Grade 05: Mathematics

In this unit, 'Numbers to 10 lakh and millions', you learn to read, write, compare and round large numbers. Lessons are short and playful, with objects, pictures and talk. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Numbers to 10 lakh and millions' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: read, write, compare and round large 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, Read 4,52,368 in the lakh system: four lakh fifty-two thousand three hundred sixty-eight. Commas split it as lakhs, thousands, then hundreds. The same number in the million system is written 452,368.
2. Step 2, Link lakh to million: 10 lakh = 1,000,000 = 1 million. So 25,00,000 is 25 lakh, which is 2,500,000 or two and a half million. Divide a lakh count by 10 to get millions.
3. Step 3, Compare 7,35,420 and 7,53,240: both have 7 lakh. Next compare ten-thousands: 3 against 5. Since 5 > 3, the second number is bigger, so 7,35,420 < 7,53,240.
4. Step 4, Round 6,48,572 to the nearest thousand: look at the hundreds digit 5, so round up to 6,49,000. To the nearest lakh, the remainder 48,572 is less than 50,000, so round down to 6,00,000.
5. Step 5, Write from words: nine lakh seven thousand and forty is 9,07,040. The empty ten-thousands place holds a 0 and the hundreds place holds 0. Reading 9,07,040 aloud checks it.
6. Step 6, Your turn: round 5,67,891 to the nearest ten thousand and write it in words. Answer: 5,70,000, since the thousands digit 7 rounds up; the original reads five lakh sixty-seven thousand eight hundred ninety-one.

Activity. About 20 to 30 minutes of class work, then one short practice task for home. Keep evidence: one page in your learning book.

Standard to meet. Read, write, compare and round large numbers: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Four operations', you learn to solve multi-step problems with all four operations. Lessons are short and playful, with objects, pictures and talk. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Four operations' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: solve multi-step problems with all four operations. 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, Problem: a shop buys 24 boxes of 35 pens, sells 650 pens, and packs the rest in packs of 5. First multiply: 24 x 35 = 24 x 30 + 24 x 5 = 720 + 120 = 840 pens.
2. Step 2, Subtract the sales: 840 - 650 = 190 pens remain. Then divide into packs of 5: 190 / 5 = 38. So 38 packs can be made, and nothing is left over because 38 x 5 = 190.
3. Step 3, Check by working backwards: 38 x 5 = 190, 190 + 650 = 840, and 840 / 24 = 35. Each reverse step returns to the number before, so the chain is correct.
4. Step 4, Order of operations: 5 + 3 x 4 means multiply first, so 3 x 4 = 12 and 5 + 12 = 17, not 32. With brackets, (5 + 3) x 4 = 32. Brackets change the order.
5. Step 5, Another multi-step problem: 6,250 - (3 x 1,200 + 475). Brackets first: 3 x 1,200 = 3,600, plus 475 = 4,075. Then 6,250 - 4,075 = 2,175.
6. Step 6, Your turn: 18 boxes of 25 pencils are bought, 130 are given away, the rest are shared among 4 classes. Answer: 18 x 25 = 450; 450 - 130 = 320; 320 / 4 = 80 each.

Activity. About 20 to 30 minutes of class work, then one short practice task for home. Keep evidence: one page in your learning book.

Standard to meet. Solve multi-step problems with all four operations: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Factors, multiples, HCF and LCM', you learn to find HCF and LCM and use them in word problems. Lessons are short and playful, with objects, pictures and talk. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Factors, multiples, HCF and LCM' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find HCF and LCM and use them in word 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, List factors of 36 and 48. 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The common factors are 1, 2, 3, 4, 6, 12, so HCF = 12.
2. Step 2, Prime factors: 36 = 2 x 2 x 3 x 3 and 48 = 2 x 2 x 2 x 2 x 3. HCF takes shared primes with the smaller powers, 2 x 2 x 3 = 12. LCM takes the larger powers, 2^4 x 3^2 = 144.
3. Step 3, Check with the product rule: HCF x LCM = product of the numbers. 12 x 144 = 1,728 and 36 x 48 = 1,728. They match, so both answers are right.
4. Step 4, HCF word problem: ribbons of 36 cm and 48 cm are cut into equal pieces with no waste. The longest piece is the HCF, 12 cm. That gives 3 + 4 = 7 pieces in all.
5. Step 5, LCM word problem: two bells ring every 12 and every 18 minutes. Multiples of 12: 12, 24, 36. Multiples of 18: 18, 36. They ring together again after 36 minutes, which is the LCM.
6. Step 6, Your turn: find the HCF and LCM of 24 and 40. Answer: HCF is 8 (24 = 8 x 3, 40 = 8 x 5) and LCM is 24 x 40 / 8 = 120.

Activity. About 20 to 30 minutes of class work, then one short practice task for home. Keep evidence: one page in your learning book.

Standard to meet. Find HCF and LCM and use them in word problems: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Fractions', you learn to add, subtract and multiply fractions and mixed numbers. Lessons are short and playful, with objects, pictures and talk. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Fractions' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: add, subtract and multiply fractions and mixed 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 unlike fractions: 2/3 + 3/4. The common denominator is 12. 2/3 = 8/12 and 3/4 = 9/12. Sum = 17/12, which is the mixed number 1 5/12.
2. Step 2, Subtract mixed numbers: 3 1/2 - 1 3/4. Convert to improper fractions: 7/2 and 7/4. Use denominator 4: 14/4 - 7/4 = 7/4 = 1 3/4. Check: 1 3/4 + 1 3/4 = 3 2/4 = 3 1/2.
3. Step 3, Multiply fractions: 2/5 x 3/4 = (2 x 3) / (5 x 4) = 6/20. Simplify by dividing top and bottom by 2 to get 3/10. Cancel first when you can: 2 and 4 share a factor 2.
4. Step 4, Multiply mixed numbers: 1 1/2 x 2 2/3. Change to 3/2 x 8/3. The 3s cancel, leaving 8/2 = 4. Check by estimating: 1.5 x 2.7 is about 4.
5. Step 5, Word problem: a tank is 3/4 full and 2/5 of the water is used. Used = 2/5 x 3/4 = 6/20 = 3/10 of the tank. Left = 3/4 - 3/10 = 15/20 - 6/20 = 9/20.
6. Step 6, Your turn: find 5/6 - 1/4 and 2 1/4 x 4/9. Answer: 10/12 - 3/12 = 7/12; and 9/4 x 4/9 = 1.

Activity. About 20 to 30 minutes of class work, then one short practice task for home. Keep evidence: one page in your learning book.

Standard to meet. Add, subtract and multiply fractions and mixed numbers: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Decimals', you learn to add, subtract and multiply decimals and link to money and measures. Lessons are short and playful, with objects, pictures and talk. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Decimals' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: add, subtract and multiply decimals and link to money and measures. 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 decimals: 12.45 + 3.8. Write 3.8 as 3.80 and line up the decimal points. 12.45 + 3.80 = 16.25. Keeping points aligned stops place-value mistakes.
2. Step 2, Subtract: 10 - 6.37. Write 10 as 10.00. Then 10.00 - 6.37 = 3.63. Check by adding: 3.63 + 6.37 = 10.00, which returns the start.
3. Step 3, Multiply: 2.5 x 1.2. Ignore the points: 25 x 12 = 300. The two numbers have 1 + 1 = 2 decimal places, so the answer is 3.00 = 3. Estimate: 2.5 x 1 is 2.5, and a bit more is 3.
4. Step 4, Money problem: rice costs 24.60 per kg and you buy 3.5 kg. 24.6 x 3 = 73.8 and 24.6 x 0.5 = 12.3. Total 86.1, so you pay 86.10.
5. Step 5, Measures link: 2.75 m = 275 cm because 1 m = 100 cm. 0.6 kg = 600 g. A rope of 4.5 m cut into pieces of 0.75 m gives 4.5 / 0.75 = 6 pieces, since 0.75 x 6 = 4.5.
6. Step 6, Your turn: find 7.25 + 4.9 and 1.5 x 0.8. Answer: 7.25 + 4.90 = 12.15; and 15 x 8 = 120 with two decimal places gives 1.20 = 1.2.

Activity. About 20 to 30 minutes of class work, then one short practice task for home. Keep evidence: one page in your learning book.

Standard to meet. Add, subtract and multiply decimals and link to money and measures: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Percentages and ratio', you learn to link fractions, decimals and percentages and simple ratio. Lessons are short and playful, with objects, pictures and talk. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Percentages and ratio' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: link fractions, decimals and percentages and simple ratio. 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, Link fractions, decimals and percentages. Percent means out of 100. So 1/2 = 50/100 = 0.5 = 50%. Draw a hundred square, shade 50 squares and write all three forms beside it. Each form names the same part.
2. Step 2, Convert common fractions. 3/4 = 75/100 = 0.75 = 75%. 1/5 = 20/100 = 0.2 = 20%. For 7/20, multiply top and bottom by 5 to get 35/100, which is 0.35 and 35%. Always make the denominator 100 when you can.
3. Step 3, Find a percentage of an amount. 25% is a quarter, so 25% of 80 = 80 / 4 = 20. 10% of 150 = 150 / 10 = 15. For 15%, add 10% and 5%: 15 + 7.5 = 22.5. Break any percentage into easy pieces.
4. Step 4, Write a simple ratio. A bag has 3 red beads and 5 blue beads, so red to blue is 3 : 5. The whole bag has 3 + 5 = 8 parts, so red is 3/8 of the beads. Keep the order the same as the words.
5. Step 5, Share in a ratio. Share 24 sweets in the ratio 1 : 3. There are 4 parts, so one part is 24 / 4 = 6. The shares are 6 and 18. Check: 6 : 18 simplifies to 1 : 3 by dividing both by 6.
6. Step 6, Your turn: write 0.6 as a fraction and a percentage, then share 40 stickers in the ratio 3 : 2. Answer: 0.6 = 3/5 = 60%. There are 5 parts, one part is 8, so the shares are 24 and 16 stickers.

Activity. About 20 to 30 minutes of class work, then one short practice task for home. Keep evidence: one page in your learning book.

Standard to meet. Link fractions, decimals and percentages and simple ratio: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Perimeter, area and volume', you learn to find area of rectangles and triangles and volume of cuboids. Lessons are short and playful, with objects, pictures and talk. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Perimeter, area and volume' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find area of rectangles and triangles and volume of cuboids. 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, Area of a rectangle 14 cm by 9 cm: length x width = 14 x 9 = 126 cm2. Perimeter is a different quantity: 2 x (14 + 9) = 46 cm. Area uses square units.
2. Step 2, Area of a triangle with base 12 cm and height 7 cm: half x base x height = 1/2 x 12 x 7 = 6 x 7 = 42 cm2. The height must be at right angles to the base.
3. Step 3, Check the triangle: it is half of a 12 by 7 rectangle. That rectangle has area 84 cm2 and half of 84 is 42 cm2, which agrees.
4. Step 4, Volume of a cuboid 8 cm by 5 cm by 4 cm: length x width x height = 8 x 5 x 4 = 160 cm3. Check: bottom layer has 8 x 5 = 40 cubes, 4 layers give 160.
5. Step 5, Capacity link: a tank 50 cm by 40 cm by 30 cm has volume 50 x 40 x 30 = 60,000 cm3. Since 1,000 cm3 = 1 litre, the tank holds 60 litres.
6. Step 6, Your turn: find the area of a triangle with base 10 cm and height 9 cm, and the volume of a 6 by 5 by 3 cm cuboid. Answer: 1/2 x 10 x 9 = 45 cm2; 6 x 5 x 3 = 90 cm3.

Activity. About 20 to 30 minutes of class work, then one short practice task for home. Keep evidence: one page in your learning book.

Standard to meet. Find area of rectangles and triangles and volume of cuboids: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Angles and shapes', you learn to measure and draw angles and classify polygons. Lessons are short and playful, with objects, pictures and talk. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Angles and shapes' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: measure and draw angles and classify polygons. 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, Classify angles: 65 degrees is acute (less than 90), 90 is a right angle, 130 is obtuse (between 90 and 180), 180 is a straight angle, and 250 is reflex (more than 180).
2. Step 2, Measure with a protractor: place its centre on the vertex and its zero line along one arm. Read the scale that starts at 0 on that arm. If the angle opens like a corner narrower than a square, read the smaller number, e.g. 65.
3. Step 3, Draw 110 degrees: draw a base line, mark the vertex, centre the protractor and zero on the base. Mark a dot at 110 on the scale, remove the protractor and join the vertex to the dot. It is obtuse.
4. Step 4, Find a missing angle: a triangle has angles 50 and 70. The angles of a triangle add up to 180, so the third angle is 180 - 50 - 70 = 60 degrees. In a quadrilateral they total 360.
5. Step 5, Classify polygons: pentagon 5 sides, hexagon 6 sides, octagon 8 sides. A regular polygon has equal sides and equal angles. A quadrilateral with angles 90, 80, 100 has fourth angle 360 - 270 = 90.
6. Step 6, Your turn: find the third angle of a triangle with angles 35 and 85, and name a polygon with 8 sides. Answer: 180 - 120 = 60 degrees; an octagon.

Activity. About 20 to 30 minutes of class work, then one short practice task for home. Keep evidence: one page in your learning book.

Standard to meet. Measure and draw angles and classify polygons: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Data and averages', you learn to find mean, median and mode and read line graphs. Lessons are short and playful, with objects, pictures and talk. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Data and averages' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find mean, median and mode and read line 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: quiz scores 12, 15, 9, 15, 14. Add them: 12 + 15 + 9 + 15 + 14 = 65. Divide by the 5 scores: 65 / 5 = 13. The mean is 13.
2. Step 2, Median: order the scores 9, 12, 14, 15, 15. The middle one is 14. For an even count, 8, 10, 12, 20, take the two middle values: (10 + 12) / 2 = 11.
3. Step 3, Mode: in 9, 12, 14, 15, 15 the value 15 appears twice, more than any other. So the mode is 15. A set can have more than one mode or none at all.
4. Step 4, Read a line graph: rainfall for Jan to Apr is 20, 35, 50, 40 mm. The line goes up then down. The highest point is 50 in March, and the fall from March to April is 10 mm.
5. Step 5, Use the mean: the mean of 4 months of rainfall is (20 + 35 + 50 + 40) / 4 = 145 / 4 = 36.25 mm. Check: 36.25 is between the lowest 20 and the highest 50.
6. Step 6, Your turn: find the mean, median and mode of 6, 8, 8, 10, 13. Answer: sum 45, mean 9; median 8 (middle of five); mode 8.

Activity. About 20 to 30 minutes of class work, then one short practice task for home. Keep evidence: one page in your learning book.

Standard to meet. Find mean, median and mode and read line graphs: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Algebra beginnings', you learn to use a letter for an unknown and solve simple equations. Lessons are short and playful, with objects, pictures and talk. You see it, say it, then do it alone.

• Show it with objects or a picture: build or draw the idea in 'Algebra beginnings' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use a letter for an unknown 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, Use a letter: a pencil costs x dirhams, so 5 pencils cost 5x. If a pencil costs 3, 5x = 15. The letter stands for a number we may not know yet.
2. Step 2, Solve x + 7 = 15. To undo adding 7, subtract 7 from both sides: x = 15 - 7 = 8. Check: 8 + 7 = 15, which is true.
3. Step 3, Solve 3n = 21. To undo multiplying by 3, divide both sides by 3: n = 7. Check: 3 x 7 = 21. Do the same operation on both sides to keep the equation balanced.
4. Step 4, Two-step equation: 2y + 3 = 17. Subtract 3 first: 2y = 14. Then divide by 2: y = 7. Check: 2 x 7 + 3 = 14 + 3 = 17. Correct.
5. Step 5, Word problem: a number plus 12 is 30. Write n + 12 = 30, so n = 30 - 12 = 18. Check: 18 + 12 = 30. Always test the answer in the original sentence.
6. Step 6, Your turn: solve 3x - 4 = 11. Answer: add 4 to both sides to get 3x = 15, then divide by 3 so x = 5. Check: 3 x 5 - 4 = 11.

Activity. About 20 to 30 minutes of class work, then one short practice task for home. Keep evidence: one page in your learning book.

Standard to meet. Use a letter for an unknown and solve simple equations: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

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.

This course: Grade 05 Mathematics is at this level. Its units include Numbers to 10 lakh and millions; Four operations; Factors, multiples, HCF and LCM; Fractions and more.

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.

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.