Representing Numbers 0-99

We have seen how numbers up to 9 can be represented, starting from concrete form gradually becoming more and more abstract. We give below a sequence of representing quantities more than 9 in the same manner. Chapter 7.2 gives a visual representation of the various ways.

1. Sound/ Spoken Representation

   1. First learn to count in bundles using following words– Ten, Twenty, Thirty, Forty …….. Ninety

   2. Then learn to count from Ten to Nineteen as these sounds are not logical as sounds of numbers above Twenty (Please see a detailed explanation in Chapter 14.2 on Language & Mathematics).

   3. Then combine both bundles & sticks while counting. If there are 3 bundles and 4 sticks the counting sequence would be Ten, Twenty, Thirty, Thirty One, Thirty Two, Thirty Three and then Thirty Four.

2. Bundle & Sticks Representation

   1. Presenting a 2 digit number using bundles & Sticks

   2. Forty Five as 4 Bundles & 5 Sticks

   3. Twenty as 2 Bundles

3. Finger Representation–Small two digit numbers, less than 55, can be represented with fingers of both hands, with one hand representing Bundles and the other Sticks. (The right hand can be used for bundles & the left for sticks. This way for the observer, the number of bundles would appear on the left as in the written form). In the next chapter we will see how numbers more than 55 can be represented.

   1. Finger representation will deepen their understanding of Place Value

   2. It would also help in mental math by strengthening visualisation (Refer to Chapter 11.7 on addition fluency)

4. Picture Representation – pictures of bundles & sticks or sheets, strips & pieces representing the number.

5. Numeral Representation – 15,23,38,49 etc. Numerals have different shapes which vary from language to language. When a child is learning 2 languages then numbers could be represented with numerals of both languages.

6. Word Representation – One, Two, Three etc. Number words are also written in various ways from language to language. When a child is learning 2 languages then numbers could be written in both languages.

   1. Following the logic of Place Value, just 30 “word cards” would be sufficient to build all numbers up to thousand. (Refer to Chapter 7.5 for details)

7. Arithmetic Representation – 45 as 30+15 or 5 X 9 etc. Arithmetic representation continues up to High School in terms of more complex ways of representing numbers.

   1. Combining Number Sense with arithmetic representations helps students become fluent in number operations. Eg 34 + 47 can be mentally manipulated as 34 + 40 + 7 -> 74 +6 +1 -> 80 + 1 -> 81

8. Number Line Representation – drawing a number line divided into tens and/or hundreds as required and marking the number as a point on the line

Children should be able to represent numbers from Ten to Ninety-Nine in all these representations. They should be able to match one representation with another.

Advantages of a logical number word system

The system of writing numbers with numerals (453, 6808 etc) is now standard all over the world in international science & commerce.

But in terms of number sounds, each language uses different words (One, Ek, Ondru etc). Many of them have different patterns underlying these words which create problems for children trying to remember them.

For example, in Hindi the counting sounds (in equivalent terms) are like fifty one, fifty two … until fifty eight. But fifty nine is spoken as sixty less one, followed by sixty. This sudden break in the pattern makes it difficult for children mastering them in their early years. Even numbers from eleven to nineteen are not in the same pattern. Some of them even sound like three ten, four ten etc.

In contrast the Tamil number words are logical even in the numbers from eleven to nineteen. They sound like ten one, ten two, ten three etc right up to ten nine.

Recently many researchers have attributed the competence of Chinese students in math to the logical structure of the number words right from eleven to ninety-nine.

Why are Eleven & Twelve different?

The decimal place value system was adopted universally only about 500 years back. Before that each culture evolved their own number names.

In many cultures some numbers were used with cultural interpretations that they had distinct names. For example, two was called pair, couple and five was called a fist (panja in Hindi)

Most cultures seem to have twelve or dozen as a common unit. We still buy fruits in dozens.

So names evolved to specify these quantities from one to twelve.

When the decimal place value system was adopted, it brought a uniform naming system.

Three & ten became thirteen, four & ten became fourteen etc.

All numbers from 20 upwards used a uniform naming system.

But eleven & twelve retained the old system of number names.