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A. Numbers
By the end of 8th grade, students should know that:
- There have been systems for writing numbers other than the Arabic system of place values based on tens.
- A number line can be extended on the other side of zero to represent negative numbers.
- Numbers can be written in different forms, depending on how they are being used.
- The operations + and - are inverses of each other one undoes what the other does; likewise x and ÷ .
- The expression a/b can mean different things: a parts of size 1/b each, a divided by b, or a compared to b.
- Numbers can be represented by using sequences of only two symbols (such as 1 and 0, on and off); computers work this way.
- Computations (as on calculators) can give more digits than make sense or are useful.
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B. Symbolic Relationships
By the end of 8th grade, students should know that:
- An equation containing a variable may be true for just one value of the variable.
- Mathematical statements can be used to describe how one quantity changes when another changes.
- Graphs can show a variety of possible relationships between two variables.
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C. Shapes
By the end of 8th grade, students should know that:
- Some shapes have special properties: triangular shapes tend to make structures rigid, and round shapes give the least possible boundary for a given amount of interior area.
- Lines can be parallel, perpendicular, or oblique.
- Shapes on a sphere like the earth cannot be depicted on a flat surface without some distortion.
- The graphic display of numbers may help to show patterns such as trends, varying rates of change, gaps, or clusters.
- It takes two numbers to locate a point on a map or any other flat surface.
- The scale chosen for a graph or drawing makes a big difference in how useful it is.
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D. Uncertainty
By the end of 8th grade, students should know that:
- How probability is estimated depends on what is known about the situation.
- Probabilities are ratios and can be expressed as fractions, percentages, or odds.
- The mean, median, and mode tell different things about the middle of a data set.
- Comparison of data from two groups should involve comparing both their middles and the spreads around them.
- The larger a well-chosen sample is, the more accurately it is likely to represent the whole.
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E. Reasoning
By the end of 8th grade, students should know that:
- Some aspects of reasoning have fairly rigid rules for what makes sense; other aspects don't.
- Practical reasoning, such as diagnosing or troubleshooting almost anything, may require many-step, branching logic.
- Sometimes people invent a general rule to explain how something works by summarizing observations.
- People are using incorrect logic when they make a statement such as "If A is true, then B is true; but A isn't true, therefore B isn't true either.
- A single example can never prove that something is always true, but sometimes a single example can prove that something is not always true.
- An analogy has some likenesses to but also some differences from the real thing.
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