Grade 12 Commerce: Mathematics

In this unit, 'Relations, functions and matrices', you learn to operate on matrices 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 'Relations, functions and matrices' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: operate on matrices 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, Add matrices: A = [[1, 2], [3, 4]] and B = [[0, 1], [2, 3]]. A + B adds matching entries: [[1, 3], [5, 7]]. And 2A = [[2, 4], [6, 8]]. Same order is needed.
2. Step 2, Multiply: AB uses rows of A with columns of B. Entries: 1x0 + 2x2 = 4; 1x1 + 2x3 = 7; 3x0 + 4x2 = 8; 3x1 + 4x3 = 15. So AB = [[4, 7], [8, 15]].
3. Step 3, Order matters: BA = [[3, 4], [11, 16]], which differs from AB. For BA, entry (1,1) = 0x1 + 1x3 = 3. So matrix multiplication is not commutative.
4. Step 4, Transpose: the transpose of A swaps rows and columns, [[1, 3], [2, 4]]. Business use: a matrix of unit sales per shop times a column of prices gives total revenue per shop.
5. Step 5, Functions: f(x) = 2x + 1 is one-one since different x give different y. But f(x) = x^2 on all real numbers is not one-one because f(2) = f(-2) = 4.
6. Step 6, Your turn: multiply [[2, 0], [1, 3]] by [[1, 4], [2, 0]]. Answer: row 1 gives [2, 8], row 2 gives [1 + 6, 4 + 0] = [7, 4], so the product is [[2, 8], [7, 4]].

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

In this unit, 'Calculus: differentiation and integration', you learn to find derivatives and simple integrals. 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 'Calculus: differentiation and integration' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find derivatives and simple integrals. 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, Differentiate: d/dx of 3x^4 - 5x^2 + 2 = 12x^3 - 10x. A constant differentiates to zero and each power drops by one.
2. Step 2, Marginal cost: C(x) = 0.5x^2 + 20x + 100 has C'(x) = x + 20. At x = 10 units the marginal cost is 30, the extra cost of making the 11th unit, roughly.
3. Step 3, Maximise revenue: R(x) = 50x - x^2. R'(x) = 50 - 2x = 0 gives x = 25. R(25) = 1,250 - 625 = 625. Second derivative -2 is negative, so it is a maximum.
4. Step 4, Integrate: the integral of 3x^2 + 4x dx = x^3 + 2x^2 + C. Check by differentiating: 3x^2 + 4x, which returns the original expression.
5. Step 5, Definite integral: from 1 to 3 of 2x dx = [x^2] from 1 to 3 = 9 - 1 = 8. This is the area under y = 2x from x = 1 to x = 3, a trapezoid with parallel sides 2 and 6 and width 2: (2 + 6)/2 x 2 = 8.
6. Step 6, Your turn: differentiate 4x^3 - 2x + 7 and integrate 6x^2 - 2. Answer: 12x^2 - 2; and 2x^3 - 2x + C.

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

In this unit, 'Probability and linear programming', you learn to calculate probability and optimise. 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 'Probability and linear programming' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: calculate probability and optimise. 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, Conditional probability: P(A and B) = 0.2 and P(B) = 0.5, so P(A given B) = 0.2/0.5 = 0.4. The conditional probability uses only the outcomes in B.
2. Step 2, Linear programming set-up: maximise profit Z = 3x + 2y with x + y <= 4, x <= 3, x >= 0 and y >= 0. Draw the feasible region and list its corner points.
3. Step 3, Corners: (0, 0), (3, 0), (3, 1) where x = 3 meets x + y = 4, and (0, 4). Check (3, 1): 3 + 1 = 4 and x = 3, both lines hold. The feasible region is the quadrilateral joining them.
4. Step 4, Evaluate Z: at (0, 0) Z = 0; at (3, 0) Z = 9; at (3, 1) Z = 9 + 2 = 11; at (0, 4) Z = 8. The maximum is 11 at (3, 1). The best point is always a corner.
5. Step 5, Check feasibility: (3, 1) satisfies x + y = 4 <= 4 and x = 3 <= 3. A nearby interior point like (2, 1) gives Z = 8, which is less, confirming the corner is best.
6. Step 6, Your turn: maximise Z = 5x + 4y with x + y <= 6, x <= 4, x, y >= 0. Answer: corners (0,0), (4,0), (4,2), (0,6) give 0, 20, 28, 24, so the maximum is 28 at (4, 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. Calculate probability and optimise: at least 70 percent on an exam-style task, with the method shown.

In this unit, 'Index numbers and time series', you learn to compute index numbers and trends. 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 'Index numbers and time series' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: compute index numbers and trends. 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, Price relative: an item costs 40 in the base year and 50 now. Index = 50/40 x 100 = 125. So prices are 25 percent higher than the base year.
2. Step 2, Laspeyres index: base prices 10 and 20, base quantities 5 and 3, current prices 12 and 25. Sum of p1q0 = 12 x 5 + 25 x 3 = 60 + 75 = 135. Sum of p0q0 = 50 + 60 = 110.
3. Step 3, Index = 135/110 x 100 = 122.73. The basket that cost 110 would now cost 135, a rise of about 22.7 percent. It uses base-year quantities as weights.
4. Step 4, Moving average: yearly sales 20, 24, 28, 26, 30. Three-year averages are (20 + 24 + 28)/3 = 24, (24 + 28 + 26)/3 = 26 and (28 + 26 + 30)/3 = 28. They rise steadily, showing the trend.
5. Step 5, Check: the averages 24, 26, 28 rise by 2 each time, while the raw sales 20, 24, 28, 26, 30 dip at 26. So the moving average has smoothed out the short-term dip and shows the upward trend.
6. Step 6, Your turn: base prices 8 and 5, base quantities 2 and 4, current prices 10 and 6. Find the Laspeyres index. Answer: (20 + 24)/(16 + 20) x 100 = 44/36 x 100 = 122.22.

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. Compute index numbers and trends: 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 12 Commerce Mathematics is at this level. Its units include Relations, functions and matrices; Calculus: differentiation and integration; Probability and linear programming; Index numbers and time series.

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.