Number Expanders
 Description of number expanders (including an electronic version to try) How to make and use a number expander (including templates for printing) Using a decimal number expander Activity 1: Expansions, expansions Activity 2: When do zeros matter? Activity 3: Multiplication and division by 10

### Description of number expanders

A number expander is a simple aid to show the many ways of writing a number using expanded notation. For example, it can show that 236 is equal to:

 236 units (ones) or 2 hundreds and 3 tens and 6 units (ones) or 2 hundreds and 36 units (ones) or 23 tens and 6 units (ones)

A number expander can also be used for showing various representations of decimals. Play with an electronic number expander to see how 3.145 can be expanded.

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### How to make and use a number expander

Number expanders are made by folding strips of paper. Children will enjoy the challenge of folding them correctly - somewhat like a fan. Print the template or let children make their own from scratch. The folding instructions are on the templates. Children write their own numbers on the number expander, one digit per blank space. Open the number expander up in many ways to see the possibilities. Every combination is correct!

Three templates ready to print are provided. They are suitable for:

 whole number place value hundreds to units (ones) early primary whole number place value millions to units (ones) upper primary decimal place value units (ones) to thousandths upper primary

These pictures show various expansions of 3 174 682.

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### Using a decimal number expander

This works in the same way as for whole numbers but a decimal point made of Blu-Tak (or similar sticky material) must be used when the words are hidden. It is removed when the words are shown.

Various expansions of 3.145 are shown below:

Ready-made number expanders can be purchased. (an order form from one Australian supplier). Note that in the decimal versions, the decimal point is actually printed on the number expander and needs to be covered with a finger to see the different expansions.

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### Activity 1: Expansions, expansions!

Students work in pairs and write their own decimal number on a number expander. They then write down as many expansions as they can. Expansions can be written in words and/or fractions, e.g. 0.342 is 34 hundredths + 2 thousandths OR 34/100 + 2/1000. Groups could then swap number expanders and write down as many expansions as possible in a given period of time (say 2 minutes). At the end of this time, the new expansions are checked by the original group who made the number.

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### Activity 2: When do zeros matter?

Students work in pairs. Instruct them to write a number made of only 2 non-zero digits, with zeros written in all the other columns. They then swap number expanders with another group. All groups now try to interpret the number on the expander they have been given; writing the number on paper and deciding which zeros are essential, and which can be omitted. This activity is suggested because there are many special difficulties with zero, as explained throughout this resource.

Example: 0000.307000 may then be interpreted as 0.307

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### Activity 3: Multiplication and division by 10

Students work in pairs with 2 number expanders. On the first they write any number that has non-zero digits in the ones (units) and tenths columns. Ask them to show the expansion as a number of tenths. (For example: 2.6 would be expanded to 26 tenths). Then ask them to multiply the number by 10 (260 tenths). This result should be written on the second number expander (260 tenths) which is then (after rearrangement ) seen to be 26.0, or 26. So 2.6 x 10 was found to be 26 by virtue of arithmetic of whole numbers!

Ask them to make up more examples for multiplication by 10, 100 and 1000 followed by division by powers of 10. (Note that the parallel division examples need zeros on the right of the number, so either they start with numbers like 50 or be prepared to append zeros e.g. 4.5 =4.50)