nctm number line · 4/21/12 3 fractions,+the+numberline,+&+the+!...
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Importance of the Number Line in Understanding Fractions
Jon Wray Sorsha Mulroe
Lauri Susi
Number Line
Ê Measurement
Ê Locate and compare fractions
Ê Equivalence
Ê Translate between fractions, decimals and percents
Ê Improve understanding of procedures
Models for Fraction Concepts
Ê Area Model – Fraction is the part of the area covered, as it relates to the whole unit
Ê Length or Number Line – The location of a point in relation to 0 and other values on the number line.
Ê Set Model – The count of objects in the subset, as it relates to the defined whole
Source: Van de Walle et al. 2013
The Number Line’s Importance…
Ê The number line is a significantly more sophisticated measurement model (Bright, Behr, Post, & Wachsmuth, 1988)
Ê Many researchers have found [the number line] to be an essential model that should be emphasized more in the teaching of fractions (Clarke et al., 2008, Flores et al., 2006; Siegler, 2010; Usiskin, 2007, Watabe, 2006)
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Research on Effective Instruction
Recommendation 1. Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts.
Recommendation 2. Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.
Recommendation 3. Help students understand why procedures for computations with fractions make sense. Siegler et al., 2010
Research on Effective Instruction
Recommendation 4. Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-‐multiplication as a procedure to use to solve such problems.
Recommendation 5. Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them.
Siegler et al., 2010
Recommendation 2
Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward. Ê Use measurement activities and number lines to help students understand
that fractions are numbers, with all the properties that numbers share.
Ê Provide opportunities for students to locate and compare fractions on number lines.
Ê Use number lines to improve students’ understanding of fraction equivalence, fraction density (the concept that there are an infinite number of fractions between any two fractions), and negative fractions.
Ê Help students understand that fractions can be represented as common fractions, decimals, and percentages, and develop students’ ability to translate among these forms.
7 Siegler et al., 2010
The Number Line: A Central Representational Tool
Key tool to learn that fractions are numbers. “A fraction (or a decimal) is more than a shaded part of an area, a part of a pizza, or a representation using base-‐ten blocks; a fraction (or a decimal) is also a number with a specific location on a number line.”
-‐ Shaughnessy (2011) Number line diagram -‐ A diagram of the number line used to represent numbers and support reasoning about them. In a number line diagram for measurement quantities, the interval from 0 to 1 on the diagram represents the unit of measure for the quantity.
-‐ CCSS-‐M (2010) Visual fraction model -‐ A tape diagram, number line diagram, or area model
-‐ CCSS-‐M (2010)
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Fractions, the Number Line, & the
Ê 3.NF.2 -‐ Understand a fraction as a number on the number line; represent fractions on a number line diagram.
Ê 3.NF.3 -‐ Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
Ê 4.NF.6 -‐ 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.
Ê 4.MD.2 -‐ 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. CCSSO & NGA, 2010
More on the Number Line & the
Ê 6.RP.3 -‐ Use ratio and rate reasoning to solve real-‐world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Ê 6.NS.6 -‐ Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
Ê 7.NS.1 -‐ Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
CCSSO & NGA, 2010
What students need to know
Ê Fractions are numbers with magnitude that represent quantities.
Ê Fractions are not just between zero and one, they live between all the numbers on the number line
Ê Equivalent fractions describe the same magnitude
Ê There are an infinite number of fractions between any two whole numbers
Ê Understand the relative size of fractions; estimate and compare
Ê How to represent fractions as decimals and percents
Doing What Works – http://dww.ed.gov
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Multiple Representations of Fractions
Ê Representing Unit Fractions on a Number Line Ê Jonathan Brendefur
Free Number Line Tools http://www.conceptuamath.com/fractions.html
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Conceptua Math: Free Tools
Ê 5 different number line tools Ê Identifying Fractions Ê Estimate on a Number Line
Ê Place on a Number Line
Ê Multiply Fractions
Ê Divide Fractions
Developing Teachers, Teaching Students
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Elementary Mathematics Specialists and Teacher Leaders (ems&tl) Project
Ê Received a $2,000 grant through the Elementary Mathematics Specialists and Teacher Leaders (ems&tl) Project
Ê Project Goal: Increase student understanding of fractions through purposeful professional development that is based on assessing teacher understanding of fractions
A Professional Development Priority
Recommendation 5. Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them.
Siegler et al., 2010
Pilot School: Bryant Woods ES
Ê Title I School, Howard County
Ê Columbia, Maryland
Ê 53.9% African American
Ê 25.3% White
Ê 10% Hispanic
Ê 2.8% Asian
Ê 45.3% Free/Reduced Lunch
Professional Development Plan
• Focus professional development on both content and pedagogy, 11 teachers in grades 2-‐5, 2 math interventionists.
• Administer Fractions Assessment developed by Jenny Ward, New Zealand • Findings: Only 38% of teachers answered Question 7 (number line)
correctly.
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Resources for Professional Development
Best Practices Approach: concrete – pictorial – abstract
Fractions: Student Pre and Post-‐Test Results
Shawn: Modeling Fractions on a Number Line
Getting Started
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Conceptua Math Number Line Tools
http://www.conceptuamath.com/fractions.html
Questions?
Contact Us
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