The Developmental Progression of Rational Number Understanding
Using the Number Line
Stephanie Hanson, Diana Chang, Michelle Chee, Niki Dowlat Singh, Allison Musson, and Joan Moss
Dr. Eric Jackman Institute of Child Study, Ontario Institute for Studies in Education, University of Toronto
PURPOSE
To explore how children’s understanding of rational numbers
develops over the elementary grades (using the number line)
• Fractions are typically taught using the pie strategy. This limits
students’ abilities to comprehend how fractions compare to one
another and relate to whole numbers
• We were interested in exploring how manipulating fractions on the
number line can improve students‘ conceptual understanding
SUPPORTING LITERATURE
METHODS
RESULTS/RECOMMENDATIONS
JUNIOR KINDERGARTEN
EARLY YEARS
Participants
• 4 JK students from JICS (1 male, 3 females)
Materials
• A strip of Bristol board for the number line
• Cut-outs of houses labeled with whole numbers 1 to 5
Protocol
• Place house number 2, when house numbers 1, 3, and 5 are
present
• Show where I would end up if I were driving from house number
1 to house number 2 and only got halfway there
Results
• Students had varying knowledge of half, and only one was
able to demonstrate one quarter and three quarters
• Students paid some attention to even spacing, but in relation
to the house directly next to the one they were placing
• None of the students could identify any numbers between 0
and 1
• Three students were able to correctly place the house on the
halfway point between 0 and 1, but they could not identify
what the house would be called
GRADE ONE
Placing fractions on a number line is crucial to students’
understanding because it allows students to:
•further develop their understanding of fraction size
•see that the interval between two fractions can be further partitioned
•see that the same point on the number line represents an infinite
number of equivalent fractions
(Lewis, p43)
In the Japanese Curriculum, the number line model is used to help
students recognize fractions as numbers and learn how they are
related to whole numbers
(Watanabe, 2007)
Participants
• 4 Grade One students from JICS (2 males, 2 females)
Materials
• A strip of Bristol board for the number line from 0 to 1
• Cut-outs of houses (unlabeled)
Protocol
• What if someone wants to build a new house halfway between
house 0 and house 1, where would the house be built?
• If a car is driving from house 0 to house 1 and they break down
here (at ¼ and then ¾), where did they end up?
GRADE THREE
Participants
• 4 Grade Three students from JICS (2 males, 2 females)
Materials
• Two strips of Bristol board for the number lines labeled 0 to 1 (one
calibrated by tenths, and one uncalibrated)
• Pointers labeled with fractions
Protocol
• Can you show me where ½ is on the number line? Can you locate
¾ on the number line? What about 7/8? Do you think that it is
between ½ and ¾ or ¾ and 1? Why?
GRADE FIVE/SIX
Participants
• 4 Grade Five/Six students from JICS (2 males, 2 females)
Materials
• Two strips of Bristol board for the number lines labeled 0 to 1 (one
calibrated by tenths, and one uncalibrated)
• Pointers labeled with fractions
Protocol
• Where is 2/6ths located on the number line? Place a pointer
there. Now can you find 4/12ths. Can you think of another fraction
that is the same is 2/6 and 4/12?
Recommendations
• Introduce students to rational number by integrating these
ideas into every-day activities
• Explicitly teach students from a younger age that there are
many numbers between 0 and 1 and demonstrate this on a
number line
• Expose students to the written representation of fractions
earlier so that they can develop a conceptual understanding
of it
PRIMARY/JUNIOR
Results
• Students were generally able to identify and compare
benchmark fractions on the number line
• Use of partitioning (concretely with hands) during
explanations by some students
• Generally limited understanding of equivalency
Recommendations
• Build conceptual understanding of part-whole relationships
(e.g., using fraction strips) and how they are related to the
written representation of fractions
• Help students develop an understanding that two equivalent
fractions can occupy the same point on the number line
through layering and fraction wall activities