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A. Numbers
By the end of 5th grade, students should know that:
- The meaning of numerals in many-digit numbers depends on their positions.
- In some situations, "0" means none of something, but in others it may be just the label of some point on a scale.
- When people care about what is being counted or measured, it is important for them to say what the units are (three degrees Fahrenheit is different from three centimeters, three miles from three miles per hour).
- Measurements are always likely to give slightly different numbers, even if what is being measured stays the same.
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B. Symbolic Relationships
By the end of 5th grade, students should know that:
- Mathematical statements using symbols may be true only when the symbols are replaced by certain numbers.
- Tables and graphs can show how values of one quantity are related to values of another.
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C. Shapes
By the end of 5th grade, students should know that:
- Length can be thought of as unit lengths joined together, area as a collection of unit squares, and volume as a set of unit cubes.
- If 0 and 1 are located on a line, any other number can be depicted as a position on the line.
- Graphical display of numbers may make it possible to spot patterns that are not otherwise obvious, such as comparative size and trends.
- Many objects can be described in terms of simple plane figures and solids.
- Areas of irregular shapes can be found by dividing them into squares and triangles.
- Scale drawings show shapes and compare locations of things very different in size.
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D. Uncertainty
By the end of 5th grade, students should know that:
- Some predictions can be based on what is known about the past, assuming that conditions are pretty much the same now.
- Statistical predictions (as for rainy days, accidents) are typically better for how many of a group will experience something than for which members of the group will experience it and better for how often something will happen than for exactly when.
- Summary predictions are usually more accurate for large collections of events than for just a few.
- Spreading data out on a number line helps to see what the extremes are, where they pile up, and where the gaps are.
- A small part of something may be special in some way and not give an accurate picture of the whole.
- Events can be described in terms of being more or less likely, impossible, or certain.
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E. Reasoning
By the end of 5th grade, students should know that:
- One way to make sense of something is to think how it is like something more familiar.
- Reasoning can be distorted by strong feelings.
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