Activity 2C: Fractions
Category 1: Numerical Representations and Relationships
Activity 2C: Comparing Fractions
Objective: The students will compare fractions.
Procedure:
 This lesson is written using unmarked fraction squares for models. However,
fraction bars, strips, towers, or circles could also be used.
 Before using the concrete models, develop the student’s informal ideas and
thinking for comparing fractions.
 Have the students compare 3/6 and 5/6. They need to think about having 3 of
something and also 5 of the same thing (sixths). Therefore, 5 is more of the
same size parts than 3 so 5/6 > 3/6. Give the students several other examples.
 Have the students work in pairs to compare fractions with the same denominator,
but different numerators. One student will model the fraction ⅓ using the
unmarked fraction squares. The other student will model ⅔ with the unmarked
fraction squares. The blue piece is the whole region. The students should put
the model of ⅔ under the model of ⅓. Give the students a number line from zero
to 1. Have each student put a dot on the number line for the fraction they are
modeling. Are both fractions smaller than 1? (yes) Place the number line beside
their models. What is the denominator of both fractions? (3) Since both of the
denominators are the same, thirds, you must look at the numerators to decide
which fraction is larger. Look at your models and number line. Which fraction
has the larger numerator? (⅔). Record the equation comparing these fractions.
You can write ⅔ > ⅓ or ⅓ < ⅔.
1
3
0
58
< (is less than)
1
3
2
3
2
3
1
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Have the students follow the same procedure to compare fourths, fifths, sixths,
eighths, tenths, and twelfths where the denominators are the same, but the
numerators are different.
Next, the students are going to compare fractions that have the same
numerators, but different denominators.
Before using the concrete models, develop the student’s ideas and thinking for
comparing fraction with the same number of parts but where the parts are
different sizes.
Have the students compare 3/4 and 3/6. If the whole is divided into 6 parts, the
parts will be smaller than if the whole is only divided into 4 parts. Therefore, if
the 4 parts are larger and you have 3, it will be more than the 3 of the smaller
parts that are sixths (3/4 > 3/6). Use an example of candy bars. One bar is divided
into 6 pieces. The other bar is divided in 4 pieces. From which bar would you
want to take 3 pieces? You might even divide two candy bars to illustrate.
Have the students work in pairs to compare fractions with the same numerators,
but different denominators. One student will model the fraction ⅓ using the
unmarked fraction squares. The other student will model ¼ with the unmarked
fraction squares. The blue piece is the whole region. The students should put
the model of ¼ under the model of ⅓. Give each student a number line from
zero to 1. One number line will be divided into thirds and the other number line
will be divided into fourths. Have each student put a dot on the number line at
the fraction they are modeling. Are both fractions smaller than 1? (yes) Place
the number lines beside their models. Are the denominators of both fractions the
same number? (No, one is 3 and one is 4.) Are the numerators of both fractions
the same number? (yes, 1) When the numerators are the same number and the
denominators are different numbers, this means that the equal parts are also
different. You use the denominator to compare the fractions. Look at your
models. Which fraction has the largest part, thirds or fourths? (thirds) Look at
the two number lines. Which fraction is closer to 1? (thirds) Record the
equation comparing these fractions. You can write ⅓ > ¼ or ¼ < ⅓.
1
3
> (is greater than)
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1
4
59
1
3 > (is greater than)
0
0
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60
1
2
3
1
3
1
4
1
4
2
4
3
4
1
Have the students follow the same procedure to compare halves, thirds, fourths,
fifths, sixths, eighths, tenths, and twelfths where the numerators are the same,
but the denominators are different.
Have them do examples where the numerators are not just 1, such as 3/6 to 3/8.
Next the students are going to compare fractions where both the numerators and
denominators are different. Before using the concrete models, develop the
student’s informal ideas and thinking for comparing these fractions: 3/7 to 5/8 and
5/ to 7/ . These fraction pairs do not use either of the previous thought
4
8
processes. For these fractions, students will use benchmarks of 0, 1/2, and 1 to
decide which fraction is larger. These benchmark numbers can be used for
making size judgements with fractions.
When comparing 3/7 to 5/8, the students need to decide if 3/7 is less than one half
or more than one half. Since 3 is not quite a half of seven, 3/7 is less than one
half. However, 5/8 is more than one half of 8, so it is more than one half.
Therefore, 5/8 is the larger fraction.
Give the students fractions such as 4/6, 5/9, 3/10, 7/21, 3/18, 9/15, etc. Have them decide
if the fractions is more or less than one half. Then give them two fractions to
compare using the one half benchmark.
In comparing 5/4 to 7/8, the students are going to use the benchmark of 1. The
fraction 5/4 is more than 1 and the fraction 7/8 is less than 1. Therefore, 5/4 is the
larger fraction.
Give the students fractions such as 7/6, 3/10, 14/12, 3/2 13/15, 8/24, etc. Have them
decide if the fractions are more or less than one. Then give them two fractions to
compare using the benchmark of one.
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Have the students work in pairs to compare the fractions 3/6 to 6/8. One student
will model the fraction 3/6 using the unmarked fraction squares. The other
student will model 6/8 with the unmarked fraction squares. The blue piece is the
whole region. The students should put the model of 3/6 under the model of 6/8.
Give each student a number line from zero to 4. One number line will be divided
into thirds in the section from 0 to 1 and the other number line will be divided into
eighths in the section from 0 to 1. Have each student put a dot on the number
line at the fraction they are modeling. Are both fractions smaller than 1? (yes)
Place the number lines beside their models. Look at your models. Which
fraction is larger? (six eighths) Look at the two number lines. Which fraction is
closer to 1? (six eighths) Record the equation comparing these fractions. You
can write 3/6 < 6/8 or 6/8 > 3/6.
3
6
0
0
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2
6
1
6
2
8
6
8
< (is less than)
3
6
4
8
5
6
4
6
6
8
6
6 or 1
8
8 or 1
1
Do many other examples using the unmarked fraction squares and the number
lines where both the numerators and denominators are different numbers. Be
sure to include examples where one of the fractions is an improper fraction. This
will give the students an opportunity to see that one fraction could be between 0
and 1 and the other fraction could be between 1 and 2, 2 and 3, or 3 and 4.
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