•Research has shown that “procedural knowledge,
such as algorithms for operations, is often taught
without contexts or concepts, implying to the
learner that algorithms are an ungrounded code
only mastered through memorization.
Introducing algorithms before conceptual
understanding is established, or without linking
the algorithm to conceptual knowledge, creates
a curriculum that tends to be perplexing for
children to master or appreciate.” • Memorization without understanding often leads
to misapplication of the algorithm.
MAKING CONNECTIONS •Concepts must be placed in context.
In most cases, working abstractly with numbers does not foster
understanding unless a foundation has
been previously laid so that a child
can make connections. •Putting concepts into situations with
which children are familiar is crucial.
Meaningful learning depends on
connecting the new concept to the
existing knowledge base in some way.
FACTIONS IN CONTEXT •When we say “½” we are implicitly referring to
½ of something. • Every fraction has a “whole” or base of
reference associated with it. Contexts can help
one to focus on what that “whole” is. • Consider ½ of a 12-inch pizza and ½ of 16-inch
pizza. Are they the same? Different? Explain.
•Being aware of the “whole” to which a
fraction refers will be important in working
with fractions. • Pictorial representations of fractions will also
be an important tool to ground student
understanding. The notion of sharing
something equally among people is an
intuitive way to introduce the concept of
WHAT ARE RATIONAL NUMBERS?
(WHAT ARE FRACTIONS?) •Rational numbers (fractions)
are those that can be written
as a comparison of two
integers , a/b, b≠0.
MODELING RATIONAL NUMBERS
•Identifying the Whole and Separating It into Equal
Parts (Pictorial Representation)
2/3 : Dividing a whole into equal size parts and choosing two of those parts • Using Two Integers to Describe Part of a Whole
3 slices of pizza / 8 slices of pizza (whole pizza) • Using Fraction Language
“halves” “thirds” “fourths”
RATIONAL NUMBERS VS. FRACTIONS
•A rational number is the relationship
represented by an infinite set of
ordered pairs, each of which
describes the same quantity. •A fraction is a symbol, a/b, where a
and b are numbers and b ≠ 0. Here,
a is the numerator of the fraction
and b is the denominator of the
REPRESENTING AND DESCRIBING FRACTIONS
•Write definitions and draw pictorial
representations for the following fractions.
TWO TYPES OF FRACTIONS
•When the numerator of a fraction
is less than the denominator, the
fraction is called a proper
fraction. •When the numerator of a fraction
is greater than or equal to the
denominator, the fraction is
called an improper fraction.
•Take a piece of paper and fold it in half. Think
about the fractions represented by each
rectangle formed. • Fold the paper in half again. What fractions
can be represented now? • Fold the paper in half once again. Discuss
different fraction interpretations of the
rectangles formed. • What is the significance of this activity? (What
concept is being introduced?)
•Two fractions, a/b and c/d, are equivalent fractions iff ad = bc. •Fundamental Law of Fractions
Given a fraction a/b and a
number c ≠ 0, a/b = ac/bc.
•A fraction representing a
rational number is in simplest
form when the numerator and denominator are both integers
that are relatively prime and the
denominator is greater than
FAIR SHARE ACTIVITY
•A. For each of the following problems, imagine that
you have the given number of brownies to share
equally among a certain number of people. Find out
how many (or how much of a) brownies each person
gets. • Explain your process and reasoning. In any stage of the
process, if you talk about or use a fraction, be sure to
write the expression for the fraction. Be sure to label
any diagrams with appropriate fraction notation. Write
your answer as a fraction or sum of fractions that
expresses your process (not just the final answer). • 3 people share 4 brownies • 4 people share 7 brownies • 4 people share 2 brownies • 3 people share 2 brownies
FOUR MEANINGS OF ELEMENTARY FRACTIONS 1) Part of a Whole
2 slices of a pizza cut into 8 equal pieces
2) Part of a Group or Set
3/5 of a group of 20 people prefer juice over
3) Position on a Number Line
A scarf 3 ½ feet long made from a length of silk 5
1 chocolate cream pie split between four people
Elementary fractions will most likely not deal with
rational values represented by negative fractions,
nor irrational fractions.
•A decimal is a symbol that uses a
base-ten place-value system with
tenths and multiples of tenths to
represent a number. A decimal point is used to identify the ones
WAYS TO EXPRESS DECIMALS
•Expanded notation •As a fraction
1) Express 31.25 in expanded
2) Write 0.75 and 1.3 as simplified
CONVERTING FRACTIONS TO DECIMALS •Using the Fundamental Law of
Fractions • Multiply the numerator and
denominator by some value that will
produce a product in the denominator
that can be written as a power of 10.
CONVERTING FRACTIONS TO DECIMALS
• Examples: Use the Fundamental Law of Fractions
to convert each fraction to a decimal.
CONVERTING FRACTIONS TO DECIMALS •Using Division •Divide the numerator by the
denominator using the standard
algorithm for division.
CONVERTING FRACTIONS TO DECIMALS
• Examples: Use division to
change each fraction to a
TYPES OF DECIMALS
• When using division to change from fractions to
decimals, the remainder determines the type of
decimal. • If the remainder finally becomes 0, then the
resulting decimal has a fixed number of places
and is called a terminating decimal. With a
terminating decimal, the denominator can be
expressed as a power of ten (using the
Fundamental Law of Fractions). • If the remainder will never become 0, then the
decimal in the quotient has a digit or group of
digits that will repeat over and over. This is
called a repeating decimal. • Thus, every rational number can be expressed
as terminating or repeating decimal.
•A rational number is expressed in scientific notation when it is
written as a product where one factor is a decimal greater than or
equal to 1 and less than 10 and the
other factor is a product of 10.
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