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Prime Time Investigation 4-Linking Multiplication and Addition: The Distributive Property
Throughout this Unit, students have been looking at the multiplicative structure of a number. This Investigation explores
the relationship between the multiplicative and additive structures of numbers. That is, numbers can be written as a
product of factors or as a sum of terms.
Multiplicative and additive structures are explored through the introduction of the Distributive Property. The
Distributive Property states that for any numbers a, b, and c, a(b+c)=ab+ac. The factored form, a(b+c), and the expanded
form, ab+ac, are equivalent expressions. They represent the same quantity.
Working with equivalent numerical expressions provides an opportunity to introduce the Order of Operations
convention. The Investigation ends with several applications in which the students have to decide which operations are
needed to solve problems.
Prime Time Investigation 4.1-Reasoning with Even and Odd Numbers
In Investigation 4.1, students review even (a number that has 2 as a
factor or is a multiple of 2) and odd (a number that does not have 2
as a factor) numbers by representing them with rectangular arrays of
square tiles (an example is at right). Students use tiles and the
multiplicative and additive structures of numbers to classify numbers
as even or odd. Students work on cases in which they reason about
the sum (the answer to an addition problem) or product (the answer
to a multiplication problem) of combinations of even and odd numbers.
At the end of Problem 4.1, students make conjectures (an educated guess about an observed pattern or relationship)
about the sums and products of even and odd numbers and provide justifications using tiles or other methods. When
students use rectangular arrays to show that the sum of two even numbers is even, they are informally using the
Distributive Property. Discussions about the approaches that students used to prove their various conjectures provide
an introduction to the Distributive Property in Problem 4.2.
Prime Time Investigation 4.2-Using the Distributive Property
In Investigation 4.2, we shift the focus to equivalent expressions and seeing how the Distributive Property can be used
to write equivalent expressions. Students will notice that a number can be expressed both as a product and as a sum. To
help make this connection, students relate the Distributive Property to areas of rectangles. They work on areas of
rectangular shapes divided into two rectangles, and they distribute multiplication over addition by using the additive
property of area. At the end of this Problem, students will see the advantage of using the Distributive Property to
represent and solve a problem.
In Investigation 4.2, students use the Distributive Property in several ways. They first explore it by representing areas of
rectangles two ways. This allows them to practice writing the computations that are the conceptual basis of the
Distributive Property. Students then explore the property in context. The Distributive Property will be revisited in later
units and in seventh and eighth grade and is important for students’ success in Algebra.
See the example below for details on how to use the Distributive Property to write equivalent expressions.
Prime Time Investigation 4.3-Ordering Operations
Investigation 4.3 revisits the Distributive Property and
introduces Order of Operations. Using the Distributive
Property, students write equivalent expressions for a
number in factored form and expanded form (see
right). They justify the equivalency of the different forms
using area models.
Students work on number sentences with more than
one operation and decide which operation to perform
first. At the end of this Investigation, students will be
able to compute the value of an expression correctly
using the Order of Operations.
An example of using Order of Operations is below
32 + 5(2 + 3) − 25
Work within parenthesis
(2+3) = 5
32 + 5(5) − 25
Compute all exponents
32 = 3 × 3 = 9
9 + 5(5) − 25
Do all Multiplication/Division in order from left to right
5(5) = 25
9 + 25 − 25
Do all Addition/Subtraction in order from left to right
9 + 25 = 34
34 − 25
=9
Prime Time Investigation 4.4-Chosing an Operation
This Investigation brings together the last two Investigations and provides practice in deciding which operations are
needed to solve a problem and how to write a mathematical sentence to represent a problem.
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