Hands Up: engaging students as a resource for teaching
This teaching outline is based on an article by Brian Tickle, Handy hints for teaching number
in the new syllabus (Primary Curriculum Support, 2003, Vol 8, No 3).
Why place value?
Algorithms are introduced after students have gained a firm understanding of basic
concepts including place value, and have developed mental strategies for computing with
two- and three-digit numbers.
Mathematics K–6 syllabus (2002, p. 40)
Why do students need a firm understanding of place value in order to discover and develop
efficient mental computation strategies? Let’s consider what may have been accepted as
indicators of an understanding of place value in the past.
·
The student can locate and indicate ones, tens and hundreds columns in three-digit
numbers.
·
The student knows the values of numbers according to the columns in which they
are placed. For example the students says, “The eight is in the hundreds column, so
it means 800”.
·
The student can line digits up in columns in the manner of the formal algorithm.
Are these really indicators of a deep understanding of place value?
The short answer is no. These examples may indicate an understanding of the “language of
position in numbers” but they do not indicate an understanding of place value that can be
used to solve problems. A more reliable indicator of a firm understanding of place value
would be the ability to discover, use and explain the ‘jump strategy’ or the ‘split strategy.’
Students who demonstrate these strategies understand that ten is both ten ones and one
ten. They are able to use this working knowledge of place value to solve problems involving
two-digit numbers.
Making the concept of place value visible
When students solve problems in mathematics, teachers frequently ask questions like, How
did you work that out? Is there another way? Many students benefit greatly from hearing
oral explanations of strategies from their peers, but for students who are not strong auditory
learners such explanations may be of little value. Many teachers already use a variety of
materials to help to make the concept of place value ‘visible’ to students. Such materials
include ten frames, unifix cubes, Multi-base Arithmetic Blocks.
An excellent resource to help students to see the concept of place value is the students’
own fingers – not to count by ones, but to show groups of tens and ones. Most students
instantly recognise the fingers of two hands as a powerful representation of ten. This
representation has the advantage that the ten can be viewed as one ten, as ten ones, or as
the combinations of ten e.g. 7 + 3. The following activities could be used to make the
concept of place value ‘visible’ to students.
© State of New South Wales through the NSW Department of Education and Training, 2007. This work may be freely
reproduced and distributed for most purposes, however some restrictions apply. See the copyright notice on Curriculum
Support site or contact the Department.
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1. Ask the students if anyone can come to the front of the class and show twelve fingers. A
typical reaction in a Kindergarten class is for a large number of students to raise their
hands and then lower them when they realise they don’t have enough fingers. Some
creative solutions may include showing ten fingers and two toes or ‘flashing’ ten fingers
and then displaying two of those fingers again. Explain that you really want to see twelve
fingers ‘all at once’. Usually a student will realise that this can be done with the help of a
friend.
If two students manage to show the twelve fingers, help them (if necessary) to display the
fingers as shown below with palms facing forward so that the student showing ten fingers is
standing to the right (i.e. the left of the audience) of the student who is showing two fingers:
Ask the class to check that there are twelve fingers. Encourage students to count like this:
10… 11, 12.
It is important at this stage to explain to students the preferred method of displaying less
than ten fingers. Encourage students to show numbers less than ten by starting with the
little finger of their right hand (i.e. the extreme left for the audience).e.g.
1
3
7
Representing numbers less than ten in this way will help students to see how many more
are needed to make five or ten. For example, in the representation of 7 shown above,
students should clearly see that 5 + 2 = 7 and that three more (the fingers folded down on
the left hand) are needed to make ten.
2. When the class agrees that the two students at the front of the class are showing twelve
fingers, ask the student showing two fingers to return to his or her seat. Explain that the
other student should remain because ‘We are going to need that ten.’ Now ask if
someone can come out and make fourteen fingers. When the student joins the one
showing ten and makes this pattern, ask the class to check: 10…. 11, 12, 13, 14.
A student may say, ‘It’s 14, because it’s 10 and 4.’
3. Ask two students to show 16 fingers. Ask another two students to show 18 fingers. Ask
the two students to stand close to the board. Write a large ‘1’ on the board above the
student who is displaying ten fingers. Write a large ‘8’ above the student who is
displaying 8 fingers. Discuss why 18 is written this way. Students may say, ‘It’s 1 ten
and 8 ones.’ Ask how many more are needed to make 20.
© State of New South Wales through the NSW Department of Education and Training, 2007.
http://www.curriculumsupport.education.nsw.gov.au/index.htm
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4. Ask the two students who are displaying 18 fingers to show 20 fingers. Ask the class to
check. A student may say, ‘It’s 20 because it’s 2 tens.’
5. Now ask another student to come out and build on the 20 to make 24.
Ask the class to check. Encourage them to count like this: 10, 20, 24. Now ask them to
count from ‘the other end’: 4, 14, 24. The students are now counting by ten off the
decade, a skill used by students who use the jump strategy for addition and subtraction.
6. Now ask another student to build on the 24 to make 34.
This is a good time to introduce a ‘Mexican wave’ effect. The first student (on the
audience’s left) raises his or her ten fingers. The class says ‘10’. The next student raises
ten fingers (‘20’). The third student raises ten fingers (‘30’) and the last student raises 4
fingers (‘34’). Ask students to do the wave again starting from ‘the other end’ i.e.
counting by ten off the decade.
7. Once students have the idea of making two-digit numbers with groups of tens and ones,
the ‘wave’ can become a short daily activity involving different numbers and different
students. A good extension is to ask a student to make (say) 73 but ask first, ‘How many
people will you need?’ A student who can say, ‘I’ll need 8, because I need 7 to make the
70 and one more to make the 3, is demonstrating a firm understanding of tens and
ones.
This activity can be made into a game like musical chairs. Students move around an open
space while the teacher plays music. The teacher stops the music and announces a twodigit number. Students must arrange themselves to form the number with their fingers using
groups of tens and ones as quickly as they can. Anyone who is not part of a group is ‘out’.
© State of New South Wales through the NSW Department of Education and Training, 2007.
http://www.curriculumsupport.education.nsw.gov.au/index.htm
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Counting on and off the decade
A simple variation is to link the physical activity and the oral-aural work with the symbolic, as
in Figure 1. Students can check that the number formed is correct by counting from the left
in groups of ten before adding the three. Following this by counting from the right starts with
the “three” and helps students to increment the total by ten each time.
Figure 1 Counting off the decade
This activity is clearly designed to enable students to use the associative property of
addition and to learn to repeatedly increment the total by ten. You may prefer to simply write
the number on the board and invite a student make it!
Making the strategies visible
Why do some students persist with inefficient strategies such as counting by ones? One
reason may be that they only see ten as ten ones. Students who can see ten as ten ones
and one ten have a distinct advantage when it comes to mental computation. This is
because most efficient strategies for dealing with two-digit numbers involve multiples of ten.
The following activities could be used from Early Stage 1 to Stage 3 to help students to
discover, use, explain and compare strategies for adding and subtracting two-digit numbers:
© State of New South Wales through the NSW Department of Education and Training, 2007.
http://www.curriculumsupport.education.nsw.gov.au/index.htm
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1. Ask some students to form ‘34’ using the groupings described above.
Ask another group of students to display 20 fingers.
Ask the class how many fingers there are altogether. Ask how they worked out the
answer. If a student says, ‘I put the tens together’, ask the students who are showing
the 20 to join the tens in the 34.
2. Ask students to form groups in which both numbers are not multiples of ten e.g. 43 + 25
43
53
63
68
A student may say, ‘I started from 43 and went up like this: 53, 63. Then I added the five.
That’s 68.’ (This is an explanation of the jump strategy.) Ask the students who are displaying
the ‘25’ to do the ‘wave’ as the class counts 53, 63, 68.
A student who uses a more sophisticated version of the jump strategy may say, “I started at
43 and added on 20. That’s 63. Then I added the 5. That’s 68.” Ask the students who are
showing the two tens in the ‘25’ to raise their hands when the student who is explaining
says, ‘I added on 20.’
40 + 20 = 60
3+5=8
Another student may say, “I put the tens together. That’s 40 + 20 = 60. Then I put the ones
together. That’s 3 + 5 = 8. So it’s 60 + 8. That’s 68”. This is an explanation of the split
strategy. Ask the students who are displaying the fingers to move in accordance with the
student’s explanation.
© State of New South Wales through the NSW Department of Education and Training, 2007.
http://www.curriculumsupport.education.nsw.gov.au/index.htm
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3. To demonstrate that subtraction problems can be solved using similar strategies to
those used for addition, ask some students to display 31 fingers.
Ask students how they could solve 31 – 12. A student may say, “Just take off the 11”.
(The two students on the audience’s right drop their hands.) “That’s 20. And then you
need to take off one more, so it’s 19”.
4. It is possible to use groupings of fingers like these to show problems like 48 + 37. In
fact, the compensation method may even be demonstrated in this way e.g. add 2 fingers
to the 48 and compensate by removing them from the 37. The question then becomes
50 + 35, which is easier to solve because 50 is a multiple of ten. Examples of the
compensation strategy may be found on p. 40 of the Mathematics K--6 syllabus (2002).
A deep understanding of tens and ones helps students to discover, use, explain and
compare efficient mental computation strategies for dealing with two-digit numbers. Using
students’ fingers to represent groups of tens and ones helps to make the concept of place
value visible for students. This, in turn, can help them to develop a range of efficient mental
strategies for adding and subtracting two-digit numbers.
© State of New South Wales through the NSW Department of Education and Training, 2007.
http://www.curriculumsupport.education.nsw.gov.au/index.htm
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