Making Sense of Rational Numbers

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Making Sense of Rational Numbers

• Linda Jensen Sheffield
• Sheffield_at_nku.edu
• http//www.nku.edu/mathed

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• Questions to deepen understanding
• Who? What? When? Where? Why? and How?
• Who uses fractions and decimals? Everyone!
• What or what if? What patterns do I see? What
  generalizations might I make from the patterns?
  What proof do I have? What are the chances? What
  is the best answer, the best method of solution,
  the best strategy to begin with ? What if I
  change one or more parts of the problem?
• When? When would I use this? When does this work?
  When does this not work?
• Where? Where did that come from? Where should I
  start? Where might I go for help?
• Why or why not? Why does that work? If it does
  not work, why not?
• How? How is this like other problems or patterns
  that I have seen? How does it differ? How does
  this relate to "real-life" situations or models?
  How many solutions are possible? How many ways
  might I use to represent, simulate, model, or
  visualize these ideas? How many ways might I
  sort, organize, and present this information?

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• Role of a Student Mathematician
• ? Repeat/rephrase
• ? Agree/disagree...and tell why
• ? Add on to...
• ? Wait, think, and go deeper
• ? Talk to a partner
• ? Record reasoning and questions
• Our role as teachers is to
• ?ask questions that encourage mathematical
  creativity and reasoning
• ?elicit, engage and challenge each students
  thinking
• ?listen carefully to students ideas
• ?ask students to clarify and justify their ideas
• ?attach mathematical notation and language to
  students ideas
• decide when to provide information, clarify,
  model, lead or let students struggle
• ?monitor and encourage participation
• Adapted from Project M3 Mentoring Mathematical
  Minds

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Egyptian Unit Fractions

• Write the fraction 1/2 as the sum of two other
  unit fractions (fractions with 1 as the
  numerator).

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Think deeply about simple things.
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Find another way.

• Write the fraction 1/2 as the sum of two other
  unit fractions that is different than the first
  way you did it.
• (See p. 20 Bits and Pieces I)

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Try different unit fractions.

• Choose a different unit fraction and write it as
  the sum of two unit fractions.
• Can you do this in more than one way?
• Generalize your method for any unit fraction.

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• Question the answers dont just answer the
  questions.

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Patterns in Unit Fractions

• Study the following
• 1/2 1/3 1/6
• 1/3 1/4 1/12
• 1/4 1/5 1/20
• What patterns do you note?
• Predict how you might get 1/5 as the sum of two
  unit fractions. Check out your prediction.
• How might you get 1/n as the sum of two
  fractions?
• Can you prove your hypothesis?

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• The creative mathematics begins after the
  original problem has been solved.

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Designing Algorithms

• Develop your own algorithm for each of the
  following
• Addition (see p. 48 53, Bits and Pieces II)
• Subtraction (see p. 48 53, Bits and Pieces II)
• Multiplication (see p. 59 63, Bits and Pieces
  II)
• Division (see p. 87, Bits and Pieces II)

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Does this work?

• Jermaine said that to multiply two mixed numbers
  you should multiply the whole numbers together
  and then multiply the fractions together. Then
  add the two products together.
• Do you agree? Why or why not? Use diagrams and
  real-life examples in your explanation as well as
  equations and words.

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Does this work?

• Samantha said that you can divide two fractions
  by dividing the numerators and dividing the
  denominators straight across. She says that there
  is no need to invert.
• Do you agree? Why or why not? Use diagrams and
  real-life examples in your explanation as well as
  equations and words.

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Homework

• Study the following equations
• 1/2 - 1/3 1/2 x 1/3
• 1/3 - 1/4 1/3 x 1/4
• 1/4 - 1/5 1/4 x 1/5
• What pattern do you notice?
• Does this pattern hold for all fractions? If so,
  explain why this works. If not, for what types of
  fractions does this work? Justify your responses.
• List two related questions to explore and answer
  your questions completely.

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The principal activities of brains are making
changes in themselves.
Marvin L. Minsky (from Society of the Mind, 1986)

• And we as teachers have the power to impact the
  creation of new minds.
• Use your power wisely.