Questions about infinity, division by zero, and primes and composites, combine with discussions about finite and infinite sequences and series and searches for patterns to open up the full richness of a mathematical world. Estimation should be a routine part of mathematics classes. Students should be regularly engaged in estimating both quantities and the results of operations. They should respond to questions that arise naturally during the course of the class with answers which demonstrate confident and well-conceived use of estimation strategies and sense of number. Technology plays an important role in number sense at these grade levels.
Calculators can be wonderful exploration tools when examining numerical relationships.
Many insights about the relationships between fractions and decimals, for instance, can be attained by simply dividing the numerators of fractions by their denominators. Generalizations about what kinds of fractions produce what kinds of decimals start to flow very freely in such open-ended explorations. Computer software can also be very useful. Spreadsheets, for example, can show a great many ratios on the screen at the same time. The cumulative progress indicators for grade 8 appear below in boldface type.