### Number Lines - Elementary Mathematics

```Amy LeHew
Elementary Math Facilitator Meeting
October 2012
4/
5 OR
5/
4
Solutions?
Last time we looked closely at types of visual
fraction models and considered the
connection between the task and visual
model used to solve it.
What visual fraction model is used by most
students in our schools?
What visual fraction model is used the least?
Take a look at the 3rd grade fraction standards


Highlight the phrase “visual fraction model”
everywhere it appears.
Underline the phrase “number line”
everywhere it appears.
Develop understanding of fractions as numbers.
1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b
equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
2. Understand a fraction as a number on the number line; represent fractions on a number line
diagram.
Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the
whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the
endpoint of the part based at 0 locates the number 1/b on the number line.
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0.
Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b
on the number line.
3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about
their size.
Understand two fractions as equivalent (equal) if they are the same size, or the same point on a
number line.
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the
fractions are equivalent, e.g., by using a visual fraction model.
Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1
at the same point of a number line diagram.
Compare two fractions with the same numerator or the same denominator by reasoning about
their size. Recognize that comparisons are valid only when the two fractions refer to the same
whole. Record the results of comparisons with the symbols >, =, or <, and justify the
conclusions, e.g., by using a visual fraction model.
Turn and make a statement to your partner
about something you learned while
highlighting/underlining.
Support understanding of important properties
of fractions
◦ Numerical unit
◦ Relationship between whole numbers and fractions
◦ Density of rational numbers (infinite # between any
two)
◦ A number can be named infinitely many ways
http://lmr.berkeley.edu/docs/Pt3-Ch13NCTM%20Yearbook07-4.pdf

Most widely used fraction model? Area
◦ Partition pizzas, brownies, etc.

Limitations of area models
◦ Count pieces without attending to the whole (don’t
distinguish between fractional part of a set from
continuous quantity (area)).
How can number lines help students
understand fraction concepts that are often
obscured by area models?
Of the students who were successful, they
counted
“two-sixths” for the point depicted in
the figure below: it is “six spaces and
that’s two.”



One-third is three pieces and one thing
There isn’t another fraction name
Don’t know
1
15 or
How do you know?
1
9
“The smaller the
denominator, the bigger the
fraction”
Read the classroom scenario on pages 20-22
◦ When you finish, reflect silently on the following:
 Do student in your building consider fractions as
numbers? Do teachers?
 What other generalizations do students and teachers
make about fractions? Are some of these
Consider the following two statements. Declare
Always True, Sometimes True
If sometimes true, show an example and a counter
example.
1. The larger the denominator, the
smaller the fraction.
2. Fractions are always less than one.
3. Finding a common denominator is the
only way to compare fractions with
different denominators.
Which is larger
6 The smaller
the number the
bigger the pieces.
5/
6 OR
7/
8
?
7/
5/
5/
If the
denominator is
smaller, the piece
is bigger
6
5/
The 1/6
piece is bigger
than 1/8
6
Since 8 is
bigger than 6
and 7 is bigger
than 5
8
This one (7/8 )
because it has
more pieces


How do we ensure students make meaning of
the numerator and denominator?
How can we make sure students think of
fractions as numbers
Using your blank number line sheet and
Cuisenaire rods, label the first line in halves
2nd line in Thirds
3rd line in Fourths
4th line in Sixths
5th line in Twelfths
0
1
What number is halfway between zero and
one-half?
What number is one-fourth more than onehalf?
What number is one-sixth less than one?
What number is one-third more than one?
What number is halfway between one-twelfth
and three-twelfths?
What would you call a number that is halfway
between zero and one-twelfth?




How might this impact student’s
understanding of fractions as numbers on a
number line? (using the Cuisenaire rods to make
a number)
How does a number line diagram help students
make meaning of the numerator and denominator?
This activity found on page 23-26
Look at page 27+ for using 2 wholes
Time has run out for the tortoise and the hare!
The Hare jumped five-eighths of the way
The tortoise inched along to three-fourths
of the way.
Who is winning?
How would constructing a number line diagram
to solve this task help students make
meaning of the numerator and denominator?
With fractions? Yea or Na
Use < = > to compare the following sum:
(do not add) *You must represent your
thinking on a number line diagram.
1/
2 +
1/
4
______
1/
3 +
1/
5
3.NF Closest to 1/2
3.NF Comparing sums of unit fractions
3.NF Find 1
3.NF Find 2/3
3.NF Locating Fractions Greater than One on
the Number Line
3.NF Locating Fractions Less than One on the
Number Line
3.NF Which is Closer to 1?
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
Bring 3rd grade unit 7: Finding Fair Shares
Bring 4th grade unit 6: Fraction Cards and
Decimal Squares
```