What is the Empty
Number Line?
The empty number line was originally
proposed as a model for addition and
subtraction by researchers from the
Netherlands in the 1980s. A number line
with no numbers or markers, essentially
the empty number line is a visual
representation for recording and sharing
students’ thinking strategies during the
actual process of mental computation.
Before using an empty number line
children need to show a secure
understanding of numbers to 100. Prior
experiences counting on and back using
numbered lines, recall of addition and
subtraction facts for all numbers to ten
and the ability to add/subtract a multiple
of ten to or from any two-digit number
are all important prerequisite skills.
Reference: http://www.k-5mathteachingresources.com/empty-number-line.html
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Introducing the Empty Number Line
During this stage children practice
moving to given numbers in the least
number of jumps. Possible questions
may include:


How can we go from 0
least number of jumps
ones?
How can we go from 0
least number of jumps
tens and ones?
to 59 in the
of tens and
to 189 in the
of hundreds,
Students should be encouraged to share
different strategies and discuss which
strategy is the most efficient. For
example, when jumping from 0 to 59
one student could make five jumps of
ten and nine jumps of one, while another
may make 6 jumps of ten to 60 and then
jump back one to 59.
Using the Empty Number Line to
Solve Addition and Subtraction
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Problems
During the next phase students solve
addition and subtraction problems and
draw the jumps to explain their thinking
process. Students are free to choose
what type of jumps they will use. Again
the focus is on sharing different
strategies in order to lead children to use
the empty number line efficiently when
adding or subtracting any pairs of
numbers.
Possible questions include:




How can we go from 27 to 53 in a
small number of jumps? Who has
another way?
How can we go from 62 to 45 in a
small number of jumps? Who has
another way?
How can we solve 34+23?(counting
on without crossing the tens
boundary)
How can we solve 37 + 25?(counting
on crossing the tens boundary)
Reference: http://www.k-5mathteachingresources.com/empty-number-line.html
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



How can we solve 47 - 23? (counting
back without crossing the tens
boundary)
How can we solve 42-25? (counting
back crossing the tens boundary)
How can we solve 82- 47? (counting
up)
How can we solve 157 + 36?
Writing equations horizontally forces
students to look at the numerals,
whereas written vertically students tend
to immediately turn to the procedural
algorithm.
One of the interesting things about
mental calculations is that we do not all
think the same way. The empty number
line allows students to see the variety of
ways that the same question can be
solved. For example, to solve 157 + 36
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one student may begin at 157, add 30,
then 6 while another may start at 157
and break the 36 into 3 and 33. This
turns the question into the problem of
adding 33 to 160.
Using the Empty Number Line to
Solve Word Problems
Once students are confident with using
the number line for showing their
thinking strategy they can use it to
support them while solving a range of
problems in different contexts (e.g.
elapsed time, money, measurement).
Addition (2 digit + 2 digit)
A sunflower is 47 cm tall. It grows
another 25cm. How tall is it?
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Subtraction (2 digit – 2 digit)
I need 72 dollars to buy a skateboard. I
have 39 dollars already. How many more
dollars do I need to save?
39 is placed near the start of the empty
number line and 72 near the end. We
can count up in 'friendly' jumps to reach
72. First a jump of 1 to reach 40
(multiples of ten are easy numbers to
jump to and from), then a jump of 30 to
reach 70 and finally a jump of 2 to reach
our target of 72. I need to save 33 more
dollars.
Subtraction (2 digit - 2 digit)
A piece of string is 42cm long. If you cut
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off 25cm how much will be left?
42 is placed towards the end of the
empty number line. 20 is subtracted to
get to 22, then 5 is subtracted to get to
17.
Subtraction (3 digit – 3 digit)
There are 543 people on the subway
platform. 387 board a train. How many
people are left on the platform?
We start at 387 and count up in 'friendly'
jumps to reach 543. First a jump of 13
to reach 400 (multiples of ten and
hundred are easy numbers to jump to
and from) then a jump of 100 to reach
500 and finally a jump of 43 to reach the
target of 543. 156 people are left on the
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platform.
Multiplication:
Example: Use an empty number line to
calculate 19 x 4
20 (a ‘friendly’ number) is one more
than 19. We can do 4 jumps of 20 to 80
and then jump back 4 spaces to 76. 19 x
4 = 76
How could you use this idea of
multiplying by 20 and adjusting to solve
these problems?: 21 x 6; 19 x 8; 21 x 4
Can you use the idea of adjusting by one
to calculate multiples of 29 and 31?
Division (2 digit by 1 digit)
6 cupcakes fit in a box. How many boxes
can be filled with 85 cupcakes?
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We need to find out many sixes there
are in 85. To solve this problem we could
count up in sixes one group at a time
but this would not be efficient. We need
to think about easy, but bigger ‘chunks’
of 6 such as 6x10(60). We can add 6
chunks of 6 as 60 and then add a further
4 chunks of 6 (24) to total 84, leaving 1
spare cupcake. 14 boxes can be filled.
Division (3 digit by 2 digit)
If a box holds 28 apples, how many
boxes can be filled with 350 apples?
To solve this problem we need to find
out how many 28s there are there in
350. We can think about ‘friendly’
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chunks of 28 such as 28 x 10 (280).First
the 10 chunks of 28 (280) are added
then two more ‘chunks’ of 28 (56) are
added mentally to leave 14 chocolates.
This is not enough to fill a further box so
12 boxes can be filled.
Given ongoing opportunities to use the
empty number line children will begin to
solve problems mentally by picturing the
empty number line in their heads.
Regular use increases students’
confidence in their ability to use
numbers flexibly and leads more easily
towards mental calculations without
paper.
Reference: http://www.k-5mathteachingresources.com/empty-number-line.html
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