Entry Task for Rational Number
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Mathematics Mathematics Initiative Initiative Office of Superintendent NWMC Workshop NWMC Workshop October 12, 2007 October 12, 2007 Slide 1 1. Represent 3/5 in as many ways as possible 2. List concerns you see in your classroom with regards to rational numbers
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Entry Task for Rational Number. Represent 3/5 in as many ways as possible List concerns you see in your classroom with regards to rational numbers. 4 out of 3 people have trouble with fractions. Rational Number WHAT WE CAN LEARN FROM STUDENT WORK. Northwest Mathematics Conference - PowerPoint PPT Presentation
Transcript of Entry Task for Rational Number
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 1
1. Represent 3/5 in as many ways as possible
2. List concerns you see in your classroom with regards to rational numbers
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 2
4 out of 3 people have trouble with fractions
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 3
Northwest Mathematics Conference
Bellevue, Washington October 12, 2007
presented by
Mary HolmbergLynda Eich
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 4
Mathematics is a language and science of patterns
Mathematical content must be embedded in the mathematical processes
For all students to learn significant mathematics, content must be taught and assessed in meaningful situations
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 5
GOALS:
Define rational number Analyze student work Connect research to
conceptual understanding of fractions
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 6
“Of all the topics in the school curriculum, fractions, ratios, and proportions arguably hold the distinction of being
the most protracted in terms of development,
the most difficult to teach, the most mathematically complex, the most cognitively challenging, the most essential to success in higher
mathematics and science…”
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 7
Diagnostic Tool
Common Errors Incomplete Knowledge Incorrect Knowledge
Formative Assessment
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 8
A rational number – “fraction” is a number that can be expressed as a ratio of two integers.
Different ways to interpret rational numbers: Part- whole meaning Quotient meaning Ratio meaning Operator meaningStudents must understand the different
interpretations of rational numbers as well as how the different interpretations interrelate
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 9
Do the problem Analyze student work Discussion of student
work Review research
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 10
Explain or show how to change 3 2/5 to an improper fraction using words, numbers, and/or pictures.
Explain or show how to change 17/4 to mixed number using words, numbers, and/or pictures.
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 11
What did the student do?What does the student understand?
(evidence)What questions would you want to ask the
student to learn more about their thinking?
What does the student have yet to learn?What would be your next learning goal for
this student?
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 12
Video clip Let students develop rules -“There is absolutely no reason ever to provide a
rule about multiplying the whole number by the bottom number and adding the top number. Nor should students need a rule about dividing the bottom number into the top to convert fractions to mixed numbers. These rules will readily be developed by the students but in their own words and with complete understanding. “Van de Walle p. 69
Changing mixed numbers to improper fractions
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 13
Wanda really likes cake. She decides that a serving should be of a cake. She has 4 cakes. How many servings does she have? (Schifter, Bastable, Russell, 1999, p. 69)
Explain how you got your answer using words, numbers, and/or pictures.
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 14
What did the student do?What does the student understand?
(evidence)What questions would you want to ask the
student to learn more about their thinking?
What does the student have yet to learn?What would be your next learning goal for
this student?
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 15
Delay using algorithm procedures until students are ready
Guidelines for developing computational strategies: Begin with simple contextual tasks Connect the meaning of fraction computation
with whole -number computation Let estimation and informal methods play a
big role in development of strategies Explore each of the operations using models
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 16
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 17
Brandon was give the task to order the following fractions on a number line starting with the greatest fraction:
3/8 4/75/10
What is the greatest fraction?
Explain how you know using words, numbers, and/or pictures.
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 18
What did the student do?What does the student understand?
(evidence)What questions would you want to ask the
student to learn more about their thinking?
What does the student have yet to learn?What would be your next learning goal for
this student?
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 19
Concepts vs. rules Develop number sense with fractions Which fraction is greater? 4/5 or 4/9 4/7 or 5/7 3/8 or 4/10 5/3 or 5/8 2/4 or
14/28 Conceptual thought patterns for comparison
More of the same-sized parts Same number of parts but parts of different sizes More and less than one-half or one whole Distance from one-half or one whole Equivalence Numerical transformations
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 20
The school cafeteria made seventeen submarine sandwiches to share among four groups of students. Since there weren’t the same number of kids in each group, the sandwiches were divided according to the picture below. Each group shared the sandwiches equally without any leftovers. Several of the kids complained that it hadn’t been fair—that some kids got more to eat. (Fosnot, Dolk, 2002, p. 3)
Determine what portion of a sandwich each student will get in each group.
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 21
What did the student do?What does the student understand?
(evidence)What questions would you want to ask the
student to learn more about their thinking?
What does the student have yet to learn?What would be your next learning goal for
this student?
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 22
Development of student’s reasoning about fractions
Making meaning by linking quotients to divided quantities
Fair Sharing/Partitioning Starts in kindergarten and continues Real life situation Importance skills are developed
Exploring the mathematical properties of fractions as number
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 23
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 24
Pick an idea that came up today that you found particularly interesting. What is your current thinking about this idea?
Where are you and/or your school now with regard to this idea?
What are one or two things that you will go back and pursue to move yourself and/or your school along with this idea?
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 25
Mathematics InitiativeMathematics Initiative
Office of Superintendent
of Public Instruction
NWMC WorkshopNWMC Workshop
October 12, 2007October 12, 2007
Slide 26
Cuoco, Albert A., and Curcio, Frances R., The Roles of Representation in School Mathematics, 2001 Yearbook, Reston: The National Council of Teachers of Mathematics, Inc., 2001.
Curcio, Frances R., and Bezuk, Nadine S., Understanding Rational Numbers and Proportions, Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5-8, Reston: The National Council of Teachers of Mathematics, Inc., 1994.
Fosnot, Catherine Twomey, and Dolk, Maarten, Young Mathematicians At Work, constructing Fractions, Decimals, and Percents, Portsmouth: Heinemann, 2002.
Gregg, Jeff, and Underwood, Diana, “Measurement and Fair-Sharing Models for Dividing Fractions,” Mathematics Teaching in the Middle School, May 2007, 490-496.
Kilpatrick, Jeremy, Swafford, Jane, and Findell, Bradford, Adding It Up, Helping Children Learn Mathematics., Washington, DC: National Academy Press, 2001.
Lester, Frank K, Jr., Second Handbook of Research on mathematics Teaching and Learning, Charlotte, NC: Information Age Publishing, Inc., 2007
Litwiller, Bonnie, and Bright, George, Making Sense of Fractions, Ratios, and Proportions, 2007 Yearbook, Reston: The National Council of Teachers of Mathematics, Inc., 2002.
Rhynard, Karen, and Aurand, Eric, “Why Are Fractions so Difficult?” Texas Mathematics Teacher, Spring 2004, 14-19.
Van de Walle, John A. and Lovin, LouAnn H., Teaching Student-Centered Mathematics, Grades 5-8, Boston: Pearson Education, Inc., 2006, 66 – 105.
Watanabe, Tad, “Representations in Teaching and Learning Fractions,” Teaching Children Mathematics, April 2002, 457-463.
