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NCTM Interactive InstituteAnaheim, CA July 27-29, 2015
Annemarie Newhouse6th Grade Mathematics Teacher
South Euclid Lyndhurst City SchoolsSouth Euclid OH
Fractions: It’s All About Equivalency
Workshop Agenda
• Grandma’s and the Acrobats – discussion of SMP 4
• Fraction Progression Grades 3-5
– Jesse’s Train - fractions as numbers
– Wipe out – Fractional Relationships
– Comparing Fractions Task Cards – number lines
– Computation – fraction towers to add/subtract
• Resources
– Illuminations, Illustrative Mathematics, Mangahigh.com, NCTM Teaching Children Mathematics
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Standards for Mathematical Practice
• “The Standards for Mathematical Practice are the heart and soul of the Common Core State Standards for Mathematics…”
• The SMP highlight the level to which the math content must be know, focusing on the application, reasoning, communication, and representation of the content.
-John SanGiovanni and Susan O’cConnell, Putting the Practices Into Action
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Grandma and the Acrobats
Each grandma has equal strength.
Each acrobat has equal strength.
Who wins the final round?
Show your solution in two different ways
Using words, numbers, and pictures.
From 50 Problem-solving Lessons Grades
1-6 by Marilyn Burns Math Solutions Press
2003
Grandmas and SMP 4
• “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace…” (CCSSM, 2010)
• Modeling Mathematics vs Modeling WITH Mathematics.
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Grandmas and Acrobats Student Work
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SMP #4 Model with Mathematics
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Model with Mathematics
• Models are representations of abstract math ideas
• Students model mathematics through numbers, symbols, objects, diagrams, and graphic representations
• We visualize, simplify, and make sense of mathematics through modeling
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Putting the Practices into Action – Susan O’Connell
and John SanGiovanni
Mathematically Proficient students…
• Model math ideas and problems in varied ways
• Analyze models to draw conclusions and solve problems
• Make sense of mathematics through modeling.
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NCTM Process Standards
• Problem Solving Standard
– using mathematics “to solve problems arising in everyday life society, and the workplace” (CCSS 2010, p.7)
• Communication Standard
– By communicating their modeling activities, students improve their ability to communicate ideas and deepen their understanding of the concepts being studied
12Connecting the NCTM Process Standards & the
CCSSM Practices - NCTM 2013
NCTM Process Standards and SMP 4
• Representation Standard
– “The term representation refers both to process and product – in other words, to the act of capturing a mathematical concept or relationship in some form and to the form itself.” – NCTM, 2000, p.67
- Modeling itself can be understood as a form of
representation
• Connections Standard
– Students should experience problems with everyday contexts that are meaningful to children.
13Connecting the NCTM Process Standards & the
CCSSM Practices - NCTM 2013
Fraction Progression 3-5
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IllustrativeMathematics.org
Jesse’s Train
• What relationships can you find when you explore the Cuisenaire Rods?
• Work in your group, using the clues and Cuisenaire Rods to create Jesse’s Train
– Is there more than one solution?
– Can the problem be interpreted in more than one way?
http://bit.ly/16caAud (template for rods)
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Jesse’s Train Clue Cards
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Extra Clues
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Jesse’s Train- Rod Fractional Values
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Teaching Children Mathematics Feb 2015
Jesse’s Train Sample Solution
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Moving beyond area models
Fractions are numbers – students can use what they know about whole numbers to extend their thinking to fractions.
• The number line model allows students to explore fractions as measures.
– Relative size of fractions (3/4 is a bigger number than ½)
– Fractions measure specific intervals (eraser is ¾ of an inch wide)
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Comparing Fractions
• Task Cards for students to explore fractions
Goals:
1. Build understanding of fractions as numbers with a definite magnitude
2. Increase students’ understanding of fractions as measures
3. Develop fractional sense
21Moving beyond Brownies and Pizza –
Freeman and Jorgensen TCM March 2015
Comparing Fractions Problem Sets
Work with a partner or group to solve the problems on the cards.
• What do you notice about the relationships between the fractions?
• What do you notice about the context the problems were situated in?
• How did the real world context help students develop an understanding of fractions as measures?
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Sequence of Fractional Relationships
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Kayla’s Solution to Which is larger, 2/3 or 3/5?
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Diego’s Solution toWhich is larger, 2/3 or 3/5 ?
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Wipeout – Fractional Relationships
Three Moves in the Game
Roll the die and:
1. Remove the pattern block that the fraction on the die represents,
2. Trade in any of your remaining pattern blocks for an equivalent portion, or
3. Do nothing.
Objective is to be first player to discard all blocks
26Teaching Mathematics by Marylin Burns Dice (1/2,1/3,1/3,1/6,1/6,1/6)
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Fraction Addition and Subtraction
Game from ETA Hand2Mind
Hands-On Standards, Common
Core Fractions, Grade 5
Fraction Feud – Calculation Nation
• Fraction Feud
http://calculationnation.nctm.org/Games/Game.aspx?GameId=791A122F-BFCB-4B10-9DEC-217D5AAFB6AF
Check out some other games in Calculation Nation
Sign up as a teacher and have students sign up
http://calculationnation.nctm.org/Games/
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Disclaimer
The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.
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