Grade 06: Mathematics

In this unit, 'Knowing our numbers', you learn to read, round and estimate large numbers and use the Roman system. 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 'Knowing our numbers' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: read, round and estimate large numbers and use the Roman system. 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 in the Indian system: 3,45,67,890 is three crore forty-five lakh sixty-seven thousand eight hundred ninety. In the international system the same number is 34,567,890.
2. Step 2, Estimate a sum: 4,897 + 3,112. Round each to the nearest thousand: 5,000 + 3,000 = 8,000. The exact sum is 8,009, very close to the estimate, so it is sensible.
3. Step 3, Roman numerals: XLIV = XL + IV = 40 + 4 = 44. And 1,994 = 1000 + 900 + 90 + 4 = M + CM + XC + IV = MCMXCIV. A smaller symbol before a larger one is subtracted.
4. Step 4, Roman rules: V, L and D are never repeated, and I, X, C, M repeat at most 3 times. So 4 is IV, not IIII, and 9 is IX. Check XIX = 10 + 9 = 19.
5. Step 5, Round to compare: a city has 4,86,215 people and another 5,12,780. Rounded to the nearest lakh these are 5,00,000 and 5,00,000, so round to the nearest ten thousand: 4,90,000 and 5,10,000. The second is larger by about 20,000.
6. Step 6, Your turn: write 2,019 in Roman numerals and estimate 7,962 - 3,018. Answer: MMXIX (2000 + 10 + 9); estimate 8,000 - 3,000 = 5,000, and the exact answer is 4,944.

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. Read, round and estimate large numbers and use the Roman system: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Whole numbers and number play', you learn to use properties, factors, multiples, HCF and LCM, divisibility tests. 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 'Whole numbers and number play' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use properties, factors, multiples, HCF and LCM, divisibility tests. 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, Use properties: 25 x 48. Write 48 as 50 - 2 and use the distributive property: 25 x 50 - 25 x 2 = 1,250 - 50 = 1,200. Check: 25 x 48 is 25 x 4 x 12 = 100 x 12 = 1,200.
2. Step 2, Divisibility tests on 3,726: digit sum 3 + 7 + 2 + 6 = 18, which is a multiple of 9, so 3,726 is divisible by 3 and 9. It ends in 6, so it is divisible by 2, and so by 6.
3. Step 3, Check 4: the last two digits 26 are not divisible by 4, so 3,726 is not divisible by 4. Verify: 3,726 / 6 = 621 exactly, since 6 x 621 = 3,726.
4. Step 4, Prime factors of 84: 84 = 2 x 42 = 2 x 2 x 21 = 2 x 2 x 3 x 7. For 70: 2 x 5 x 7. Both share 2 and 7, so HCF(84, 70) = 14.
5. Step 5, LCM of 42 and 70: 42 = 2 x 3 x 7 and 70 = 2 x 5 x 7. Take each prime once with the highest power: 2 x 3 x 5 x 7 = 210. Check: HCF 14 x LCM 210 = 2,940 = 42 x 70.
6. Step 6, Your turn: test 1,512 for 9 and 4, and find the HCF and LCM of 18 and 24. Answer: digit sum 9 so divisible by 9; 12 is divisible by 4 so yes; HCF 6, LCM 72.

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 properties, factors, multiples, HCF and LCM, divisibility tests: 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 compare and operate on fractions and decimals. 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: compare and operate on fractions and decimals. 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, Compare 3/4 and 5/6. The common denominator is 12: 3/4 = 9/12 and 5/6 = 10/12. Since 10 > 9, 5/6 is the larger fraction.
2. Step 2, Add: 2/5 + 1/3 = 6/15 + 5/15 = 11/15. Subtract mixed numbers: 1 3/4 - 5/8 = 7/4 - 5/8 = 14/8 - 5/8 = 9/8 = 1 1/8.
3. Step 3, Convert to a decimal: 3/8 means 3 divided by 8 = 0.375, since 8 x 0.375 = 3. So 3/8 = 0.375. Also 7/20 = 35/100 = 0.35.
4. Step 4, Add decimals: 6.75 + 2.8 = 6.75 + 2.80 = 9.55. Subtract: 5 - 2.36 = 5.00 - 2.36 = 2.64. Align decimal points, and add zeros to make the same number of places.
5. Step 5, Word problem: A learner runs 2.4 km and walks 3/4 km. Change 3/4 to 0.75. Total = 2.4 + 0.75 = 3.15 km. Check with fractions: 2 2/5 + 3/4 = 2 8/20 + 15/20 = 3 3/20 = 3.15.
6. Step 6, Your turn: find 3/10 + 2/5 and 8.4 - 3.75. Answer: 3/10 + 4/10 = 7/10; and 8.40 - 3.75 = 4.65.

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

In this unit, 'Integers', you learn to use negative numbers on a number line and add and subtract integers. 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: use negative numbers on a number line and add and subtract integers. 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, Locate on a number line: -3 is 3 steps left of 0. Moving right is positive, left is negative. So -3 + 7 means start at -3 and move 7 right, landing on 4.
2. Step 2, Subtract: 5 - 9. Start at 5 and move 9 left, passing 0. That is 5 steps to 0 and 4 more, so 5 - 9 = -4. Check: -4 + 9 = 5.
3. Step 3, Two negatives: -6 - 2 means start at -6 and move 2 more left, giving -8. The number gets smaller, so -8 < -6.
4. Step 4, Subtracting a negative: -4 - (-7) = -4 + 7 = 3. Taking away a negative acts like adding the positive. Check with the number line: from -4 move 7 right to land on 3.
5. Step 5, Temperature problem: morning temperature is -5 degrees and noon is 8 degrees. Rise = 8 - (-5) = 8 + 5 = 13 degrees. On a thermometer, count 5 up to 0 and 8 more, total 13.
6. Step 6, Your turn: find -8 + 3 and 6 - (-4). Answer: -8 + 3 = -5 (move 3 right from -8), and 6 - (-4) = 6 + 4 = 10.

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 negative numbers on a number line and add and subtract integers: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Ratio and proportion', you learn to write ratios, simplify and solve unitary 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 'Ratio and proportion' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: write ratios, simplify and solve unitary 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, Write a ratio. A class has 12 boys and 18 girls, so boys to girls is 12 : 18. The order matters, so girls to boys would be 18 : 12. A ratio compares two quantities of the same kind and has no unit after it.
2. Step 2, Simplify by dividing both numbers by their highest common factor. For 12 : 18 the HCF is 6, so 12 / 6 = 2 and 18 / 6 = 3. The simple form is 2 : 3. Check by multiplying back: 2 x 6 = 12 and 3 x 6 = 18.
3. Step 3, Make the units match before simplifying. For 50 cm to 2 m, change 2 m to 200 cm. The ratio is 50 : 200. Divide both by 50 to get 1 : 4. Equivalent ratios such as 2 : 3, 4 : 6 and 10 : 15 all describe the same comparison.
4. Step 4, Use the unitary method: find the cost of one first. If 5 kg of apples cost 45 coins, then 1 kg costs 45 / 5 = 9 coins. So 8 kg cost 8 x 9 = 72 coins. Finding one unit makes any quantity easy.
5. Step 5, Solve a proportion problem. Paint is mixed blue to yellow as 2 : 3 and 20 litres are needed. That is 5 parts, so 1 part is 4 litres, giving 8 litres blue and 12 litres yellow. A recipe with 3 cups flour for 12 biscuits needs 5 cups for 20.
6. Step 6, Your turn: 4 pencils cost 24 coins. Find the cost of 7 pencils, then simplify the ratio 28 : 42. Answer: 1 pencil costs 6 coins, so 7 cost 42 coins. The HCF of 28 and 42 is 14, so the ratio is 2 : 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. Write ratios, simplify and solve unitary problems: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Basic geometry and shapes', you learn to use a ruler, compass and protractor to draw and measure. 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 'Basic geometry and shapes' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use a ruler, compass and protractor to draw and measure. 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, Measure a line with a ruler. Put the 0 mark exactly at one end, not the edge of the ruler. If the other end falls 7 small marks after 7 cm, read 7.7 cm, since each small mark is 1 mm. Draw a 7.5 cm line and label its ends A and B.
2. Step 2, Measure an angle with a protractor. Put the centre hole on the vertex and the base line along one arm. Start counting from the 0 on that arm. If the other arm passes 65, the angle is 65 degrees. Check it is acute, so the inner scale makes sense.
3. Step 3, Draw a 50 degree angle. Draw a 6 cm line, put the protractor centre on one end, make a small dot at 50 on the scale starting from 0 on the line, and remove the protractor. Join the dot to the end. Label the angle and measure again to check.
4. Step 4, Draw a circle with a compass. Set the compass width to 3 cm using the ruler, press the point on the centre and turn the pencil end smoothly. The radius is 3 cm and the diameter is 6 cm. Safety: the metal point is sharp, so hold the top and keep it capped when not in use.
5. Step 5, Construct a triangle with sides 5 cm, 4 cm and 3 cm. Draw the 5 cm base, then with the compass draw an arc of 4 cm from one end and an arc of 3 cm from the other. Where they cross is the third corner. Measure the angle opposite the 5 cm side to find 90 degrees.
6. Step 6, Your turn: draw an angle of 110 degrees, say what type it is, and draw a circle with diameter 8 cm. Answer: the angle is obtuse because it is more than 90 degrees, and the compass is set to a radius of 4 cm.

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 a ruler, compass and protractor to draw and measure: 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 perimeter and area of squares, rectangles and composite shapes. 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 perimeter and area of squares, rectangles and composite shapes. 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, Rectangle 12 cm by 7 cm: perimeter = 2 x (12 + 7) = 38 cm. Area = 12 x 7 = 84 cm2. Perimeter is a length in cm and area is in square cm.
2. Step 2, Square of side 9 cm: perimeter = 4 x 9 = 36 cm. Area = 9 x 9 = 81 cm2. Reverse it: a square with perimeter 40 cm has side 40 / 4 = 10 cm and area 100 cm2.
3. Step 3, Composite shape: an L-shape is a 10 cm by 6 cm rectangle with a 4 cm by 3 cm corner removed. Area = 10 x 6 - 4 x 3 = 60 - 12 = 48 cm2.
4. Step 4, Perimeter of the L-shape: a corner notch does not change the perimeter, since the outer boundary is shifted but the same total length. So perimeter = 2 x (10 + 6) = 32 cm.
5. Step 5, Word problem: a 20 m by 13 m plot needs fencing and turf. Fence = perimeter = 2 x (20 + 13) = 66 m. Turf = area = 20 x 13 = 260 m2. Different units for different questions.
6. Step 6, Your turn: find the perimeter and area of a rectangle 25 m by 16 m. Answer: perimeter 2 x 41 = 82 m; area 25 x 16 = 400 m2.

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 perimeter and area of squares, rectangles and composite shapes: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Algebra introduction', you learn to use letters for numbers, write expressions and solve simple equations. 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 'Algebra introduction' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use letters for numbers, write expressions and solve simple equations. 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, Write expressions: a pencil costs p dirhams. Five pencils cost 5p. A child has k stickers and gets 7 more: k + 7. A square of side s has perimeter 4s.
2. Step 2, Evaluate: if p = 3, then 5p = 15. If k = 12, then k + 7 = 19. Replace the letter with the number and then calculate.
3. Step 3, Solve x + 9 = 20. Subtract 9 from both sides: x = 20 - 9 = 11. Check: 11 + 9 = 20, so the equation balances.
4. Step 4, Solve 3n = 27. Divide both sides by 3: n = 9. Square problem: perimeter 36 cm, so 4s = 36 and s = 36 / 4 = 9 cm. Check: 4 x 9 = 36.
5. Step 5, Word problem: I think of a number, double it and add 5, and get 19. So 2x + 5 = 19. Subtract 5: 2x = 14, then divide by 2: x = 7.
6. Step 6, Your turn: write an expression for 3 notebooks at n each plus a pen costing 4, and solve 2x + 5 = 19 by checking 7. Answer: 3n + 4; and 2 x 7 + 5 = 19, so x = 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. Use letters for numbers, write expressions and solve simple equations: at least 8 of 10 marks on a short task, with one line of reasoning for each answer.

In this unit, 'Data handling and symmetry', you learn to draw bar graphs, find mean and identify 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 'Data handling and symmetry' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: draw bar graphs, find mean and identify 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, Draw a bar graph: marks of 5 students are 14, 18, 12, 16, 20. Choose the scale 0 to 20 in steps of 2, draw equal-width bars with gaps, and label both axes and the title.
2. Step 2, Find the mean: add 14 + 18 + 12 + 16 + 20 = 80, then divide by 5. The mean is 16. Check that 16 lies between the lowest 12 and the highest 20.
3. Step 3, Read from the graph: the highest bar is 20 and the lowest is 12, so the range is 20 - 12 = 8. Four of the five students scored 14 or above, since only 12 is below 14.
4. Step 4, Lines of symmetry: a rectangle has 2 (one vertical, one horizontal), a square has 4, an equilateral triangle has 3, and a circle has infinitely many. The letter A has 1 vertical line.
5. Step 5, Test a line: fold a paper rectangle along its diagonal and the corners do not match, so a diagonal is not a line of symmetry for a rectangle. Only the lines through midpoints of opposite sides work.
6. Step 6, Your turn: find the mean of 7, 9, 12, 8, 14 and the lines of symmetry of a regular pentagon. Answer: sum 50, mean 10; and a regular pentagon has 5 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 bar graphs, find mean and identify 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 06 Mathematics is at this level. Its units include Knowing our numbers; Whole numbers and number play; Fractions and decimals; Integers 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.