Number & Operations
National Council of Teachers
of Mathematics
Standards 2000
URL: http://standards.nctm.org/document/chapter6/numb.htm
For all grades, NCTM standards focus
on students being able to: "understand numbers,
ways of representing numbers, relationships among
numbers, and number systems; understand meanings
of operations and how they relate to one another;"
and to "compute fluently and make reasonable
estimates."
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In grades 6–8 all students should— |
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work flexibly with fractions, decimals,
and percents to solve problems; |
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understand and use ratios and proportions
to represent quantitative relationships;
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select appropriate methods and tools
for computing with fractions and decimals
from among mental computation, estimation,
calculators or computers, and paper
and pencil, depending on the situation,
and apply the selected methods; |
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develop and analyze algorithms for computing
with fractions, decimals, and integers
and develop fluency in their use; |
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develop and use strategies to estimate
the results of rational-number computations
and judge the reasonableness of the
results; |
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develop, analyze, and explain methods
for solving problems involving proportions,
such as scaling and finding equivalent
ratios.
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Fractris
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The Problem: Students fill up rows
in a Tetris-like game by making combinations
of certain fractions that add up to 1.
Where’s the Math: This problem
deals with basic addition of fractions.
However, some of the questions also encourage
students to investigate different ways to
get the fractions to add up to 1. By multiplying
all the fractions by 12, all the fractions
are converted to integers, and the problem
becomes finding all the ways to get the
numbers 1-6 to add up to 12. This involves
combinatorics and number theory.
NCTM Standards: Number
& operations
Role of Components: AgentSheets,
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Pythagoras' Mystery Tablet
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The Problem: Students use the applet
to compute the area of a square based on
its side length. They then use this information
to determine whether it is possible to get
exact side lengths for certain areas.
Where’s the Math: The goal
of this problem was to get students to investigate
the concept of irrational numbers through
a familiar concept like area. Students realized
that certain areas, like 4, had an exact
side length, while others, like 2, did not.
This led to questions about the categorization
of such numbers, based on how "easy"
it was to get an exact area. Without actually
introducing the mathematical nomenclature,
students discovered the existence of certain
irrational numbers.
Standards: Number
& operations
Role of Components: Java text field,
Java text area, invisible "model"
component
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