5 ms·

For one, I think that the pizza slices, by being in 2 dimensions, represents the fraction as portion to a (specific) whole without representing the absolute mag

by jmilloy 13y ago

For one, I think that the pizza slices, by being in 2 dimensions, represents the fraction as portion to a (specific) whole without representing the absolute magnitude of the fraction. Both features are important.

e12e 13y ago
If the pizza has an area of unit size (1) - how does a fraction not have absolute magnitude? And what does the fact that the pizza is in two dimensions have to do with it? If the pizza is modelled as a cylinder with unit volume - any slice has both volume/x volume, and area/x area -- where x is the area of the slice? I'm not trying to be snarky, I would just like you to expand on what you mean? I can see that pizza slices is a poor framework for working with general fractions -- and I agree that a number line is a fine way of visualizing magnitude... perhaps I'm just not familiar enough with what "teaching fractions as pizza slices" entails...?
- jmilloy 13y ago
  Teaching fractions as pizza slices entails dividing the circle into 6ths, say, and shading in 5 slices[1]. The wedge in the circle shows relative magnitude; it depicts the fraction as a portion (numerator) of the whole (denominator) regardless of how big the circle is. The magnitude of the fraction is depicted only relative to 1. On a number line, the magnitude of the fraction is depicted relative to any number. In this sense, we capture the "absolute magnitude" of the fraction, by placing on a line relative to the integers. ---1-----0-----1-----2-----3--- ^ ^ 5/6 13/6 An individual fraction, as a single-dimensional magnitude, is best represented in an infinite single-dimensional space, where magnitude increases forever in exactly one direction. A circle, on the other hand, has a finite amount of space. Maybe you're right: perhaps it is the infinite nature of the number line that lends itself to "absolute magnitude" rather than the fact that it is one-dimensional. On the other hand (this is increasingly pedantic): While we could depict 1d magnitude "infinitely" in 2d with concentric circles of area 1, 2, 3, etc, using 2d area to depict a 1d magnitude is not intuitive. The natural alternative is to subdivide the space into units and show multiple discrete units. In 2d, 13/6ths would be two whole pizzas and one 1/6th-filled pizza. But in 1d, the units can be placed adjacent to each other according to the direction of increasing magnitude. In this sense, 1d lends itself to a natural depiction of absolute 1d-magnitude, but 2d does not. [1] http://www.primarygames.com/fractions/images/pizza_5-6.gif http://www.primarygames.com/fractions/images/pizza_5-6.gif
  - deleted 13y ago
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by jmilloy 13y ago

e12e 13y ago

If the pizza has an area of unit size (1) - how does a fraction not have absolute magnitude? And what does the fact that the pizza is in two dimensions have to do with it? If the pizza is modelled as a cylinder with unit volume - any slice has both volume/x volume, and area/x area -- where x is the area of the slice? I'm not trying to be snarky, I would just like you to expand on what you mean? I can see that pizza slices is a poor framework for working with general fractions -- and I agree that a number line is a fine way of visualizing magnitude... perhaps I'm just not familiar enough with what "teaching fractions as pizza slices" entails...?

jmilloy 13y ago
Teaching fractions as pizza slices entails dividing the circle into 6ths, say, and shading in 5 slices[1]. The wedge in the circle shows relative magnitude; it depicts the fraction as a portion (numerator) of the whole (denominator) regardless of how big the circle is. The magnitude of the fraction is depicted only relative to 1. On a number line, the magnitude of the fraction is depicted relative to any number. In this sense, we capture the "absolute magnitude" of the fraction, by placing on a line relative to the integers. ---1-----0-----1-----2-----3--- ^ ^ 5/6 13/6 An individual fraction, as a single-dimensional magnitude, is best represented in an infinite single-dimensional space, where magnitude increases forever in exactly one direction. A circle, on the other hand, has a finite amount of space. Maybe you're right: perhaps it is the infinite nature of the number line that lends itself to "absolute magnitude" rather than the fact that it is one-dimensional. On the other hand (this is increasingly pedantic): While we could depict 1d magnitude "infinitely" in 2d with concentric circles of area 1, 2, 3, etc, using 2d area to depict a 1d magnitude is not intuitive. The natural alternative is to subdivide the space into units and show multiple discrete units. In 2d, 13/6ths would be two whole pizzas and one 1/6th-filled pizza. But in 1d, the units can be placed adjacent to each other according to the direction of increasing magnitude. In this sense, 1d lends itself to a natural depiction of absolute 1d-magnitude, but 2d does not. [1] http://www.primarygames.com/fractions/images/pizza_5-6.gif http://www.primarygames.com/fractions/images/pizza_5-6.gif
- deleted 13y ago
  [deleted]