Grade 11 Commerce: Mathematics

In this unit, 'Sets, relations and functions', you learn to use set notation 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, relations and functions' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use set notation 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, Set operations: A = {1, 2, 3, 4, 5} and B = {2, 4, 6}. Union A U B = {1, 2, 3, 4, 5, 6}, intersection = {2, 4}, difference A - B = {1, 3, 5}. List each element once.
2. Step 2, Count with the formula n(A U B) = n(A) + n(B) - n(A n B) = 5 + 3 - 2 = 6. This agrees with the six elements listed. The formula avoids counting the overlap twice.
3. Step 3, Survey problem: of 60 students, 35 like tea, 28 like coffee and 10 like both. Either drink: 35 + 28 - 10 = 53. So 60 - 53 = 7 like neither.
4. Step 4, Function values: f(x) = 2x + 3 on the domain {1, 2, 3} gives f(1) = 5, f(2) = 7, f(3) = 9, so the range is {5, 7, 9}. Each input has exactly one output, so f is a function.
5. Step 5, Domain: for f(x) = 1/(x - 2), the denominator must not be zero, so the domain is all real numbers except 2. And f(x) = x^2 - 1 gives f(-2) = 4 - 1 = 3.
6. Step 6, Your turn: if n(A) = 20, n(B) = 15 and n(A U B) = 28, find n(A n B), and find f(4) for f(x) = 3x - 1. Answer: 20 + 15 - 28 = 7; f(4) = 11.

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 and functions: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Trigonometry and sequences', you learn to use sequences, series and basic trigonometry. 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 'Trigonometry and sequences' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use sequences, series and basic trigonometry. 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, Arithmetic progression: a = 5 and d = 3. The 8th term is a + 7d = 5 + 21 = 26. The sum of 8 terms is 8/2 x (5 + 26) = 4 x 31 = 124.
2. Step 2, Geometric progression: 2, 6, 18, ... has a = 2 and r = 3. The 5th term = 2 x 3^4 = 162. Sum of 5 terms = 2(3^5 - 1)/(3 - 1) = 242. Check: 2 + 6 + 18 + 54 + 162 = 242.
3. Step 3, Business use: 10,000 invested at 10 percent compound interest grows as a GP with ratio 1.1. After 3 years the amount is 10,000 x 1.1^3 = 13,310.
4. Step 4, Trigonometry: if tan theta = 3/4 in the first quadrant, the hypotenuse is 5, so sin theta = 3/5 and cos theta = 4/5. Convert degrees to radians with 180 degrees = pi, so 60 degrees = pi/3.
5. Step 5, Check the GP sum formula with a short case: 3 terms of 2, 6, 18 sum to 2(27 - 1)/2 = 26, and 2 + 6 + 18 = 26. Matching small cases build trust in the formula.
6. Step 6, Your turn: find the sum of the first 6 terms of the GP 3, 6, 12, ... Answer: S_6 = 3(2^6 - 1)/(2 - 1) = 3 x 63 = 189.

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 sequences, series and basic trigonometry: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Complex numbers and quadratic equations', you learn to solve quadratics and work with complex numbers. 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: solve quadratics and work with complex numbers. 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, Add and multiply complex numbers: (3 + 2i) + (1 - 4i) = 4 - 2i. And (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i^2 = 2 + i + 3 = 5 + i, using i^2 = -1.
2. Step 2, Conjugate and modulus: the conjugate of 3 - 4i is 3 + 4i. The modulus of 3 - 4i is the root of (9 + 16) = root of 25 = 5. Also (3 - 4i)(3 + 4i) = 9 + 16 = 25, which equals the modulus squared.
3. Step 3, Quadratic with complex roots: x^2 + 2x + 5 = 0 has discriminant 4 - 20 = -16. So x = (-2 plus or minus 4i)/2 = -1 plus or minus 2i.
4. Step 4, Check: substitute x = -1 + 2i. x^2 = 1 - 4i + 4i^2 = -3 - 4i. Then x^2 + 2x + 5 = (-3 - 4i) + (-2 + 4i) + 5 = 0. The root works.
5. Step 5, Real roots: x^2 - 5x + 6 = 0 factorises as (x - 2)(x - 3), giving x = 2 or 3 with discriminant 25 - 24 = 1 > 0. A positive discriminant gives real roots, a negative one gives complex roots.
6. Step 6, Your turn: find (4 + i)(4 - i) and solve x^2 - 2x + 10 = 0. Answer: 16 + 1 = 17; discriminant 4 - 40 = -36, so x = 1 plus or minus 3i.

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 quadratics and work with complex numbers: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Permutations, combinations and the binomial theorem', you learn to 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 'Permutations, combinations and the binomial theorem' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: 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, Factorial and arrangements: 5! = 5 x 4 x 3 x 2 x 1 = 120. The 5 distinct letters of MATHS can be arranged in 120 ways. For LEVEL, with E twice and L twice, it is 5!/(2! x 2!) = 120/4 = 30.
2. Step 2, Permutations: P(7, 2) = 7 x 6 = 42 ways to choose a president and a secretary from 7 people, where order matters. Combination: C(8, 3) = 8 x 7 x 6 / 6 = 56 where order does not matter.
3. Step 3, Committee problem: choose 3 from 5 men and 4 women with 2 men and 1 woman. C(5, 2) x C(4, 1) = 10 x 4 = 40 ways. Multiply the independent choices.
4. Step 4, Binomial theorem: (x + 2)^4 = x^4 + 4x^3 x 2 + 6x^2 x 4 + 4x x 8 + 16 = x^4 + 8x^3 + 24x^2 + 32x + 16. Check with x = 1: 3^4 = 81 and 1 + 8 + 24 + 32 + 16 = 81.
5. Step 5, Check C(8, 3) = C(8, 5) = 56. Choosing 3 to take is the same as choosing 5 to leave. This symmetry helps with larger numbers, e.g. C(10, 8) = C(10, 2) = 45.
6. Step 6, Your turn: find C(10, 2) and the number of 4-digit codes from digits 1 to 6 with no repeats. Answer: 45; and P(6, 4) = 6 x 5 x 4 x 3 = 360.

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. Count arrangements: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Straight lines and introduction to limits', you learn to use line equations and simple limits. 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 introduction to limits' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use line equations and simple limits. 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, Equation of a line: through (1, 2) and (3, 8) the slope is (8 - 2)/(3 - 1) = 3. Using point-slope, y - 2 = 3(x - 1), so y = 3x - 1. Check with (3, 8): 9 - 1 = 8.
2. Step 2, Parallel and perpendicular: the line 2x + 3y = 6 has slope -2/3. A parallel line has the same slope. A perpendicular line has slope 3/2 because (-2/3) x (3/2) = -1.
3. Step 3, Intercept form: the line x/4 + y/3 = 1 meets the axes at (4, 0) and (0, 3). Multiply by 12 to get 3x + 4y = 12. Check (4, 0): 12 = 12.
4. Step 4, Limit by factorising: lim as x approaches 2 of (x^2 - 4)/(x - 2). Direct substitution gives 0/0, so factorise: (x - 2)(x + 2)/(x - 2) = x + 2, which tends to 4.
5. Step 5, Simple limit: lim as x approaches 3 of (2x + 1) = 7 by substitution. Business use: cost C = 50 + 8x is a line with fixed cost 50 and slope 8 per unit, so 20 units cost 210.
6. Step 6, Your turn: write the line through (0, -1) with slope 2, and find the limit of (x^2 - 1)/(x - 1) as x approaches 1. Answer: y = 2x - 1; 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. Use line equations and simple limits: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Statistics and probability', you learn to find mean, variance and 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 mean, variance and 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, Mean and deviations: data 4, 8, 6, 10, 12 has mean 40/5 = 8. Deviations are -4, 0, -2, 2, 4. Their squares are 16, 0, 4, 4, 16, which add up to 40.
2. Step 2, Variance = 40/5 = 8 and standard deviation = root 8 = about 2.83. Check: the sizes of the deviations are 4, 0, 2, 2, 4, and a typical size near 2.83 is believable for this spread.
3. Step 3, Addition rule: P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. Then P(A or B) = 0.5 + 0.4 - 0.2 = 0.7. Subtracting the overlap avoids double counting.
4. Step 4, Two dice: a sum of 7 happens in 6 ways, (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), out of 36. So P = 6/36 = 1/6.
5. Step 5, Independent events: if P(A) = 0.6 and P(B) = 0.3 are independent, then P(A and B) = 0.6 x 0.3 = 0.18, and P(A or B) = 0.6 + 0.3 - 0.18 = 0.72.
6. Step 6, Your turn: find the variance of 2, 4, 6, 8. Answer: mean 5, squared deviations 9, 1, 1, 9 sum to 20, so variance 20/4 = 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. Find mean, variance and 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 Commerce Mathematics is at this level. Its units include Sets, relations and functions; Trigonometry and sequences; Complex numbers and quadratic equations; Permutations, combinations and the binomial theorem 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.