Educators should be aware that, with the exception of the Grade 9 mathematics course, 2021 (MTH1W), the 2005 Mathematics curriculum for Grade 10 and the 2007 Mathematics curriculum for Grades 11–12 remain in effect. All secondary mathematics courses for Grades 10–12 will continue to be based on those documents. All references to Grade 9 that appear in The Ontario Curriculum, Grades 9 and 10: Mathematics, 2005 and The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007 have been superseded by The Ontario Curriculum, Grade 9: Mathematics, 2021. Addenda have been issued to the Grade 10 MPM2D and MFM2P courses, to be implemented for the 2022–23 school year.

secondary

# Mathematics

## Glossary

The definitions provided in this glossary are specific to the curriculum context in which the terms are used.

### A

#### algebraic expression

A collection of one or more terms involving variables, numbers, and operations. For example, the algebraic expression 5m has one term, and 6xxy – 8 has three terms. See also term, variable.

#### appreciation

An increase in value of an asset or a currency.

#### assumption

A premise that a person believes to be true.

### B

#### base

A factor or the value in a power that is being repeatedly multiplied. For example, in the power 35, 3 is the base. See also exponent, power.

#### box plot

A graphic representation of the spread of a data set. A rectangle (box) shows the spread of the central half of the distribution, with the first quartile on the left edge, the third quartile on the right edge, and the median as a line within the box. Lines (whiskers) extend from the sides of the box to the lowest and highest values that are not outliers. Potential outliers are marked with a symbol beyond the whiskers. Also known as box-and-whisker plot. See also median, quartile values.

#### budget

An estimate or plan to manage income and expenses over a set period; for example, many people have a weekly or monthly budget. See also expenses, income.

### C

#### Cartesian plane

A two-dimensional coordinate system divided into quadrants by a horizontal axis (x-axis) and a vertical axis (y-axis) intersecting at a point called the origin. The location of any point (x, y) on the x-y plane is described relative to the origin (0, 0). For example, the point (3, 4) is located within the first quadrant, 3 units to the right of the y-axis and 4 units above the x-axis.

#### characteristic

A distinguishing trait or quality. For example, any line defined by = k has the characteristic of being a vertical line parallel to the y-axis on the x-y plane.

#### coding

The process of writing computer programming instructions.

#### composite shape

A shape composed of two or more basic shapes.

#### conceptual understanding

A deep understanding of mathematical ideas that goes beyond isolated facts and procedures to recognizing the connections between and usefulness of mathematical ideas in various contexts. For example, having a conceptual understanding of place value helps in understanding the various procedures involved when doing operations such as multiplying multi-digit or decimal numbers.

#### cone

A three-dimensional object with a circular base and a curved surface that tapers proportionally to an apex. See also three-dimensional object.

#### constant

A part of an algebraic expression that does not change. For example, in the expression x + y = k, k represents a constant and x and y are the variables. When k is equal to 1, 1 will remain the same, and the values of x and y can vary as long as they have a sum of 1. See also algebraic expression, variable.

#### constraint

A restriction placed on the parameters upon which a subroutine or program works or in a loop to define the scope of the problem.

### D

#### density

The concept that between any given two real numbers, there will always be another real number. Thus, there are infinitely many real numbers between any two real numbers. See also infinity, real number.

#### dependent variable

A variable whose value depends on the value of another variable for a particular situation. For example, in the expression d = vt, distance (d) depends on velocity (v) and time (t). In graphing, the dependent variable is usually represented on the vertical axis of a Cartesian plane. See also independent variable, variable.

#### depreciation

A decrease in value of an asset or a currency.

#### dispersion

The spread of values in a data set.

### E

#### earning

Obtaining money in return for labour or services. See also income.

#### equation

A mathematical statement that has equivalent expressions on either side of an equal sign.

#### evaluate

To determine a value for an expression.

#### expenses

Things that one spends money on; for example, most adults’ expenses include food, shelter, utilities, and entertainment.

#### exponent

The value in a power defining the operation on the base. For example, the exponent 3, in the power 53, defines that three 5s are multiplied together. See also base, power.

#### exponential decay

A decrease in a quantity over time, in which the quantity is diminished by a consistent fraction or percentage over a period of time. For example, when quantity n decreases by half, the exponential decay can be written as n × 0.5, n × 0.5 × 0.5, n × 0.5 × 0.5 × 0.5, n × 0.5 × 0.5 × 0.5 × 0.5, ... to show the initial value of n being halved in each time period. See also exponential growth.

#### exponential growth

An increase in a quantity over time, in which the quantity is increased by a consistent multiple over a period of time. For example, when quantity n doubles, the exponential growth can be written as n × 2, n × 2 × 2, n × 2 × 2 × 2, n × 2 × 2 × 2 × 2, ... to show the initial value of n being doubled in each time period. See also exponential decay.

#### expression

A numeric or algebraic representation of a quantity. An expression may include numbers, variables, and operations; for example, 3 + 7, 2– 1. See also algebraic expression.

### F

#### fraction

A number in the form $$\frac{a}{b}$$, in which the numerator a and the denominator b are integers and b ≠ 0. For example, $$\frac{1}{2}$$, $$\frac{17}{10}$$, $$\frac{3}{3}$$, and $$\frac{−1}{4}$$ are all fractions. See also rational number.

### G

#### generalize

To make a statement that is consistent with all specific instances.

#### geometric property

An attribute that remains the same for a class of objects or shapes. For example, an attribute for any parallelogram is that its opposite sides are of equal length.

#### growing pattern

A pattern that involves an increase from term to term. A growing pattern that has a constant increase from term to term, such as 3, 7, 11, 15, … , is an example of a linear growing pattern. A growing pattern that does not have a constant increase from term to term, such as 3, 6, 12, 21, … is an example of a non-linear growing pattern. See also shrinking pattern.

### I

#### income

Money that an individual receives in exchange for work or from investments. See also earning.

#### independent variable

A variable for which values are not dependent on the values of other variables. For example, in the expression b = 2 + c, c is the independent variable because it does not depend on another variable for its value. In graphing, the independent variable is usually represented on the horizontal axis of a Cartesian plane. See also dependent variable.

#### inequality

The relationship between two expressions or values that are not equal, indicating with a sign whether one is less than (<), greater than (>), or not equal to (≠) another. An inequality can include an equal component such as less than or equal to (≤) and greater than or equal to (≥). For example, b means a is less than b and ≥ b means a is greater than or equal to bSee also equation.

#### infinity

The state of having no end or limit. For example, the set of even numbers or the set of rational numbers cannot be counted. A pattern or expression is said to approach infinity if the value can always be made larger than any given value.

#### initial value

The value of the dependent variable when the independent variable is equal to zero. For example, in the expression = 50 + 2b, when = 0, = 50. See also dependent variable, independent variable, rate of change.

#### integer

Any one of the numbers …, –4, –3, –2, –1, 0, +1, +2, +3, +4, .... Integers are the entire set of whole numbers and their opposites (negative numbers).

#### irrational number

A real number that is not a rational number. Thus, it is a number that cannot be represented as a fraction, and when expressed as a decimal it does not terminate or repeat; for example, $$\sqrt{5}$$, pi.

### K

#### knowledge systems

Knowledge systems are developed over time by specific groups of people in particular locations around the world and passed on from generation to generation. A range of knowledge systems, including the rich diversity of Indigenous knowledge systems, shares related world views on key core values, beliefs, and practices and reflects the depth of locally held knowledge that is often rooted in a culture and place.

Understanding various knowledge systems is necessary for more than culturally relevant or responsive education because for some cultures and communities, such as diverse Indigenous peoples from around the world, the knowledge is connected to the land and part of a collective history and contemporary knowledge and perspectives.

### L

#### limit

The long-term behaviour of a pattern or function, or the result as the number of terms increases. For example, the limit of the value of $$\frac{a}{b}$$ (for positive values of a and b) as b approaches 0 is infinity.

#### linear relation

A relation between two variables that appears as a straight line when graphed on a coordinate system. May also be referred to as a linear function.

### M

#### mathematical model

1) A representation of a mathematical idea. For example, a number line is a model of a mathematical idea, as it shows the order and magnitude of numbers.
2) A mathematical solution to a complex real-life situation, created through the mathematical modelling process.

#### mathematical modelling process

An iterative and interconnected process of using mathematics to represent, analyse, make predictions, and provide insight into real-life situations. This process involves four components: understanding the problem, analysing the situation, creating a mathematical model, and analysing and assessing the model.

#### mathematical processes, the

The set of interconnected actions involved in doing mathematics. In the Ontario mathematics curriculum, the seven mathematical processes are problem solving, reasoning and proving, representing, reflecting, selecting tools and strategies, connecting, and communicating.

#### mean

One of the measures of central tendency. The mean represents the value that each piece of data would have if the data were evenly distributed. It can be calculated by adding up all the numbers and then dividing the result by the number of numbers in the set. For example, the mean of 10, 20, and 60 is (10 + 20 + 60) ÷ 3 = 30. Also called average. See also measures of central tendency, median, mode.

#### measurement system

A collection of measurement units and rules that define these units’ relationships to each other.

#### measures of central tendency

A set of measures that represent the approximate centre of a set of data. Mean, median, and mode are all measures of central tendency. See also mean, median, mode.

#### median

One of the measures of central tendency. The median is the middle value of a list of numbers sorted in ascending or descending order. For example, 14 is the median for the set of numbers 7, 9, 14, 21, 39. If there is an even number of data values, then the median is the average of the two middle values. See also mean, measures of central tendency, mode.

#### mixed number

A number that is composed of an integer and a fraction; for example, $$–8\frac{1}{4}$$.

### N

#### non-linear relation

A relation between two variables that does not appear as a straight line when graphed on a coordinate system.

#### number system

A way in which number relationships are defined. For example, the base ten number system includes the digits 0 to 9, and the relationship of one place value to the next is a multiple of 10.

### P

#### parameter

A special type of variable used in the definition of a subprogram, which defines the values that are inputted when the subprogram is executed.

#### percent

A ratio with a second term of 100. A percent is expressed using the symbol %. For example, 30% means 30 out of 100. A percent can be represented by a fraction with a denominator of 100; for example, 30% = $$\frac{30}{100}$$.

#### point of intersection

The point at which two or more lines or curves cross. Two lines may have one point of intersection, no points of intersection, or an infinite number of points that intersect.

#### population

The total number of individuals or items under consideration in a surveying or sampling activity.

#### power

A number written in exponential form. For example, in the power 25, the base is 2 and the exponent is 5. See also base, exponent.

#### prism

A three-dimensional object with two parallel and congruent faces. A prism is named by the shape of its bases, for example, rectangle-based prism, triangle-based prism. See also three-dimensional object.

#### probability

The likelihood that an event will occur. Probability is often represented as a percentage between 0 and 100 or as a decimal between 0 and 1.

#### procedural fluency

The ability to use procedures in an accurate, efficient, and flexible way to solve problems.

### Q

#### quantitative data

Data that is numerical and acquired through counting or measuring; for example, number of sides of a three-dimensional object or amount of rainfall in a season.

#### quartile values

Values that divide a sequenced data set into four parts, each of which represents 25% of the data falling within that range. For example, 25% of the data is below the point that defines the first quartile, which is the middle number between the smallest number in the data set and the median. See also box plot.

#### quotient

The result of a division.

### R

#### rate

A comparison, or a type of ratio, of two measurements with different units; for example, 100 km/h, 10 kg/m3, 20 L/100 km. See also ratio.

#### rate of change

The change in one variable relative to the change in another. The slope of a line represents a constant rate of change. See also slope.

#### ratio

A comparison of quantities with the same units. A ratio can be expressed in ratio form or in fraction form, for example, 3:4 or $$\frac{3}{4}$$.

#### rational number

A number that can be expressed as a quotient of two integers where the divisor is not zero. It can also be represented as a decimal number that either repeats or terminates. For example, $$\frac{1}{3}$$ or 0.3333..., $$\frac{17}{10}$$ or 1.7, $$\frac{3}{3}$$ or 1, and $$\frac{−1}{4}$$ or −0.25 are rational numbers. See also fraction, quotient.

#### real number

A rational or irrational number. See also rational number, irrational number.

#### reflection

A transformation that flips points over a line, such as the x-axis or y-axis, such that the reflected point and the original point are the same distance perpendicularly from the line of reflection.

#### regression

A statistical method for determining the relationship between the dependent variable and the independent variable for a set of data.

#### relation

An identified relationship between two variables that may be expressed as a table of values, a graph, or an equation.

### S

#### sample

A subset of a population. See also population, subset.

#### scatter plot

A graph designed to show a relationship between corresponding numbers from two sets of data measurements associated with a single object or event; for example, a graph of data about students’ marks and the corresponding amounts of study time. Drawing a scatter plot involves plotting ordered pairs on a coordinate grid.

#### scientific notation

A way of expressing a very large or very small number in terms of a decimal number between 1 and 10 multiplied by a power of 10. For example, 690 890 000 000 is 6.9089 × 1011 in scientific notation, and 0.000279 is 2.79 × 10−4. See also power.

#### set

A collection of elements that fulfil specific criteria.

#### shrinking pattern

A pattern that involves a decrease from term to term. A shrinking pattern such as −3, −7, −11, −15 is linear since there is a constant decrease of 4 from term to term. This type of shrinking pattern is also known as a decreasing pattern. A shrinking pattern such as 40, 20, 10, 5, 2.5 is an example of a shrinking pattern that is non-linear, since the decrease from term to term is not constant. See also growing pattern.

#### simplify

To create an equivalent fraction by dividing the numerator and denominator by their greatest common factor, or by performing operations to create fewer terms in an algebraic expression.

#### slope

A measure of the steepness of a line, calculated as the rate of the rise (vertical change between two points) to the run (horizontal change between the same two points).

#### source

A place where data is obtained. Types of sources include original (primary) sources, such as observations, conversations, and measurements, and secondary sources, such as magazines, newspapers, government documents, and databases.

### T

#### term

A single number, variable, or combination of numbers and variables involving no addition or subtraction.

#### three-dimensional object

An object that has the dimensions of length, width, and depth.

#### transformation

A change in a set of points that results in a different position on a coordinate system. Transformations include translations, reflections, rotations, and dilations.

#### translation

A transformation that moves every point in a set the same distance, in the same direction.

### U

#### unit

A quantity used as a standard of measurement.

#### unit fraction

Any fraction that has a numerator of 1, for example, $$\frac{1}{2}$$, $$\frac{1}{3}$$, or $$\frac{1}{4}$$. Every fraction can be decomposed into unit fractions. For example, $$\frac{3}{4}$$ is 3 one-fourth units, or $$\frac{3}{4}$$ = $$\frac{1}{4}$$ + $$\frac{1}{4}$$ + $$\frac{1}{4}$$.

### V

#### variable

A symbol, letter, or words representing a value that can vary depending on the context. For example, in an equation x + = 3, if the variable x changes, the value of variable y will change accordingly. In coding, words are often used instead of a letter, such as FirstNumber + SecondNumber = 3. In coding, a variable is a temporary storage location for data such as a numerical value or a series of characters, and the values stored in that location vary depending on the commands given by the program. In a data set, a variable such as height may take on a different value for each member of a population.

### X

#### x-intercept

The value of x for a point (x, y) on the x-axis when y is zero.