TURNING THE BAR EXAMS INTO AN EXACT SCIENCE

According to an article I read, the following are the basic steps grade three students normally follow in dealing with errors and uncertainties in measured values:

• Students will apply a wide variety of mathematical concepts, processes, and reasoning skills to investigate, evaluate, justify and solve a broad range of problems in various content areas and everyday situations.
  1. Formulate a problem, determine information required to solve the problem, choose methods for obtaining this information, and set limits for acceptable solutions.
  2. Demonstrate that there may be multiple ways to solve a problem and explain why this is so.
  3. Understand that there is no one right way to solve mathematical problems but that different methods (e.g., working backward from a solution, using a similar problem type, identifying a pattern) have different advantages and disadvantages.
  4. Transfer strategies from a prior problem to a new situation.
  5. Use trial and error and the process of elimination to solve problems.
  6. Verify the correctness and reasonableness of simple mathematical results.
• Students will select and use a wide variety of tools and technology to support and validate mathematical results.
  1. Represent and examine mathematical situations using concrete materials and computers.
  2. Use a four-function or fraction calculator to confirm computations and to explore patterns.
  3. Use a variety of standard tools (e.g., rulers, clocks, measuring tapes, thermometers) and non-standard objects (e.g., counters, sticks, bolts), to measure mathematical and physical objects in the environment.
• Students will understand and apply numbers, ways of representing numbers, relationships among numbers, and number systems.
  1. Connect physical, verbal, and symbolic representations of whole numbers, fractions and mixed numbers.
  2. Use drawings, diagrams, and models to show the concept of fractions as part of a whole and part of a set.
  3. Explain how numbers are used in various ways, including counting, ordering, representing quantities, measuring, labeling, and indicating location.
  4. Explain how numbers are used in various ways, including counting, ordering, representing quantities, measuring, labeling, and indicating location.
  5. Apply place‑value concepts and numeration to describe, compare, count, order, and group numbers.
  6. Explain the connections between operations.
  7. Use concrete objects to count, order, group, and demonstrate one-to-one correspondence with whole numbers beyond 100.
  8. Identify patterns in number sequences (identify even and odd numbers, count by 2s, 3s, 5s, 10s, and 25s).
  9. Read, write, and order numbers to 10,000.
• Students will estimate, compute, and assess reasonableness of solutions.
  1. Demonstrate proficiency with and memorize addition and subtraction facts through 20 and multiplication facts through 10.
  2. Add and subtract single- and multi-digit whole numbers with regrouping.
  3. Apply addition and subtraction in a variety of situations (such as computing perimeter, extending functions).
  4. Multiply multi-digit whole numbers by single-digit numbers.
  5. Divide two-digit whole numbers by single-digit numbers.
  6. Demonstrate the concept of multiplication as repeated addition and arrays; demonstrate the concept of division as repeated subtraction and as sharing.
  7. Understand and appropriately use the vocabulary of estimation (such as about, near, between).
  8. Use a variety of mental computational methods, strategies, and estimation skills to find solutions and to determine the reasonableness of calculated answers, including those involving concrete and abstract items and situations, such as time and money.
  9. Determine the value of a set of host country currency and U.S. currency.
  10. Read, write, add, and subtract with decimal notation in situations involving money.
• Students will estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.
  1. Estimate before measuring to determine the reasonableness of a solution.
  2. Estimate and measure length, time, temperature, and weight to the nearest unit using customary, metric, and nonstandard measurement.
  3. Compare and order measurable characteristics (for example, time, temperature, length, weight, capacity, area, perimeter) of different objects on the same dimensions.
  4. Tell time to the minute with both analog and digital clocks.
  5. Determine elapsed time to the hour using AM and PM.
  6. Find the perimeter and area of rectangles with direct methods, including using concrete objects as tools.
  7. Recognize the need for a uniform unit of measure.
• Students will use algebraic methods to represent, analyze, and solve abstract and practical mathematical situations involving patterns and functional relationships.
  1. Recognize, reproduce, extend, create, and describe repeating and increasing patterns and sequences using a variety of materials.
  2. Use tables, graphic organizers, verbal rules, and open sentences to describe patterns and other relationships.
  3. Generate and solve simple functions by identifying and applying addition and subtraction patterns.
  4. Generate, write, and solve open sentences using informal methods (such as using manipulatives, drawing, or acting out the solution).
  5. Use concrete objects and symbols to model the concepts of variables, expressions, equations, and inequalities (for example, find the missing number, symbol, or operation sign).
  6. Identify and describe numeric patterns and make predictions based on them (e.g., 1 bicycle = 2 wheels, so 6 bicycles = how many wheels).
• Students will use spatial reasoning and apply the properties and relationships of geometric figures to represent, investigate, analyze, and solve problems.
  1. Use comparative directional and positional words (such as above, inside, left, horizontal, middle).
  2. Describe, name, and label related geometric two- and three-dimensional shapes (such as circle and sphere, square and cube, triangle and pyramid, rectangle and prism).
  3. Draw two-dimensional geometric shapes and construct rectangles, squares, and triangles using tools (such as geoboards, grid paper, ruler, compass), including representation of side, top, and bottom views of the object.
  4. Construct three-dimensional geometric shapes, including boxes and triangular prisms.
  5. Identify and describe geometric figures in the environment.
  6. Identify and create examples of line symmetry.
  7. Order simple geometric figures by size.
  8. Estimate and determine the perimeter and area of geometric figures using manipulatives; demonstrate conservation of area.
  9. Describe, identify, and model slides, flips, and turns with geometric figures.
• Students will use statistical methods to question, collect, organize, analyze, and represent data in order to make decisions and predictions.
  1. Pose a question and collect data by observing, measuring, surveying, and counting.
  2. Construct, read, interpret, and label graphs, including pictographs, simple bar and line graphs, and pie charts.
  3. Interpret data by looking for patterns and relationships, determining range, considering cause and effect, then drawing conclusions and answering related questions.
• Students will understand and apply basic concepts of probability.
  1. Predict and measure the outcome of events, and explain why the results of an experiment may not match predicted outcomes.
  2. Use concepts of certainty, fairness, and chance to discuss the probability of actual events