OCP-MATH10C · Knowledge Layer · CONTROLLED PILOT
Math 10C Knowledge Layer
Alberta Grade 10 Mathematics course-first tutoring portal aligned to the Biology 30 delivery model.
M10C-A: Measurement
Measurement connects numbers to the physical world. Learners first separate 1-D length, 2-D area, and 3-D volume, then attach the correct square or cubic units. With that foundation they compute perimeter, area, surface area, and volume, decompose composite figures and solids, convert units, and always finish with a precision and reasonableness check.
Essential knowledge definitions
| Term | Definition | Example |
|---|---|---|
| measurement | Assigning a number and a unit to a quantity by comparing it to a standard. | A desk measured as 1.2 m long. |
| dimension | The number of independent directions needed to describe a figure: length is 1-D, area 2-D, volume 3-D. | A line is 1-D; a rectangle is 2-D. |
| perimeter | The total distance around the boundary of a 2-D figure. | Perimeter of a 3 m by 4 m rectangle is 14 m. |
| area | The amount of surface a 2-D figure covers, measured in square units. | Area of a 3 m by 4 m rectangle is 12 m^2. |
| surface area | The total area of all the faces (surfaces) of a 3-D solid. | Surface area of a cube with edge 2 cm is 24 cm^2. |
| volume | The amount of space a 3-D solid occupies, measured in cubic units. | Volume of a 2 cm cube is 8 cm^3. |
| composite figure | A 2-D shape formed by combining simpler shapes. | An L-shape made from two rectangles. |
| composite solid | A 3-D object formed by combining simpler solids. | A cylinder topped by a cone (a silo). |
| unit conversion | Rewriting a measurement in a different unit using a conversion factor. | 2.5 km = 2500 m. |
| precision | How finely a measurement is reported, shown by its decimal places / significant figures. | 12.0 cm is more precise than 12 cm. |
| reasonableness | Judging whether an answer makes sense for the situation and its units. | A 15 cm pencil is reasonable; 15 m is not. |
| square units vs cubic units | Area uses square units (2-D); volume uses cubic units (3-D). | cm^2 for area, cm^3 for volume. |
Vocabulary / symbols
- measurement: Assigning a number and a unit to a quantity by comparing it to a standard
- dimension: The number of independent directions needed to describe a figure: length is 1-D, area 2-D, volume 3-D
- perimeter (P): The total distance around the boundary of a 2-D figure
- area (A): The amount of surface a 2-D figure covers, measured in square units
- surface area (SA): The total area of all the faces (surfaces) of a 3-D solid
- volume (V): The amount of space a 3-D solid occupies, measured in cubic units
- composite figure: A 2-D shape formed by combining simpler shapes
- composite solid: A 3-D object formed by combining simpler solids
- unit conversion: Rewriting a measurement in a different unit using a conversion factor
- precision: How finely a measurement is reported, shown by its decimal places / significant figures
- reasonableness: Judging whether an answer makes sense for the situation and its units
- square units vs cubic units: Area uses square units (2-D); volume uses cubic units (3-D)
Formulas / symbols
P = 2(l + w)— Perimeter of a rectangle.A = l x w— Area of a rectangle.A = (1/2) b h— Area of a triangle.A = pi r^2— Area of a circle.SA = 6 s^2— Surface area of a cube.V = l w h— Volume of a rectangular prism.V = pi r^2 h— Volume of a cylinder.V = (1/3) pi r^2 h— Volume of a cone.
Prerequisite ladders
- Area readiness: dimension → perimeter → area → square units vs cubic units (Learners must separate 1-D distance from 2-D area before area work.)
- Volume readiness: area → surface area → volume → square units vs cubic units (Volume builds on area and the 2-D vs 3-D unit distinction.)
- Composite readiness: area → composite figure → volume → composite solid (Composite work decomposes complex shapes into known parts.)
- Conversion readiness: measurement → unit conversion → precision → reasonableness (Conversions require correct units, precision, and a reasonableness check.)
Worked examples
Area of a rectangle
Problem: Find the area of a rectangle 3 m by 4 m.
- Area = length x width
- = 3 m x 4 m
- = 12 (square metres)
Answer: 12 m^2
Surface area of a cube
Problem: Find the surface area of a cube with edge 2 cm.
- A cube has 6 equal square faces
- Face area = 2 x 2 = 4 cm^2
- SA = 6 x 4
Answer: 24 cm^2
Volume of a cylinder
Problem: Find the volume of a cylinder with r = 3 m, h = 5 m.
- V = pi r^2 h
- = pi (3)^2 (5)
- = 45 pi
Answer: approximately 141.4 m^3
Composite solid volume
Problem: A cylinder (r = 2, h = 6) is topped by a cone (h = 3). Find the volume.
- V_cylinder = pi r^2 h = pi(4)(6) = 24 pi
- V_cone = (1/3) pi r^2 h = (1/3) pi(4)(3) = 4 pi
- Total = 28 pi
Answer: approximately 87.96 m^3
Visual models
- Area vs volume unit model: Distinguish square units from cubic units. Elements: unit-square grid, unit-cube stack, labels cm^2 and cm^3. Interaction: Learner counts squares vs cubes and labels the correct units.
- Composite solid decomposition model: Break a composite solid into known solids. Elements: cylinder body, cone cap, dashed separation line. Interaction: Learner splits the solid and totals the part volumes.
Misconception contrasts
| Misconception | Correct concept | Why it matters |
|---|---|---|
| Area and perimeter are the same thing. | Perimeter is the distance around (linear units); area is the space inside (square units). | They measure different attributes of a figure. |
| Volume can be reported in cm^2. | Volume uses cubic units such as cm^3. | Volume is a 3-D measure. |
| To find volume, add the dimensions. | Multiply the dimensions (for a prism, l x w x h). | Volume is a product of lengths, not a sum. |
| 12 cm and 12.0 cm show the same precision. | 12.0 cm is precise to the tenth of a centimetre. | The trailing decimal communicates measurement precision. |
M10C-B: Trigonometry
Trigonometry links angles to side ratios in right triangles. Learners first label the hypotenuse and the opposite and adjacent sides relative to a chosen angle, then apply sine, cosine, or tangent. Inverse functions recover angles from ratios. Word problems begin with a labelled diagram and end with a reasonableness check (the hypotenuse is longest, angles under 90 degrees).
Essential knowledge definitions
| Term | Definition | Example |
|---|---|---|
| right triangle | A triangle containing one 90-degree angle. | A triangle with legs 3 and 4 and hypotenuse 5. |
| hypotenuse | The side opposite the right angle; the longest side of a right triangle. | In a 3-4-5 triangle the hypotenuse is 5. |
| opposite side | The side directly across from the reference angle. | For the angle at the base, the vertical leg is opposite. |
| adjacent side | The side next to the reference angle that is not the hypotenuse. | For the base angle, the horizontal leg is adjacent. |
| sine | The ratio opposite/hypotenuse for an angle in a right triangle. | sin(t) = 7/12 when opposite = 7, hypotenuse = 12. |
| cosine | The ratio adjacent/hypotenuse for an angle. | cos(t) = 5/13. |
| tangent | The ratio opposite/adjacent for an angle. | tan(t) = 5/12. |
| inverse trigonometry | Functions (sin^-1, cos^-1, tan^-1) that return the angle from a known ratio. | cos^-1(0.5) = 60 degrees. |
| angle of elevation | The angle measured upward from the horizontal to a line of sight. | Looking up to a treetop at 35 degrees. |
| angle of depression | The angle measured downward from the horizontal to a line of sight. | Looking down from a cliff to a boat at 20 degrees. |
| diagram construction from word problem | Drawing and labelling a right triangle from a described situation. | Sketching a ladder against a wall and labelling the angle and sides. |
| reasonableness of trig answers | Checking that side and angle results fit the triangle. | The hypotenuse must be the longest side; angles are under 90 degrees. |
Vocabulary / symbols
- right triangle: A triangle containing one 90-degree angle
- hypotenuse: The side opposite the right angle; the longest side of a right triangle
- opposite side: The side directly across from the reference angle
- adjacent side: The side next to the reference angle that is not the hypotenuse
- sine (sin): The ratio opposite/hypotenuse for an angle in a right triangle
- cosine (cos): The ratio adjacent/hypotenuse for an angle
- tangent (tan): The ratio opposite/adjacent for an angle
- inverse trigonometry: Functions (sin^-1, cos^-1, tan^-1) that return the angle from a known ratio
- angle of elevation: The angle measured upward from the horizontal to a line of sight
- angle of depression: The angle measured downward from the horizontal to a line of sight
- diagram construction from word problem: Drawing and labelling a right triangle from a described situation
- reasonableness of trig answers: Checking that side and angle results fit the triangle
Formulas / symbols
sin(t) = opposite / hypotenuse— Sine ratio.cos(t) = adjacent / hypotenuse— Cosine ratio.tan(t) = opposite / adjacent— Tangent ratio.t = sin^-1(x), cos^-1(x), tan^-1(x)— Inverse trig to find an angle.
Prerequisite ladders
- Ratio readiness: right triangle → hypotenuse → opposite side → sine (Ratios require correct side identification first.)
- Solve a side: sine → diagram construction from word problem → reasonableness of trig answers (Learners set up a ratio from a diagram, then check the result.)
- Solve an angle: tangent → inverse trigonometry → reasonableness of trig answers (Finding an angle needs an inverse function and a sanity check.)
- Applied trigonometry: diagram construction from word problem → angle of elevation → angle of depression → reasonableness of trig answers (Applications rely on correct elevation/depression modelling.)
Worked examples
Write a sine ratio
Problem: opposite = 7, hypotenuse = 12. Write sin(t).
- sin = opposite/hypotenuse
- = 7/12
Answer: 7/12
Find an angle
Problem: cos(t) = 0.5. Find t.
- Use the inverse cosine
- t = cos^-1(0.5)
Answer: 60 degrees
Ladder height
Problem: A 6 m ladder leans at 70 degrees to the ground. Find the height reached.
- sin 70 = height/6
- height = 6 sin 70
Answer: approximately 5.64 m
Tree height
Problem: An observer 20 m away sees the top at 35 degrees elevation. Find the height.
- tan 35 = h/20
- h = 20 tan 35
Answer: approximately 14.0 m
Visual models
- Right-triangle ratio labeller: Label sides relative to a chosen angle. Elements: right-angle marker, reference angle, opposite/adjacent/hypotenuse labels. Interaction: Learner drags labels onto the correct sides for the chosen angle.
Misconception contrasts
| Misconception | Correct concept | Why it matters |
|---|---|---|
| Opposite and adjacent are interchangeable. | Opposite is across from the angle; adjacent is beside it (not the hypotenuse). | Each ratio depends on the reference angle. |
| Use sine for every triangle problem. | Choose the ratio from the sides you have (SOH-CAH-TOA). | The correct ratio depends on which sides are known. |
| You can get an angle directly from a ratio. | Use an inverse function to convert a ratio into an angle. | sin(t) gives a ratio; sin^-1 gives the angle. |
| The hypotenuse can be a short side. | The hypotenuse is always the longest side, opposite the right angle. | This is a built-in reasonableness check. |
M10C-C: Factors, Products and Roots
This unit builds fluency with exponents, radicals, and polynomial structure. Learners use exponent laws and simplify radicals, expand products with the distributive property and special-product patterns, then factor by reversing those patterns. The unifying idea is equivalence: expanded and factored forms are the same expression written differently, and expanding is the built-in check for factoring.
Essential knowledge definitions
| Term | Definition | Example |
|---|---|---|
| factor | A quantity that divides another exactly; also, each part of a product. | 3 and 4 are factors of 12. |
| product | The result of multiplying quantities. | The product of 3 and 4 is 12. |
| prime factor | A factor that is a prime number. | The prime factors of 12 are 2 and 3. |
| exponent | A number showing how many times a base is used as a factor. | In 2^3, the exponent is 3. |
| exponent law | A rule for combining powers, such as a^m x a^n = a^(m+n). | x^3 x x^4 = x^7. |
| radical | An expression using a root symbol, such as a square root. | sqrt(50) is a radical. |
| square root | A value that, multiplied by itself, gives the original number. | sqrt(25) = 5. |
| polynomial | An expression of terms with whole-number exponents joined by + or -. | x^2 + 3x - 5. |
| binomial | A polynomial with exactly two terms. | x + 4. |
| trinomial | A polynomial with exactly three terms. | x^2 + 7x + 12. |
| distributive property | a(b + c) = ab + ac; multiply across a sum. | 3(x + 2) = 3x + 6. |
| factoring | Rewriting an expression as a product of factors. | x^2 + 7x + 12 = (x + 3)(x + 4). |
| common factor | A factor shared by all terms of an expression. | 3x is the common factor of 6x^2 + 9x. |
| special products | Patterns such as (a + b)^2 = a^2 + 2ab + b^2. | (x + 5)^2 = x^2 + 10x + 25. |
| equivalent expressions | Expressions that have the same value for every input. | x^2 + 7x + 12 and (x + 3)(x + 4). |
Vocabulary / symbols
- factor: A quantity that divides another exactly; also, each part of a product
- product: The result of multiplying quantities
- prime factor: A factor that is a prime number
- exponent (a^n): A number showing how many times a base is used as a factor
- exponent law: A rule for combining powers, such as a^m x a^n = a^(m+n)
- radical (√): An expression using a root symbol, such as a square root
- square root: A value that, multiplied by itself, gives the original number
- polynomial: An expression of terms with whole-number exponents joined by + or -
- binomial: A polynomial with exactly two terms
- trinomial: A polynomial with exactly three terms
- distributive property: a(b + c) = ab + ac; multiply across a sum
- factoring: Rewriting an expression as a product of factors
- common factor: A factor shared by all terms of an expression
- special products: Patterns such as (a + b)^2 = a^2 + 2ab + b^2
- equivalent expressions: Expressions that have the same value for every input
Formulas / symbols
a^m x a^n = a^(m+n)— Product of powers.(a^m)^n = a^(mn)— Power of a power.(ab)^n = a^n b^n— Power of a product.sqrt(ab) = sqrt(a) x sqrt(b)— Radical of a product.(a + b)^2 = a^2 + 2ab + b^2— Perfect-square trinomial.
Prerequisite ladders
- Power readiness: exponent → exponent law → radical → square root (Radicals and roots build on exponent meaning.)
- Product readiness: factor → product → distributive property → special products (Expanding products relies on distribution patterns.)
- Factoring readiness: factor → common factor → factoring → equivalent expressions (Factoring reverses expansion and preserves value.)
- Polynomial readiness: polynomial → binomial → trinomial → factoring (Naming polynomials supports choosing a factoring method.)
Worked examples
Product of powers
Problem: Simplify x^3 x x^4.
- Same base: add exponents
- 3 + 4 = 7
Answer: x^7
Simplify a radical
Problem: Write sqrt(50) in simplest form.
- 50 = 25 x 2
- sqrt(25 x 2) = sqrt(25) x sqrt(2)
- = 5 sqrt(2)
Answer: 5 sqrt(2)
Expand a binomial product
Problem: Expand (x + 4)(x - 7).
- Distribute each term
- x^2 - 7x + 4x - 28
- combine like terms
Answer: x^2 - 3x - 28
Factor a trinomial
Problem: Factor x^2 + 7x + 12.
- Find two numbers with product 12 and sum 7
- 3 and 4
- write as a product
Answer: (x + 3)(x + 4)
Visual models
- Algebra-tile expand/factor model: Show expansion and factoring as area. Elements: unit tiles, x tiles, x^2 tiles, rectangle frame. Interaction: Learner builds a rectangle to expand, then reads factors from its sides.
Misconception contrasts
| Misconception | Correct concept | Why it matters |
|---|---|---|
| Multiply the exponents when multiplying same bases. | Add the exponents: a^m x a^n = a^(m+n). | Multiplying powers repeats the base. |
| A square root distributes over addition. | sqrt(a + b) is not sqrt(a) + sqrt(b). | Roots do not distribute over sums. |
| (x + 5)^2 = x^2 + 25. | (x + 5)^2 = x^2 + 10x + 25. | The middle term 2ab is required. |
| Factoring changes the value of an expression. | Factoring produces an equivalent expression. | The factored form expands back to the original. |
M10C-D: Relations and Functions
Relations and functions describe how one quantity depends on another. Learners identify independent and dependent variables, domain and range, and move fluently between ordered pairs, tables, graphs, and equations. For linear relations they read slope and intercepts and, most importantly, interpret what those numbers mean in a real context.
Essential knowledge definitions
| Term | Definition | Example |
|---|---|---|
| relation | A set of ordered pairs linking inputs to outputs. | {(1,2), (2,4), (3,6)}. |
| function | A relation in which every input has exactly one output. | y = 2x is a function. |
| independent variable | The input variable, usually x, that is chosen or controlled. | Time t in d = 60t. |
| dependent variable | The output variable, usually y, that depends on the input. | Distance d in d = 60t. |
| domain | The set of all input (x) values of a relation. | Domain of {(1,2),(2,4)} is {1, 2}. |
| range | The set of all output (y) values of a relation. | Range of {(1,2),(2,4)} is {2, 4}. |
| ordered pair | A pair (x, y) giving an input and its output. | (3, 6). |
| table of values | A table listing inputs and matching outputs. | x: 0,1,2 and y: 3,5,7. |
| graph | A visual plot of ordered pairs on a coordinate plane. | A straight line through plotted points. |
| slope | The rate of change of a line: rise over run. | Slope of a line through (2,5) and (6,13) is 2. |
| y-intercept | The y-value where a graph crosses the y-axis (x = 0). | In y = -2x + 7, the y-intercept is 7. |
| x-intercept | The x-value where a graph crosses the x-axis (y = 0). | In y = 2x - 4, the x-intercept is 2. |
| linear relation | A relation with a constant rate of change; its graph is a straight line. | y = 3x + 2. |
| rate of change | How much the output changes per unit change in input. | Cost rising $0.10 per minute. |
| contextual interpretation | Explaining what slope and intercepts mean in a real situation. | In C = 0.10n + 20, 20 is the base cost. |
Vocabulary / symbols
- relation: A set of ordered pairs linking inputs to outputs
- function: A relation in which every input has exactly one output
- independent variable: The input variable, usually x, that is chosen or controlled
- dependent variable: The output variable, usually y, that depends on the input
- domain: The set of all input (x) values of a relation
- range: The set of all output (y) values of a relation
- ordered pair: A pair (x, y) giving an input and its output
- table of values: A table listing inputs and matching outputs
- graph: A visual plot of ordered pairs on a coordinate plane
- slope (m): The rate of change of a line: rise over run
- y-intercept (b): The y-value where a graph crosses the y-axis (x = 0)
- x-intercept: The x-value where a graph crosses the x-axis (y = 0)
- linear relation: A relation with a constant rate of change; its graph is a straight line
- rate of change: How much the output changes per unit change in input
- contextual interpretation: Explaining what slope and intercepts mean in a real situation
Formulas / symbols
m = (y2 - y1)/(x2 - x1)— Slope from two points.y = mx + b— Slope-intercept form of a line.
Prerequisite ladders
- Relation readiness: relation → ordered pair → independent variable → dependent variable (Learners must read pairs and variable roles first.)
- Representation readiness: table of values → graph → relation → function (Connecting tables and graphs supports the function idea.)
- Linear readiness: linear relation → slope → y-intercept → x-intercept (Line features build on the linear-relation idea.)
- Interpretation readiness: slope → rate of change → contextual interpretation → y-intercept (Meaning in context requires slope and intercept fluency.)
Worked examples
Slope from two points
Problem: Find the slope through (2, 5) and (6, 13).
- slope = (y2 - y1)/(x2 - x1)
- = (13 - 5)/(6 - 2)
- = 8/4
Answer: 2
Read slope and intercept
Problem: State the slope and y-intercept of y = -2x + 7.
- Compare to y = mx + b
- m = -2, b = 7
Answer: slope -2, y-intercept 7
Equation from a table
Problem: Write the equation for (0,3), (1,5), (2,7).
- Rate of change = 2 per step
- y-intercept at x = 0 is 3
Answer: y = 2x + 3
Interpret in context
Problem: In C = 0.10n + 20, explain 20 and 0.10.
- 20 is the value when n = 0 (base cost)
- 0.10 is the change per unit n
Answer: 20 = base cost; 0.10 = cost per unit
Visual models
- Slope-intercept grapher: Connect equation, table, and graph. Elements: coordinate grid, plotted points, line through points, intercept markers. Interaction: Learner changes slope/intercept and watches the line and table update.
Misconception contrasts
| Misconception | Correct concept | Why it matters |
|---|---|---|
| Independent and dependent variables can be swapped freely. | The dependent variable depends on the independent one; the roles are fixed by the situation. | Cause/order determines the roles. |
| Domain and range mean the same thing. | Domain is inputs (x); range is outputs (y). | They describe different sets. |
| Every relation is a function. | A relation is a function only if each input has one output. | Repeated inputs with different outputs break the rule. |
| The slope is the number after the plus sign. | The slope is the coefficient of x; the constant is the y-intercept. | Confusing them misreads the line. |