# Grade Ten Math Curriculum

# Grade Ten Math Curriculum

## Introduction

Grade 10 mathematics builds on the elementary program, relying on the same fundamental principles on which that program was based. It is founded on the premise that students learn mathematics most effectively when they have a thorough understanding of mathematical concepts and procedures, and when they build that understanding through an investigative approach, as reflected in the inquiry model of learning. This curriculum is designed to help students build a solid conceptual foundation in mathematics that will enable them to apply their knowledge and skills and further their learning successfully. In grade 10, students familiarize themselves further with mathematical concepts while also initiating the process of their college applications, shortlisting career options. The information provided in this page are identical to the Official Grade Ten Math Curriculum established by the Ministry of Education.

#### Table of Contents

## Overall and Specific Expectations

The mathematics courses in the grade 10 curriculum are offered in two types, academic and applied, which are defined as follows:

**Academic courses**

These develop students’ knowledge and skills through the study of theory and abstract problems. These courses focus on the essential concepts of a subject and explore related concepts as well. They incorporate practical applications as appropriate.

**Applied courses**

These courses focus on the essential concepts of a subject and develop students’ knowledge and skills through practical applications and concrete examples. Familiar situations are used to illustrate ideas, and students are given more opportunities to experience hands-on applications of the concepts and theories they study.

The expectations identified for each course describe the knowledge and skills that students are expected to acquire, demonstrate, and apply in their class work, on tests, and in various other activities on which their achievement is assessed and evaluated.

## Academic Courses: Principles of Mathematics

This course enables students to broaden their understanding of relationships and extend their problem-solving and algebraic skills through investigation, the effective use of technology, and abstract reasoning. Students will explore quadratic relations and their applications; solve and apply linear systems; verify properties of geometric figures using analytic geometry; and investigate the trigonometry of right and acute triangles. Students will reason mathematically and

communicate their thinking as they solve multi-step problems.

The overall expectations are divided in three main categories each of which contains several subcategories. Every subcategory lists specific expectations for academic courses of grade 10 in more detail.

### Quadratic Relations of the form y=ax^2+bx+c

**A. Investigating the Basic Properties of Quadratic Relations:**

**Students will**

**1. **collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology (e.g., concrete materials, scientific probes, graphing calculators), or from secondary sources (e.g., the Internet, Statistics Canada); graph the data and draw a curve of best fit, if appropriate, with or without the use of technology.

**2. **determine, through investigation with and without the use of technology, that a quadratic relation of the form can be graphically represented as a parabola, and that the table.

**3. **identify the key features of a graph of a parabola (i.e., the equation of the axis of symmetry, the coordinates of the vertex, the *y*-intercept, the zeros, and the maxi- mum or minimum value), and use the appropriate terminology to describe

**4. **compare, through investigation using technology, the features of the graph of and the graph of , and determine the meaning of a negative exponent and of zero as an exponent (e.g., by examining patterns in a table of values for ; by applying the exponent rules for multiplication and division).

**B. Relating the graph of and its transformations:Students will**

**1.**identify, through investigation using technology, the effect on the graph of of transformations (i.e., translations, reflections in the *x*-axis, vertical stretches, or compressions) by considering separately each parameter *a*, *h*, and *k *[i.e., investigate the effect on the graph of * * of *a*, *h*, and *k* in *,* , and ].

**2.**explain the roles of *a*, *h*, and *k* in , using the appropriate terminology to describe the transformations, and identify the vertex and the equation of the axis of

**3.**sketch, by hand, the graph of * *by applying transformations to the graph of .

**4**.determine the equation, in the form , of a given graph of a

**C.Solving quadratic equations:Students will**

**1. **expand and simplify second-degree polynomial expressions [e.g., using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

**2. **factor polynomial expressions involving common factors, trinomials, and differences of squares [e.g., ], using a variety of tools (e.g., concrete materials, computer algebra systems, paper and pencil) and strategies (e.g., patterning).

**3. **determine, through investigation, and describe the connection between the factors of a quadratic expression and the *x*-intercepts [i.e., the zeros of the graph of the corresponding quadratic relation, relation, expressed in the form ].

**4. **interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the *x*-intercepts of the corresponding

**5. **sketch or graph a quadratic relation whose equation is given in the form , using a variety of methods (e.g., sketching using intercepts and symmetry; sketching by completing the square and applying transformations; graphing using technology).

**6. **explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numerical example; follow a demonstration of the algebraic development [student reproduction of the development of the general case is not required]).

**7. **solve quadratic equations that have real roots, using a variety of methods (i.e., factoring, using the quadratic formula, graphing).

**D.Solving problems involving quadratic relations:Students will**

**1. **determine the zeros and the maximum or minimum value of a quadratic relation from its graph (i.e., using graphing calculators or graphing software) or from its defining equation (i.e., by applying algebraic techniques).

**2. **solve problems arising from a realistic situation represented by a graph or an equation of a quadratic relation, with and without the use of technology (e.g., given the graph or the equation of a quadratic relation representing the height of a ball over elapsed time, answer questions such as the following: What is the maximum height of the ball? After what length of time will the ball hit the ground? Over what time interval is the height of the ball greater than 3m?).

### Analytic Geometry

**A.Using linear systems to solve problems:Students will**

** **

**1. **solve systems of two linear equations involving two variables, using the algebraic method of substitution or elimination.

**2. **solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method.

**B.Solving problems involving properties of line segments:Students will**

**1. **develop the formula for the midpoint of a line segment, and use this formula to solve problems (e.g., determine the coordinates of the midpoints of the sides of a triangle, given the coordinates of the vertices, and verify concretely or by using dynamic geometry software).

**2. **develop the formula for the length of a line segment, and use this formula to solve problems (e.g., determine the lengths of the line segments joining the midpoints of the sides of a triangle, given the coordinates of the vertices of the triangle, and verify using dynamic geometry software).

**3. **develop the equation for a circle with centre (0, 0) and radius *r*, by applying the formula for the length of a line segment.

**4. **determine the radius of a circle with center (0, 0), given its equation; write the equation of a circle with centre (0, 0), given the radius; and sketch the circle, given the equation in the form .

**5. **solve problems involving the slope, length, and midpoint of a line segment (e.g., determine the equation of the right bisector of a line segment, given the coordinates of the endpoints; determine the distance from a given point to a line whose equation is given, and verify using dynamic geometry software).

**C.Using analytic geometry to verify geometric properties:Students will**

** **

**1. **determine, through investigation (e.g., using dynamic geometry software, by paper folding), some characteristics and properties of geometric figures (e.g., medians in a triangle, similar figures con- structed on the sides of a right triangle).

**2. **verify, using algebraic techniques and analytic geometry, some characteristics of geometric figures (e.g., verify that two lines are perpendicular, given the coordinates of two points on each line; verify, by determining side length, that a triangle is equilateral, given the coordinates of the vertices).

**3. **plan and implement a multi-step strategy that uses analytic geometry and algebraic techniques to verify a geometric property (e.g., given the coordinates of the vertices of a triangle, verify that the line segment joining the midpoints of two sides of the triangle is parallel to the third side and half its length, and check using dynamic geometry software; given the coordinates of the vertices of a rectangle, verify that the diagonals of the rectangle bisect each other).

### Trigonometry

**A.Investigating similarity and solving problems involving similar triangles:Students will**

** **

**1. **verify, through investigation (e.g., using dynamic geometry software, concrete materials), the properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides).

**2. **describe and compare the concepts of similarity and congruence

**3. **solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying).

**B.Solving problems involving the trigonometry of right triangles:Students will**

** **

**1. **determine, through investigation (e.g., using dynamic geometry software, concrete materials), the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios.

**2.**determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean

**3.**solve problems involving the measures of sides and angles in right triangles in real- life applications (e.g., in surveying, in navigating, in determining the height of an inaccessible object around the school), using the primary trigonometric ratios and the Pythagorean

**C.Solving problems involving the trigonometry of acute triangles:Students will**

** **

**1. **explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that the ratio of the side lengths equals the ratio of the sines of the opposite angles; follow the algebraic development of the sine law and identify the application of solving systems of equations [student reproduction of the development of the formula is not required]).

**2. **explore the development of the cosine law within acute triangles (e.g., use dynamic geometry software to verify the cosine law; follow the algebraic development of the cosine law and identify its relationship to the Pythagorean theorem and the cosine ratio [student reproduction of the development of the formula is not required]).

**3. **determine the measures of sides and angles in acute triangles, using the sine law and the cosine law.

**4. **solve problems involving the measures of sides and angles in acute triangles.

### Spatial Sense

**A. Geometric and Spatial Reasoning**Students will describe and represent shape, location, and movement by applying geometric properties and spatial relationships to navigate the world around them. They will

**1. Geometric Reasoning**

- sort three-dimensional objects and two-dimensional shapes according to one attribute at a time and identify the sorting rule being used.
- construct three-dimensional objects, and identify two-dimensional shapes contained within structures and objects.
- construct and describe two-dimensional shapes and three-dimensional objects that have matching halves.

**2. Location and Movement**

- describe the relative locations of objects or people, using positional language.
- give and follow directions for moving from one location to another.

**B. Measurement**Students will compare, estimate, and determine measurements in various contexts. They will

**1. Attributes **

- identify measurable attributes of two-dimensional shapes and three-dimensional objects, including length, area, mass, capacity, and angle.
- compare several everyday objects and order them according to length, area, mass, and capacity.

**2. Time**

- read the date on a calendar, and use a calendar to identify days, weeks, months, holidays, and seasons.

### Applied Courses: Foundation of Mathematics

This course enables students to consolidate their understanding of linear relations and extend their problem-solving and algebraic skills through investigation, the effective use of techno- logy, and hands-on activities. Students will develop and graph equations in analytic geometry; solve and apply linear systems, using real-life examples; and explore and interpret graphs of quadratic relations. Students will investigate similar triangles, the trigonometry of right triangles, and the measurement of three-dimensional figures. Students will consolidate their mathematical skills as they solve problems and communicate their thinking.

The overall expectations are divided in three main categories each of which contains several subcategories. Every subcategory lists specific expectations for applied courses of grade 10 in more detail.

### Measurement and trigonometry

**A.Solving problems involving similar triangles:Students will**

** **

**1. **verify, through investigation (e.g., using dynamic geometry software, concrete materials), properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides).

**2. **determine the lengths of sides of similar triangles, using proportional reasoning.

**3. **solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying).

**B.****Solving problems involving the trigonometry of right triangles:Students will**

** **

**1. **determine, through investigation (e.g., using dynamic geometry software, concrete materials), the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios.

**2. **determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem.

**3. **solve problems involving the measures of sides and angles in right triangles in real- life applications (e.g., in surveying, in navigation, in determining the height of an inaccessible object around the school), using the primary trigonometric ratios and the Pythagorean theorem.

**4. **describe, through participation in an activity, the application of trigonometry in an occupation (e.g., research and report on how trigonometry is applied in astronomy; attend a career fair that includes a surveyor and describe how a surveyor applies trigonometry to calculate distances; job shadow a carpenter for a few hours and describe how a carpenter uses trigonometry).

**C.Solving problems involving surface area and volume, using the imperial and metric systems of measurement:Students will**

** **

**1. **use the imperial system when solving measurement problems (e.g., problems involving dimensions of lumber, areas of carpets, and volumes of soil or concrete).

**2. **perform everyday conversions between the imperial system and the metric system (e.g., millilitres to cups, centimetres to inches) and within these systems (e.g., cubic metres to cubic centimetres, square feet to square yards), as necessary to solve problems involving measurement.

**3. **determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a square-based pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles).

**4. **solve problems involving the surface areas of prisms, pyramids, and cylinders, and the volumes of prisms, pyramids, cylinders, cones, and spheres, including problems involving combinations of these figures, using the metric system or the imperial system, as appropriate.

### Modelling linear relations

**A.Manipulating and solving algebraic equations:Students will **

**1. **solve first-degree equations involving one variable, including equations with fractional coefficients (e.g. using the balance analogy, computer algebra systems, paper and pencil).

**2. **determine the value of a variable in the first degree, using a formula (i.e., by isolating the variable and then substituting known values; by substituting known values and then solving for the variable) (e.g., in analytic geometry, in measurement).

**3. **express the equation of a line in the form *,* given the form.

**B.Graphing and writing equations of lines:Students will**

** **

**1. **connect the rate of change of a linear relation to the slope of the line and define the slope as the ratio .

**2. **dentify, through investigation, as a common form for the equation of a straight line, and identify the special cases .

**3. **identify, through investigation with technology, the geometric significance of *m* and *b *in the equation .

**4. **identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism), using graphing technology to facilitate investigations, where appropriate.

**5. **graph lines by hand, using a variety oftechniques (e.g., graph *y* *= *3*x – *4 using the *y*-intercept and slope; graph using the *x*– and *y*-intercepts).

**6. **determine the equation of a line, given its graph, the slope and *y*-intercept, the slope and a point on the line, or two points on the line.

**C.Solving and interpreting systems of linear equations:Students will**

** **

**1. **determine graphically the point of inter- section of two linear relations (e.g., using graph paper, using technology).

**2. **solve systems of two linear equations involving two variables with integral coefficients, using the algebraic method of substitution or elimination.

**3. **solve problems that arise from realistic situations described in words or represented by given linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method.

### Quadratic Relations of the form y=ax^2+bx+c

**A.Investigating the Basic Properties of Quadratic Relations:**

**Students will**

**1. **collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology (e.g., concrete materials, scientific probes, graphing calculators), or from secondary sources (e.g., the Internet, Statistics Canada); graph the data and draw a curve of best fit, if appropriate, with or without the use of technology.

**2. **determine, through investigation with and without the use of technology, that a quadratic relation of the form can be graphically represented as a parabola, and that the table.

**3. **identify the key features of a graph of a parabola (i.e., the equation of the axis of symmetry, the coordinates of the vertex, the *y*-intercept, the zeros, and the maxi- mum or minimum value), and use the appropriate terminology to describe

**4. **compare, through investigation using technology, the features of the graph of and the graph of , and determine the meaning of a negative exponent and of zero as an exponent (e.g., by examining patterns in a table of values for ; by applying the exponent rules for multiplication and division).

**B. Relating the graph of and its transformations:Students will **

**1. **identify, through investigation using technology, the effect on the graph of of transformations (i.e., translations, reflections in the *x*-axis, vertical stretches, or compressions) by considering separately each parameter *a*, *h*, and *k *[i.e., investigate the effect on the graph of * * of *a*, *h*, and *k* in *,* , and ].

**2. **explain the roles of *a*, *h*, and *k* in , using the appropriate terminology to describe the transformations, and identify the vertex and the equation of the axis of

**3. **sketch, by hand, the graph of * *by applying transformations to the graph of .

**4**. determine the equation, in the form , of a given graph of a

**C.Solving quadratic equations:Students will**

**1. **expand and simplify second-degree polynomial expressions [e.g., using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

**2. **factor polynomial expressions involving common factors, trinomials, and differences of squares [e.g., ], using a variety of tools (e.g., concrete materials, computer algebra systems, paper and pencil) and strategies (e.g., patterning).

**3. **determine, through investigation, and describe the connection between the factors of a quadratic expression and the *x*-intercepts [i.e., the zeros of the graph of the corresponding quadratic relation, relation, expressed in the form ].

**4. **interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the *x*-intercepts of the corresponding

**5. **sketch or graph a quadratic relation whose equation is given in the form , using a variety of methods (e.g., sketching using intercepts and symmetry; sketching by completing the square and applying transformations; graphing using technology).

**6. **explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numerical example; follow a demonstration of the algebraic development [student reproduction of the development of the general case is not required]).

**7. **solve quadratic equations that have real roots, using a variety of methods (i.e., factoring, using the quadratic formula, graphing).

**D.Solving problems involving quadratic relations:Students will**

**1. **determine the zeros and the maximum or minimum value of a quadratic relation from its graph (i.e., using graphing calculators or graphing software) or from its defining equation (i.e., by applying algebraic techniques).

**2. **solve problems arising from a realistic situation represented by a graph or an equation of a quadratic relation, with and without the use of technology (e.g., given the graph or the equation of a quadratic relation representing the height of a ball over elapsed time, answer questions such as the following: What is the maximum height of the ball? After what length of time will the ball hit the ground? Over what time interval is the height of the ball greater than 3m?).

## List of Skills

## Evaluation

Objective evaluation is believed to be one of the most essential parts of teaching mathematics. In Genius Math, we use different tools and methods to evaluate the mathematical knowledge of students and their progress. Our evaluation process consists of three stages: before teaching sessions, during teaching sessions and after teaching sessions.

**Initial Assessment Test**

Before starting our teaching sessions, we administrate an assessment test to obtain some insights on the strengths and weaknesses of students and their previous math knowledge. This key information helps us to come up with a special plan for every single student.**Standard Problems**

During teaching sessions, we use a combination of different resources providing standard problems that are designed by famous mathematicians all over the world to improve the problem-solving skills of students. Among those resources are Math Kangaroo Contests, CEMC (University of Waterloo), AMC (American Mathematics Competitions), and even IMO (International Mathematics Olympiad), the latter might be considered for those who want to tackle more challenging problems or prepare for math olympiads. We use these problems to design homework, quizzes, and tests for our students based on their grades, needs and goals. As a matter of fact, such problems can be used to unveil the depth of students’ mathematical understanding.**Final Assessment Test**When teaching sessions are over, students are asked to take another assessment test aiming to show their real progress in mathematics.

## Most Common Challenging Topics

The followings are among the most common challenges students face in grade 10:

- Graphing quadratic relations
- Factoring quadratic polynomials
- Solving quadratic equations
- Slope and equation of a line
- Systems of linear equations
- Trigonometry

## What We Can Offer

- To review and practice their class notes and handouts
- To be helped with their homework, quizzes, and tests
- To improve their math skills in general
- To level up (e.g., moving from B- to B+)
- To get A+
- To learn topics beyond curriculum
- To prepare for math competitions