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Following benchmarks were found| Mathematics |
| | | Standard 1. Uses a variety of strategies in the problem-solving process
| | | | | Level II [Grade: 3-5] |
| | | | | 1. | Uses a variety of strategies to understand problem situations (e.g., discussing with peers, stating problems in own words, modeling problem with diagrams or physical objects, identifying a pattern) |
| | | | | 2. | Represents problems situations in a variety of forms (e.g., translates from a diagram to a number or symbolic expression) |
| | | | | 3. | Understands that some ways of representing a problem are more helpful than others |
| | | | | 4. | Uses trial and error and the process of elimination to solve problems |
| | | | | 5. | Knows the difference between pertinent and irrelevant information when solving problems |
| | | | | 6. | Understands the basic language of logic in mathematical situations (e.g., "and," "or," "not") |
| | | | | 7. | Uses explanations of the methods and reasoning behind the problem solution to determine reasonableness of and to verify results with respect to the original problem |
| | | | | 8. | Understands basic valid and invalid arguments (e.g., counter examples, irrelevant approaches) |
| | | Standard 2. Understands and applies basic and advanced properties of the concepts of numbers
| | | | | Level II [Grade: 3-5] |
| | | | | 1. | Understands basic number theory concepts (e.g., prime and composite numbers, factors, multiples, odd and even numbers, divisibility) |
| | | | | 2. | Understands equivalent forms of basic percents, fractions, and decimals (e.g., 1/2 is equivalent to 50% is equivalent to .5) and when one form of a number might be more useful than another |
| | | | | 3. | Understands the basic difference between odd and even numbers |
| | | | | 4. | Understands the basic meaning of place value |
| | | | | 5. | Understands the relative magnitude and relationships among whole numbers, fractions, decimals, and mixed numbers |
| | | | | 6. | Uses models (e.g., number lines, two-dimensional and three-dimensional regions) to identify, order, and compare numbers |
| | | Standard 3. Uses basic and advanced procedures while performing the processes of computation
| | | | | Level II [Grade: 3-5] |
| | | | | 1. | Multiplies and divides whole numbers |
| | | | | 2. | Adds, subtracts, multiplies, and divides decimals |
| | | | | 3. | Adds and subtracts simple fractions |
| | | | | 4. | Uses specific strategies (e.g., front-end estimation, rounding) to estimate computations and to check the reasonableness of computational results |
| | | | | 5. | Performs basic mental computations (e.g., addition and subtraction of whole numbers) |
| | | | | 6. | Determines the effects of addition, subtraction, multiplication, and division on size and order of numbers |
| | | | | 7. | Understands the properties of and the relationships among addition, subtraction, multiplication, and division (e.g., reversing the order of two addends does not change the sum; division is the inverse of multiplication) |
| | | | | 8. | Solves real-world problems involving number operations (e.g., computations with dollars and cents) |
| | | | | 9. | Knows the language of basic operations (e.g., "products," "multiplication") |
| | | Standard 4. Understands and applies basic and advanced properties of the concepts of measurement
| | | | | Level II [Grade: 3-5] |
| | | | | 1. | Understands the basic measures perimeter, area, volume, capacity, mass, angle, and circumference |
| | | | | 2. | Selects and uses appropriate tools for given measurement situations (e.g., rulers for length, measuring cups for capacity, protractors for angle) |
| | | | | 3. | Knows approximate size of basic standard units (e.g., centimeters, feet, grams) and relationships between them (e.g., between inches and feet) |
| | | | | 4. | Understands relationships between measures (e.g., between length, perimeter, and area) |
| | | | | 5. | Understands that measurement is not exact (i.e., measurements may give slightly different numbers when measured multiple times) |
| | | | | 6. | Uses specific strategies to estimate quantities and measurements (e.g., estimating the whole by estimating the parts) |
| | | | | 7. | Selects and uses appropriate units of measurement, according to type and size of unit |
| | | Standard 5. Understands and applies basic and advanced properties of the concepts of geometry
| | | | | Level II [Grade: 3-5] |
| | | | | 1. | Knows basic geometric language for describing and naming shapes (e.g., trapezoid, parallelogram, cube, sphere) |
| | | | | 2. | Understands basic properties of figures (e.g., two- or three-dimensionality, symmetry, number of faces, type of angle) |
| | | | | 3. | Predicts and verifies the effects of combining, subdividing, and changing basic shapes |
| | | | | 4. | Understands that shapes can be congruent or similar |
| | | | | 5. | Uses motion geometry (e.g., turns, flips, slides) to understand geometric relationships |
| | | | | 6. | Understands characteristics of lines (e.g., parallel, perpendicular, intersecting) and angles (e.g., right, acute) |
| | | | | 7. | Understands how scale in maps and drawings shows relative size and distance |
| | | Standard 6. Understands and applies basic and advanced concepts of statistics and data analysis
| | | | | Level II [Grade: 3-5] |
| | | | | 1. | Understands that data represent specific pieces of information about real-world objects or activities |
| | | | | 2. | Understands that spreading data out on a number line helps to see what the extremes are, where the data points pile up, and where the gaps are |
| | | | | 3. | Understands that a summary of data should include where the middle is and how much spread there is around it |
| | | | | 4. | Organizes and displays data in simple bar graphs, pie charts, and line graphs |
| | | | | 5. | Reads and interprets simple bar graphs, pie charts, and line graphs |
| | | | | 6. | Understands that data come in many different forms and that collecting, organizing, and displaying data can be done in many ways |
| | | | | 7. | Understands the basic concept of a sample (e.g., a large sample leads to more reliable information; a small part of something may have unique characteristics but not be an accurate representation of the whole) |
| | | Standard 7. Understands and applies basic and advanced concepts of probability
| | | | | Level II [Grade: 3-5] |
| | | | | 1. | Understands that the word "chance" refers to the likelihood of an event |
| | | | | 2. | Recognizes events that are sure to happen, events that are sure not to happen, and events that may or may not happen (e.g., in terms of "certain," "uncertain," "likely," "unlikely") |
| | | | | 3. | Understands that when predictions are based on what is known about the past, one must assume that conditions stay the same from the past event to the predicted future event |
| | | | | 4. | Understands that statistical predictions are better for describing what proportion of a group will experience something (e.g., what proportion of automobiles will be involved in accidents) rather than which individuals within the group will experience something, and how often events will occur (e.g., how many sunny days will occur over a year) rather than exactly when they will occur |
| | | | | 5. | Uses basic sample spaces (i.e., the set of all possible outcomes) to describe and predict events |
| | | Standard 8. Understands and applies basic and advanced properties of functions and algebra
| | | | | Level II [Grade: 3-5] |
| | | | | 1. | Recognizes a wide variety of patterns (e.g., basic linear patterns such as [2, 4, 6, 8 . . .] ; simple repeating and growing patterns) and the rules that explain them |
| | | | | 2. | Understands that the same pattern can be represented in different ways (e.g., geometrically or numerically; the pattern of numbers [7, 14, 21, 28 . . .] is equivalent to the mathematical relationship 7 × n) |
| | | | | 3. | Knows that a variable is a letter or symbol that stands for one or more numbers |
| | | | | 4. | Understands the basic concept of an equality relationship (i.e., an equation is a number sentence that shows two quantities that are equal) |
| | | | | 5. | Solves simple open sentences involving operations on whole numbers (e.g., ? + 17 = 23) |
| | | | | 6. | Knows basic characteristics and features of the rectangular coordinate system (e.g., the horizontal axis is the X axis and the vertical axis is the Y axis) |
| | | Standard 9. Understands the general nature and uses of mathematics
| | | | | Level II [Grade: 3-5] |
| | | | | 1. | Understands that numbers and the operations performed on them can be used to describe things in the real world and predict what might occur |
| | | | | 2. | Understands that mathematical ideas and concepts can be represented concretely, graphically, and symbolically |
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