About This Book This book presents ideas for introducing and teaching with a number of different linear representations of number. It is important to realise that different models are best suited to the teaching of different skills and understandings at different levels. Calibrated number lines, for example, are particularly powerful, but formal, representations for number suited to children who are developing more formal knowledge of number and operations. The table below suggests appropriate levels and concepts for each representation. Representation Number Track, Number grid
Grade Level K-6
Concept Counting forwards and backward in ones; reciting the number sequence in different ranges. Investigating number patterns formed by skip counting, lowest common multiples.
Empty number line for estimation and number value
F-6
Judging the value of numbers, e.g. being able to “peg” 5 in the middle of a washing line strung between 0 and 10, or 0.5 in the middle between 0 and 1.
Empty number line for computation
2-6
Calibrated number line
2-6
Reflecting upon/explaining mental calculation strategies which involving adding or subtracting for example in 1s, 10s or 100s (which can be shown as jumping forwards or backwards along the number line). E.g. 45 + 21 as 45… 55, 65, 66. Measurement, number recognition, counting in ones, demonstrating calculation strategies based on counting and grouping, e.g. bridge ten, showing multiplication as repeated addition, showing division as repeated subtraction.
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2
3
4
1 5
5
2 6
6
3 7
7
8
9
10
4 8
9 10 11 12 13 14 15 16
Please note that although number tracks may be appropriate for some children as early as 4 years of age, it is recommended that children are competent in counting concrete objects in any number range before being introduced to any type of number line representation. For example, children would be able to count collections of up to 10 objects, observing counting principles, before engaging with any of the activities on the following pages which deal with number tracks up to 10.
Number Lines
6
© P. Swan
Life-sized Number Tracks The first number line that students might be introduced to would be one made from children. Counting numbers in the target range to be learned could be written on card and given out to the students, who put themselves in order and then “number off” by reciting the counting numbers in order.
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2
3
4 5
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9
10
Number off! •
starting from one, take turns to say the number on your card, one after the other
•
take turns to say the number on your card, but perform an action as you say your number (hold the card up high/jump/sit down). This provides practice in coordinating one action with one number word, as for 1:1 correspondence when counting
•
an additional person walks along the line, from 1-10, tapping each person on the shoulder and saying their number as they do so.
•
Repeat all of the above in the reverse order (counting backwards)
10
9
Number Lines
8
5
7 6
10
4
3
2
1
© P. Swan
Number Lines
36
© P. Swan
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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