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A. Numbers
By the end of 12th grade, students should know that:
- Comparison of numbers of very different size can be made approximately by expressing them as nearest powers of 10.
- Numbers can be written with bases different from ten (which people probably use because of their 10 fingers).
- When calculations are made with measurements, a small error in the measurements may lead to a large error in the results.
- The effects of uncertainties in measurements on a computed result can be estimated.
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B. Symbolic Relationships
By the end of 12th grade, students should know that:
- In some cases, the more of something there is, the more rapidly it may change (as the number of births is proportional to the size of the population).
- Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly.
- Any mathematical model, graphic or algebraic, is limited in how well it can represent how the world works.
- Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another.
- When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed.
- The reasonableness of the result of a computation can be estimated from what the inputs and operations are.
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C. Shapes
By the end of 12th grade, students should know that:
- Distances and angles that are inconvenient to measure directly can be found from measurable distances and angles using scale drawings or formulas.
- There are formulas for calculating the surface areas and volumes of regular shapes.
- Geometric shapes and relationships can be described in terms of symbols and numbers and vice versa.
- Different ways to map a curved surface (like the earth's) onto a flat surface have different advantages.
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D. Uncertainty
By the end of 12th grade, students should know that:
- Even when there are plentiful data, it may not be obvious what mathematical model to use to make predictions from them or there may be insufficient computing power to use some models.
- When people estimate a statistic, they may also be able to say how far off the estimate might be.
- The middle of a data distribution may be misleading when the data are not distributed symmetrically, or when there are extreme high or low values, or when the distribution is not reasonably smooth.
- The way data are displayed can make a big difference in how they are interpreted.
- Both percentages and actual numbers have to be taken into account in comparing different groups; using either category by itself could be misleading.
- Considering whether two variables are correlated requires inspecting their distributions, such as in two-way tables or scatterplots.
- The larger a well-chosen sample of a population is, the better it estimates population summary statistics.
- A physical or mathematical model can be used to estimate the probability of real-world events.
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E. Reasoning
By the end of 12th grade, students should know that:
- To be convincing, an argument needs to have both true statements and valid connections among them.
- Logic requires a clear distinction among reasons: A reason may be sufficient to get a result, but perhaps is not the only way to get there; or, a reason may be necessary to get the result, but it may not be enough by itself; some reasons may be both sufficient and necessary.
- Wherever a general rule comes from, logic can be used in testing how well it works.
- Once a person believes in a general rule, he or she may be more likely to notice cases that agree with it and to ignore cases that don't.
- Very complex logical arguments can be made from a lot of small logical steps.
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