Number Sense Series: A Sense of 'ten' and Place Value

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You can read an updated version of this article in our Number Sense and Place Value Feature.

Once a basic number sense has developed for numbers up to ten (see Developing Early Number Sense) a strong 'sense of ten' needs to be developed as a foundation for both place value and mental calculations. (This is not to say that young children do not have an awareness of much larger numbers. Indeed, there is no reason why children should not explore larger numbers while working in depth on 'tenness').

Ten-Frames

Ten-Frames are two-by-five rectangular frames into which counters are placed to illustrate numbers less than or equal to ten, and are therefore very useful devices for developing number sense within the context of ten. The use of ten-frames was developed by researchers such as Van de Walle (1988) and Bobis (1988). Various arrangements of counters on the ten frames can be used to prompt different mental images of numbers and different mental strategies for manipulating these numbers, all in association with the numbers' relationship to ten.

For example, examine the three ten-frames below. What numbers are illustrated? What does the particular arrangement of the counters prompt you to think about the numbers? What can you say about each number's relationship to ten?

Frame A:

There are five counters; perhaps seen as a sub-groups of three and two, either by looking at the clusters at either end of the frame, or by looking at the number in the top and bottom rows.

Frame B:

Again there are five counters; perhaps seen as three in top row and three in the bottom, or as four and one, or two and two and one. It is also noticeable that there are five empty boxes remaining, in a similar shape to the full boxes. This prompts the awareness that 'five and five make ten'.

Frame C:

This arrangement strongly illustrates the idea that 'five and five make ten'. It also suggests the idea that half of ten is five. This type of thinking would not occur if the five counters were presented without the context of the ten-frame.

Plenty of activities with ten-frames will enable children to automatically think of numbers less than ten in terms of their relationship to ten, and to build a sound knowledge of the basic addition/subtraction facts for ten which are an integral part of mental calculation. For example, a six year old child, when shown the following ten-frame, immediately said, "There's eight because two are missing."

This child had a strong sense of ten and its subgroups and was assisted by the frame of reference provided by the ten-frame. Once this type of thinking is established, the ten-frame is no longer needed. Although dealing with whole numbers initially, the 'part-part-whole' thinking about numbers supports the understanding of fractions, in particular tenths.

Place Value

'Ten' is of course the building block of our Base 10 numeration system. Young children can usually 'read' two-digit numbers long before they understand the effect the placement of each digit has on its numerical value. For example, a 5 year-old might be able to correctly read 62 as sixty-two and 26 as twenty-six, and even know which number is larger, without understanding why the numbers are of differing values.

Ten-frames can provide a first step into understanding two-digit numbers simply by the introduction of a second frame. Placing the second frame to the right of the first frame, and later introducing numeral cards, will further assist the development of place-value understanding.

Games

Ten-Frame Flash (5-7 years) 4 players

Materials: A dozen ten-frames with dot arrangements on them, a blank ten-frame for each child, counters.

Rules: One child shows a ten-frame for a count of three, then hides it while the other children place counters in the same positions on their frames from memory. The 'flasher' shows the card again and helps each child check his/her display. After three cards the next child becomes the 'flasher' and so on, until everyone has had a turn.

Variations/Extensions:

1. Points can be awarded for each correct response. The child with the most points wins.

Twenty (5-7 years) 3-4 players

Materials: Blank ten-frames (2 per child), counters, dice

Rules: Each child takes a turn to roll a die, places that number of counters onto his/her ten-frames, then announces the total number of counters on the frames. The winner is the first player to fill all twenty spaces.

Variations/Extensions:

Each turn could include placing the correct numeral cards under the frames.
Each player can also announce the number of counters needed to reach twenty. The exact number must be rolled to win the game.

Guess What (6-8 years) 2 players

Materials: Blank ten-frames, counters, a large hard-cover books to form a barrier between pairs of children.

Rules: One player secretly arranges some counters on a ten-frame. The other player asks questions that can be answered yes or no, trying to gain enough clues to work out the arrangement of counters. For example: Is the top row full? Are there 8 counters? Is there an empty box in the bottom row?

Variations/Extensions:

1. As players become more skilled, the number of questions can be counted. The player asking fewer questions wins.

References

Van de Walle, J. (1988). The early development of number relations. Arithmetic Teacher. Vol.35, February, 15-21.

Bobis, J. (1996). Visualisation and the development of number sense with kindergarten children. In Mulligan, J. & Mitchelmore, M. (Eds.) Children's Number Learning: A Research Monograph of the Mathematics Education Group of Australasia and the Australian Association of Mathematics Teachers. Adelaide: AAMT

Note: An article, called Subitizing: What is it? Why Teach it? (by Douglas Clements) in the March 1999 issue of Teaching Children Mathematics covers similar material to this month's and last month's article. The journal is published by the National Council of Teachers of Mathematics (USA) and can ordered via their website www.nctm.org