- Subchapter A
- Numbers
- For Grades: K-2
- Learning Goal 1
- Numbers can be used to count things, place them in order, measure them, or name them.

- Learning Goal 2
- Sometimes in describing things there is a need to use numbers between whole numbers.

- Learning Goal 4
- Simple graphs can help to tell about observations.

- Learning Goal 5
- An important kind of relationship between things is one thing being part of a whole.

- Learning Goal 6
- The first digit of a two-digit number describes how many sets of 10 there are in the number.

- Learning Goal 7
- A quantity is stated as a number and a label, such as 4 inches or 7 blocks.

- Learning Goal 1
- For Grades: 3-5
- Learning Goal 1
- The meaning of a digit in a many-digit number depends on its position.

- Learning Goal 2
- In some situations, "0" means none of something, but in others it may be just the label of some point on a scale, such as a number line.

- Learning Goal 3
- Specifying a quantity requires both a number and a unit.

- Learning Goal 4
- Measurements are always likely to give slightly different numbers, even if what is being measured stays the same.

- Learning Goal 5
- Fractions are numbers used to represent part of something.

- Learning Goal 6
- Symbols are used to signify which operations to perform on numbers. The most common are +, -, x, and ÷.

- Learning Goal 7
- It is possible (and often useful) to estimate quantities without determining them exactly.

- Learning Goal 1
- For Grades: 6-8
- Learning Goal 1
- The system of using the Arabic numerals 0-9 is just one way of representing numbers. The very old Roman numerals are now used primarily for dates, clock faces, or ordering chapters in a book.

- Learning Goal 2
- A number line can be extended on the other side of zero to represent negative numbers. Negative numbers allow subtraction of a bigger number from a smaller number to make sense, and are often used when something can be measured on either side of some reference point (time, ground level, temperature, budget).

- Learning Goal 3a
- The same number can be written in different forms, depending on its intended use.

- Learning Goal 3b
- How a quantity is expressed depends on how precise the measurement is and how precise an answer is needed.

- Learning Goal 4
- The operations + and - are inverses of each other—one undoes what the other does; likewise x and ÷.

- Learning Goal 5
- A number expressed in the form a/b can mean different things: a parts of size 1/b each, a divided by b, or a compared to b.

- Learning Goal 6
- Numbers can be represented by using sequences of only two symbols (such as 1 and 0, on and off); computers work this way.

- Learning Goal 7
- Computations (as on calculators) can give more digits than make sense or are useful.

- Learning Goal 8
- Some interesting relationships between two variables include the variables always having the same difference or the same ratio.

- Learning Goal 9
- Exponents can be used to represent how many times a number is to be multiplied by itself. For example, 43 = 4 x 4 x 4.

- Learning Goal 1
- For Grades: 9-12
- Learning Goal 1
- Comparison of numbers of very different size can be made approximately by expressing them as nearest powers of ten.

- Learning Goal 2
- Numbers can be written with bases other than ten. The simplest base, 2, uses just two symbols (1 and 0, or on and off).

- Learning Goal 3
- When calculations are made with measurements, a small error in the measurements may lead to a large error in the results.

- Learning Goal 4
- The effects of uncertainties in measurements on a computed result can be estimated.

- Learning Goal 1

- For Grades: K-2

- Numbers
- Subchapter B
- Symbolic Relationships
- For Grades: K-2
- Learning Goal 1
- Similar patterns may show up in many places in nature and in the things people make.

- Learning Goal 2
- Sometimes changing one thing causes changes in something else. In some situations, changing the same thing in the same way has the same result.

- Learning Goal 1
- For Grades: 3-5
- Learning Goal 1
- Mathematical statements using symbols may be true only when the symbols are replaced by certain numbers.

- Learning Goal 2
- Tables and graphs can show how values of one quantity are related to values of another.

- Learning Goal 1
- For Grades: 6-8
- Learning Goal 1
- An equation containing a variable may be true for just one value of the variable.

- Learning Goal 2
- Rates of change can be computed from differences in magnitudes and vice versa.

- Learning Goal 3
- Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease, increase or decrease in steps, or do something different from any of these.

- Learning Goal 1
- For Grades: 9-12
- Learning Goal 1a
- 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).

- Learning Goal 1b
- Sometimes the rate of change of something depends on how much there is of something else (as the rate of change of speed is proportional to the amount of force acting).

- Learning Goal 2a
- 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.

- Learning Goal 2b
- Symbolic statements can be combined to look for values of variables that will satisfy all of them at the same time.

- Learning Goal 3
- Any mathematical model, graphic or algebraic, is limited in how well it can represent how the world works. The usefulness of a mathematical model for predicting may be limited by uncertainties in measurements, by neglect of some important influences, or by requiring too much computation.

- Learning Goal 4
- Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another.

- Learning Goal 5
- 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. Sometimes the relationship may be satisfied by one value, sometimes by more than one, and sometimes not at all.

- Learning Goal 1a

- For Grades: K-2

- Symbolic Relationships
- Subchapter C
- Shapes
- For Grades: K-2
- Learning Goal 1
- Circles, squares, triangles, spheres, cubes, cylinders and other shapes can be observed in things found in nature and in things that people build.

- Learning Goal 1
- For Grades: 3-5
- Learning Goal 1
- 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.

- Learning Goal 2
- If 0 and 1 are located on a line, any other number can be depicted as a position on the line.

- Learning Goal 3
- Graphical display of quantities may make it possible to spot patterns that are not otherwise obvious, such as cycles and trends.

- Learning Goal 4
- Objects can be described in terms of their shape or the shapes of their parts.

- Learning Goal 5
- Areas of irregular shapes can be found by dividing them into squares and triangles.

- Learning Goal 6
- Scale drawings show shapes and compare locations of things very different in size.

- Learning Goal 7
- Two shapes can match exactly or be identical except for their sizes.

- Learning Goal 8
- Two lines can be parallel, perpendicular, or slanted with respect to one another.

- Learning Goal 9
- Sometimes two shapes will match if one of them is rotated or flipped.

- Learning Goal 1
- For Grades: 6-8
- Learning Goal 1
- Some of the properties an object has depend on its shape: triangular shapes tend to make structures rigid, and spheres give the least possible boundary for a given amount of interior volume.

- Learning Goal 3
- Shapes on a sphere like the earth cannot be depicted on a flat surface without some distortion. Different ways to map a curved surface (like the earth's) onto a flat surface have different advantages.

- Learning Goal 4
- The graphic display of numbers may help to show patterns such as trends, varying rates of change, gaps, or clusters that are useful when making predictions about the phenomena being graphed.

- Learning Goal 5
- It takes two numbers to locate a point on a map or any other flat surface. The numbers may be two perpendicular distances from a point, or an angle and a distance from a point.

- Learning Goal 6
- The scale chosen for a graph or drawing makes a big difference in how useful it is.

- Learning Goal 7
- For regularly shaped objects, relationships exist between the linear dimensions, surface area, and volume.

- Learning Goal 8
- Shapes can be compared in terms of concepts such as parallel and perpendicular, congruence and similarity, and symmetry.

- Learning Goal 9
- Relationships exist among the angles between the sides of triangle and the lengths of those sides. For example, when two sides of a triangle are perpendicular, the sum of the squares of the lengths of those sides is equal to the square of the third side of the triangle.

- Learning Goal 10
- Geometric relationships can be described using symbolic equations.

- Learning Goal 1
- For Grades: 9-12
- Learning Goal 1
- Distances and angles that are inconvenient to measure directly can be found from measurable distances and angles using scale drawings or formulas.

- Learning Goal 2
- When the linear dimensions of an object change by some factor, its area and volume change disproportionately: area in proportion to the square of the factor and volume in proportion to its cube. Properties of an object that depend on its area or volume also change disproportionately.

- Learning Goal 3a
- Geometric shapes and relationships can be described in terms of symbols and numbers—and vice versa.

- Learning Goal 3b
- The position of any point on a surface can be specified by two numbers.

- Learning Goal 3c
- A graph represents all the values that satisfy an equation, and if two equations have to be satisfied at the same time, the values that satisfy them both will be found where the graphs intersect.

- Learning Goal 4
- Different ways to map a curved surface (like the earth's) onto a flat surface have different advantages.

- Learning Goal 5
- Although real objects never perfectly match a geometric figure, they more or less approximate them, so that what is known about geometric figures and relationships can be applied to objects.

- Learning Goal 6
- Both shape and scale can have important consequences for the performance of systems.

- Learning Goal 1

- For Grades: K-2

- Shapes
- Subchapter D
- Uncertainty
- For Grades: K-2
- Learning Goal 1a
- Some things are more likely to happen than others.

- Learning Goal 1b
- Some events can be predicted well and some cannot.

- Learning Goal 1c
- Sometimes people aren't sure what will happen because they don't know everything that might be having an effect.

- Learning Goal 2
- Often a person can find out about a group of things by studying just a few of them.

- Learning Goal 1a
- For Grades: 3-5
- Learning Goal 1
- Some predictions can be based on what is known about the past, assuming that conditions are pretty much the same now.

- Learning Goal 2
- 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—better for how often something will happen than for exactly when.

- Learning Goal 3a
- Summary predictions are usually more accurate for large collections of events than for just a few.

- Learning Goal 3b
- Even very unlikely events may occur fairly often in very large populations.

- Learning Goal 4a
- Spreading data out on a number line helps to see what the extremes are, where they pile up, and where the gaps are.

- Learning Goal 4b
- A summary of data includes where the middle is and how much spread is around it.

- Learning Goal 5a
- A small part of something may be special in some way and not give an accurate picture of the whole.

- Learning Goal 5c
- There is a danger of choosing only the data that show what is expected by the person doing the choosing.

- Learning Goal 6
- Events can be described in terms of being more or less likely, impossible, or certain.

- Learning Goal 1
- For Grades: 6-8
- Learning Goal 1
- How probability is estimated depends on what is known about the situation. Estimates can be based on data from similar conditions in the past or on the assumption that all the possibilities are known.

- Learning Goal 2
- Probabilities are ratios and can be expressed as fractions, decimals, percentages, or odds.

- Learning Goal 3
- The mean, median, and mode tell different things about the middle of a data set.

- Learning Goal 4
- Comparison of data from two groups should involve comparing both their middles and the spreads around them.

- Learning Goal 5
- The larger a well-chosen sample is, the more accurately it is likely to represent the whole. But there are many ways of choosing a sample that can make it unrepresentative of the whole.

- Learning Goal 1
- For Grades: 9-12
- Learning Goal 1
- Even when there are plentiful data, it may not be obvious what mathematical model to use, or there may be insufficient computing power to use some models.

- Learning Goal 2
- When people estimate a statistic, they may also be able to say how far off the estimate might be due to chance.

- Learning Goal 3
- The middle of a data distribution might be misleading when the data are not distributed symmetrically, when there are extreme high or low values, or when the distribution is not reasonably smooth.

- Learning Goal 4
- The way data are displayed can make a big difference in how they are interpreted.

- Learning Goal 5
- Both percentages and actual counts have to be taken into account in comparing different groups; using either category by itself could be misleading.

- Learning Goal 6a
- Considering whether and how two variables are correlated requires inspecting their distributions, such as in two-way tables or scatterplots.

- Learning Goal 6bc
- A correlation between two variables doesn't mean that one causes the other; perhaps some other variable causes them both or the correlation might be attributable to chance alone. A true correlation means that differences in one variable imply differences in the other when all other things are equal.

- Learning Goal 7a
- The larger a well-chosen sample of a population is, the better it estimates population summary statistics.

- Learning Goal 7bc
- For a well-chosen sample, the size of the sample is much more important than the size of the population. To avoid intentional or unintentional bias, samples are usually selected by some random system.

- Learning Goal 8
- A physical or mathematical model can be used to estimate the probability of real-world events.

- Learning Goal 1

- For Grades: K-2

- Uncertainty
- Subchapter E
- Reasoning
- For Grades: K-2
- Learning Goal 1
- People are more likely to believe your ideas if you can give reasons for them.

- Learning Goal 2
- Reasoning can be distorted by strong feelings.

- Learning Goal 1
- For Grades: 3-5
- Learning Goal 1
- One way to think about something is to compare it to something more familiar.

- Learning Goal 2
- The claims people make are sometimes based on how they feel about something rather than on what they observe.

- Learning Goal 1
- For Grades: 6-8
- Learning Goal 1
- If people have generalizations that always hold, and good information about a particular situation, then logic can help them to figure out what is true about it. This kind of formal logic requires care in the use of key words such as if, then, and, not, or, all, and some.

- Learning Goal 3
- Sometimes people invent a generalization to summarize a set of observations. But sometimes people overgeneralize, imagining generalizations on the basis of too few observations.

- Learning Goal 4
- People are using incorrect logic when they assume that a statement such as "If A is true, then B is true" implies that "If A isn't true, then B must not be true either."

- Learning Goal 5
- In formal logic, a single example can never prove that a generalization is always true, but sometimes a single example can prove that a generalization is not always true. Proving a generalization to be false is easier than proving it to be true.

- Learning Goal 6
- An analogy has some likenesses to but also some differences from the real thing.

- Learning Goal 7
- Reasoning by similarities can suggest ideas to consider but can't prove them one way or the other.

- Learning Goal 1
- For Grades: 9-12
- Learning Goal 1
- A sound argument should have both true statements and valid connections among them. Formal logic is mostly about connections among statements, not about whether they are true. People sometimes use logic that begins with untrue statements, and they sometimes use poor logic even if they begin with true statements.

- Learning Goal 2
- Logic requires a clear distinction between those conditions that are necessary to get a result and those that are sufficient to get the result. Some conditions may be both necessary and sufficient.

- Learning Goal 3
- In using logic in real-world situations, one often has to deal with probabilities rather than certainties.

- Learning Goal 4
- Once a person believes a generalization, he or she may be more likely to notice cases that agree with it and to overlook cases that don't.

- Learning Goal 5
- Because computers can store, retrieve, and process large amounts of data, they can rapidly perform a long series of logic steps. They are therefore being used increasingly to help experts solve complex problems that would otherwise be very difficult or impossible to solve. Not all logic problems, however, can be solved by computers.

- Learning Goal 6
- A failure to find an exception to a generalization after reviewing a large number of instances increases the confidence in the accuracy of the generalization.

- Learning Goal 1

- For Grades: K-2

- Reasoning