Number, Number Sense, and Operations

Number and Number Systems

  1.  Use models and visual representation to develop the concept of ratio as part-to-part and part-to-whole, and the concept of percent as part-to-whole.

  2.  Use various forms of “one” to demonstrate the equivalence of fractions.

  3.  Identify and generate equivalent forms of fractions, decimals and percents.

  4.  Round decimals to a given place value and round fractions (including mixed numbers) to the nearest half.

  5.  Recognize and identify perfect squares and their roots.

  6.  Represent and compare numbers less than 0 by extending the number line and using familiar applications; e.g., temperature, owing money.

Meaning of Operations

  7.  Use commutative, associative, distributive, identity and inverse properties to simplify and perform computations.

  8.  Identify and use relationships between operations to solve problems.

  9.  Use order of operations, including use of parentheses, to simplify numerical expressions.

 10. Justify why fractions need common denominators to be added or subtracted.

11.  Explain how place value is related to addition and subtraction of decimals; e.g., 0.2 + 0.14; the two tenths is added to the one tenth because they are both tenths.

Computation and Estimation

12.  Use physical models, points of reference, and equivalent forms to add and subtract commonly used fractions with like and unlike denominators and decimals.

13.  Estimate the results of computations involving whole numbers, fractions and decimals, using a variety of strategies.

Measurement

Measurement Units

1. Identify and select appropriate units to measure angles; i.e., degrees.

2. Identify paths between points on a grid or coordinate plane and compare the lengths of the paths; e.g., shortest path, paths of equal length.

3. Demonstrate and describe the differences between covering the faces (surface area) and filling the interior (volume) of three-dimensional objects.

4. Demonstrate understanding of the differences among linear units, square units and cubic units.

Use Measurement Techniques and Tools

5. Make conversions within the same measurement system while performing computations.

6. Use strategies to develop formulas for determining perimeter and area of triangles, rectangles and parallelograms, and volume of rectangular prisms.

7. Use benchmark angles (e.g.; 45º, 90º, 120º) to estimate the measure of angles, and use a tool to measure and draw angles.

Geometry and
Spatial Sense

Study Guide

Characteristics and Properties

1. Draw circles, and identify and determine relationships among the radius, diameter, center and circumference; e.g., radius is half the diameter, the ratio of the circumference of a circle to its diameter is an approximation of Pi.

2. Use standard language to describe line, segment, ray, angle, skew, parallel and perpendicular.

3. Label vertex, rays, interior and exterior for an angle.

4. Describe and use properties of congruent figures to solve problems.

5. Use physical models to determine the sum of the interior angles of triangles and quadrilaterals.

Spatial Relationships

6. Extend understanding of coordinate system to include points whose x or y values may be negative numbers.

Visualization and Geometric Models

7. Understand that the measure of an angle is determined by the degree of rotation of an angle side rather than the length of either side.

8. Predict what three-dimensional object will result from folding a two-dimensional net, then confirm the prediction by folding the net.
    

Patterns, Functions and Algebra

Use Patterns, Relations, and Functions

1. Justify a general rule for a pattern or a function by using physical materials, visual representations, words, tables or graphs.

2. Use calculators or computers to develop patterns, and generalize them using tables and graphs.

Use Algebraic Representation

3. Use variables as unknown quantities in general rules when describing patterns and other relationships.

4. Create and interpret the meaning of equations and inequalities representing problem situations.

5. Model problems with physical materials and visual representations, and use models, graphs and tables to draw conclusions and make predictions.

Analyze Change

6. Describe how the quantitative change in a variable affects the value of a related variable; e.g., describe how the rate of growth varies over time, based upon data in a table or graph.

Data Analysis and Probability

Data Collection

1. Read, construct and interpret frequency tables, circle graphs and line graphs.

2. Select and use a graph that is appropriate for the type of data to be displayed; e.g., numerical vs. categorical data, discrete vs. continuous data.

3. Read and interpret increasingly complex displays of data, such as double bar graphs.

4. Determine appropriate data to be collected to answer questions posed by students or teacher, collect and display data, and clearly communicate findings.

5. Modify initial conclusions, propose and justify new interpretations and predictions as additional data are collected.

Statistical Methods

6. Determine and use the range, mean, median and mode, and explain what each does and does not indicate about the set of data.

Probability

7. List and explain all possible outcomes in a given situation.

8. Identify the probability of events within a simple experiment, such as three chances out of eight.

9. Use 0, 1 and ratios between 0 and 1 to represent the probability of outcomes for an event, and associate the ratio with the likelihood of the outcome.

10. Compare what should happen (theoretical /expected results) with what did happen (experimental/actual results) in a simple experiment.

11. Make predictions based on experimental and theoretical probabilities.

Help on 6th Grade Ohio Standards

Number and Number Systems

1. Decompose and recompose whole numbers using factors and exponents (e.g., 32 = 2 x 2 x 2 x 2 x 2 = 25 ), and explain why “squared” means “second power” and “cubed” means “third power.”

2. Find and use the prime factorization of composite numbers. For example:
   a. Use the prime factorization to recognize the greatest common factor (GCF).
   b. Use the prime factorization to recognize the least common multiple (LCM).
   c. Apply the prime factorization to solve problems and explain solutions.

3. Explain why a number is referred to as being “rational,” and recognize that the expression b a can mean a parts of size 1 b each, a divided by b, or the ratio of a to b.

4. Describe what it means to find a specific percent of a number, using real-life examples.

5. Use models and pictures to relate concepts of ratio, proportion and percent, including percents less than 1 and greater than 100.

6. Use the order of operations, including the use of exponents, decimals and rational numbers, to simplify numerical expressions.

7. Use simple expressions involving integers to represent and solve problems; e.g., if a running back loses 15 yards on the first carry but gains 8 yards on the second carry, what is the net gain/loss?

8. Represent multiplication and division situations involving fractions and decimals with models and visual representations; e.g., show with pattern blocks what it means to take

9. Give examples of how ratios are used to represent comparisons; e.g., part-to-part, part-to-whole, whole-to-part.

10. Recognize that a quotient may be larger than the dividend when the divisor is a fraction.

11. Perform fraction and decimal computations and justify their solutions; e.g., using manipulatives, diagrams, mathematical reasoning.

12. Develop and analyze algorithms for computing with fractions and decimals, and demonstrate fluency in their use.

13. Estimate reasonable solutions to problem situations involving fractions and decimals.

14. Use proportional reasoning, ratios and percents to represent problem situations and determine the reasonableness of solutions.

15. Determine the percent of a number and solve related problems; e.g., find the percent markdown if the original price was $140, and the sale price is $100.

Measurement Units

1. Understand and describe the difference between surface area and volume.

2. Use strategies to develop formulas for finding circumference and area of circles, and to determine the area of sectors. e.g., 1/2 circle, 2/3 circle, 1/3 circle, 1/4 circle.

3.     Estimate perimeter or circumference and area for circles, triangles and quadrilaterals, and surface area and volume for prisms and cylinders by:

a. estimating lengths using string or links, areas using tiles or grid, and volumes using cubes;

 b. measuring attributes (diameter, side lengths, or heights) and using established formulas for circles, triangles, rectangles, parallelograms and rectangular prisms.

4. Determine which measure (perimeter, area, surface area, volume) matches the context for a problem situation; e.g., perimeter is the context for fencing a garden, surface area is the context for painting a room.

5. Understand the difference between perimeter and area, and demonstrate that two shapes may have the same perimeter, but different areas or may have the same area, but different perimeters.

6. Describe what happens to the perimeter and area of a two-dimensional shape  when the measurements of the shape are changed; e.g. length of sides are doubled.

Geometry and Spatial Sense Standard

Characteristics and Properties

    1. Classify and describe two-dimensional and three-dimensional geometric figures and objects by using their properties; e.g., interior angle measures, perpendicular/parallel sides, congruent angles/sides.

   2.  Use standard language to define geometric vocabulary: vertex, face, altitude, diagonal, isosceles, equilateral, acute, obtuse and other vocabulary as appropriate.

  3.  Use multiple classification criteria to classify triangles; e.g., right scalene triangle.

  4.  Identify and define relationships between planes; i.e., parallel, perpendicular and intersecting.

Spatial Relationships

  5.  Predict and describe sizes, positions and orientations of two-dimensional shapes after transformations such as reflections, rotations, translations and dilations.

Transformations and Symmetry

  6.  Draw similar figures that model proportional relationships; e.g., model similar figures with a 1 to 2 relationship by sketching two of the same figure, one with corresponding sides twice the length of the other.

Visualization and Geometric Models

  7.  Build three-dimensional objects with cubes, and sketch the two-dimensional representations of each side; i.e., projection sets.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations, and Functions

1.  Represent and analyze patterns, rules and functions, using physical materials, tables and graphs.

  2.  Use words and symbols to describe numerical and geometric patterns, rules and functions.

Use Algebraic Representations

  3.  Recognize and generate equivalent forms of algebraic expressions, and explain how the commutative, associative and distributive properties can be used to generate equivalent forms; e.g., perimeter as 2(l + w) or 2l + 2w.

  4.  Solve simple linear equations and inequalities using physical models, paper and pencil, tables and graphs.

   5.  Produce and interpret graphs that represent the relationship between two variables.

  6.  Evaluate simple expressions by replacing variables with given values, and use formulas in problem-solving situations.

Analyze Change

  7.  Identify and describe situations with constant or varying rates of change, and compare them.

8.  Use technology to analyze change; e.g., use computer applications or graphing calculators to display and interpret rate of change.

Data Collection

1.  Read, construct and interpret line graphs, circle graphs and histograms.

  2.  Select, create and use graphical representations that are appropriate for the type of data collected.

  3.  Compare representations of the same data in different types of graphs, such as a bar graph and circle graph.

Statistical Methods

  4.  Understand the different information provided by measures of center (mean, mode and median) and measures of spread (range).

  5.  Describe the frequency distribution of a set of data, as shown in a histogram or frequency table, by general appearance or shape; e.g., number of modes, middle of data, level of symmetry, outliers.

  6.  Make logical inferences from statistical data.

Probability

7.  Design an experiment to test a theoretical probability and explain how the results may vary.