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## Grade 10 Math

## CONTENTS

## Numbers

The study of whole numbers and their properties is called Number theory. Number theory is a large and interesting area in mathematics, including studying Prime numbers, rational numbers and so on. Number theory is important because it helps you to understand and master how the numbers function which helps with logical reasoning skills.

Factors

Prime factorization

Greatest common factor

Least common multiple

GCF and LCM: word problems

Square roots

Cube roots

Classify rational and irrational numbers

Classify numbers

Compare and order rational numbers

Absolute value and opposites

Number lines

Convert between decimals and fractions

## Exponents

The exponent of a number tells us how many times to multiply that number with itself. It is written as a small number to the right and above the base number.

Exponents with integer bases

Exponents with decimal and fractional bases

Negative exponents

Multiplication with exponents

Division with exponents

Multiplication and division with exponents

Power rule

Evaluate expressions using properties of exponents

Identify equivalent expressions involving exponents

## Scientific notation

Scientific notation is a way of representing a number where that number is written in two parts: just the digits with the decimal point placed after the first digit, followed by power of 10.

Convert between standard and scientific notation

Compare numbers written in scientific notation

Multiply numbers written in scientific notation

Divide numbers written in scientific notation

## Radical expressions

The symbol √ is called radical in mathematics. Radical expression is any expression containing this symbol. The √ symbol is used to indicate square root or nth root of a number. Understanding radical expressions are important for future math works and also for real lives problems such as calculating inflation and interest.

Simplify radical expressions

Simplify radical expressions by rationalizing the denominator

Multiply radical expressions

Add and subtract radical expressions

Simplify radical expressions using the distributive property

Simplify radical expressions: mixed review

## Rational exponents

Using rational numbers as exponents. A rational exponent represents both an integer exponent and an nth root. The root is found in the denominator and the integer exponent is found in the numerator. In other words, a rational exponent is an exponent that is a fraction.

Evaluate rational exponents

Multiplication with rational exponents

Division with rational exponents

Power rule with rational exponents

Simplify expressions involving rational exponents I

Simplify expressions involving rational exponents II

## Coordinate plane

The plane containing X axis and Y axis is called coordinate plane. Cartesian coordinated can be used to pinpoint where we are on a map or graph. We can mark a point on a graph by how far along and how far up it is, the point (10,6) is 10 units along and 6 units up. Coordinate plane is exciting and important for learning math and it has important use in real life like mapping an area or arranging furniture in your room.

Coordinate plane review

Midpoints

Distance between two points

## Solve equations

Solving an equation is the process of finding a value (or values) that we can put in place of a variable which makes the equation true. Solving an equation is like solving a puzzle which means there are things we can (an cannot) do.

Model and solve equations using algebra tiles

Write and solve equations that represent diagrams

Solve one-step linear equations

Solve two-step linear equations

Solve advanced linear equations

Solve equations with variables on both sides

Solve equations: complete the solution

Find the number of solutions

Create equations with no solutions or infinitely many solutions

Solve linear equations: word problems

Solve linear equations: mixed review

## Single-variable inequalities

In mathematics sometimes we only know that something is greater or smaller than. Inequality tells us about the relative size of two values. A single-variable inequality is a mathematical statement that relates a linear expression as either less than or greater than another. Learning equations and inequalities helping to get ready for more advanced math problems.

Graph inequalities

Write inequalities from graphs

Identify solutions to inequalities

Solve one-step linear inequalities: addition and subtraction

Solve one-step linear inequalities: multiplication and division

Solve one-step linear inequalities

Graph solutions to one-step linear inequalities

Solve two-step linear inequalities

Graph solutions to two-step linear inequalities

Solve advanced linear inequalities

Graph solutions to advanced linear inequalities

Graph compound inequalities

Write compound inequalities from graphs

Solve compound inequalities

Graph solutions to compound inequalities

## Data and graphs

A collection of facts, such as numbers, measurements or observations is called data. We can create a table with the data. A diagram of values, usually shown as lines is called graph. Understanding data and the appropriate graph related to it can help interpreting data.

Interpret bar graphs, line graphs and histograms

Create bar graphs, line graphs and histograms

Interpret circle graphs

Interpret stem-and-leaf plots

Interpret box-and-whisker plots

Interpret a scatter plot

Scatter plots: line of best fit

## Relations and functions

A relation is a set of inputs and outputs, while a function is a relation with one output for each input. It works like a machine that takes something in (input) and at the end gives us something back (output). F(x) is the traditional way of expressing functions. Each function has three parts the Input, the Relationship and the Output.

Relations: convert between tables, graphs, mappings and lists of points

Domain and range of relations

Identify independent and dependent variables

Identify functions

Identify functions: vertical line test

Find values using function graphs

Evaluate a function

Evaluate a function: plug in an expression

Complete a function table from a graph

Complete a function table from an equation

Interpret the graph of a function: word problems

## Direct and inverse variation

A relationship between two variables in which one is a constant multiple of the other is called direct variation. The statement “Y varies directly as X” means that when X increases, Y increases by the same factor.

Identify proportional relationships

Find the constant of variation

Graph a proportional relationship

Write direct variation equations

Write and solve direct variation equations

Identify direct variation and inverse variation

Write inverse variation equations

Write and solve inverse variation equations

## Linear functions

A function related an input to an output. It works like a machine that takes something in (input) and at the end gives us something back (output). F(x) is the traditional way of expressing functions. Each function has three parts the Input, the Relationship and the Output. Linear equations are equations that make a straight line when graphed.

Identify linear functions

Find the slope of a graph

Find the slope from two points

Find a missing coordinate using slope

Slope-intercept form: find the slope and y-intercept

Slope-intercept form: graph an equation

Slope-intercept form: write an equation from a graph

Slope-intercept form: write an equation

Slope-intercept form: write an equation from a table

Slope-intercept form: write an equation from a word problem

Write linear functions to solve word problems

Complete a table and graph a linear function

Compare linear functions: graphs, tables and equations

Write equations in standard form

Standard form: find x- and y-intercepts

Standard form: graph an equation

Equations of horizontal and vertical lines

Graph a horizontal or vertical line

Point-slope form: graph an equation

Point-slope form: write an equation

Point-slope form: write an equation from a graph

Slopes of parallel and perpendicular lines

Write an equation for a parallel or perpendicular line

Find the distance between a point and a line

Find the distance between two parallel lines

Transformations of linear functions

## Systems of linear equations

Two or more equations containing common variables is called the system of equations. A system of equations in which every equation is linear is called system of linear equations. For any linear system, there are three possible outcomes: there is only one solution, there are infinitely solutions or there are no solutions at all. If the number of equations is more than the variables the system is called overdetermined, while if the variables are more than the equations the system is called underdetermined.

Is (x, y) a solution to the system of equations?

Solve a system of equations by graphing

Solve a system of equations by graphing: word problems

Find the number of solutions to a system of equations by graphing

Find the number of solutions to a system of equations

Classify a system of equations by graphing

Classify a system of equations

Solve a system of equations using substitution

Solve a system of equations using substitution: word problems

Solve a system of equations using elimination

Solve a system of equations using elimination: word problems

Solve a system of equations using any method

Solve a system of equations using any method: word problems

## Monomials

A polynomial with just one term is called a Monomial.

Identify monomials

Multiply monomials

Divide monomials

Multiply and divide monomials

Powers of monomials

## Polynomials

The sum or difference of terms which have variables raised to positive integer powers and which have coefficients. A polynomial can have constants, variables and exponents, but never division by variable. Even though the poly- means many the polynomials terms should be finite.

Polynomial vocabulary

Model polynomials with algebra tiles

Add and subtract polynomials using algebra tiles

Add and subtract polynomials

Add polynomials to find perimeter

Multiply a polynomial by a monomial

Multiply two polynomials using algebra tiles

Multiply two binomials

Multiply two binomials: special cases

Multiply polynomials

## Factoring

Finding what to multiply to get an expression is called factoring or factorizing. Factoring is like splitting an expression into a multiplication of simpler expressions. Factoring is important because it helps us to simplify expressions.

GCF of monomials

Factor out a monomial

Factor quadratics with leading coefficient 1

Factor quadratics with other leading coefficients

Factor quadratics using algebra tiles

Factor quadratics: special cases

Factor by grouping

Factor polynomials

## Quadratic equations

An equation where the highest exponent of the variable is two is called quadratic equation. In other words, an equation includes only second-degree polynomials. The quadratic equation is used to find the curve on a Cartesian plane.

Characteristics of quadratic functions

Complete a function table: quadratic functions

Solve a quadratic equation using square roots

Solve a quadratic equation using the zero product property

Solve a quadratic equation by factoring

Complete the square

Solve a quadratic equation by completing the square

Solve a quadratic equation using the quadratic formula

Using the discriminant

## Parabolas

Parabola is a special curve, shaped like an arch. Any point of a parabola is at an equal distance from a fixed point (the focus), and a fixed straight line (the directrix). For formal definition we say: for a given point, called focus, and a given line not through the focus, called directrix, a parabola is the locus of points such that the distance of the focus equals the distance to the directrix. Not all U-shaped curves are parabola, a parabola should satisfy the conditions listed above. Parabola has many important applications in real life, such as designing automobiles headlights to calculating the path of a ballistic missile.

Identify the direction a parabola opens

Find the vertex of a parabola

Find the axis of symmetry of a parabola

Write equations of parabolas in vertex form from graphs

Write equations of parabolas in vertex form using properties

Graph parabolas

## Functions: linear, quadratic, exponential

Identify linear, quadratic and exponential functions from graphs

Identify linear, quadratic and exponential functions from tables

Write linear, quadratic and exponential functions

Linear functions over unit intervals

Exponential functions over unit intervals

Describe linear and exponential growth and decay

## Rational functions and expressions

A function that be written as a polynomial divided by a polynomial is a rational function. It is rational because one is divided by the other, like a ratio.

Rational functions: asymptotes and excluded values

Simplify complex fractions

Simplify rational expressions

Multiply and divide rational expressions

Divide polynomials

Add and subtract rational expressions

Solve rational equations

## Circles in the coordinate plane

A circle is a two-dimensional shape made by drawing a curve that is always the same distance from a center. In other words, a circle is the locus of all points that are a fixed distance from a given point.

Find the centre of a circle

Find the radius or diameter of a circle

Write equations of circles in standard form from graphs

Write equations of circles in standard form using properties

Graph circles

## Logic

One area of mathematics that has its roots deep in philosophy is the study of logic. Mathematical log helps to detect whether a statement is valid or invalid.

Identify hypotheses and conclusions

Counterexamples

Truth tables

Truth values

Conditionals

Negations

Converses, inverses and contrapositives

Biconditionals

## Measurement

Measurement is the process of assigning a number to a physical property, like length, area, mass, etc.

Estimate metric measurements

Convert rates and measurements: metric units

Estimate imperial measurements of length

Convert rates and measurements: imperial units of length

Unit rates

Unit prices with imperial length conversions

Imperial mixed units of length

Convert between metric and imperial units of length

Scale drawings: word problems

## Constructions

Construction in Geometry means to draw shapes, angles or lines accurately. These constructions use only compass, straightedge and a pencil. Construction is important in geometry because it allows you to draw lines, angles, and polygons with the simplest tools at you hand.

Construct an angle bisector

Construct a congruent angle

## Two-dimensional figures

Two-dimensional geometry or plane geometry is about flat shapes like triangles and circles. Two-dimensional figures have only two dimensions such as width and height but no thickness. It also known as “2D”.

Polygon vocabulary

Interior angles of polygons

Perimeter

Area of triangles and quadrilaterals

Area and perimeter in the coordinate plane I

Area and perimeter in the coordinate plane II

Area and circumference of circles

Area of compound figures

Area between two shapes

Area and perimeter of similar figures

Identify trapezoids

Classify quadrilaterals

Graph quadrilaterals

Properties of parallelograms

Properties of rhombuses

## Three-dimensional figures

Having three dimensions such as Height, Width and Depth, like any real-world object is a three-dimensional figure. Three-dimensional geometry is about solid shapes like spheres or cubes. It is also known as “3D”.

Three-dimensional figure vocabulary

Parts of three-dimensional figures

Nets and drawings of three-dimensional figures

Introduction to surface area and volume

Surface area of prisms and cylinders

Surface area of pyramids and cones

Volume of prisms and cylinders

Volume of pyramids and cones

Surface area and volume of spheres

Introduction to similar solids

Surface area and volume of similar solids

Surface area and volume review

Cross-sections of three-dimensional figures

Solids of revolution

## Congruent figures

When we change a shape by using Turn, flip, slide or resize it is called transformation. If one shape can become another using Turn, Flip or Slide then shapes are congruent. Congruence keeps the size, area, angles and line lengths of the shape.

Congruent line segments

Congruence statements and corresponding parts

Solve problems involving corresponding parts

Identify congruent figures

SSS and SAS Theorems

ASA and AAS Theorems

SSS, SAS, ASA and AAS Theorems

SSS Theorem in the coordinate plane

Congruency in isosceles and equilateral triangles

Hypotenuse-Leg Theorem

## Triangles

**Triangle is a polygon with 3 sides. **

Classify triangles

Triangle Angle-Sum Theorem

Midsegments of triangles

Triangles and bisectors

Identify medians, altitudes, angle bisectors and perpendicular bisectors

Angle-side relationships in triangles

Triangle Inequality Theorem

## Similarity

When we change a shape by using Turn, flip, slide or resize it is called transformation. Two shapes are similar when one can become the other after a resize, flip, slide or turn. When two shapes are similar then the corresponding angles are equal, and the lines are in proportion. This can make life a lot easier when solving geometry problems.

Identify similar figures

Similarity ratios

Similarity statements

Side lengths and angle measures in similar figures

Similar triangles and indirect measurement

Perimeters of similar figures

Similarity rules for triangles

Similar triangles and similarity transformations

Similarity of circles

Triangle Proportionality Theorem

Areas of similar figures

## Right triangles

Right triangles are important in geometry due to their properties and the use of Pythagoras theorem.

Pythagorean Theorem

Converse of the Pythagorean theorem

Pythagorean Inequality Theorems

Special right triangles

## Trigonometry

Trigonometry is the study of triangles with emphasis on calculations involving the lengths of sides and the angles. Trigonometry is based on six functions, namely Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant. Trigonometry also includes study of the properties of these functions and their graphs. Trigonometry is an important field in mathematics with a lot of applications in real life problems, like calculating the waves and tides in oceans, creating maps, satellite systems and a lot more.

Trigonometric ratios: sin, cos and tan

Trigonometric ratios: csc, sec and cot

Find trigonometric functions of special angles

Find trigonometric functions using a calculator

Inverses of trigonometric functions

Trigonometric ratios: find a side length

Trigonometric ratios: find an angle measure

Solve a right triangle

## Transformations

When we change a shape by using Turn, flip, slide or resize it is called transformation. Transformation is an operation or operations that alter the form of a figure. Transformations are important in geometry and advanced math works.

Translations: graph the image

Translations: find the coordinates

Translations: write the rule

Reflections: graph the image

Reflections: find the coordinates

Rotate polygons about a point

Rotations: graph the image

Rotations: find the coordinates

Classify congruence transformations

Compositions of congruence transformations: graph the image

Transformations that carry a polygon onto itself

Congruence transformations: mixed review

Dilations: graph the image

Dilations: find the coordinates

Dilations: scale factor and classification

Dilations and parallel lines

## Probability

Probability is the chance of something happening or how likely it is that some event will happen. Probability is a number between 0 (not happening) to 1 (certainly happening).

Theoretical and experimental probability

Compound events: find the number of outcomes

Independent and dependent events

Factorials

Permutations

Counting principle

Permutation and combination notation

Geometric probability

## Statistics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of data. Statistics is a strong tool in everyday life to get answers about data and make concrete decisions.

Mean, median, mode and range

Quartiles

Identify biased samples

Mean absolute deviation

Variance and standard deviation