Rationale
Description of Unit Structure
This unit will fill a deficit that currently exists in my curriculum, which is the lack of a cohesive and complete progression of concepts that deal with the arithmetic of rational numbers. The progression of concepts will all be delivered in the unifying context of the number line. As expressed earlier, my students lack a sound way of thinking about operating with rational numbers.
Figure 1
This is now very clear and well organized.
As seen in the figure above, my unit, highlighted in red, will be used in succession with Jeff Rossiter’s unit, Placing Rational Numbers on the Number Line, highlighted in blue. The concepts covered in the first unit will focus on the realization and placement of rational numbers on the number line. It will be prerequisite that students first have a unified understanding of what rational numbers are and how they can be represented on the number line before moving on to adding and subtracting of rational numbers, the focus of my unit.
For this unit, the flow of concepts will be detailed in the Unit Concepts section along with a thorough description of the essential understandings for each concept. Then, an explanation of how the unit will be implemented is laid out in the Teaching Approach and Activities sections.
Why the Number Line
The use of the number line is fundamental to this unit and is intended to provide an anchoring context in which students can see and represent numbers and then make sense of operations. This unit will only address the making sense of addition and subtraction of rational numbers and will prepare my students to address multiplication and division later in the year. By “making sense”, I am referring to a student’s understanding and ability to visualize what is happening in the addition and subtraction of rational numbers. By seventh grade, most of my students exhibit many gross misconceptions about addition and subtraction. They rely on algorithms and calculators, especially when it comes to operating with integers and fractions. These misunderstandings are highlighted when they are asked questions such as, “Does your answer make sense?”, “ How do you know the sum/difference is accurate?”,” Can you draw a representation of your computation?” In order to represent the addition or subtraction of whole numbers, students often employ previously learned devices such as adding or crossing out tally marks or finger counting. These methods fail them when, for example, they are required to add fractions with unlike denominators or subtract a negative from a positive. Students seldom demonstrate a conceptual understanding and are only able to operate in the abstract using algorithms. When they are explicitly prompted to demonstrate such understanding they again rely on representations that demonstrate a limited mastery of the concepts. Some students use money to reason with decimals, pie or tile representations for fractions, or different colored chips for signed numbers, but herein lies the problem. Students lack a unified way to think about these different varieties of number and therefore view adding and subtracting of different types as very different tasks. This disjointed understanding is due to the fact that they have not come to realize how operating with these different types of numbers should be viewed as similar tasks that follow similar logic.
The number line ultimately serves the purpose of giving students the opportunity to operate with these different forms of number in a uniform way. As a result of the preceding unit, students will have a unified understanding of the different types of number in terms of distance on a number line. This is prerequisite and serves as a foundation to build on for this unit. In the following section I will lay out how each concept will be taught with the number line, what key language will be used, and what essential points need to be brought to light in order for students to reach a complete understanding.
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