The Number Line in the Common Core

Introduction

This is an introductory unit designed for my 9th grade Algebra 1 classes. The majority of students who are taking my class have not taken pre-algebra and only took middle school math, though a handful of the students took middle school pre-algebra or algebra. The unit, however, addresses concepts that should be prior knowledge to all students from the elementary years but is not. It would, therefore, be immediately applicable as an introductory unit in any secondary high school mathematics course. The unit, with very little modification, would also fit well into an elementary level class. The complexity of questions and techniques can be adjusted depending on the age or ability level of the students.

I work at a small comprehensive high school where the student body is incredibly diverse. My classroom contains students who represent a full range of academic abilities. In addition, students come from a variety of backgrounds that reflect the demographics of Washington DC as a whole. There are a large number of recent immigrants and English language learners, from a wide variety of countries and cultures. This enormous variety of backgrounds and skill levels is at the front of my mind as I design this unit.

I believe this unit will be extremely helpful for the students that I teach.  I teach in the District of Columbia public school system, an enormous and diverse urban district that serves the entire city.  Based on the data received from my beginning of the year assessments over the past two years at DCPS, a large majority of the students entering into the ninth grade are performing below grade-level in mathematics.  While some students are able to answer simple questions that only show understanding at the abstract level of the math involved, they still have not fully conceptualized foundational math skills.  In the 9^th grade the students will be asked not only to answer high-level questions about Algebra but also to write explanations about their mathematical thinking in both brief and extended constructed responses in preparation for the standardized PARCC examination.  I believe that if we start off the year with this unit where we connect students’ prior knowledge of these skills through the study of the number line, we will build a solid conceptual foundation for their continued success throughout the school year.

The focus of the unit, “The Starting Line-Up: analyzing the number line to connect foundational skills to Algebra”, develops what is considered to be prior knowledge, things the students should already know coming into my class, to the number line.  These skills include adding and subtracting, using signed whole numbers, and understanding fractions.  During this review I will use the number line and associated manipulatives as the primary tools to ensure that students have an abstract as well as a conceptual understanding of these foundational skills through discussion.  Many students already have an abstract understanding of the skills, as they are able to solve math problem sets in isolation to one concept and without any application. The plan is to push the students towards a conceptual understanding where they are able to apply their learning to a variety of situations and contexts.

The unit will start with ensuring student understanding of numbers and the conceptualization of the addition and subtraction operations.  Once the basic understanding of these operations exists, I will transfer these skills to the number line to re-explain by addressing the ideas of placing, shifting, and reflecting numbers on the number line.  Having the visualization of these operations will increase student conceptualization of the operations and they will be in the beginning stages of understanding the effects of shifts and reflections when graphing in Algebra.  Additionally, the number line will assist students in recalling the mental math skills needed to increase mathematical fluency.  Such mental skills include identifying large shifts when skip counting, and using friendly numbers.

After this introductory work, we can begin using numbers to solve equations.  We will begin our understanding by representing base-10 numbers by using base ten bars and transferring them to the number line for addition and subtraction.  By transferring back and forth from base-10 blocks and base-10 bars to ten strips, students will begin to understand the connection between basic operations, measurement, and graphing. This connection is not often addressed by teachers and can lead to misconceptions in Algebra I when solving and graphing algebraic equations.

I will progress through the instruction using the base-ten connection to addition and subtraction on the number line to encourage continued conceptual understanding from whole numbers to representing fractions by demonstrating the parallels of the arithmetic of fractions and the arithmetic of whole numbers.  Finally, I will incorporate solving problems with signed numbers on the number line by acknowledging the idea of direction and its relation to positive and negative numbers on the number line.  At the end of the unit through the use of the number line model, students will have the conceptual understanding of numbers and the number line needed to solve problems that require addition and subtraction of signed, whole numbers and fractions in the Algebra I course.

In each lesson, students will use learning strategies such as the use of manipulatives to work from concrete to pictorial to abstract comprehension of the operations on the number line.  I will encourage active learning through student-to-student discourse and intentional math discussions in the classroom, employing the use of Socratic Seminars.  These discussions increase student understanding and they also prepare students to express their understandings both orally and collaboratively.  It will be important to plan high-level questions with an understanding of how to structure the conversations to encourage debate while knowing what to listen for, what to highlight, and what misconceptions to address.  It will be important for students to hear a wide variety of strategies and thought patterns as they gain their own conceptual understanding of mathematics.