Grade 08: Mathematics

In this unit, 'Rational numbers', you learn to apply properties and represent them on a number line. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Rational numbers' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: apply properties and represent them on a number line. 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, Test closure. Adding two rationals gives a rational, as 1/2 + 1/3 = 5/6. Subtraction and multiplication also stay rational. Division is not closed, because dividing 1/2 by 0 is undefined. So rationals are closed under addition, subtraction and multiplication.
2. Step 2, Test commutativity. Addition: 1/2 + 1/3 = 5/6 and 1/3 + 1/2 = 5/6, so order does not matter. Subtraction is different: 5 - 2 = 3 but 2 - 5 = -3. So addition and multiplication are commutative, while subtraction and division are not.
3. Step 3, Test associativity and distributivity. (1/2 + 1/3) + 1/4 = 5/6 + 1/4 = 13/12, and 1/2 + (1/3 + 1/4) = 1/2 + 7/12 = 13/12. Also -2/3 x (3/4 + 1/2) = -2/3 x 5/4 = -5/6, and -1/2 + (-1/3) = -5/6.
4. Step 4, Find identities and inverses. The additive identity is 0, and the additive inverse of -7/9 is 7/9 since their sum is 0. The multiplicative identity is 1, and the reciprocal of -3/5 is -5/3. Zero has no reciprocal.
5. Step 5, Represent on a number line. To mark -3/4, divide the unit between 0 and -1 into 4 equal parts and move 3 parts left of zero. To find a rational between 1/4 and 1/2, take the mean: (1/4 + 1/2) / 2 = 3/8. There are infinitely many between any two.
6. Step 6, Your turn: find a rational number between 1/5 and 1/3, and say where -5/4 lies on a number line. Answer: the mean is (3/15 + 5/15) / 2 = 4/15, and -5/4 lies one quarter beyond -1, between -1 and -2.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Apply properties and represent them on a number line: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Linear equations in one variable', you learn to solve equations with variables on both sides. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Linear equations in one variable' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: solve equations with variables on both sides. 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 5x - 7 = 3x + 9. Move 3x to the left by subtracting it from both sides: 2x - 7 = 9. Add 7: 2x = 16. Divide by 2: x = 8.
2. Step 2, Check x = 8: left side 5 x 8 - 7 = 33, right side 3 x 8 + 9 = 33. Both sides equal 33, so x = 8 is correct. Always substitute into the original equation.
3. Step 3, With brackets: 3(x + 4) = 2x + 19. Expand the left side: 3x + 12 = 2x + 19. Subtract 2x: x + 12 = 19, so x = 7. Check: 3 x 11 = 33 and 2 x 7 + 19 = 33.
4. Step 4, With fractions: x/3 + 2 = x/2 - 1. Multiply every term by the LCM 6: 2x + 12 = 3x - 6. Then 12 + 6 = 3x - 2x, so x = 18. Check: 6 + 2 = 8 and 9 - 1 = 8.
5. Step 5, Word problem: a father is 3 times as old as his son. After 10 years he will be twice as old. With son's age x: 3x + 10 = 2(x + 10), so x = 10. Son 10, father 30. Check: 40 = 2 x 20.
6. Step 6, Your turn: solve 7x + 4 = 3x + 28. Answer: 4x = 24, so x = 6. Check: 7 x 6 + 4 = 46 and 3 x 6 + 28 = 46.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Solve equations with variables on both sides: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Understanding quadrilaterals', you learn to use angle sum and properties of parallelograms and special quadrilaterals. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Understanding quadrilaterals' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use angle sum and properties of parallelograms and special quadrilaterals. 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, Angle sum: a quadrilateral has angles 85, 95, 110 and x. The four angles add to 360, so x = 360 - 290 = 70 degrees. The sum 360 comes from splitting it into two triangles.
2. Step 2, Parallelogram angles: if angle A = 65, then opposite angle C = 65, and each adjacent angle is 180 - 65 = 115, so B = D = 115. Check: 65 + 115 + 65 + 115 = 360.
3. Step 3, Parallelogram diagonals: they bisect each other. If AC = 18 cm and BD = 14 cm, they meet at O with AO = 9 cm and BO = 7 cm. Opposite sides are equal and parallel.
4. Step 4, Rhombus: its diagonals are perpendicular and bisect each other. With diagonals 16 and 12, each half is 8 and 6, so a side is the root of 64 + 36 = 100, which is 10 cm. Perimeter 40 cm.
5. Step 5, Rectangle test: a rectangle's diagonals are equal. A parallelogram with sides 6 and 8 and diagonals both 10 is a rectangle, since 36 + 64 = 100 gives a right angle at the corner.
6. Step 6, Your turn: adjacent angles of a parallelogram are (2x + 10) and (3x - 20). Find x and the angles. Answer: 5x - 10 = 180, x = 38; the angles are 86 and 94.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Use angle sum and properties of parallelograms and special quadrilaterals: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Data handling', you learn to use pie charts, histograms and probability. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Data handling' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use pie charts, histograms 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, Pie chart angles: 600 students choose sports. Cricket 240, football 150, swimming 120, others 90. Cricket angle = 240 / 600 x 360 = 144 degrees. Football = 150 / 600 x 360 = 90.
2. Step 2, Finish and check: swimming = 120 / 600 x 360 = 72 degrees and others = 90 / 600 x 360 = 54. Sum 144 + 90 + 72 + 54 = 360. The angles must always add up to 360.
3. Step 3, Histogram: marks grouped as 0-10, 10-20, 20-30, 30-40 with frequencies 4, 9, 12, 5. Bars touch because the classes are continuous. The tallest bar, 20-30, holds 12 of the 30 students.
4. Step 4, Probability: a bag has 5 red, 3 blue and 2 green balls, 10 in all. P(blue) = 3/10. P(not red) = 5/10 = 1/2. Probabilities of all outcomes add to 1: 5/10 + 3/10 + 2/10 = 1.
5. Step 5, Die: P(prime) when rolling one die counts 2, 3, 5, so 3/6 = 1/2. P(a number greater than 4) counts 5 and 6, so 2/6 = 1/3. Probability is favourable outcomes divided by total outcomes.
6. Step 6, Your turn: of 40 students, 10 prefer tea; find its pie-chart angle, and P(even number) on a die. Answer: 10 / 40 x 360 = 90 degrees; P(even) = 3/6 = 1/2.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Use pie charts, histograms and probability: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Squares, square roots, cubes and cube roots', you learn to find roots by factors and division. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Squares, square roots, cubes and cube roots' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find roots by factors and division. 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, Recall perfect squares: 1, 4, 9, 16, 25, 36 and so on, where 15 x 15 = 225. A number is a perfect square if its prime factors can be paired. A square never ends in 2, 3, 7 or 8, a fast test to rule numbers out.
2. Step 2, Find a square root by prime factors. Divide 324 by 2 twice to get 81, then 81 = 3 x 3 x 3 x 3, so 324 = 2^2 x 3^4. Take one from each pair: 2 x 3 x 3 = 18. Check: 18 x 18 = 324.
3. Step 3, Find a square root by division. For 529 pair the digits from the right as 5 | 29. The largest square under 5 is 4, so write 2, subtract to leave 1, and bring down 29 to make 129. Double 2 to get 4, and 43 x 3 = 129. So the root is 23.
4. Step 4, Find a cube root by factors. 1728 = 2^6 x 3^3. Group the factors in threes: (2 x 2 x 2) x (2 x 2 x 2) x (3 x 3 x 3). Take one from each triple: 2 x 2 x 3 = 12. Check: 12 x 12 x 12 = 1728.
5. Step 5, Make a perfect square or cube. 180 = 2^2 x 3^2 x 5, and 5 is unpaired, so multiply by 5 to get 900, which is 30 squared. And 54 = 2 x 3^3, so multiply by 4 to get 216, which is 6 cubed.
6. Step 6, Your turn: find the square root of 676 by factors, then the cube root of 3375. Answer: 676 = 2^2 x 13^2, so the root is 2 x 13 = 26. And 3375 = 3^3 x 5^3, so the cube root is 3 x 5 = 15.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Find roots by factors and division: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Comparing quantities', you learn to use compound interest, discount and tax. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Comparing quantities' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use compound interest, discount and tax. 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, Find a discount. A jacket has a marked price of 1200 with a 15% discount. Discount = 1200 x 15 / 100 = 180. Selling price = 1200 - 180 = 1020. A discount is always taken from the marked price.
2. Step 2, Add tax. A bill of 1000 has 18% tax added. Tax = 1000 x 18 / 100 = 180, so the total is 1180. To go backwards, if 840 includes 5% tax, divide by 1.05: 840 / 1.05 = 800 before tax.
3. Step 3, Find compound interest with A = P(1 + R/100)^n. Take P = 10000, R = 10% and n = 2 years. A = 10000 x 1.1 x 1.1 = 12100. Compound interest = 12100 - 10000 = 2100, while simple interest would only be 2000.
4. Step 4, Compound half-yearly. For 8000 at 10% a year for 1 year, the rate per half-year is 5% and there are 2 periods. A = 8000 x 1.05^2 = 8000 x 1.1025 = 8820. Interest = 8820 - 8000 = 820.
5. Step 5, Combine markup and discount. A trader buys for 800 and marks it up 25%, so the marked price is 1000. After a 10% discount the sale price is 900. Profit = 900 - 800 = 100, which is 100 / 800 = 12.5% of cost.
6. Step 6, Your turn: find the compound interest on 5000 at 8% a year for 2 years. Answer: A = 5000 x 1.08 x 1.08 = 5832, so the interest is 5832 - 5000 = 832.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Use compound interest, discount and tax: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Algebraic expressions and identities', you learn to multiply expressions and use standard identities. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Algebraic expressions and identities' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: multiply expressions and use standard identities. 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, Multiply two binomials: (2x + 3)(x - 4) = 2x(x) + 2x(-4) + 3(x) + 3(-4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12. Every term in one bracket meets every term in the other.
2. Step 2, Square of a sum: (3x + 2)^2 = (3x)^2 + 2(3x)(2) + 2^2 = 9x^2 + 12x + 4. Check with x = 1: 5^2 = 25 and 9 + 12 + 4 = 25.
3. Step 3, Square of a difference: (5y - 3)^2 = 25y^2 - 30y + 9. Check with y = 1: 2^2 = 4 and 25 - 30 + 9 = 4. The middle term takes a minus sign.
4. Step 4, Difference of squares: (a + b)(a - b) = a^2 - b^2. Use it for numbers: 102 x 98 = (100 + 2)(100 - 2) = 10,000 - 4 = 9,996. And 99^2 = (100 - 1)^2 = 10,000 - 200 + 1 = 9,801.
5. Step 5, Find a missing term: x^2 + 10x + 25 is (x + 5)^2 since 2 x x x 5 = 10x and 5^2 = 25. Check with x = 1: 36 = 6^2.
6. Step 6, Your turn: expand (x + 7)(x - 7) and (4m + 1)^2. Answer: x^2 - 49; and 16m^2 + 8m + 1.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Multiply expressions and use standard identities: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Mensuration', you learn to find surface area and volume of cubes, cuboids and cylinders. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Mensuration' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find surface area and volume of cubes, cuboids and cylinders. 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, Cube of side 5 cm: total surface area = 6 x side^2 = 6 x 25 = 150 cm2 and volume = 5^3 = 125 cm3. Six faces, each a square of area 25.
2. Step 2, Cuboid 8 cm by 5 cm by 4 cm: surface area = 2(lb + bh + hl) = 2(40 + 20 + 32) = 2 x 92 = 184 cm2. Volume = 8 x 5 x 4 = 160 cm3.
3. Step 3, Cylinder with radius 7 cm and height 10 cm, using pi = 22/7: curved surface = 2 x 22/7 x 7 x 10 = 440 cm2. Total surface = 2 x 22/7 x 7 x (7 + 10) = 44 x 17 = 748 cm2.
4. Step 4, Cylinder volume: pi r^2 h = 22/7 x 7 x 7 x 10 = 22 x 7 x 10 = 1,540 cm3. Check: the total surface 748 = 440 + 2 x 154 (two circular ends), and 440 + 308 = 748.
5. Step 5, Word problem: a cylindrical tank of radius 7 m and height 10 m holds 1,540 cubic metres. Since 1 cubic metre = 1,000 litres, it holds 1,540,000 litres.
6. Step 6, Your turn: find the volume and curved surface of a cylinder with radius 3.5 cm and height 8 cm. Answer: volume 22/7 x 12.25 x 8 = 308 cm3; curved surface 2 x 22/7 x 3.5 x 8 = 176 cm2.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Find surface area and volume of cubes, cuboids and cylinders: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Exponents and powers', you learn to use laws of exponents with negative powers. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Exponents and powers' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use laws of exponents with negative powers. 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, Meet a negative exponent. a^-n means 1 / a^n. So 2^-3 = 1 / 2^3 = 1/8, and 10^-2 = 1/100 = 0.01. A negative power does not make the number negative; it means a reciprocal.
2. Step 2, Add exponents when multiplying, even if one is negative. 3^-2 x 3^5 = 3^(-2 + 5) = 3^3 = 27. Check: 1/9 x 243 = 27. The rule a^m x a^n = a^(m + n) holds for all integers m and n.
3. Step 3, Handle a fraction raised to a negative power by flipping it. (2/3)^-2 = (3/2)^2 = 9/4. The reciprocal turns the power positive. Compute the numerator and denominator separately: 3^2 = 9 and 2^2 = 4.
4. Step 4, Subtract exponents when dividing. 5^-3 / 5^-5 = 5^(-3 - (-5)) = 5^2 = 25. Be careful with the double negative, which becomes plus. Check: 1/125 divided by 1/3125 = 3125 / 125 = 25.
5. Step 5, Simplify (2^-1 + 3^-1)^-1. First, 2^-1 = 1/2 and 3^-1 = 1/3, so the bracket is 5/6. A power of -1 flips it, giving 6/5. Standard form with negatives: 0.0000052 = 5.2 x 10^-6, moving the point 6 places right.
6. Step 6, Your turn: evaluate 4^-2 x 4^4 and write 0.00071 in standard form. Answer: 4^(-2 + 4) = 4^2 = 16, and 0.00071 = 7.1 x 10^-4.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Use laws of exponents with negative powers: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Direct and inverse proportion', you learn to solve problems with proportion. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Direct and inverse proportion' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: solve problems with proportion. 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, Spot direct proportion. If 3 notebooks cost 36 coins and 5 cost 60 coins, then cost divided by number is 12 each time. When one quantity rises, the other rises by the same factor, so y / x = k stays constant.
2. Step 2, Solve a direct problem. A car travels 240 km in 3 hours at a steady speed. Speed = 240 / 3 = 80 km/h. In 5 hours it travels 80 x 5 = 400 km. Or use a proportion: 240 / 3 = x / 5, so x = 400.
3. Step 3, Spot inverse proportion. If more workers do the job, it takes fewer days. Six workers need 12 days, so the work is 6 x 12 = 72 worker-days. With 9 workers the time is 72 / 9 = 8 days. Here x times y stays constant.
4. Step 4, Decide which type a table shows. For x = 2, 4, 8 and y = 12, 6, 3 the products are all 24, so it is inverse. For x = 1, 2, 3 and y = 5, 10, 15 the ratios are all 5, so it is direct.
5. Step 5, Use a map scale, a direct proportion. If 2 cm on a map shows 5 km, then 1 cm shows 2.5 km. A 7.5 cm distance is 7.5 x 2.5 = 18.75 km. Always write the units on each side of the proportion.
6. Step 6, Your turn: 4 painters finish a wall in 9 days. How long would 6 painters take at the same rate? Answer: the work is 4 x 9 = 36 painter-days, and 36 / 6 = 6 days.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Solve problems with proportion: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Factorisation and graphs', you learn to factorise expressions and plot linear graphs. You work with notes, a worked line and a self-mark. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Factorisation and graphs' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: factorise expressions and plot linear graphs. 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, Common factor: 6x^2 + 9x. The HCF of 6 and 9 is 3 and both terms have x, so the common factor is 3x. Divide: 6x^2 / 3x = 2x and 9x / 3x = 3. Result: 3x(2x + 3).
2. Step 2, Factorise x^2 + 7x + 12: find two numbers that multiply to 12 and add to 7, which are 3 and 4. So (x + 3)(x + 4). Check by expanding: x^2 + 4x + 3x + 12 = x^2 + 7x + 12.
3. Step 3, Identities: x^2 - 49 = (x - 7)(x + 7) as a difference of squares. And 4x^2 - 12x + 9 = (2x - 3)^2, since 2 x 2x x 3 = 12x and 3^2 = 9.
4. Step 4, Plot y = 2x + 1. Table: x = -1 gives y = -1, x = 0 gives 1, x = 1 gives 3, x = 2 gives 5. Mark (-1, -1), (0, 1), (1, 3), (2, 5) and join them in a straight line.
5. Step 5, Read the graph: the line cuts the y-axis at 1 (the intercept) and rises 2 for each 1 across (slope 2). Check that the point (3, 7) lies on the line: 2 x 3 + 1 = 7.
6. Step 6, Your turn: factorise x^2 - 5x + 6 and find y for y = x - 2 at x = 3 and x = 0. Answer: (x - 2)(x - 3); and y = 1 at x = 3, and y = -2 at x = 0.

Activity. About 30 to 40 minutes of class work, then one short practice task for home. Keep evidence: a worked task and a corrected copy.

Standard to meet. Factorise expressions and plot linear graphs: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

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.

This course: Grade 08 Mathematics is at this level. Its units include Rational numbers; Linear equations in one variable; Understanding quadrilaterals; Data handling and more.

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.

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.