Some children misinterpret decimals by making false analogies with other ideas. Common sources of confusion are with money, sport and reminders in division.

 Analogy with money Analogy with sport Analogy with remainders in division Different ways of writing the remainder

### Money

With money, although 26 cents is indeed 26 hundredths of a dollar, people often do not think about \$5.26 as 5 whole dollars and a fractional part of a dollar. They think of 5 dollars and 26 cents in terms of two parallel systems of units; the dollars and the cents. Although they know there are 100 cents in a dollar, the relationship of one unit as a part of the other is not uppermost in their consciousness.

Whilst teaching decimals with reference to money may be a good start, it is not sufficient. It does not help many students get a good idea of the relationship of the decimal part to the whole and it does not help understanding the general place value relationships for other than 2 decimal places.

One clue to find out whether students are relying too heavily on money is to see how they deal with decimals with more than two places. Can they, for example, describe the relative size of 0.5376 and 0.5324?

Analogy with money causing confusion of column names
Sometimes analogies with money cause confusion for children about the name of the place value columns especially between tens and tenths. For example, in the number 7.89, the 8 stands for 8 tenths, whereas in the price \$7.89, the 8 stands for eighty cents (eight tens). Veronica in Year 5, for example, seemed to routinely think about this. She read the number 0.3 as "oh point three, that's thirty" and 3.5 as "three point five, that's fifty".

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### Sport

Many sports appear to use decimal notation to report the game. For example, when a cricket scorer records that a player has bowled 5.2 overs, this is interpreted as 5 overs (each of 6 balls) and 2 additional balls. Although each ball is in a sense one sixth of an over, this relationship is not how people mostly interpret it. They think of two systems of whole numbers, the overs and the balls, not fractional parts. This is similar to the situation with money, with the additional problem that it is not a base ten system. (More information)

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### Remainders in division

In the upper years of primary school, children learn three ways of dealing with remainders in division. Firstly they learn to leave the remainder as a whole number. Secondly they learn to express the remainder as a fraction. Thirdly they express the remainder as a decimal, possibly rounded off.

Not surprisingly, these different treatments will be confused by some children whose knowledge of these procedures has not been consolidated or who have not appreciated the different contexts in which these different ways of treating the remainder are appropriate.

A common error in division made by upper primary children and lower secondary school children is to present the remainder as a decimal.

This might lead some children to think that the decimal part of a number is how many are "left over", without any consideration for place value.

This is illustrated in an interview with Veronica, who was given a card marked with "23 divided by 7" and asked to use a calculator to find the answer. Veronica had some difficulty comparing her expected remainder of 2 with the long "remainder" on the calculator display.

 Interviewer: What does that number say? Veronica: 3.285714286. (read out digit by digit) Interviewer: So what do you think about the remainder for this question? Veronica: Well, its pretty big. Interviewer: Its pretty big; its a big remainder? Veronica: Yeah. 285714286 (read digit by digit). But they don't have that many remainders? Interviewer: Right, how many remainders would it have? Veronica: Only just 2.

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### Examples of contexts illustrating different ways of writing the remainder

 Which way of dealing with the remainder is the best in these cases? a. A school has to pick teams of three people to represent it at the district sports carnival. Seventeen children are available to play. How many teams can the school enter? b. Seventeen metres of ribbon are to be used to decorate three identical banners. How much can be used on each? c. Seventeen pizzas are to be divided equally amongst three basketball teams. How much will each one get?

 Answers a. 5 remainder 2 (5 teams with 2 children left over) b. 5.67m (decimal notation allows easy measurement) c. 5 pizzas and two thirds of a pizza (pizzas may be cut into thirds)

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