• 6
Top Mathematicians
• Shape and Space
• 6.SS.1
Demonstrate an understanding of angles by:
identifying examples of angles in the environment
classifying angles according to their measure
estimating the measure of angles using 45°, 90° and 180° as reference angles
determining angle measures in degrees
drawing and labelling angles when the measure is specified.

Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Provide examples of angles found in the environment.
- Classify a given set of angles according to their measure, e.g., acute, right, obtuse, straight, reflex.
- Sketch 45°, 90° and 180° angles without the use of a protractor, and describe the relationship among them.
- Estimate the measure of an angle using 45°, 90° and 180° as reference angles.
- Measure, using a protractor, given angles in various orientations.
- Draw and label a specified angle in various orientations using a protractor.
- Describe the measure of an angle as the measure of rotation of one of its sides.
- Describe the measure of angles as the measure of an interior angle of a polygon.
• 6.SS.2
Demonstrate that the sum of interior angles is:
180° in a triangle

Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Explain, using models, that the sum of the interior angles of a triangle is the same for all triangles.
- Explain, using models, that the sum of the interior angles of a quadrilateral is the same for all quadrilaterals.
• 6.SS.3
Develop and apply a formula for determining the:
perimeter of polygons
area of rectangles
volume of right rectangular prisms.

Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Explain, using models, how the perimeter of any polygon can be determined.
- Generalize a rule (formula) for determining the perimeter of polygons, including rectangles and squares.
- Explain, using models, how the area of any rectangle can be determined.
- Generalize a rule (formula) for determining the area of rectangles.
- Explain, using models, how the volume of any right rectangular prism can be determined.
- Generalize a rule (formula) for determining the volume of right rectangular prisms.
- Solve a given problem involving the perimeter of polygons, the area of rectangles and/or the volume of right rectangular prisms.
• 6.SS.4
Construct and compare triangles, including:
scalene
isosceles
equilateral
right
obtuse
acute
in different orientations.

Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Sort a given set of triangles according to the length of the sides.
- Sort a given set of triangles according to the measures of the interior angles.
- Identify the characteristics of a given set of triangles according to their sides and/or their interior angles.
- Sort a given set of triangles and explain the sorting rule.
- Draw a specified triangle, e.g., scalene.
- Replicate a given triangle in a different orientation and show that the two are congruent.
• 6.SS.5
Describe and compare the sides and angles of regular and irregular polygons.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Sort a given set of 2-D shapes into polygons and non-polygons, and explain the sorting rule.
- Demonstrate congruence (sides to sides and angles to angles) in a regular polygon by superimposing.
- Demonstrate congruence (sides to sides and angles to angles) in a regular polygon by measuring.
- Demonstrate that the sides of a regular polygon are of the same length and that the angles of a regular polygon are of the same measure.
- Sort a given set of polygons as regular or irregular and justify the sorting.
- Identify and describe regular and irregular polygons in the environment.
• 6.SS.6
Perform a combination of translation(s), rotation(s) and/or reflection(s) on a single 2-D shape, with and without technology, and draw and describe the image.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Demonstrate that a 2-D shape and its transformation image are congruent.
- Model a given set of successive translations, successive rotations or successive reflections of a 2-D shape.
- Model a given combination of two different types of transformations of a 2-D shape.
- Draw and describe a 2-D shape and its image, given a combination of transformations.
- Describe the transformations performed on a 2-D shape to produce a given image.
- Model a given set of successive transformations (translation, rotation and/or reflection) of a 2-D shape.
- Perform and record one or more transformations of a 2-D shape that will result in a given image.
• 6.SS.7
Perform a combination of successive transformations of 2-D shapes to create a design, and identify and describe the transformations.
- Analyze a given design created by transforming one or more 2-D shapes, and identify the original shape and the transformations used to create the design.
- Create a design using one or more 2-D shapes and describe the transformations used.
• 6.SS.8
Identify and plot points in the first quadrant of a Cartesian plane using whole number ordered pairs.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Label the axes of the first quadrant of a Cartesian plane and identify the origin.
- Plot a point in the first quadrant of a Cartesian plane given its ordered pair.
- Match points in the first quadrant of a Cartesian plane with their corresponding ordered pair.
- Plot points in the first quadrant of a Cartesian plane with intervals of 1, 2, 5, or 10 on its axes, given whole number ordered pairs.
- Draw shapes or designs, given ordered pairs in the first quadrant of a Cartesian plane.
- Determine the distance between points along horizontal and vertical lines in the first quadrant of a Cartesian plane.
- Draw shapes or designs in the first quadrant of a Cartesian plane and identify the points used to produce them.
• 6.SS.9
Perform and describe single transformations of a 2-D shape in the first quadrant of a Cartesian plane (limited to whole number vertices).
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Identify the coordinates of the vertices of a given 2-D shape (limited to the first quadrant of a Cartesian plane).
- Perform a transformation on a given 2-D shape and identify the coordinates of the vertices of the image (limited to the first quadrant).
- Describe the positional change of the vertices of a given 2-D shape to the corresponding vertices of its image as a result of a transformation (limited to first quadrant).
• Statistics & Probability
• Patterns and Relations
• 6.PR.1
Demonstrate an understanding of the relationships within tables of values to solve problems.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Generate values in one column of a table of values, given values in the other column and a pattern rule.
- State, using mathematical language, the relationship in a given table of values.
- Create a concrete or pictorial representation of the relationship shown in a table of values.
- Predict the value of an unknown term using the relationship in a table of values and verify the prediction.
- Formulate a rule to describe the relationship between two columns of numbers in a table of values.
- Identify missing elements in a given table of values.
- Identify errors in a given table of values.
- Describe the pattern within each column of a given table of values.
- Create a table of values to record and reveal a pattern to solve a given problem.
• 6.PR.2
Represent and describe patterns and relationships using graphs and tables.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Translate a pattern to a table of values and graph the table of values (limit to linear graphs with discrete elements).
- Create a table of values from a given pattern or a given graph.
- Describe, using everyday language, orally or in writing, the relationship shown on a graph.
• 6.PR.3
Represent generalizations arising from number relationships using equations with letter variables.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Write and explain the formula for finding the perimeter of any regular polygon.
- Write and explain the formula for finding the area of any given rectangle.
- Develop and justify equations using letter variables that illustrate the commutative property of addition and multiplication, e.g., a + b = b + a or a × b = b × a.
- Describe the relationship in a given table using a mathematical expression.
- Represent a pattern rule using a simple mathematical expression, such as 4d or 2n + 1.
• 6.PR.4
Demonstrate and explain the meaning of preservation of equality concretely, pictorially and symbolically.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Model the preservation of equality for addition using concrete materials, such as a balance or using pictorial representations and orally explain the process.
- Model the preservation of equality for subtraction using concrete materials, such as a balance or using pictorial representations and orally explain the process.
- Model the preservation of equality for multiplication using concrete materials, such as a balance or using pictorial representations and orally explain the process.
- Model the preservation of equality for division using concrete materials, such as a balance or using pictorial representations and orally explain the process.
- Write equivalent forms of a given equation by applying the preservation of equality and verify using concrete materials, e.g., 3b = 12 is the same as 3b + 5 = 12 + 5 or 2r = 7 is the same as 3(2r) = 3(7).
• Number
• 6.N.1
Demonstrate an understanding of place value for numbers:
greater than one million
less than one thousandth.

Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Explain how the pattern of the place value system, e.g., the repetition of ones, tens and hundreds, makes it possible to read and write numerals for numbers of any magnitude.
- Provide examples of where large numbers and small decimals are used, e.g., media, science, medicine, technology.
• 6.N.2
Solve problems involving large numbers, using technology.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Identify which operation is necessary to solve a given problem and solve it.
- Determine the reasonableness of an answer.
- Estimate the solution and solve a given problem.
• 6.N.3
Demonstrate an understanding of factors and multiples by:
determining multiples and factors of numbers less than 100
identifying prime and composite numbers
solving problems involving multiples.

Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Identify multiples for a given number and explain the strategy used to identify them.
- Determine all the whole number factors of a given number using arrays.
- Identify the factors for a given number and explain the strategy used, e.g., concrete or visual representations, repeated division by prime numbers or factor trees.
- Provide an example of a prime number and explain why it is a prime number.
- Provide an example of a composite number and explain why it is a composite number.
- Sort a given set of numbers as prime and composite.
- Solve a given problem involving factors or multiples.
- Explain why 0 and 1 are neither prime nor composite.
• 6.N.4
Relate improper fractions to mixed numbers.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Demonstrate using models that a given improper fraction represents a number greater than 1.
- Express improper fractions as mixed numbers.
- Express mixed numbers as improper fractions.
- Place a given set of fractions, including mixed numbers and improper fractions, on a number line and explain strategies used to determine position.
• 6.N.5
Demonstrate an understanding of ratio, concretely, pictorially and symbolically.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Provide a concrete or pictorial representation for a given ratio.
- Write a ratio from a given concrete or pictorial representation.
- Express a given ratio in multiple forms, such as “three to five”, 3:5, 3 to 5, or 3/5.
- Identify and describe ratios from real-life contexts and record them symbolically.
- Explain the part/whole and part/part ratios of a set, e.g., for a group of 3 girls and 5 boys, explain the ratios 3:5, 3:8 and 5:8.
- Solve a given problem involving ratio.
• 6.N.6
Demonstrate an understanding of percent (limited to whole numbers) concretely, pictorially and symbolically.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Explain that “percent” means “out of 100.”
- Explain that percent is a ratio out of 100.
- Use concrete materials and pictorial representations to illustrate a given percent.
- Record the percent displayed in a given concrete or pictorial representation.
- Express a given percent as a fraction and a decimal.
- Identify and describe percents from real-life contexts, and record them symbolically.
- Solve a given problem involving percents.
• 6.N.7
Demonstrate an understanding of integers, concretely, pictorially and symbolically.
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Extend a given number line by adding numbers less than zero and explain the pattern on each side of zero.
- Place given integers on a number line and explain how integers are ordered.
- Describe contexts in which integers are used, e.g., on a thermometer.
- Compare two integers, represent their relationship using the symbols <, > and =, and verify using a number line.
- Order given integers in ascending or descending order.
• 6.N.8
Demonstrate an understanding of multiplication and division of decimals (1-digit whole number multipliers and 1-digit natural number divisors).
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Place the decimal point in a product using front-end estimation, e.g., for 15.205 m × 4, think 15 m × 4, so the product is greater than 60 m.
- Place the decimal point in a quotient using front-end estimation, e.g., for $25.83 ÷ 4, think$24 ÷ 4, so the quotient is greater than \$6.
- Correct errors of decimal point placement in a given product or quotient without using paper and pencil.
- Predict products and quotients of decimals using estimation strategies.
- Solve a given problem that involves multiplication and division of decimals using multipliers from 0 to 9 and divisors from 1 to 9.
• 6.N.9
Explain and apply the order of operations, excluding exponents, with and without technology (limited to whole numbers).
Achievement Indicators
Students who have achieved this outcome(s) should be able to:
- Demonstrate and explain with examples why there is a need to have a standardized order of operations.
- Apply the order of operations to solve multi-step problems with or without technology, e.g., computer, calculator.