rather easy to see how the tiles can be used to

help students better internalize multiplication, division, and factoring.

Given any number of tiles, they can always be arranged to form a rectangle, often in

many different ways.

Key Point

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For example, given 12 tiles we can arrange them in a rectangular way in any one of six

ways, and each of the 6 ways is a segue to

multiplication, division and factorization…

1 × 12 = 12

12 × 1 = 12

1 row of 12 tiles each

12 rows of 1 tile each1

note1 Notice that even though 1 × 12 = 12 × 1, one row of twelve tiles does not look like

one column of 12 tiles. They are equal in the sense that the cost is the same if you buy 1 pen for $12 or 12 pens for $1 each, but the two transactions are quite different.

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Other rectangular combinations for 12 tiles would be…

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2 × 6 = 122 rows of 6 tiles each




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When we write 3 × 4 = 12, the traditional vocabulary is to refer to 12 as the product of 3 and 4, and to refer to

3 and 4 as factors of 12.

Visualizing What a Factor Is

However, by agreeing to represent whole numbers in terms of square tiles, there is a more visual way to view what a factor of a number is.

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For example…

Visualizing What a Factor Is

4 is a factor of 12 because 12 tiles can be arranged into a rectangular

array that consists of 4 rows (or columns) each with 3 tiles.

7 is a factor of 28 because 28 tiles can be arranged into a rectangular

array that consists of 7 rows (or columns) each with 4 tiles.

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1 is a factor of any number because any number can be arranged in a rectangular

array that consists of only 1 row.

Two Special Cases

For example, 13 tiles can be arranged in 1 row that consists of all 13 tiles.

13

1

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Every number is a factor of itself.

Two Special Cases

For example, 12 tiles can be arranged into a rectangular array

consisting of 12 rows, each with 1 tile.

12

1

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The fact that n = n × 1 = 1 × n for any number n gives us an interesting insight as

to how numbers can be written as a product of different factors.

Note

1 tile already exists in the form of a rectangular array. Namely, it is already an array that has 1 row that consists of

a single tile.

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Other than for 1, every other set of tiles can be arranged as a rectangular array in at least 2 ways, namely either as 1 row or

1 column (in the language of factors and products this simply states that

n = n × 1 = 1 × n).

Note

However, for some sets of tiles these are the only 2 ways in which they can be

arranged in a rectangular array, and for other sets of tiles there are additional ways.

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For example…

Note

As a rectangular array, 2 tiles can be arranged only as 2 rows

or 2 columns each with 1 tile.

On the other hand, while 6 tiles can be arranged as 6 rows or 6

columns each with 1 tile; they can also be arranged as 3 rows

or columns, each with 2 tiles.

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A number greater than 1 is called a prime number if its only factors are 1 and itself.

In terms of tiles, a number is prime if the tiles can be represented as a rectangular

array in exactly 2 ways.2

note2 The tile definition of prime number eliminates 1 from being a prime number because

1 can be represented in only 1 way as a rectangular array.

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Definition

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A number greater than 1 that is not a prime number is called a composite number.

The number 1 is neither prime nor composite. It is referred to as a unit.

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Definition

In terms of tiles, a number is composite if the tiles can be represented as a

rectangular array in more than 2 ways.

next We will talk about prime and composite numbers in greater detail in future presentations.


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2 × 2

For now the important point is that by introducing

properties of numbers in terms of tiles, even the

youngest students can begin to internalize important mathematical concepts

before they are encountered more abstractly later.

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Multiplication Using Tiles © Math As A Second Language All Rights Reserved next #3 Taking the Fear out of Math