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June 2010. using anchor problems in singapore math Yeap Ban Har Scarsdale Teachers Institute New York USA. Problem 1.

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Presentation Transcript
slide1

June 2010

usinganchorproblems

insingaporemath

Yeap Ban Har

Scarsdale Teachers Institute

New York USA

slide2

Problem 1

Arrange cards numbered 1 to 10 so that the trick shown by the instructor can be done. In this problems, students get to talk about the positions of the cards using ordinal numbers. They get to use ordinal numbers in two contexts – third card from the left and third card from the top. The problem itself is too tedious to describe using words. Just come for the institute next time round!

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The above is the solution. What if the cards used are numbered 1 to 9? 1 to 8? 1 to 7? 1 to 6? 1 to 5? 1 to 4?

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Problem 2

This is a game for two players. The game can start with any number of paper clips, say, 24. The rule is that a player removes exactly 1 or 2 clips when it is his / her turn. The winner is the player who removes the last lot of paper clips. Find out a way to win the game.

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It turns out that the winning strategy is to leave your opponent with a multiple of three. What if the rule is changed to removing exactly 1, 2 or 3 clips? The winning strategy is to leave your opponent with a multiple of four. Did you notice a pattern?

Problem 2

In this anchor problem and its variant(s), the concept of multiples emerges. Anchor problem does that – to emerge a mathematical concept.

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Problem 3

The initial textbook problem was changed into the second one. Anchor problems and its variants allow students to encounter a range of problems with mathematical variability (Dienes). In this case students encounter both arithmetic and algebraic problems, and also problems with continuous as well as discrete quantities. Although both problems involve the same bar models, the mathematics involved are different.

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273

Najib

Pascal

297

Goggle

Gittar

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Problem 4

  • Mrs Liu spent 1/5 of her monthly salary on a handbag,  of the remainder on a vacuum cleaner and saved the rest. She saved $1890. Find her monthly salary.
  • = half
  • = three-eighths
  • = four-sevenths

The anchor problem was varied systematically so that students get to learn different skills in bar modeling. Notice the different bar modeling skills needed to solve the problem when  is changed from half to three-eighths to four-sevenths.

This problem is taken from a Singapore school’s mock examination to prepare students for the national examination at the end of grade six.

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Problem 5

Think of a 4-digit number, say, 1104. Jumble the digits up to form a different number, say, 0411. Fin d the difference between the two numbers (1104 – 411). Write the difference on a piece of paper. Circle any digit. Tell me the digits you did not circle and I will tell you the one you circled.

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Problem 6

Piece the pieces together so that two adjacent values are equal. Initially students can be asked to simply piece any two pieces together. Later they can be asked to form a single ‘snaking’ figure. Finally they can be challenge to form a square.

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Problem 7

Place digits 0 to 9 in the five spaces to make a correct multiplication sentence – no repetition. 2-digit number multiplied by 1-digit number to give a 2-digit product.

x

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Summary

anchorproblems

roles
Roles
  • To emerge the idea of multiples
  • To provide opportunities to use ordinal numbers in varied situations
  • To provide opportunity to solve arithmetic and algebraic problems that involve continuous and discrete quantities
  • To provide opportunity to learn different skills in using bar models
  • To provide opportunity to do drill-and-practice while
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References

theoriesandmodels

polya
Polya

Polya

Problem-Solving Stages

  • Understand
  • Plan
  • Do
  • Look Back
newman
Newman

Newman

Difficulties in Word Problem Solving

  • Read
  • Comprehend
  • Know Strategies
  • Transform
  • Do Procedures
  • Interpret Answers
dienes
Dienes

Dienes

  • Principle of Variability
    • Mathematical Variability
    • Perceptual Variability
krutetskii
Krutetskii

Krutetskii

  • The ability to pose a ‘natural’ question as a form of mathematical ability

Krutetski (1976). The psychology of mathematical abilities in school children. Chicago, IL:

University of Chicago.

big ideas
Big Ideas

Big Ideas

  • Visualization
  • Patterning
  • Number Sense
  • Metacognition
  • Communication
mathematics curriculum framework

Beliefs

Interest

Appreciation

Confidence

Perseverance

Monitoring of one’s own thinking

Self-regulation of learning

Attitudes

Metacognition

Numerical calculation

Algebraic manipulation

Spatial visualization

Data analysis

Measurement

Use of mathematical tools

Estimation

Mathematical Problem Solving

Reasoning, communication & connections

Thinking skills & heuristics

Application & modelling

Skills

Processes

Concepts

Numerical

Algebraic

Geometrical

Statistical

Probabilistic

Analytical

Mathematics Curriculum Framework