CONSTRUCTING NUMBER LINES:
YOUNG STUDENTS EXPLORE
CONCEPTS OF UNITS, MEASUREMENT,
AND PROPORTION
M4YC @ Humberwood Downs
Introductions
Our Learning Journey
Day 1
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Focus on Number Sense
Difficulties/Challenges
Big Concepts
Number Knowledge Test
Questions for Clinical
Interviews
Number Knowledge Test
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A test that is designed to measure the intuitive
knowledge of numbers that an average child
has available at the age levels of 4, 6, 8, and
10 years.
Administered by teachers, graduate students,
and researchers from the Dr. Eric Jackman
Institute of Child Study, University of Toronto
Number Knowledge Test
Clinical Interviews
Clinical Interviews
Research Lessons
Human Number Line
Three students are asked to line up side by side and
the class is asked "Who is in the middle?" The number
of students is increased, so one student is in the
middle, e.g., 5, 7. The students reflect on strategies
such as, having the same number of students on each
side determines the middle. Then a challenge is posed
with having 5 students. The students are asked "Who
is in the middle?" At this point the students reflect on
the fact that a person is not the middle, but a space,
with the same reasoning that having the same number
of students on each side determines the middle. In
Kindergarten we used a marker to show the person in
the middle, such as a picture of a sun placed above
the person in the middle.
Research Lessons
Tim Hortons’ Lesson
The Tim Hortons’ exploratory lesson was one of
the early lessons to consolidate the students’
concept of "middle as halfway". The students
enjoy "Timbits," and this was used as prior
knowledge to extend their understanding of
"middle" and to link " middle" and
halfway to distance. Butcher paper was used
to represent the road from the school to the
zoo. A stop had to be made halfway so we
could buy some Timbits. Students were to
estimate and mark the middle on the road,
and then use the folding strategy to prove or
disprove their estimate.
Research Lessons
Connecting to the Hundred Chart
In an effort to connect the understanding of
a hundred chart to a number line, we
used the hundred chart to find the middle,
using the strategy of the same number of
"things" on each side determines the
middle. The challenge was posed when
we compared finding the middle with and
without the zero. The students had many
reflections including the "main middle",
indicating a group of numbers (4,5,6) are
in the middle of a number line from 0-10,
but one was in the "main middle" (5).
Research Lessons
Connecting to the Hundred Chart
Research Lessons
The Clothesline
The clothesline lesson was one of the early
exploratory lessons to give students practice in
locating "middle", "proportional spacing", and
to consolidate their understanding of "middle as
halfway" and to develop spatial reasoning. A
clothesline was made with numbered 3x5 cards
and clothes pegs. Students were told that a
strong wind came and blew all the clothes off the
line and only zero and ten remained on the
line. The number five card was replaced on the
line, and students were asked to help put the
clothes back on the line. The emphasis of this
lesson was locating middle and equal spacing of
the numbered cards on the clothes line.
Research Lessons
The Freezie Lesson
The Freezie lesson was to provide students practice in
locating "middle" to determine "half of a
whole". A freezie was shared for Ms. Hassen and
Ms. James. The students instructed the teacher to cut
the freezie in two, so the teacher purposefully cut it
in 1/4 and 3/4. The students were furious. They
said, "No, Ms. James, you made a mistake, you must
give Ms. Hassen a half." The students then
suggested ways to find the middle of the freezie. It
was then measured with a ruler to locate, the
"middle" and to find a "half". The students, Ms
Hassen, and myself then had a freezie treat.
Research Lessons
Solving Addition & Subtraction Problems
Children worked with counters and with the number
line. The number line displays counting and
measurement simultaneously. Using the number
line gives students a visual image of the
operation being done and a better
understanding of the answer. Also, the number
line helps students experiment with decomposing
numbers. Benefits:
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Over time, this will enable students to become more
capable in performing mental computations.
Numbers can be decomposed and the subunits or
smaller amounts can be added or subtracted in
varying orders, yet still be equivalent.
Research Lessons
Solving Addition & Subtraction Problems
Research Lessons
Making Rulers
Research Lessons
Making Rulers
What we learned
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The linear representation of numbers, which includes
presenting numbers on a number line, can access
foundational mathematical concepts including;
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ordinality,
spatial awareness and magnitude,
concept of middle.
Originally focused on counting with precision. Then spatial
awareness was greatly influenced by their understanding
of middle. Finally students paid more attention to space
as well as number counting
What we learned
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Number line offers insights and supports:
Counting and counting on, counting down
 Sequencing
 Understanding units as equal intervals of
distance
 Use of benchmarks
 Sense of proportional reasoning
 Understanding measurement concepts
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Researchers study the development
of estimation on number line
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Many JK students, even those who can count
perfectly from 1 to 10, do not understand
the rank order of the numbers’ magnitudes.
These children’s number line estimates
correlate only minimally with the
magnitudes of the numbers they are
estimating.
Researchers study the development
of estimation on number line
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Even after children learn the rank order of
numbers’ magnitudes, they still do not
immediately represent the magnitudes as
increasing linearly. Their number line estimates
often increase logarithmically with the size of
the number being estimated.
By grade 2 children with experience their
magnitude estimates increase linearly
Today’s Lesson
Now it’s your turn to create your VERY OWN ruler!
You will have the opportunity to make three rulers, all will
begin with 0.
You can choose how long your ruler is going to be AND
even more exciting – you’re going to be able to choose
what number you want as your end point on ONE of
your rulers. The other two will be 0 – 10.
The only “rule” is that you have to use a marker to write
the zero and the end number. Everything in between you
can use a pencil.
Let’s Try!
Questions for Observers
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What specific strategies did you notice as students were building their rulers?
Do students focus on equal spacing? What strategies do they use to determine
the intervals?
When making their own rulers, were students able to use an ordinal counting
strategy in tandem with their spacing strategy?
Did you notice any students using the middle as a benchmark to create the
spaces? If so, how did they find middle? How did children show their
understanding of half?
Did you see evidence of students using gestures and actions to express their
mathematical thinking to a peer, the group, etc.? Please describe what you saw.
Is there a difference between boys and girls with their accuracy in spacing
numbers on the number line/ruler?
Do children change the orientation of their rulers?
Did you see children imitating their peers? Did the imitating strategy seem to
help students?