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Notes for Session 2, Part A


Note 2

Take a few minutes to read the information about conservation, transitivity, and unit iteration. Whereas adults conserve measures, we can sometimes become confused (as with the tangram activity in Session 1) by a visual image. Transitivity is used in algebra and geometry (for example, as justification for steps in a proof) as well as in measurement, when comparing the equality of a number of measures. Examining the concept of units leads us to consider the kind of units that are used when we count versus when we measure.

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Note 3

If you are working in a group, discuss Problems A1-A3 together. When discussing Problem A2, consider the fact that young children first learn about numbers using discrete quantities. How does that differ from measurement, which is never exact (discrete), as we can infinitely divide continuous quantities?

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Note 4

Rational numbers are what is known as a dense set: A dense set is such that for any two elements you choose, you can always find another element of the same type between the two.

To learn more about the concept of density, go to Learning Math: Number and Operations, Session 2.

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Note 5

To learn more about rational numbers and the part-whole interpretation of fractions, go to Learning Math: Number and Operations, Session 8.

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Note 6

If you are working in a group, work in pairs on both parts of Problem A6. First use the fraction given to find one unit; then consider how you can use partitioning and equivalence to locate the desired fraction.

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Note 7

The compensatory principle is an important mathematical idea. The idea of an inverse relationship between the size of a unit and the number of units can be examined numerically (e.g., the area of a surface that is 1 m2 can also be expressed as 10,000 cm2). An inverse relationship can also be shown graphically. A linear inverse relationship produces a straight line that is drawn diagonally from the upper left to the lower right in the first quadrant. Be sure to reflect on or discuss other inverse relationships when working on Problem A8.

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