Number Concepts, Theory, Sense
- Read, write, round, compare, and order any whole or decimal number.
EXT: Read and write numbers in scientific notation.
- Use order of operations and properties to solve problems.
- Investigate the concept of square numbers, square roots, and exponents. - - Find the square of a given number.
EXT: Develop techniques for estimating square roots.
- Identify, model, compare, order, and round fractions and mixed numbers.
- Find factors, multiples, primes, opposite numbers, and prime factorization using a variety of methods including factor trees.
EXT: Write prime factorizations in exponential form.
- Express numbers as fractions, decimals, ratios, and percents, and recognize the relationships; convert from one form to another.
EXT: Make predictions and solve problems using equivalent ratios.
- Estimate percent in problem-solving situations using percent bar models. Set up proportions using percent bar models to solve for an unknown.
- Distinguish between terminating and repeating decimals and write repeating decimals using the appropriate notation.
- Estimate sums and differences. Add and subtract whole numbers, decimal numbers, fractions, and mixed numbers to solve problems.
- Estimate products. Multiply whole numbers with 3 digits in each factor, decimal numbers, fractions, and mixed numbers.
- Estimate quotients. Divide whole numbers (including those with zero in the quotient) and decimal numbers through thousandths by divisors through thousandths (with no more than 1 non-zero digit), writing remainders as fractions or decimals.
EXT: Use short division to divide by single-digit divisors.
- Multiply and divide by powers of ten.
- Estimate quotients. Divide whole numbers, fractions, and mixed numbers by fractions and mixed numbers.
- Find the percent of a number using the percent bar model.
EXT: Determine truth in statistics or advertising and adjust the misleading information.
- Find what percent one number is of another.
EXT: Find the original price when given the discount.
- Estimate and determine length, weight/mass, area, and liquid volume/capacity, using standard and nonstandard units of measure.
EXT: Estimate length using indirect measurement.
- Solve measurement problems by making and interpreting scale drawings; comparing (using "ballpark" approximations) and/or converting customary and metric units; and using tables of equivalents for length, capacity, and weight.
- Develop and apply formulas to find perimeter/circumference and area of rectangles, triangles, parallelograms, and circles.
EXT: Investigate the change in area and perimeter as the dimensions of a figure change; use a spreadsheet to record.
- Estimate angle measures using 45š, 90š, and 180š as referents. Identify, construct, name, and measure angles. Find the sum of the angles for any triangle.
EXT: Develop strategies for finding the sum of the angles for any polygon.
- Investigate and develop procedures and formulas for volume of rectangular prisms, pyramids, cones, and cylinders.
EXT: Explore how changing the dimensions (height and/or circumference) of a cylinder affects the volume.
- Determine and/or compare times in different time zones. Find elapsed time.
- Apply knowledge of basic geometric forms (e.g., lines, segments, angles, polygons, circles, and parts of circles) in problem solving and constructions.
- Classify and define two-dimensional figures (e.g., concave/convex, number of sides, regular/irregular).
- Identify and draw congruent and similar figures.
EXT: Investigate the Golden Ratio as applied to geometric figures.
- Measure and draw angles using a protractor, compass, and straight edge. Construct the perpendicular bisector of a line segment and the bisector of an angle using a straight edge and a compass.
- Identify, draw, and define slides, flips, and rotations of polygons.
EXT: Determine the angle of rotation.
- Compare, contrast, classify, define, and sketch three-dimensional figures (e.g., cones, prisms, pyramids, cylinders). Identify and verbalize the relationships among vertices, faces, and edges of solids.
EXT: Make 2-dimensional drawings of 3-dimensional solids, including dotted lines to indicate unseen edges.
- Investigate the relationship between volume and surface area by building models of prisms. EXT: Plan and design a product container (e.g., cereal box, perfume bottle). Construct a model, indicating the scale used and share with the class.
- Determine possible nets for solid figures (e.g., pyramids, prisms, cylinders).
- Collect data and display it in an appropriate graphic representation (tables, charts, picto-, line, bar, or circle graph; spreadsheets; or computer-generated graphs).
- Collect data from a representative portion of a group using sampling and make predictions and/or recommendations about the larger group using that sample.
- Analyze the data from surveys using percentages and circle graphs.
- Determine the mean, median, mode, and range of a set of data and report the central tendency using the most appropriate measure.
EXT: Find the mean, median, and mode from data given in a graph.
- Represent data using Venn diagrams, stem-and-leaf plots, box-and-whisker plots, and scatter plots.
- Determine the probability of single-outcome (independent) events, recognizing that the probability of certain events is 1, impossible events is 0, and all others lie between 0 and 1.
EXT: Explore and describe how the probability changes when each outcome is dependent upon previous outcomes.
- Determine all possible outcomes for single- and multiple-outcome events using a graphic representation (tree diagram, organized list, etc.); recognize that in some cases order is important and in some it is not.
- Compare experimental probability with theoretical probability. Use simulations to determine probability, applying the concept of randomness.
- Solve simple equations for missing addends or missing factors.
- Represent growth patterns by using variables and expressions.
- Identify and extend addition and multiplication patterns including perfect squares, trinumbers, quadrinumbers, and hexinumbers.
- Multiply and divide by tens and multiples of ten by using patterns, multiplication properties, and mental math.
- Choose and write equations with 1 variable to solve real-life problems (e.g., x - 5 = 32).
EXT: Write a word problem to match a given equation.
- Complete a chart showing input and output when given a function rule.
- Pose problems from everyday life situations and from within and outside mathematics.
EXT: Pose and set up a project problem, working with a small group.
- Solve problems using a logical procedure (a plan).
- Identify information that is available but is not needed to solve a problem. -- Identify additional information needed to solve a problem.
- Select and use appropriate materials and tools needed to solve a problem (e.g., graph paper, manipulative, calculator).
- Simplify problem solving using computer software (e.g., spreadsheet, database, graphing program, simulation).
- Develop and apply strategies (e.g., act it out, build a model, draw a picture or diagram, guess and check, make a chart or table, make a list, make a graph, use a pattern, use logical reasoning, solve a simpler problem, work backward) to solve a wide variety of nonroutine and multistep routine problems.
EXT: Publish a class book, Problem Solving Strategies, that defines and explains the strategies and provides problems that appropriately exemplify the strategy.
- Solve a problem using a formula.
- Solve problems by working collaboratively with peers; entertain others' points of view.
- Share, explain (verbalize/record), and justify (defend) reasoning during and after solving a problem.
- Verify and interpret results with respect to the original problem situation.
- Compare and analyze solution paths (process used to solve problems).
- Generalize solutions and strategies to new problem situations including problems without numbers (e.g., If I want to know which is the best buy, what information do I need and then what should I do?).
- Acquire confidence in using mathematics meaningfully to solve problems.