Number+Line

=Overview= Wertsch (1993) wrote, "...human action typically employs "mediational means" such as tools and language, and that these mediational means shape the action in essential ways" (loc. 175). One tool that humans can use to mediate actions is the number line. A number line can be used with numbers already written on it as shown in figure 1 or it can be empty, as shown in figure 2.


 * [[image:Screen Shot 2014-01-28 at 6.51.08 AM.png width="362" height="323" align="center" caption="Figure 1: Two Number Lines With Numbers on Them. "]] || [[image:empty number line.png width="480" height="240" align="center" caption="Figure 2: An Empty Number Line"]] ||

On this wikipage, you will find:
 * 1) A concept map about number lines (Abby, Spring 2014)
 * 2) 3 ideas about number lines for learning
 * 3) Common Core Learning Standards that include number lines
 * 4) Links to virtual number lines
 * 5) New York State Math assessment questions that include number lines
 * 6) Lesson Plan(s)

2.) Three Important Ideas to Consider when Using Number Lines for Learning

 * Diezmann & Lowrie (2006) found that students need explicit teaching about the number line
 * Heffer (2011) said, "it [number line] should be understood as a model for reasoning, teaching and understanding concepts and properties in mathematics (p. 865)
 * Gravemeijer (1994) found that empty number lines are closely aligned with young children's mental strategies (as cited in Bobis, 2007, p. 411)

3.) Number Lines in the Common Core Learning Standards
The Common Core Learning Standards for Math require that students use a number line as a mathematical tool 14 times. I have broken down the references to number lines by grade level, math domain, the standard in that domain and the performance expectation associated with the standard. When you review the expectation, you will notice that there are no references to number lines in K - 1 and a large number in 3rd grade. This does not mean that you should not use a number line to teach with in grades K and 1 and that the only place to use a number line should be in the referenced standards.

Grade 2

 * Measurement & Data
 * Relate addition and subtraction to length.
 * Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

Grade 3

 * Number & Operations—Fractions
 * Develop understanding of fractions as numbers
 * Understand a fraction as a number on the number line; represent fractions on a number line diagram.
 * Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
 * Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
 * Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
 * Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
 * Measurement & Data
 * Solve problems involving measurement and estimation
 * Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

Grade 4

 * Number & Operations—Fractions
 * Understand decimal notation for fractions, and compare decimal fractions
 * Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
 * Measurement & Data
 * Solve problems involving measurement and conversion of measurements.
 * Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Grade 5

 * Geometry
 * Graph points on the coordinate plane to solve real-world and mathematical problems.
 * Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

**4.) Virtual Number Lines**
Examine a number of virtual number lines found on web sites. >>>
 * 1) @http://illuminations.nctm.org/Activity.aspx?id=3510
 * 2) @http://illuminations.nctm.org/Activity.aspx?id=4148
 * 3) @http://nlvm.usu.edu/en/nav/frames_asid_197_g_2_t_1.html
 * 4) @http://nlvm.usu.edu/en/nav/frames_asid_180_g_2_t_1.html
 * 5) @http://nlvm.usu.edu/en/nav/frames_asid_265_g_2_t_1.html
 * 6) @http://nlvm.usu.edu/en/nav/frames_asid_107_g_2_t_1.html
 * 7) @http://nlvm.usu.edu/en/nav/frames_asid_334_g_2_t_1.html

**5.) Number lines on the New York State Tests**
Analyze four NYS exam questions that featured a number line. media type="custom" key="25044420" align="center"

**6.) Lesson Plan(s)**

 * 1.) Standards**
 * ELA Standards:**
 * **Domain:** Writing
 * **Standard:** Text Types and Purposes
 * Performance Expectation: Write informative/explanatory texts to examine a topic and convey ideas and information clearly. ||


 * Math** **Standards:**
 * **Domain:** Numbers & Operations in Base Ten
 * **Standard:** Use place value understanding and properties of operations to perform multi-digit arithmetic.
 * **Performance Expectations:** Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.


 * 2.) Grade:** 3


 * 3.) Content Knowledge:**


 * 4.) Essential Question:** How can you show addition and subtraction using a number line?


 * 5.) URL:** @http://nlvm.usu.edu/en/nav/frames_asid_107_g_2_t_1.html


 * 6.) Student Learning Objectives**:
 * 1) Write an explanatory text that describes how to use a number line to show addition and subtraction using evidence from a math simulation.


 * 7.) Checking for Understanding:**
 * Writing**
 * 1) The paragraph will consist of 1 - 2 sentences
 * 2) Students will use standard English conventions (capitalization, punctuation)
 * Math**
 * 1) Students will use evidence from the simulation to describe how to use a number line to show both addition and subtraction.
 * Student Work**
 * Self-Assessment**
 * 1) Students will use the Windshield Strategy to declare their understanding of knowing how to add and subtract using a time line.
 * CLEAR = I get it! I thoroughly understand the concept.
 * BUGGY = I understand it for the most part, but a few things are still unclear.
 * MUDDY = I don’t get it at all.

In this section, I will describe the Universal Design for Learning Teaching Methods that I plan to use in my my lesson.
 * 8.) UDL Support:**
 * ** To Support Recognition Learning Networks ** - I would support my students' recognition learning "because students aren't all on equal footing when it comes to recognizing such patterns, teachers need to provide differentiated instruction" (Rose & Meyer, 2002,[])
 * **Support Background Knowledge** - "When we learn, we incorporate new knowledge into old knowledge. In neural network terms, new learning is integrated into networks that have been shaped by previous learning. Consequently, what the brain already knows can influence what it will learn from a new example or experience." (2002,[])
 * The students are already aware of how to add and subtract therefore they are accessing prior knowledge when they begin adding and subtracting using a number line. Students will be able to use their background knowledge when learning how to add and subtract using a number line.