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.