Ontario Grade 10 Academic math (MPM2D)

Data, parabolas, transformations, quadratic equations, and applications.

• Q1 · Identify quadratic patterns using second differences (published written lesson)
• Q2 · Collect quadratic data and draw a curve of best fit (published written lesson)
• Q3 · Identify the vertex, axis, intercepts, and extrema of a parabola (written lesson not published)
• Q4 · Compare quadratic and exponential graphs and interpret zero and negative exponents (published written lesson)
• Q5 · Investigate translations, reflections, stretches, and compressions of y = x² (written lesson not published)
• Q6 · Explain the parameters and vertex of y = a(x − h)² + k (published written lesson)
• Q7 · Sketch a quadratic graph from vertex form (published written lesson)
• Q8 · Determine a vertex-form equation from a parabola graph (published written lesson)
• Q9 · Expand and simplify second-degree polynomial expressions (published written lesson)
• Q10 · Factor quadratics using common factors, trinomials, and differences of squares (published written lesson)
• Q11 · Connect quadratic factors to graph zeros (written lesson not published)
• Q12 · Interpret real and non-real quadratic roots using graphs (written lesson not published)
• Q13 · Complete the square without fractional coefficients (written lesson not published)
• Q14 · Sketch a quadratic from standard form using suitable methods (written lesson not published)
• Q15 · Explore how the quadratic formula is developed (written lesson not published)
• Q16 · Solve real-root quadratic equations by factoring, formula, or graphing (written lesson not published)
• Q17 · Find zeros and maximum or minimum values from a graph or equation (written lesson not published)
• Q18 · Solve realistic problems modelled by quadratic relations (written lesson not published)

Linear systems, line segments, circles, geometric verification, and the 2022 slope addendum.

• G1 · Find slope from two points and write a line equation (written lesson not published) (2022 transition addendum)
• G2 · Use slopes of parallel and perpendicular lines (written lesson not published) (2022 transition addendum)
• G3 · Translate among common forms of a line equation (written lesson not published) (2022 transition addendum)
• G4 · Solve two-variable linear systems by substitution or elimination (written lesson not published)
• G5 · Model and solve real situations with linear systems (written lesson not published)
• G6 · Develop and use the midpoint formula (written lesson not published)
• G7 · Develop and use the distance formula for line segments (written lesson not published)
• G8 · Develop the equation of a circle centred at the origin (written lesson not published)
• G9 · Find a circle radius, write its equation, and sketch its graph (written lesson not published)
• G10 · Solve problems using slope, distance, and midpoint (written lesson not published)
• G11 · Investigate properties of geometric figures (written lesson not published)
• G12 · Verify geometric properties with algebra and coordinates (written lesson not published)
• G13 · Plan a multi-step coordinate proof of a geometric property (written lesson not published)

Similar triangles, right-triangle ratios, and sine and cosine laws for acute triangles.

• T1 · Use sine, cosine, and tangent in right triangles (written lesson not published)
• T2 · Investigate equal angles and proportional sides in similar triangles (written lesson not published)
• T3 · Compare similarity and congruence (written lesson not published)
• T4 · Solve realistic problems with similar triangles (written lesson not published)
• T5 · Find right-triangle sides and angles using ratios and Pythagoras (written lesson not published)
• T6 · Apply right-triangle trigonometry to real situations (written lesson not published)
• T7 · Explore the development of the sine law for acute triangles (written lesson not published)
• T8 · Explore the development of the cosine law for acute triangles (written lesson not published)
• T9 · Find acute-triangle sides and angles with sine or cosine law (written lesson not published)
• T10 · Solve real problems involving acute triangles (written lesson not published)

For y = x² − 4x + 3, find the outputs at x = 0, 1, 2, 3, 4.

Worked method: Substitute each x: the outputs are 3, 0, −1, 0, 3. The first differences are −3, −1, 1, 3; the second differences are 2, 2, 2. The constant second difference is a clue that this is quadratic.

Find the equation of the line through (1, 3) and (3, 7).

Worked method: The rise is 7 − 3 = 4 and the run is 3 − 1 = 2, so the slope is 4 ÷ 2 = 2. Use y = 2x + b and (1, 3): 3 = 2 + b, so b = 1. The line is y = 2x + 1.

A right triangle has legs 6 cm and 8 cm. Find the angle opposite the 6 cm leg.

Worked method: For that angle, opposite ÷ adjacent = 6 ÷ 8 = 0.75. Therefore tan θ = 0.75 and θ ≈ 36.9°. Check that the angle is below 45° because the opposite leg is shorter than the adjacent leg.

For y = x² + 2x, find the outputs when x = 0, 1, 2, 3 and their second differences.

Find the equation of the line through (0, −2) and (2, 4).