**Fractions /Rational Numbers**

**Grades K-1**

Although the Grade 2 unit is the first unit in the rational numbers strand,
students develop ideas about

fractions informally both outside of school and through activities in
Kindergarten and Grade 1. For

example, as kindergarteners and first graders use the Shapes software, and work
on activities such as Fill

the Hexagons and Pattern Block Fill-In, they begin to find relationships among
the pattern block

shapes–two trapezoids equal one hexagon (or a trapezoid is 1/2 of a hexagon),
six triangles equal one

hexagon, three triangles equal one trapezoid, etc.

Young students also encounter fractions as they measure
and compare quantities or lengths. The first

grade curriculum capitalizes on this in the unit Fish Lengths and Animal Jumps
(Measurement), asking

students to consider lengths that fall between whole numbers of units.

**Grade 2**

Second graders develop an understanding of what fractions
are and how they can be used to name

quantities. They learn that fractions are quantities that are equal parts of a
whole whether the whole is a

single object or a set of objects. Students work with 1/2, 1/3, 1/4, 2/3, 2/4,
and 3/4 of single objects such

as blocks, rectangles, squares and flags. They work with 1/2, 1/3, and ¼ of sets
of objects such as

balloons, sandwiches and other objects shared among a group of people.

Students learn how fractions are expressed in words—one
half, two thirds—and represented using

numbers—for example, 1/2, 2/3. They learn that the denominator represents the
number of equal parts

in the whole and that the numerator represents the number of the equal parts
being considered, though

they are not expected to use the words denominator and numerator when describing
fractions. Students

also learn the notation for mixed numbers through dividing sets. For example, if
two girls share three

sandwiches, each girl gets 1 1/2 sandwiches.

**Emphases**

What Fractions Are

• Understanding fractions as equal parts of a whole

• Using terms and notation

**Benchmarks
**• Identify 1/2, 1/3, and 1/4 of a region

• Find 1/2 of a set of objects

• Recognize that a fraction divides the whole into equal parts

**Grade 3**

Students use a variety of contexts to understand,
represent, and combine fractions. These include

rectangles representing “brownies,” hexagonal pattern block “cookies,” and
groups of objects.

Students work with halves, fourths, eighths, thirds, and
sixths as they learn how fractions represent equal

parts of a whole. They learn the meanings of the numerator and denominator of a
fraction, so that when

comparing unit fractions (fractions with a numerator of 1), they understand that
the larger the

denominator the smaller the part of the whole: 1/6 is smaller than ½ of the same
whole. Students also

gain experience with common equivalencies, for example, that 3/6 and 2/4 are
both equal to 1/2. Using

these equivalents in contexts, students find combinations of fractions that are
equivalent to a whole or to

another fraction. For example,

1/2 + 2/6 + 1/6 = 1

1/3 + 1/6 = 1/2

Students are introduced to decimal fractions (0.50 and
0.25), using the context of money, and gain

familiarity with fraction and decimal equivalents involving halves and fourths.

**Emphases**

Rational Numbers

• Understanding the meaning of fractions (halves, fourths,
eighths, thirds, sixths) and decimal

fractions (0.50, 0.25) as equal parts of a whole (an object, an area, a set of
objects)

• Using representations to combine fractions (halves, fourths, eighths, thirds,
and sixths)

**Benchmarks**

• Divide a single whole or a quantity into equal parts and
name those parts as fractions or mixed

numbers

• Identify equivalent fractions (e.g.

• Find combinations of fractions that are equal to 1 and to other fractions (e.g.

**Grade 4**

The major focus of the work on rational numbers in Grade 4
is on building students’ understanding of

the meaning, order, and equivalencies of fractions and decimals. Students
continue to focus on the

meaning of fractions as equal parts of a whole. They extend their images of
equal parts to accommodate

fractions that are greater than 1. Students work with fractions in the context
of area (equal parts of a

rectangle), as a group of things (e.g., a fractional part of the class), and on
a number line. They work

with fractions that represent halves, thirds, fourths, fifths, sixths, eighths,
tenths, and twelfths.

Students are introduced to decimal fractions in tenths and
hundredths as an extension of the place value

system they have studied for whole numbers. They relate decimals to equivalent
decimals and fractions

(for example, when they represent 0.25 as part of a rectangle, they can see how
it is equal to ¼ and to 2

½ tenths). Students draw on their mental images of fractions and decimals and on
their knowledge of

fraction and decimal equivalencies and relationships to reason about fraction
comparisons, to order

fractions on a number line, and to add fractions and decimals using
representations.

**Emphases**

Rational Numbers

• Understanding the meaning of fractions and decimal fractions

• Comparing the values of fractions and decimal fractions

Computation with Rational Numbers

• Using representations to add rational numbers

**Benchmarks**

• Identify fractional parts of an area

• Identify fractional parts of a group (of objects, people, etc.)

• Read, write, and interpret fraction notation

• Order fractions with like and unlike denominators

• Read, write, and interpret decimal fractions in tenths and hundredths

**Grade 5**

The major focus of the work on rational numbers in grade 5 is on understanding
relationships among

fractions, decimals, and percents. Students make comparisons and identify
equivalent fractions,

decimals, and percents, and they develop strategies for adding and subtracting
fractions and decimals.

In a study of fractions and percents, students work with
halves, thirds, fourths, fifths, sixths, eighths,

tenths, and twelfths. They develop strategies for finding percent equivalents
for these fractions so that

they are able to move back and forth easily between fractions and percents and
choose what is most

helpful in solving a particular problem, such as finding percentages or
fractions of a group.

Students use their knowledge of fraction equivalents,
fraction-percent equivalents, the relationship of

fractions to landmarks such as ½, 1, and 2, and other relationships to decide
which of two fractions is

greater. They carry out addition and subtraction of fractional amounts in ways
that make sense to them

by using representations such as rectangles, rotation on a clock, and the number
line to visualize and

reason about fraction equivalents and relationships.

Students continue to develop their understanding of how
decimal fractions represent quantities less than

1 and extend their work with decimals to thousandths. By representing tenths,
hundredths, and

thousandths on rectangular grids, students learn about the relationships among
these numbers—for

example, that one tenth is equivalent to ten hundredths and one hundredth is
equivalent to ten

thousandths—and how these numbers extend the place value structure of tens that
they understand from

their work with whole numbers.

Students extend their knowledge of fraction-decimal
equivalents by studying how fractions represent

division and carrying out that division to find an equivalent decimal.

They compare, order, and add decimal fractions (tenths,
hundredths, and thousandths) by carefully

identifying the place value of the digits in each number and using
representations to visualize the

quantities represented by these numbers.

**Emphases**

Rational Numbers

• Understanding the meaning of fractions and percents

• Comparing fractions

• Understanding the meaning of decimal fractions

• Comparing decimal fractions

Computation with Rational Numbers

• Adding and subtracting fractions

• Adding decimals

**Benchmarks**

• Use fraction-percent equivalents to solve problems about the percentage of a
quantity

• Order fractions with like and unlike denominators

• Add fractions through reasoning about fraction equivalents and relationships

• Read, write, and interpret decimal fractions to thousandths

• Order decimals to the thousandths

• Add decimal fractions through reasoning about place value, equivalents, and
representations