Mathematics

Mathematical skills and knowledge support the interpretation of diverse quantitative and spatial information and can be applied to solving both theoretical and practical problems. With mathematics, abstract ideas can be visualized, represented, and explained. Mathematics is a powerful tool that can be used to simplify and solve complicated real-life problems.

- objects
- pictures
- words
- numerals

Express a quantity in different ways.

Relate a numeral to a specific quantity.

Canadian money includes

- nickels
- dimes
- quarters
- loonies
- toonies
- five-dollar bills
- ten-dollar bills
- twenty-dollar bills
- fifty-dollar bills
- hundred-dollar bills

Quantity in the world is represented in multiple ways, including with money.

Represent quantities using numerals.

Identify a quantity of 0 in familiar situations.

Express the value of each coin and bill within

The number line is a spatial interpretation of quantity.

Each natural number is associated with exactly one point on the number line.

Represent quantities using natural numbers.

Relate a natural number to its position on the number line.

The digits 0 to 9 indicate the number of groups in each place in a number.

The value of each place in a number is the product of the digit and its place value.

Numbers can be composed in various ways using place value.

Numbers can be rounded in context when an exact count is not needed.

A zero in the leftmost place of a natural number does not change the value of the number.

The

The

Place value determines the value of a digit based on its place in a number, relative to the ones place.

Place value is used to read and write numbers.

Relate the values of adjacent places.

Determine the value of each digit in a natural number.

Express natural numbers using words and numerals.

Express various compositions of a natural number using place value.

Round natural numbers to various places.

Compare and order natural numbers.

Count and represent the value of a collection of nickels, dimes, and quarters as cents.

Count and represent the value of a collection of loonies, toonies, and bills as dollars.

Compare French and English symbolic representations of monetary values.

Multiplying or dividing a number by 10 corresponds to moving the decimal point one position to the right or left, respectively.

A point is used for decimal notation in English.

A comma is used for decimal notation in French.

Numbers, including decimal numbers, can be composed in various ways using place value.

A zero placed to the right of the last digit in a decimal number does not change the value of the number.

Decimal numbers are fractions with denominators of 10, 100, etc.

The separation between wholes and parts can be represented using decimal notation.

Patterns in place value are used to read and write numbers, including wholes and parts.

Relate the values of adjacent places, including tenths and hundredths.

Relate place value to multiplication by 10 and division by 10.

Determine the value of each digit in a number, including tenths and hundredths.

Express numbers, including decimal numbers, using words and numerals.

Express various compositions of a number, including decimal numbers, using place value.

Compare decimal notation expressed in English and in French.

Round numbers to various places, including tenths.

Compare and order numbers, including decimal numbers.

Express a monetary value in cents as a monetary value in dollars using decimal notation.

A quantity remains the same no matter the order in which the objects are counted (counting principle: order irrelevance).

A quantity can be determined by counting each object in a set once and only once (counting principle: one-to-one correspondence).

The last number used to count represents the quantity (counting principle: cardinality).

Any quantity of like or unlike objects can be counted as a set (counting principle: abstraction).

Counting more than one object at a time is called skip counting.

A quantity can be determined by counting more than one object in a set at a time.

Count backward from 20 to 0

Skip count to 100, forward by 5 and 10, starting

Skip count to 20, forward by 2, starting

Count within 1000, forward and backward by 1, starting at any number.

Skip count by 20, 25, or 50, starting

Determine the value of a collection of coins or bills of the same denomination by skip counting.

- more
- less
- same
- enough
- too many
- too few

A quantity can be described in relation to a purpose or need.

Describe a quantity in relation to a purpose or need using comparative language.

Solve problems in familiar situations by counting.

- equal
- not equal
- greater than
- less than

The unequal sign

Describe a quantity relative to another quantity.

Represent equal quantities symbolically.

Represent unequal quantities symbolically.

The greater than sign

Describe a natural number as greater than, less than, or equal to another natural number using words or symbols.

Compose quantities

Addition is a process of combining quantities to find a sum.

Subtraction is a process of finding the difference between quantities.

The order in which two quantities are added does not affect the sum (commutative property).

The order in which two quantities are subtracted affects the difference.

Addition of 0 to any number, or subtraction of 0 from any number, results in the same number (zero property).

Estimation can be used when an exact sum or difference is not needed and to check if an answer is reasonable.

Standard algorithms for addition and subtraction are conventional procedures based on place value.

Standard algorithms are universal tools for addition and subtraction and may be used for any natural numbers independently of their nature.

Estimate sums and differences.

Model regrouping by place value for addition and subtraction.

Explain the standard algorithms for addition and subtraction of natural numbers.

Add and subtract natural numbers using standard algorithms.

Solve problems using addition and subtraction.

Estimation can be used to verify a sum or difference.

Assess the reasonableness of a sum or difference by estimating.

Solve problems using addition and subtraction, including problems involving money.

Addition and subtraction strategies include

- counting on
- counting back
- decomposition
- compensation

The subtraction sign

The equal sign

Add and subtract within 20.

Express addition and subtraction symbolically.

Solve problems using addition and subtraction in joining, separating, or comparing situations.

Model transactions with money, limited to dollar values within

Add and subtract numbers within 100.

Solve problems using addition and subtraction of countable quantities or measurable lengths.

Model transactions with money, limited to dollar values within

In a part-part-whole relationship, the sum represents the whole and the difference represents a missing part.

Fact families are groups of related addition and subtraction number facts.

Recognize families of related addition and subtraction number facts.

Recall addition number facts, with addends to 10, and related subtraction number facts.

Grouping involves partitioning a quantity into groups of a certain size.

Demonstrate conservation of number when sharing or grouping.

An odd quantity will have a remainder of one when partitioned into two equal groups or groups of two.

Describe a quantity as even or odd.

Partition a set of objects by sharing or grouping, with or without remainders.

Multiplication is repeated addition.

Multiplication by two is doubling and multiplication by three is tripling.

Division is a process of sharing or grouping to find a quotient.

The order in which two quantities are multiplied does not affect the product (commutative property).

The order in which two numbers are divided affects the quotient.

Multiplication or division

Relate multiplication to repeated addition.

Relate multiplication to skip counting.

Investigate multiplication by 0.

Model a quotient by partitioning a quantity into equal groups with or without remainders.

Visualize and model products and quotients as arrays.

A prime number has factors of only itself and one.

A composite number has other factors besides one and itself.

Zero and one are neither prime nor composite.

A number is a multiple of any of its factors.

Prime factorization represents a number as a product of prime numbers.

The order in which three or more numbers are multiplied does not affect the product (associative property).

The order in which numbers are divided affects the quotient.

Numbers can be multiplied or divided in parts (distributive property).

Any natural number can be represented uniquely as a product of prime numbers, including repeated prime numbers.

Any factor of a number can be determined from its prime factorization.

Describe a number as prime or composite.

Recognize multiples of numbers within 100.

Determine the greatest common factor (greatest common divisor) of

Compose a product in multiple ways, including with more than two factors.

Represent composite numbers as products of prime numbers.

Relate composite factors of a number to its prime factorization.

Compare the prime factorization of two natural numbers.

- repeated addition
- multiplying in parts
- compensation

- repeated subtraction
- partitioning the dividend

The division symbol

The equal sign

A remainder is the quantity left over after division.

Multiplication and division strategies can be supported by addition and subtraction.

Multiply and divide

Express multiplication and division symbolically.

Explain the meaning of the remainder in various situations.

Solve problems using multiplication and division in sharing or grouping situations.

Standard algorithms facilitate multiplication and division of natural numbers that have multiple digits.

Multiply and divide

Examine standard algorithms for multiplication and division.

Multiply and divide

Express a quotient with or without a remainder according to context.

Solve problems using multiplication and division.

Fact families are groups of related multiplication and division number facts.

Recognize families of related multiplication and division number facts.

Recall multiplication number facts, with factors to 10, and related division facts.

Partition an even set of objects into two equal groups.

One-quarter is one of four equal parts.

Describe one of two equal groups as one-half and one of four equal groups as one-quarter.

Describe a whole set of objects as a composition of halves and as a composition of quarters.

A whole quantity can be a whole set of objects or a whole object that can be partitioned.

Each fraction is associated with a point on the number line.

Fractions can represent part-to-whole relationships.

Describe a whole as a fraction, limited to denominators of 12 or less.

Model fractions of a whole, limited to denominators of 12 or less.

Express fractions symbolically.

Relate a fraction less than one to its position on the number line, limited to denominators of 12 or less.

Compare fractions to benchmarks of 0, ,

Decimal numbers are fractions with denominators of 10, 100, etc.

Relate fractions and equivalent decimal numbers, limited to tenths and hundredths, to their positions on the number line.

Fractions can be compared by considering the number of parts or the size of parts.

The size of the parts and the number of partitions in the whole are inversely related.

Compare the same fraction of different-sized wholes.

Compare different fractions with the same denominator.

Compare different fractions with the same numerator.

Multiplication

Division

The numerator and denominator of a fraction in simplest form have no common factors.

The most efficient way to express a fraction in simplest form is using the greatest common factor of the numerator and denominator.

Exactly one of infinitely many equivalent fractions is in simplest form.

Represent fractions equivalent to a given fraction symbolically.

Relate the position of equivalent fractions on the number line.

Relate multiplying the numerator and denominator of a fraction by the same number to multiplying

Recognize a fraction where the numerator and denominator have a common factor.

Relate dividing the numerator and denominator of a fraction by the same number to dividing

Express a fraction in simplest form.

Natural numbers can be expressed as improper fractions with a denominator of 1.

Decimals can be expressed as fractions with the place value of the last non-zero digit of the decimal number as the denominator.

Fractions can represent quotients.

A fraction with the same numerator and denominator represents a quotient of 1.

Model improper fractions.

Express improper fractions symbolically.

Relate fractions, including improper fractions, and equivalent decimal numbers to their positions on the number line.

Convert an improper fraction to a mixed number using division.

Convert between fractions and decimal numbers.

Compare and order fractions, including improper fractions.

Fractions with common denominators are multiples of the same unit fraction.

Express a fraction as repeated addition of a unit fraction.

Relate repeated addition of a unit fraction to multiplication of a natural number by a unit fraction.

Add and subtract fractions within one whole, limited to common denominators of 12 or less.

Solve problems involving fractions, limited to common denominators of 12 or less.

The product of the denominators of two fractions provides a common denominator.

The most efficient way to express two fractions with common denominators is using the least common multiple of the two denominators.

Addition and subtraction of fractions can be used to solve problems in real-life situations, such as cooking and construction.

Recognize two fractions where the denominators have a common factor or multiple.

Express two fractions with common denominators.

Add and subtract fractions.

Solve problems using addition and subtraction of fractions.

The left and right sides of an equation are interchangeable.

Two expressions are equal if they represent the same number.

- Multiplication and division are performed before addition and subtraction.
- Multiplication and division are performed in order from left to right.
- Addition and subtraction are performed in order from left to right.

Create various expressions of the same number using one or more operations.

Determine an unknown value on the left or right side of an equation, limited to equations with one operation.

Solve problems using equations, limited to equations with one operation.

Investigate preservation of equality by adding, subtracting, multiplying, or dividing the same number on both sides of an equation without an unknown value.

Apply preservation of equality to determine an unknown value in an equation, limited to equations with one operation.

Solve problems using equations, limited to equations with one operation.

Two-dimensional shapes include

- squares
- circles
- rectangles
- triangles

- cubes
- prisms
- cylinders
- spheres

Identify two- and three-dimensional shapes.

Investigate three-dimensional shapes by rolling, stacking, or sliding.

Describe a shape using words such as flat, curved, straight, or round.

Sort shapes according to one attribute and describe the sorting rule.

shapes include

- squares
- circles
- rectangles
- triangles

- cubes
- prisms
- cylinders
- spheres
- pyramids
- cones

A shape can be composed of two or more shapes.

A shape is symmetrical if it can be decomposed into matching halves.

Model two-dimensional shapes.

Sort shapes according to one attribute and describe the sorting rule.

Compose and decompose two- or three-dimensional shapes.

Identify shapes within two- or three-dimensional composite shapes.

Investigate symmetry of two-dimensional shapes by folding and matching.

- sides
- vertices
- faces or surfaces

Three-dimensional shapes may have faces that are two-dimensional shapes.

A shape can be visualized as a composition of other shapes.

Relate the faces of three-dimensional shapes to two-dimensional shapes.

Create a picture or design with shapes from verbal instructions, visualization, or memory.

Parallel lines or planes are always the same distance apart.

Perpendicular lines or planes intersect at a right angle.

Familiar representations of a right angle may include

- the corner of a piece of paper
- the angle between the hands on an analog clock at 3:00
- a capital
letter L

- triangles
- quadrilaterals
- pentagons
- hexagons
- octagons

Geometric properties define a class of polygon.

Describe geometric properties of regular and irregular polygons.

Sort polygons according to geometric properties and describe the sorting rule.

Classify polygons as regular or irregular using geometric properties.

Two or more angles that compose 90° are complementary angles.

Two or more angles that compose 180° are supplementary angles.

Quadrilaterals include

- squares
- rectangles
- parallelograms
- trapezoids
- rhombuses

- equilateral
- isosceles

- right
- obtuse
- acute

Geometric properties define a hierarchy for classifying shapes.

Identify relationships between angles within a polygon, including equal, supplementary, complementary, and sum of interior angles, by measuring.

Identify relationships between the faces of three-dimensional models of prisms, including parallel or perpendicular, by measuring.

Classify triangles as equilateral, isosceles, or neither using geometric properties related to sides.

Classify triangles as right, acute, or obtuse using geometric properties related to angles.

Classify quadrilaterals in a hierarchy according to geometric properties.

First Nations, Métis, and Inuit translate, rotate, and reflect shapes in the creation of cultural art.

Describe geometric attributes of two- and three-dimensional shapes in various orientations.

Recognize translation, rotation, or reflection of shapes represented in First Nations, Métis, or Inuit art inspired by the natural world.

- translations
- rotations
- reflections

Rigid transformations can be used to illustrate geometric properties of a polygon.

Verify geometric properties of polygons by translating, rotating, or reflecting using hands-on materials or digital applications.

- length
- area
- capacity
- mass

- height
- width
- depth
- diameter

The length of empty space between two points is called distance.

Familiar contexts of distance include

- distance between objects or people
- distance between home and school
- distance between towns or cities

Length remains the same if an object is repositioned but may be named differently.

Recognize the diameter of a circle as a length.

Compare and order objects according to length.

Describe distance in familiar contexts.

Iterating is the process of measuring a length by repeating one copy of a unit without gaps or overlaps.

Length can be measured more efficiently using a measuring tool that shows iterations of a unit.

The unit can be chosen based on the length to be measured.

Length can be measured with non-standard units or standard units (e.g., centimetres).

Standard units enable a common language around measurement.

Length is measured with equal-sized units that themselves have length.

The size of the unit and the number of units in the length are inversely related.

Compare and order measurements of different lengths measured with the same non-standard units, and explain the choice of unit.

Compare measurements of the same length measured with different non-standard units.

Measure length with standard units by tiling or iterating with a centimetre.

Compare and order measurements of different lengths measured with centimetres.

The basic unit of length in the metric system is the metre.

Metric units are named using prefixes that indicate the relationship to the basic unit (e.g., for length, the prefix centi- indicates there are 100 centimetres in a metre).

Metric units are abbreviated for convenience (e.g., metre is abbreviated

Standard measuring tools show iterations of a standard unit from an origin.

The other, older, system of measurement that is also commonly used in the United States and Canada is sometimes called the imperial system and uses “Canadian units.”

“Canadian or imperial” units that are still commonly used include miles, yards, feet, inches, acres, pounds, quarts, pints, and ounces. You may encounter these in hospitals (birth announcements), housing and property (square footage/acreage), cooking and drink (pounds,ounces, quarts, pints), some roads and cars (miles, mileage, miles per hour, gallons), railways, and other contexts where integration with the United States is important.

The perimeter of a polygon is the sum of the lengths of its sides.

An alternative system, the imperial system, still partly in use, uses “Canadian units” (sometimes called “imperial units”). This system is important to know about because it provides core numeracy for current everyday life, understanding works of the past, and literacy concerning culture and trade with our biggest trading partner, the United States.

Length can be expressed in various units according to context and desired precision.

Length remains the same when decomposed or rearranged.

Relate centimetres to metres.

Justify the choice of centimetres or metres to measure various lengths.

Measure lengths of straight lines and curves, with centimetres or metres, using a standard measuring tool.

Express length in centimetres or metres.

Convert commonly used units of measure between metric and Canadian (imperial) units within 100.

Determine perimeter of polygons.

Determine the length of an unknown side given the perimeter of a polygon.

Units that tile fit together without gaps or overlaps.

The unit can be chosen based on the area to be measured.

Area can be measured with non-standard units or standard units (e.g., square centimetres).

The area of a rectangle equals the product of its perpendicular side lengths.

Area may be interpreted as the result of motion of a length.

An area remains the same when decomposed or rearranged.

Area is quantified by measurement.

Area is measured with equal-sized units that themselves have area and do not need to resemble the region being measured.

The area of a rectangle can be perceived as square-shaped units structured in a two-dimensional array.

Recognize the rearrangement of area in First Nations, Métis, or Inuit design.

Compare non-standard units that tile to non-standard units that do not tile.

Measure area with non-standard units by tiling.

Measure area with standard units by tiling with a square centimetre.

Visualize and model the area of various rectangles as two-dimensional arrays of square-shaped units.

Determine the area of a rectangle using multiplication.

Solve problems involving area of rectangles.

- longer
- taller
- shorter
- heavier
- lighter
- bigger
- smaller
- big enough
- too big
- too small

The size of two objects can be compared directly.

The size of an object can be described in relation to a purpose or need.

Order objects according to length, area, mass, or capacity.

Describe the size of an object in relation to another object, using comparative language.

Describe the size of an object in relation to a purpose or need, using comparative language.

Order objects according to length, area, mass, or capacity.

Describe the size of an object in relation to another object, using comparative language.

A common referent for a centimetre is the width of the tip of the little finger.

Estimate length by visualizing the iteration of a referent for a centimetre.

Investigate First Nations, Métis, or Inuit use of the land in estimations of length.

A referent is a personal or familiar representation of a known length.

A common referent for a metre is the distance from a doorknob to the floor.

Estimate length by comparing to a benchmark of a centimetre or metre.

Estimate length by visualizing the iteration of a referent for a centimetre or metre.

Estimate an area by visualizing a referent for a square centimetre.

Estimate an area by rearranging or combining partial units.

- corners
- bends
- turns or rotations
- intersections
- slopes

The end point of a line segment or ray is called a vertex.

An angle can be interpreted as the motion of a length rotated about a vertex.

Recognize situations in which an angle can be perceived as motion.

Compare two angles indirectly with a third angle by superimposing.

Estimate which of two angles is greater.

Angles can be classified according to their measure

- acute angles measure less than 90°
- right angles measure 90°
- obtuse angles measure between 90° and 180°
- straight angles measure 180°

Angle is measured with equal-sized units that themselves are angles.

Angle measurement is based on the division of a circle.

Describe an angle as acute, right, obtuse, or straight.

Relate angles of 90°, 180°, 270°, and 360° to fractions of a circle.

A referent is a personal or familiar representation of a known angle.

Estimate angles by comparing to benchmarks of 45°, 90°, 180°, 270°, and 360°.

Estimate angles by visualizing referents for 45°, 90°, 180°, 270°, and 360°.

The elements of a pattern can include

- sounds
- objects
- pictures
- symbols
- actions

Repeating patterns have one or more elements that repeat.

A pattern core is a sequence of one or more elements that repeat as a unit.

Recognize change or constancy between elements in a repeating pattern.

Identify the pattern core, up to three elements, in a repeating pattern.

Predict the next elements in a repeating pattern.

Create a repeating pattern with a pattern core of up to three elements.

Cycles include

- seasons
- day/night
- life cycles
- calendars

- sounds
- objects
- pictures
- symbols
- actions

A cycle is a repeating pattern that repeats in the same way forever.

Investigate cycles found in nature that inform First Nations, Métis, or Inuit practices.

Identify the pattern core, up to four elements, in a cycle.

Identify a missing element in a repeating pattern or cycle.

Describe change and constancy in repeating patterns and cycles.

Create different representations of the same repeating pattern or cycle, limited to a pattern core of up to four elements.

Extend a sequence of elements in various ways to create repeating patterns.

Pascal’s triangle is a triangular arrangement of numbers that illustrates multiple repeating, growing, and symmetrical patterns.

A pattern is more evident when the elements are represented, organized, aligned, or oriented in familiar ways.

Examine the representation, organization, alignment, or orientation of patterns in First Nations, Métis, or Inuit design.

Investigate pattern in Pascal’s triangle.

Create and express growing patterns using sounds, objects, pictures, or actions.

Explain the change and constancy in a given non-numerical growing pattern.

Extend a non-numerical growing pattern.

Finite sequences, such as a countdown, have a definite end.

Infinite sequences, such as the natural numbers, never end.

Sequences may be finite or infinite.

Describe position in a sequence using ordinal numbers.

Differentiate between finite and infinite sequences.

The Fibonacci sequence is an increasing sequence that occurs in nature.

Different representations can provide new perspectives of the increase or decrease of a sequence.

Create and explain increasing or decreasing sequences, including numerical sequences.

Express a numerical sequence to represent a concrete or pictorial sequence.

Determine any missing term in a skip-counting sequence using multiplication.

Describe the change from term to term in a numerical sequence using mathematical operations.

Guess the next term in a sequence by inferring the pattern from the previous terms.

A skip-counting sequence is an example of an arithmetic sequence.

A geometric sequence progresses through multiplication.

A geometric sequence begins at a number other than zero.

A geometric sequence has a constant ratio between consecutive terms.

Describe the initial term and the constant change in an arithmetic sequence.

Express the first five terms of an arithmetic sequence related to a given initial term and constant change.

Describe the initial term and the constant change in a geometric sequence.

Express the first five terms of a geometric sequence related to a given initial term and constant change.

- first
- next
- then
- last
- yesterday
- today
- tomorrow

Describe daily events as occurring yesterday, today, or tomorrow.

First Nations, Métis, and Inuit experience time through sequences and cycles in nature, including cycles of seasons and stars.

Cycles from a calendar include days of the week and months of the year.

Time can be perceived as a cycle.

Describe observable changes that indicate a cycle of time.

Relate cycles of seasons and stars to First Nations, Métis, or Inuit practices.

Identify cycles from a calendar.

Comparative language for describing duration can include

- longer
- shorter
- sooner
- later

Duration is the measure of an amount of time from beginning to end.

Duration can be measured in various units according to context.

Describe the duration between or until significant events using comparative language.

Describe the duration of events using non-standard units.

Relate First Nations’ winter counts to duration.

The basic unit of time is the second.

One second is of a minute.

One minute is of an hour.

Analog and digital clocks represent time of day.

Time of day can be expressed as a duration relative to 12:00 in two 12-hour cycles.

Time of day can be expressed as a duration relative to 0:00 in one 24-hour cycle in some contexts, including French-language contexts.

Relate minutes past the hour to minutes until the next hour.

Describe time of day as a.m. or p.m. relative to 12-hour cycles of day and night.

Tell time using analog and digital clocks.

Express time of day in relation to one 24-hour cycle according to context.

- quarter past the hour
- half past the hour
- quarter to the hour

Express time of day using fractions.

Determine duration in minutes using a clock.

Apply addition and subtraction strategies to the calculation of duration.

Convert between hours, minutes, and seconds.

Compare the duration of events using standard units.

Solve problems involving duration.

- years
- months
- weeks
- days
- hours
- minutes
- seconds

Describe the duration between or until significant events using standard units of time.

Pose questions about people, things, events, or experiences in the learning environment.

Gather data by sharing answers to questions.

First-hand data is data collected by the person using the data.

Collect first-hand data by questioning people within the learning environment.

Predict the answer to a statistical question.

- formulating statistical questions
- collecting data
- representing data
- interpreting data

A graph can represent data by using objects, pictures, or numbers.

Create a pictograph from a concrete graph.

Graphs can include

- pictographs
- bar graphs
- dot plots

A graph can include features such as

- a title
- a legend
- axes
- axis labels

Construct graphs to represent data.

Compare the features of pictographs, dot plots, and bar graphs.

Sources of second-hand data include

- newspapers
- maps
- databases
- websites
- social media
- stories

Representation tells a story about data.

Represent second-hand data in a dot plot or bar graph with one-to-one correspondence.

Describe the story that a representation tells about a collection of data in relation to a statistical question.

Examine First Nations, Métis, or Inuit representations of data.

Consider possible answers to a statistical question based on the data collected.

Graphs can include

- pictographs
- bar graphs
- dot plots

Different representations tell different stories about the same data.

Represent data in a graph using many-to-one correspondence.

Describe the effect of scale on representation.

Justify the choice of graph used to represent certain data.

Compare different graphs of the same data.

Interpret data represented in various graphs.