Summer 2012
BENCHMARK FRACTIONS
Resources needed: Rope number lines marked with a 0, ½, 1
and 2, various fractions recorded on index cards, clear tape.
Part 1: Students begin to develop benchmarks for fractions near
zero, one-half and one.
1. Write the following fractions on the board: 1/20, 51/100,
10/9, 13/12, 2/40, 99/100, 103/100
As the difficulty of the task depends on the fractions, begin
with fractions that are clearly close to zero, one-half or one.
2. Ask the students to work in pairs to sort the fractions into
three groups: those close to 0, close to ½, and close to 1.
3. As the students sort the fractions, ask them to explain their
decisions. Why do you think 51/100 is close to half? How
much more than a half is it? Why do you think 1/20 is close
to zero? How much more than zero is it? Why is 103/100
close to one? Is it more or less than one? How do you
know?
4. As the students explain their decisions, encourage them to
consider the size of the fractional parts and how many of
these parts are in the fraction. For example, “99/100 is 100
parts and we have 99 of them. If we had one more, it
would be 100/100 or 1 so 99/100 is very close to 1.”
5. Repeat with another list of fractions. This time use fractions
which are further away from zero, half, and one: 1/10, 5/6,
5/9, 4/9, 17/20, 13/20, 2/20, 9/20, 1/5.
Once more,
encourage students to explain their decisions.
6. Add 1/4 and 6/8 to the list of fractions. Ask students which
group they fit in. Ensure that students understand why these
fractions are exactly midway between the benchmarks.
Benchmark Fractions
Page 1 of 3
Summer 2012
Part 2: Students continue to develop their sense of the size of
fractions in relation to the benchmarks of zero, one-half and one
by coming up with fractions rather than sorting them.
1. Ask students to name a fraction that is close to one but not
more than one. Record this on the board. (for example,
5/6)
2. Next ask them to name another fraction that is closer to
one than that. Record on the board: 5/6 7/8
3. Ask the students to explain why the second fraction is closer
to 1. Encourage students to give explanations that focus
on relative size of the fractional parts.
4. Continue for several more fractions with each fraction
being closer to 1 than the previous fraction.
5. Repeat with fractions that are close to 0.
6. Ask students to work in pairs. Direct one of the pair to
record a fraction that is close to but under ½ on a piece of
paper. The other student then records a fraction that is
closer to ½ and explains why it is closer. Encourage
students to continue to record fractions that are
progressively closer to ½.
7. As the pairs work, circulate and check that they are
expressing an understanding of the relative size of the
fractional parts.
Part 3: Comparisons of fractions rely on an understanding of the
numerator and denominator in fractions and on the relative sizes
of the fractional parts. Equivalent fractions are not introduced
but if they are mentioned by students, they should be discussed.
Benchmark Fractions
Page 2 of 3
Summer 2012
1. Write 4/5 and 4/9 on the board and ask students which one
is larger. Encourage explanations that show the students
understand the fractions have the same number of parts
but the parts are of different sizes.
2. Write 4/7 and 2/7 on the board and ask students to tell
which one is larger. Encourage explanations that show the
students understand both these fractions have same size
parts and therefore the one with more parts is larger.
3. Repeat with 10/9 and 9/10. Students can draw on their
understanding of the benchmark 1.
4. Give students a number of paired fractions to make
decisions between: 7/9 and 6/9; 6/7 and 6/9; 3/8 and 4/7;
9/8 and 4/3; etc.
Part 4
Students draw on their conceptual understanding of
fraction benchmarks (0, ½, and 1) and their understanding of the
relative size of fractional parts to line fractions up on a number
line.
1. Ask students in pairs to draw eight fractions from a “hat.”
Their task is to put the fractions in order and also to line the
fractions up on a number line that is marked 0, ½, 1, and 2.
2. Ask students to write a description of how they decided on
the order for the fractions and where to place them on the
number line. To place the fractions on the number line,
students must also make estimates of fraction size in
addition to simply ordering the fractions.
3. Ask the pairs to join with another pair to see if they agree
with one another’s order and placement of fractions. As
an extension, students in this newly formed group of four
could combine all of their fractions on to a single number
line.
Benchmark Fractions
Page 3 of 3