Grade 11 Science: Mathematics

In this unit, 'Sets and relations', you learn to use set notation, relations and functions. You work with exam-style questions and written reasoning. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Sets and relations' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use set notation, relations and functions. Start with one easy question, then one with a real-life story.
• Check it: use a second method, estimate, or work backwards to test your answer.

Step by step.

1. Step 1, Cartesian product: A = {1, 2, 3} and B = {a, b}. A x B has 3 x 2 = 6 ordered pairs: (1, a), (1, b), (2, a), (2, b), (3, a), (3, b). Order matters, so (1, a) differs from (a, 1).
2. Step 2, Relation: R = {(x, y): y = x + 2} from {1, 2, 3, 4} to {3, 4, 5, 6}. Pairs: (1, 3), (2, 4), (3, 5), (4, 6). Domain {1, 2, 3, 4}, range {3, 4, 5, 6}.
3. Step 3, Venn counting: n(A) = 30, n(B) = 25, n(A n B) = 10. So n(A U B) = 30 + 25 - 10 = 45. Only A is 30 - 10 = 20, and only B is 25 - 10 = 15.
4. Step 4, Domain and range: f(x) = root of (x - 3) needs x - 3 >= 0, so the domain is x >= 3 and the range is y >= 0. For f(x) = x^2 + 1 the range is y >= 1.
5. Step 5, Power set: for {a, b, c}, the subsets are 2^3 = 8 in number: empty, 3 singletons, 3 pairs and the whole set. A set with n elements has 2^n subsets.
6. Step 6, Your turn: if n(A) = 4 and n(B) = 5 find n(A x B), and give the domain of 1/(x - 5). Answer: 20; all real numbers except 5.

Activity. About 40 to 50 minutes of class work, then one short practice task for home. Keep evidence: a timed paper and an error log.

Standard to meet. Use set notation, relations and functions: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Trigonometric functions', you learn to use identities and graphs of trigonometric functions. You work with exam-style questions and written reasoning. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Trigonometric functions' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use identities and graphs of trigonometric functions. Start with one easy question, then one with a real-life story.
• Check it: use a second method, estimate, or work backwards to test your answer.

Step by step.

1. Step 1, Radians: 150 degrees = 150 x pi/180 = 5 pi/6. And pi/4 radians = 45 degrees. The conversion factor is pi radians = 180 degrees.
2. Step 2, Identity: if sin x = 3/5 with x in quadrant II, then cos^2 x = 1 - 9/25 = 16/25, so cos x = -4/5 since cosine is negative there. Then tan x = -3/4.
3. Step 3, Compound angle: sin 75 = sin(45 + 30) = sin 45 cos 30 + cos 45 sin 30 = (root 2/2)(root 3/2) + (root 2/2)(1/2) = (root 6 + root 2)/4, about 0.966.
4. Step 4, Solve sin x = 1/2 for x in [0, 2 pi]: the principal angle is pi/6, and the second solution is pi - pi/6 = 5 pi/6. Check: sin(5 pi/6) = 1/2.
5. Step 5, Graphs: y = 2 sin x has amplitude 2 and period 2 pi, with maximum 2 at x = pi/2. y = sin 2x has period pi, so it completes one wave between 0 and pi.
6. Step 6, Your turn: find cos 2x when cos x = 3/5. Answer: cos 2x = 2 cos^2 x - 1 = 2 x 9/25 - 1 = -7/25.

Activity. About 40 to 50 minutes of class work, then one short practice task for home. Keep evidence: a timed paper and an error log.

Standard to meet. Use identities and graphs of trigonometric functions: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Principle of mathematical induction', you learn to prove statements by induction. You work with exam-style questions and written reasoning. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Principle of mathematical induction' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: prove statements by induction. Start with one easy question, then one with a real-life story.
• Check it: use a second method, estimate, or work backwards to test your answer.

Step by step.

1. Step 1, State what to prove: P(n) is 1 + 2 + ... + n = n(n + 1)/2 for every natural number n. Induction works like dominoes. Show the first falls (base case), then show that if one falls the next falls (inductive step). Then all fall.
2. Step 2, Base case n = 1. The left side is just 1. The right side is 1 x (1 + 1) / 2 = 1. Both sides agree, so P(1) is true. Check n = 2 as well: 1 + 2 = 3 and 2 x 3 / 2 = 3.
3. Step 3, Inductive hypothesis: assume P(k) is true, so 1 + 2 + ... + k = k(k + 1)/2. Now add the next term k + 1 to both sides. The right side becomes k(k + 1)/2 + (k + 1).
4. Step 4, Simplify: k(k + 1)/2 + (k + 1) = (k + 1)(k/2 + 1) = (k + 1)(k + 2)/2. This is exactly P(k + 1), since replacing n by k + 1 gives (k + 1)(k + 2)/2. By the principle of induction, P(n) holds for all natural n. For n = 10 the sum is 55.
5. Step 5, Try a divisibility statement: 7^n - 1 is divisible by 6. Base: 7 - 1 = 6. Assume 7^k - 1 = 6m. Then 7^(k+1) - 1 = 7(7^k - 1) + 6 = 42m + 6 = 6(7m + 1), which is a multiple of 6.
6. Step 6, Your turn: prove 1^2 + 2^2 + ... + n^2 = n(n + 1)(2n + 1)/6. Model answer: base n = 1 gives 1 = 1 x 2 x 3/6. Assume true for k, add (k + 1)^2 to get (k + 1)(2k^2 + 7k + 6)/6 = (k + 1)(k + 2)(2k + 3)/6, which is P(k + 1).

Activity. About 40 to 50 minutes of class work, then one short practice task for home. Keep evidence: a timed paper and an error log.

Standard to meet. Prove statements by induction: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Complex numbers and quadratic equations', you learn to operate on complex numbers and solve quadratics. You work with exam-style questions and written reasoning. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Complex numbers and quadratic equations' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: operate on complex numbers and solve quadratics. Start with one easy question, then one with a real-life story.
• Check it: use a second method, estimate, or work backwards to test your answer.

Step by step.

1. Step 1, Divide complex numbers: (3 + 4i)/(1 - 2i). Multiply top and bottom by the conjugate 1 + 2i: (3 + 4i)(1 + 2i) / 5 = (3 + 6i + 4i + 8i^2)/5 = (-5 + 10i)/5 = -1 + 2i.
2. Step 2, Check: (-1 + 2i)(1 - 2i) = -1 + 2i + 2i - 4i^2 = -1 + 4i + 4 = 3 + 4i. This returns the numerator, so the quotient -1 + 2i is correct.
3. Step 3, Polar form: z = 1 + i has modulus root 2 and argument pi/4, since tan = 1 in the first quadrant. So z = root 2 (cos pi/4 + i sin pi/4). Also |3 - 4i| = 5.
4. Step 4, Quadratic: x^2 - 2x + 5 = 0 has D = 4 - 20 = -16. So x = (2 plus or minus 4i)/2 = 1 plus or minus 2i. The roots are conjugates, as always with real coefficients.
5. Step 5, Cube roots of unity: x^2 + x + 1 = 0 gives D = 1 - 4 = -3, so x = (-1 plus or minus i root 3)/2. These are the complex cube roots of 1, and their sum with 1 is 0.
6. Step 6, Your turn: find (2 + i)^2 and the modulus of 5 + 12i. Answer: 4 + 4i + i^2 = 3 + 4i; modulus is the root of 25 + 144 = 13.

Activity. About 40 to 50 minutes of class work, then one short practice task for home. Keep evidence: a timed paper and an error log.

Standard to meet. Operate on complex numbers and solve quadratics: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Linear inequalities and permutations and combinations', you learn to solve inequalities and count arrangements. You work with exam-style questions and written reasoning. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Linear inequalities and permutations and combinations' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: solve inequalities and count arrangements. Start with one easy question, then one with a real-life story.
• Check it: use a second method, estimate, or work backwards to test your answer.

Step by step.

1. Step 1, Solve 3x - 5 < 7: add 5 to get 3x < 12, so x < 4. On a number line, draw an open circle at 4 and shade left. Test x = 3: 9 - 5 = 4 < 7, true.
2. Step 2, Double inequality: -3 <= 2x + 1 < 9. Subtract 1: -4 <= 2x < 8. Divide by 2: -2 <= x < 4. So x is in [-2, 4). Test x = 0: 1 lies between -3 and 9.
3. Step 3, Flip rule: 4 - 3x >= 10 gives -3x >= 6. Dividing by -3 reverses the sign, so x <= -2. Test x = -3: 4 + 9 = 13 >= 10, true.
4. Step 4, Arrangements: 5 people in a line can be placed in 5! = 120 ways. If 2 specific people must stand together, treat them as one block: 4! x 2! = 24 x 2 = 48 ways.
5. Step 5, Selection: choose 4 from 6 boys and 4 girls with exactly 2 girls. C(4, 2) x C(6, 2) = 6 x 15 = 90. BANANA has 6 letters with A three times and N twice, so 6!/(3! 2!) = 60.
6. Step 6, Your turn: solve 5x + 2 <= 17 and find C(9, 3). Answer: 5x <= 15 so x <= 3; C(9, 3) = 9 x 8 x 7 / 6 = 84.

Activity. About 40 to 50 minutes of class work, then one short practice task for home. Keep evidence: a timed paper and an error log.

Standard to meet. Solve inequalities and count arrangements: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Binomial theorem and sequences and series', you learn to expand binomials and sum series. You work with exam-style questions and written reasoning. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Binomial theorem and sequences and series' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: expand binomials and sum series. Start with one easy question, then one with a real-life story.
• Check it: use a second method, estimate, or work backwards to test your answer.

Step by step.

1. Step 1, Expand (1 + x)^4 using Pascal's row 1, 4, 6, 4, 1: 1 + 4x + 6x^2 + 4x^3 + x^4. Check x = 1: 16 and 1 + 4 + 6 + 4 + 1 = 16.
2. Step 2, General term: T(r+1) = C(n, r) a^(n-r) b^r. In (2 + x)^5 the coefficient of x^3 is C(5, 3) x 2^2 = 10 x 4 = 40. The third term overall has C(5, 2) = 10 as its binomial coefficient.
3. Step 3, AP sum: a = 2, d = 3, n = 15. S = 15/2 x (2 x 2 + 14 x 3) = 15/2 x 46 = 345. The 15th term is 2 + 42 = 44 and 15/2 x (2 + 44) = 345 agrees.
4. Step 4, GP sum: first term 3, ratio 2, 8 terms. S = 3(2^8 - 1)/(2 - 1) = 3 x 255 = 765. Check the 8th term: 3 x 128 = 384, and 765 is almost double 384, as expected for ratio 2.
5. Step 5, Infinite GP: 1 + 1/2 + 1/4 + ... has ratio 1/2, with |r| < 1, so the sum is a/(1 - r) = 1/(1/2) = 2. When |r| is 1 or more, no such sum exists.
6. Step 6, Your turn: find the x^2 coefficient in (1 + 2x)^3 and the sum of 6, 2, 2/3, ... to infinity. Answer: 3 x 4 = 12; and 6/(1 - 1/3) = 9.

Activity. About 40 to 50 minutes of class work, then one short practice task for home. Keep evidence: a timed paper and an error log.

Standard to meet. Expand binomials and sum series: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Straight lines and conic sections', you learn to use equations of lines, circles and conics. You work with exam-style questions and written reasoning. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Straight lines and conic sections' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use equations of lines, circles and conics. Start with one easy question, then one with a real-life story.
• Check it: use a second method, estimate, or work backwards to test your answer.

Step by step.

1. Step 1, Line through (2, -1) and (5, 5): slope = 6/3 = 2. Equation y + 1 = 2(x - 2), so y = 2x - 5. Check (5, 5): 10 - 5 = 5.
2. Step 2, Distance from a point to a line: the distance from (1, 2) to 3x + 4y - 5 = 0 is |3 + 8 - 5| / root of (9 + 16) = 6/5 = 1.2.
3. Step 3, Circle: centre (3, -2) and radius 5 gives (x - 3)^2 + (y + 2)^2 = 25. Test (6, 2): 9 + 16 = 25, so it lies on the circle.
4. Step 4, Find centre from a general equation: x^2 + y^2 - 6x + 4y - 12 = 0. Complete squares: (x - 3)^2 + (y + 2)^2 = 12 + 9 + 4 = 25. Centre (3, -2), radius 5.
5. Step 5, Parabola y^2 = 8x: 4a = 8 so a = 2, focus (2, 0), directrix x = -2. Ellipse x^2/25 + y^2/9 = 1: a = 5, b = 3, c = root of 25 - 9 = 4, foci (plus or minus 4, 0), e = 4/5.
6. Step 6, Your turn: find the focus of x^2 = 12y and the radius of x^2 + y^2 = 49. Answer: 4a = 12, a = 3, focus (0, 3); radius 7.

Activity. About 40 to 50 minutes of class work, then one short practice task for home. Keep evidence: a timed paper and an error log.

Standard to meet. Use equations of lines, circles and conics: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Limits and derivatives, introduction', you learn to find simple limits and derivatives. You work with exam-style questions and written reasoning. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Limits and derivatives, introduction' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find simple limits and derivatives. Start with one easy question, then one with a real-life story.
• Check it: use a second method, estimate, or work backwards to test your answer.

Step by step.

1. Step 1, Limit by factorising: lim as x approaches 2 of (x^2 - 4)/(x - 2) = lim (x + 2) = 4. Direct substitution gives 0/0, which signals the need to simplify first.
2. Step 2, Standard limit: lim as x approaches 0 of sin 3x / x = 3 x (sin 3x / 3x) = 3 x 1 = 3. Use the fact that sin t / t tends to 1 as t tends to 0.
3. Step 3, Power rule: d/dx of 5x^4 - 3x^2 + 7 = 20x^3 - 6x. The slope of y = x^3 at x = 2 is 3 x 4 = 12. The constant 7 differentiates to 0.
4. Step 4, First principles for f(x) = x^2: [(x + h)^2 - x^2]/h = (2xh + h^2)/h = 2x + h, which tends to 2x as h tends to 0. This confirms the power rule.
5. Step 5, Product rule: y = x^2 sin x. y' = 2x sin x + x^2 cos x. At x = pi/2 this is 2 x (pi/2) x 1 + 0 = pi. Compare with the quotient rule when dividing instead.
6. Step 6, Your turn: find the derivative of 3x^2 + 4x - 5 at x = 1, and lim of (x^2 - 1)/(x - 1) as x approaches 1. Answer: 6x + 4 = 10; the limit is x + 1 = 2.

Activity. About 40 to 50 minutes of class work, then one short practice task for home. Keep evidence: a timed paper and an error log.

Standard to meet. Find simple limits and derivatives: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Statistics and probability', you learn to find variance and calculate probability. You work with exam-style questions and written reasoning. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Statistics and probability' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find variance and calculate probability. Start with one easy question, then one with a real-life story.
• Check it: use a second method, estimate, or work backwards to test your answer.

Step by step.

1. Step 1, Variance: data 2, 4, 4, 4, 5, 5, 7, 9 has mean 40/8 = 5. Deviations are -3, -1, -1, -1, 0, 0, 2, 4. Squares add: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
2. Step 2, Variance = 32/8 = 4 and standard deviation = 2. So most values lie within 2 of the mean 5, between 3 and 7. Six of the eight values do, which is consistent.
3. Step 3, Probability of at least one head in 3 tosses: P = 1 - P(no head) = 1 - (1/2)^3 = 1 - 1/8 = 7/8. The complement method is quicker than listing 7 outcomes.
4. Step 4, Two dice: the sum 8 occurs as (2,6), (3,5), (4,4), (5,3), (6,2), five ways out of 36, so P = 5/36. Sum at most 4 has 6 ways: P = 6/36 = 1/6.
5. Step 5, Exclusive and independent: for P(A) = 0.3 and P(B) = 0.5, if exclusive then P(A or B) = 0.8. If independent then P(A and B) = 0.15 and P(A or B) = 0.65.
6. Step 6, Your turn: find the variance of 1, 3, 5, 7, and P(sum 9) on two dice. Answer: mean 4, squares 9, 1, 1, 9 give 20/4 = 5; sum 9 has 4 ways, so 4/36 = 1/9.

Activity. About 40 to 50 minutes of class work, then one short practice task for home. Keep evidence: a timed paper and an error log.

Standard to meet. Find variance and calculate probability: at least 70 percent on an exam-style task, with the method shown.

On the slide: L1 Basic · Count and name shapes

Notes: Count objects up to 20 and name a circle, square and triangle. Notes: give ten buttons and two trays; learners count aloud, move three across and say how many are left (7), then hunt the room for one shape of each kind.

On the slide: L2 Awareness · Add, subtract and measure

Notes: Add and subtract within 100 and measure a length with a ruler. Notes: work 46 + 27 = 73 on a number line, jumping 20 then 7; learners measure a pencil in centimetres and explain how they checked their answer a second way.

On the slide: L3 Foundation · Fractions and decimals

Notes: Compare fractions and decimals and use the four operations with whole numbers. Notes: shade 3/4 and 6/8 on equal strips to show they match, then add 0.5 + 0.25 = 0.75 and write it as 75 per cent of one whole.

On the slide: L4 Practitioner · Equations and percentages

Notes: Solve one-step and two-step equations and percentage problems, showing each step. Notes: practice case is 3x + 5 = 20, so x = 5, checked by substituting back; then a 15 per cent discount on 400 leaves 340, because 400 x 0.85 = 340.

On the slide: L5 Specialist · Geometry, data and graphs

Notes: Use the right-angled triangle rule, the mean and a straight-line graph to solve a problem. Notes: legs of 6 and 8 give a longest side of 10; the data 4, 6, 6, 8, 11 have mean 7; plot y = 2x + 1 and read y = 7 at x = 3.

This course: Grade 11 Science Mathematics is at this level. Its units include Sets and relations; Trigonometric functions; Principle of mathematical induction; Complex numbers and quadratic equations and more.

On the slide: L6 Advanced · Proof, calculus and algebra

Notes: Prove results and use calculus and matrices to model change. Notes: differentiate f(x) = x^3 - 3x to get 3x^2 - 3, so turning points sit at x = 1 and x = -1; the second derivative 6x shows a minimum then a maximum. Learners prove the square root of 2 is irrational.

On the slide: L7 Super expert · Research-level modelling

Notes: Formulate and defend a proof or statistical model, stating assumptions and limits. Notes: for 25 values with mean 50 and standard deviation 10, the standard error is 2 and a 95 per cent interval is about 45.9 to 54.1 (t = 2.064). Learners check residuals and write a referee-style critique.