Grade 07: Mathematics

In this unit, 'Integers', you learn to add, subtract, multiply and divide integers and use properties. 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 'Integers' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: add, subtract, multiply and divide integers and use properties. 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 with signs: (-6) x 4. A positive times a negative is negative, so the answer is -24. Think of it as 4 jumps of 6 to the left. Then (-5) x (-3) = 15, since two negatives give a positive.
2. Step 2, Divide with signs: -36 / 9 = -4, because 9 x (-4) = -36. And -48 / (-6) = 8, because (-6) x 8 = -48. Same signs give a positive quotient, different signs give a negative.
3. Step 3, Check the distributive property: (-4) x (7 + 3) = (-4) x 10 = -40. Also (-4) x 7 + (-4) x 3 = -28 + (-12) = -40. Both sides agree, which confirms the property.
4. Step 4, Word problem: the temperature drops 3 degrees every hour for 5 hours, starting at 2 degrees. Change = 5 x (-3) = -15. Final = 2 + (-15) = -13 degrees.
5. Step 5, Mixed expression: (-3) x 4 + 10 / (-2). Multiply and divide first: -12 and -5. Then add: -12 + (-5) = -17. Check by estimating signs: both parts are negative so the sum is negative.
6. Step 6, Your turn: find (-7) x (-8), 45 / (-5), and (-2) x 6 - (-9). Answer: 56; -9; and -12 + 9 = -3.

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. Add, subtract, multiply and divide integers and use properties: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Fractions and decimals', you learn to multiply and divide fractions and decimals and solve problems. 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 'Fractions and decimals' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: multiply and divide fractions and decimals and solve problems. 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 fractions: 3/4 x 2/5 = 6/20 = 3/10 after dividing top and bottom by 2. Cancelling first also works: 2 and 4 share a factor of 2, leaving 3/2 x 1/5 = 3/10.
2. Step 2, Divide fractions: 2/3 divided by 4/5 means 2/3 x 5/4, the reciprocal flipped. That is 10/12 = 5/6. Check: 5/6 x 4/5 = 20/30 = 2/3, which is the start.
3. Step 3, Mixed numbers: 2 1/2 divided by 5/6. Write 5/2 x 6/5. The 5s cancel and 6/2 = 3, so the answer is 3. Estimate: 2.5 / 0.83 is about 3.
4. Step 4, Multiply decimals: 3.6 x 0.25. Do 36 x 25 = 900 and count 1 + 2 = 3 decimal places, so 0.900 = 0.9. Divide: 7.2 / 0.8 = 72 / 8 = 9 after shifting the points one place.
5. Step 5, Word problem: 12.6 kg of rice is packed in 0.9 kg bags. 12.6 / 0.9 = 126 / 9 = 14 bags. Check: 14 x 0.9 = 12.6, so no rice is left over.
6. Step 6, Your turn: find 3/5 divided by 9/10 and 6.3 divided by 0.7. Answer: 3/5 x 10/9 = 30/45 = 2/3; and 63 / 7 = 9.

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 and divide fractions and decimals and solve problems: 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 find mean, median, mode and range and read bar 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 'Data handling' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find mean, median, mode and range and read bar 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, Mean: 8 students scored 12, 7, 9, 12, 15, 10, 12, 11. Sum = 88, and 88 / 8 = 11. The mean is 11. Pair check: 19 + 21 + 25 + 23 = 88.
2. Step 2, Median: sort to 7, 9, 10, 11, 12, 12, 12, 15. With 8 values the median is the average of the 4th and 5th: (11 + 12) / 2 = 11.5.
3. Step 3, Mode and range: 12 appears three times, more than any other value, so the mode is 12. The range is highest minus lowest = 15 - 7 = 8.
4. Step 4, Read a bar graph: bars of height 6, 9, 4 and 11 show books read in four weeks. Total = 30, mean = 7.5, and the biggest difference between weeks is 11 - 4 = 7.
5. Step 5, Compare averages: mean 11 and median 11.5 are close, so the data has no big outliers. If one score were 40, the mean would jump while the median barely moves.
6. Step 6, Your turn: find mean, median, mode and range of 4, 9, 6, 9, 12. Answer: mean 40 / 5 = 8; median 9 (sorted 4, 6, 9, 9, 12); mode 9; range 12 - 4 = 8.

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 mean, median, mode and range and read bar graphs: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Simple equations', you learn to solve and form equations in one variable. 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 'Simple equations' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: solve and form equations in one variable. 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 = 20. Subtract 5 from both sides: 3x = 15. Divide both sides by 3: x = 5. Check: 3 x 5 + 5 = 20, so the equation balances.
2. Step 2, Solve x/4 - 2 = 3. Add 2 to both sides: x/4 = 5. Multiply both sides by 4: x = 20. Check: 20 / 4 - 2 = 5 - 2 = 3.
3. Step 3, Form an equation: the sum of two consecutive numbers is 41. Let the numbers be n and n + 1. Then 2n + 1 = 41, so 2n = 40 and n = 20. The numbers are 20 and 21.
4. Step 4, Perimeter problem: a rectangle's length is 4 more than its width w and its perimeter is 36. 2(w + w + 4) = 36, so 2w + 4 = 18, w = 7. Length is 11. Check: 2 x (7 + 11) = 36.
5. Step 5, Transposing: moving a term across the equals sign changes its sign. From 5y - 9 = 16 move -9 to get 5y = 25, so y = 5. This is the same as adding 9 to both sides.
6. Step 6, Your turn: solve 5x - 7 = 18, and form and solve: 4 times a number minus 6 equals 22. Answer: x = 5; and 4n - 6 = 22 gives 4n = 28, so n = 7.

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 and form equations in one variable: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Lines and angles', you learn to use angle pairs, parallel lines and transversals. 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 'Lines and angles' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use angle pairs, parallel lines and transversals. 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, Complementary and supplementary: complementary angles add to 90 and supplementary add to 180. The complement of 38 is 90 - 38 = 52. The supplement of 125 is 180 - 125 = 55.
2. Step 2, Linear pair: two angles on a straight line are (3x + 10) and (2x + 20). They add to 180, so 5x + 30 = 180 and x = 30. The angles are 100 and 80. Check: 100 + 80 = 180.
3. Step 3, Vertically opposite angles: when two lines cross, opposite angles are equal. If one angle is 70, the one opposite is 70 and the two neighbours are 180 - 70 = 110 each.
4. Step 4, Parallel lines and a transversal: if one angle is 70, its corresponding angle is 70 and its alternate interior angle is 70. The co-interior (same side) angle is 180 - 70 = 110.
5. Step 5, Find an unknown: lines are parallel and a transversal makes alternate angles (4a + 10) and 50. Equal angles give 4a + 10 = 50, so 4a = 40 and a = 10. Check: 4 x 10 + 10 = 50.
6. Step 6, Your turn: two angles (2x + 15) and (x + 30) form a linear pair. Find x and the angles. Answer: 3x + 45 = 180, so x = 45; the angles are 105 and 75.

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 pairs, parallel lines and transversals: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'The triangle and its properties', you learn to use the angle-sum and exterior-angle properties and the Pythagoras property. 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 'The triangle and its properties' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use the angle-sum and exterior-angle properties and the Pythagoras property. 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 property: a triangle has angles 40, 65 and x. The three add up to 180, so x = 180 - 40 - 65 = 75 degrees.
2. Step 2, Exterior angle: in triangle ABC with angles A = 50 and B = 60, the exterior angle at C equals the sum of the two interior opposite angles. So it is 50 + 60 = 110. Check: interior C is 70 and 70 + 110 = 180.
3. Step 3, Pythagoras property: in a right triangle with legs 6 and 8, hypotenuse squared = 36 + 64 = 100. So the hypotenuse is 10. The triple 6, 8, 10 is a scaled 3, 4, 5.
4. Step 4, Ladder problem: a 13 m ladder leans on a wall with its foot 5 m from the wall. Height squared = 169 - 25 = 144, so the height is 12 m. Check: 5, 12, 13 is a known triple.
5. Step 5, Test for a right triangle: sides 7, 24, 25. 7^2 + 24^2 = 49 + 576 = 625 and 25^2 = 625. The squares match, so the triangle is right-angled, with the right angle opposite 25.
6. Step 6, Your turn: find x in a triangle with angles x, 2x and 3x, and the other leg when the hypotenuse is 17 and one leg is 8. Answer: 6x = 180, x = 30; other leg is the root of 289 - 64 = 225, which is 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. Use the angle-sum and exterior-angle properties and the Pythagoras property: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Congruence of triangles', you learn to test congruence with SSS, SAS, ASA and RHS. 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 'Congruence of triangles' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: test congruence with SSS, SAS, ASA and RHS. 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, SAS test: triangle ABC has AB = 5 cm, BC = 7 cm and angle B = 50. Triangle PQR has PQ = 5, QR = 7 and angle Q = 50. Two sides and the included angle match, so ABC is congruent to PQR by SAS.
2. Step 2, Use the match: with ABC congruent to PQR, corresponding parts are equal. So AC = PR, angle A = angle P, angle C = angle R. Write the correspondence in order, A to P, B to Q, C to R.
3. Step 3, SSS test: triangles with sides 4, 5, 6 and 6, 4, 5 are congruent by SSS because all three pairs of sides are equal. No angle information is needed.
4. Step 4, RHS test: two right triangles each have hypotenuse 10 cm and one leg 6 cm. They are congruent by RHS. The third side is the root of 100 - 36 = 64, so it is 8 cm in both.
5. Step 5, ASA and a failure: ASA needs two angles and the included side, such as angles 40 and 60 with the side 8 cm between them. But SSA (two sides and a non-included angle) is not enough, as different triangles can fit.
6. Step 6, Your turn: triangle one has angles 40 and 60 with the side between them 8 cm; triangle two has angles 60 and 40 with 8 cm between them. Are they congruent? Answer: yes, by ASA, and the third angle of each is 80.

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. Test congruence with SSS, SAS, ASA and RHS: 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 solve percentage, profit and loss, and simple interest problems. 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: solve percentage, profit and loss, and simple interest problems. 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 percentage. A test has 250 marks and Sam scores 40. As a percentage this is 40 / 250 x 100 = 16%. To find a part of a quantity instead, 20% of 250 is 250 x 20 / 100 = 50.
2. Step 2, Use percentage increase. A bag marked 800 rises in price by 15%. The rise is 800 x 15 / 100 = 120, so the new price is 800 + 120 = 920. A quick route is to multiply by 1.15, because 100% + 15% = 115%.
3. Step 3, Find profit percentage. A seller buys a chair for 600, which is the cost price, and sells it for 750, the selling price. Profit = 750 - 600 = 150. Profit percent = 150 / 600 x 100 = 25%, always taken on the cost price.
4. Step 4, Find loss percentage. A bat costs 500 and is sold for 450. Loss = 500 - 450 = 50. Loss percent = 50 / 500 x 100 = 10%. If the selling price is above the cost price there is a profit, and below it there is a loss.
5. Step 5, Calculate simple interest with I = P x R x T / 100. Principal 4000, rate 5% a year, time 3 years gives 4000 x 5 x 3 / 100 = 600. The amount to be repaid is principal plus interest, 4000 + 600 = 4600.
6. Step 6, Your turn: find the simple interest on 2500 at 8% a year for 2 years. Then find the selling price of an article that costs 400 sold at 20% profit. Answer: interest = 2500 x 8 x 2 / 100 = 400, and selling price = 400 + 80 = 480.

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 percentage, profit and loss, and simple interest problems: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Rational numbers', you learn to compare and operate on rational numbers. 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: compare and operate on rational 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, Recognise rational numbers. A rational number can be written as p / q with q not zero. So -3/4, 5 (which is 5/1) and 0.25 (which is 1/4) are all rational. Place -3/4 and 1/4 on a number line, with negatives to the left of zero.
2. Step 2, Compare -2/3 and -3/4. Use a common denominator of 12: -2/3 = -8/12 and -3/4 = -9/12. Since -8 is to the right of -9 on the number line, -2/3 is greater than -3/4. For negatives, the one nearer zero is larger.
3. Step 3, Add rational numbers: -1/2 + 3/4. Write -1/2 as -2/4. Then -2/4 + 3/4 = 1/4. Always bring both fractions to the same denominator first, then add the numerators, keeping their signs.
4. Step 4, Subtract by adding the opposite: 2/5 - (-3/10) = 2/5 + 3/10. Change 2/5 to 4/10. Then 4/10 + 3/10 = 7/10. Subtracting a negative number is the same as adding the positive number.
5. Step 5, Multiply and divide. (-2/3) x (9/10) = -18/30 = -3/5 after dividing by 6, and a negative times a positive is negative. For (-3/4) / (5/8), multiply by the reciprocal: (-3/4) x (8/5) = -24/20 = -6/5.
6. Step 6, Your turn: work out -5/6 + 1/3, then decide whether -1/2 is greater or less than -2/3. Answer: 1/3 = 2/6, so -5/6 + 2/6 = -3/6 = -1/2. Since -1/2 = -3/6 and -2/3 = -4/6, -1/2 is greater.

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. Compare and operate on rational numbers: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Practical geometry', you learn to construct lines, angles and triangles. 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 'Practical geometry' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: construct lines, angles and triangles. 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, Construct a perpendicular bisector. Draw a segment AB of 6 cm. Open the compass to more than 3 cm, draw arcs above and below from A, then repeat from B with the same width. Join the two crossing points. The line meets AB at its midpoint, 3 cm from each end, at 90 degrees. Safety: keep the compass point down and capped after use.
2. Step 2, Construct a 60 degree angle. Draw a ray OA. With the compass point on O draw an arc crossing OA at P. Keep the same width, put the point on P and mark a second arc on the first. Join O to that crossing. The angle is 60 degrees because the three points form an equilateral triangle.
3. Step 3, Bisect that angle. From the points where the arms cut the arc, draw two arcs of equal radius that cross inside the angle. Draw a ray from the vertex through the crossing. Each half measures 30 degrees. Check with a protractor.
4. Step 4, Construct a triangle with three sides, 5 cm, 6 cm and 7 cm. First check that two shorter sides add to more than the longest: 5 + 6 = 11, which is more than 7. Draw the 7 cm base, then arcs of 5 cm and 6 cm from its ends. Their crossing is the third vertex.
5. Step 5, Construct a triangle with two sides and the angle between them. Draw a 6 cm base PQ, make a 50 degree angle at P, mark 5 cm along that arm to R, and join Q to R. The included angle is the one between the two given sides.
6. Step 6, Your turn: can a triangle have sides 3 cm, 4 cm and 8 cm? Test the rule before you draw. Answer: no, because 3 + 4 = 7, which is less than 8, so the two arcs would never meet.

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. Construct lines, angles and triangles: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Perimeter and area', you learn to find area of parallelograms, triangles and circles. 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 'Perimeter and area' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find area of parallelograms, triangles and circles. 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, Parallelogram: base 12 cm and perpendicular height 7 cm. Area = base x height = 12 x 7 = 84 cm2. Use the height at right angles to the base, not the slanted side.
2. Step 2, Triangle: base 10 cm and height 6 cm. Area = 1/2 x 10 x 6 = 30 cm2. This is half of a parallelogram of the same base and height, 60 cm2.
3. Step 3, Circle with radius 7 cm using pi = 22/7: circumference = 2 x 22/7 x 7 = 44 cm. Area = 22/7 x 7 x 7 = 22 x 7 = 154 cm2. Check: the circle fits inside the square of side 14, area 196, and 154 is smaller, about pi/4 of 196.
4. Step 4, Circle from the diameter: diameter 20 cm gives radius 10 cm. With pi = 3.14, circumference = 3.14 x 20 = 62.8 cm and area = 3.14 x 100 = 314 cm2.
5. Step 5, Word problem: a circular park has radius 14 m. Area = 22/7 x 14 x 14 = 22 x 2 x 14 = 616 m2. Fencing the edge needs 2 x 22/7 x 14 = 88 m.
6. Step 6, Your turn: find the area of a parallelogram with base 9 cm and height 5 cm, and the circumference and area of a circle of radius 21 cm. Answer: 45 cm2; 132 cm and 1,386 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 area of parallelograms, triangles and circles: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Algebraic expressions', you learn to add and subtract expressions and find values. 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' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: add and subtract expressions and find values. 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, Identify like terms: in 3x + 2y + 5x - 4y the x terms are 3x and 5x, and the y terms are 2y and -4y. Only like terms can be combined.
2. Step 2, Add: (3x + 2y) + (5x - 4y) = (3x + 5x) + (2y - 4y) = 8x - 2y. Group the like terms, then add their numbers.
3. Step 3, Subtract: (7a - 3b + 2) - (2a + b - 5). Change every sign in the second bracket: 7a - 3b + 2 - 2a - b + 5. Collect: 5a - 4b + 7.
4. Step 4, Find a value: for 3x^2 - 4x + 1 with x = 2, work 3 x 4 - 8 + 1 = 12 - 8 + 1 = 5. Square before multiplying by 3. Try x = -1 to see: 3 + 4 + 1 = 8.
5. Step 5, Use in a problem: a rectangle has length (2x + 3) and width x. Perimeter = 2(2x + 3 + x) = 6x + 6. If x = 4, the perimeter is 6 x 4 + 6 = 30. Direct check: 2 x (11 + 4) = 30.
6. Step 6, Your turn: add (4m + 3n) and (2m - 5n), and find 2a + 3b when a = 4 and b = -1. Answer: 6m - 2n; and 8 - 3 = 5.

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. Add and subtract expressions and find values: 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 and standard form. 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 and standard form. 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, Read an exponent. In 2^5 the base is 2 and the exponent is 5, so 2^5 = 2 x 2 x 2 x 2 x 2 = 32. The exponent counts how many times the base is multiplied. Write 3^4 in full and evaluate it to get 81.
2. Step 2, Multiply powers with the same base by adding exponents: a^m x a^n = a^(m + n). So 3^2 x 3^4 = 3^6. Check: 9 x 81 = 729 and 3^6 = 729. Do not multiply the bases, so the base stays 3.
3. Step 3, Divide by subtracting exponents: a^m / a^n = a^(m - n). So 5^7 / 5^4 = 5^3 = 125. For a power of a power, multiply: (2^3)^2 = 2^6 = 64, which agrees with 8 x 8.
4. Step 4, Learn two more rules. Any non-zero number to the power 0 is 1, so 9^0 = 1. And (a x b)^m = a^m x b^m, so (2 x 3)^2 = 2^2 x 3^2 = 4 x 9 = 36, matching 6^2 = 36.
5. Step 5, Write numbers in standard form as a number from 1 to 10 times a power of 10. 45,000,000 = 4.5 x 10^7 because the decimal point moves 7 places. A small number, 0.00032, is 3.2 x 10^-4. The Earth to the Sun is about 1.5 x 10^8 km.
6. Step 6, Your turn: simplify 2^3 x 2^5 / 2^4 and write 6,300,000 in standard form. Answer: 2^(3 + 5 - 4) = 2^4 = 16, and moving the point 6 places gives 6.3 x 10^6.

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 and standard form: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Symmetry and solid shapes', you learn to draw nets and views of solids and find lines of symmetry. 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 'Symmetry and solid shapes' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: draw nets and views of solids and find lines of symmetry. 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, Nets: a cube net has 6 squares joined so that they fold into a closed box. A cuboid net has 6 rectangles, in 3 equal pairs. Fold test: a row of 4 squares with one square above and one below works.
2. Step 2, Count faces, vertices and edges: a cube has 6 faces, 8 vertices and 12 edges. Euler's check F + V - E = 2 gives 6 + 8 - 12 = 2. A triangular prism has 5 faces, 6 vertices and 9 edges: 5 + 6 - 9 = 2.
3. Step 3, Views: a stack has 3 cubes in the front row and 1 cube behind the left one. The front view shows 3 squares side by side and the top view shows 3 squares in a row plus 1 behind at the left end.
4. Step 4, Lines of symmetry: an equilateral triangle has 3, a regular hexagon has 6, a rectangle 2, and the letter H has 2 (one vertical, one horizontal). A scalene triangle has none.
5. Step 5, Check a net: a cross with 5 squares and 1 extra is a cube net only if no two squares overlap when folded. A row of 6 squares fails, since folding makes the end squares overlap.
6. Step 6, Your turn: find F + V - E for a square pyramid and say how many lines of symmetry a square has. Answer: 5 faces, 5 vertices, 8 edges gives 5 + 5 - 8 = 2; a square has 4 lines of symmetry.

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. Draw nets and views of solids and find lines of symmetry: 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 07 Mathematics is at this level. Its units include Integers; Fractions and decimals; Data handling; Simple equations 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.