Grade 04: Mathematics

In this unit, 'Large numbers', you learn to read, write, round and compare numbers to 10,000. 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 'Large numbers' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: read, write, round and compare numbers to 10,000. 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 7,308: seven thousand, three hundred and eight. Place values: 7 thousands, 3 hundreds, 0 tens, 8 ones. The zero holds the tens place. Expanded: 7000 + 300 + 0 + 8.
2. Step 2, Write from words: four thousand, six hundred and fifty is 4,650. Put 4 in thousands, 6 in hundreds, 5 in tens and 0 in ones. Check by reading 4,650 back in words.
3. Step 3, Compare 5,962 and 5,926: thousands equal (5), hundreds equal (9), tens 6 is greater than 2. So 5,962 > 5,926. Compare place by place from the left.
4. Step 4, Round 6,478 to the nearest hundred: look at the tens digit 7. It is 5 or more, so round up to 6,500. To the nearest thousand look at hundreds digit 4: round down to 6,000.
5. Step 5, Order 3,405, 3,045, 3,450, 3,504: hundreds are 4, 0, 4, 5. 3,045 is smallest. Of 3,405 and 3,450, tens 0 < 5. Order: 3,045, 3,405, 3,450, 3,504.
6. Step 6, Your turn: round 8,251 to the nearest hundred and nearest thousand, and write 9,070 in words. Answer: 8,300 and 8,000; nine thousand and seventy.

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, round and compare numbers to 10,000: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Addition and subtraction', you learn to add and subtract four-digit numbers and solve multi-step 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 'Addition and subtraction' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: add and subtract four-digit numbers and solve multi-step 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, Add 4,256 + 3,789 in columns. Ones: 6 + 9 = 15, write 5 carry 1. Tens: 5 + 8 + 1 = 14, write 4 carry 1. Hundreds: 2 + 7 + 1 = 10, write 0 carry 1. Thousands: 4 + 3 + 1 = 8. Sum 8,045.
2. Step 2, Estimate: 4,300 + 3,800 = 8,100, near 8,045. Reasonable. Subtract 7,003 - 2,846: borrow through the zeros. Ones 13 - 6 = 7, tens 9 - 4 = 5, hundreds 9 - 8 = 1, thousands 6 - 2 = 4. Answer 4,157.
3. Step 3, Check subtraction by adding: 4,157 + 2,846 = 7,003 (7 + 6 = 13, 5 + 4 + 1 = 10, 1 + 8 + 1 = 10, 4 + 2 + 1 = 7). The sum matches, so 4,157 is correct.
4. Step 4, Multi-step problem: a shop has 5,200 items. It sells 1,845 on Monday and receives 2,300 on Tuesday. After Monday: 5,200 - 1,845 = 3,355. After Tuesday: 3,355 + 2,300 = 5,655 items.
5. Step 5, Check with estimation: 5,200 - 1,800 = 3,400, then + 2,300 = 5,700. This is close to 5,655, so the working is sensible. Write each step as its own line.
6. Step 6, Your turn: a school had 6,400 books, gave away 2,175 and bought 1,850. How many now? Answer: 6,400 - 2,175 = 4,225; then 4,225 + 1,850 = 6,075 books.

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 and subtract four-digit numbers and solve multi-step problems: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Multiplication', you learn to multiply by one and two digits and use estimation. 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 'Multiplication' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: multiply by one and two digits and use estimation. 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, Estimate first by rounding. For 38 x 6, round 38 up to 40. Then 40 x 6 = 240. The real answer should be a little less, because 38 is a little less than 40. The exact answer is 228, which fits the estimate.
2. Step 2, Multiply a three-digit number by one digit. For 346 x 7 split into 300 + 40 + 6. Then 300 x 7 = 2100, 40 x 7 = 280 and 6 x 7 = 42. Add them: 2100 + 280 + 42 = 2422. Estimate 300 x 7 = 2100, close.
3. Step 3, Start two-digit multipliers by splitting the multiplier. For 24 x 13 do 24 x 10 = 240 and 24 x 3 = 72. Add: 240 + 72 = 312. Multiplying by 10 just moves each digit one place left, so it is quick.
4. Step 4, Use long multiplication for 47 x 26. Do 47 x 20 = 940, then 47 x 6 = 282. Add the rows: 940 + 282 = 1222. Estimate 50 x 30 = 1500. Both numbers were rounded up, so the exact answer is smaller than 1500, which it is.
5. Step 5, Solve a story. A shop packs 48 apples in each box and has 36 boxes. Multiply 48 x 36: 48 x 30 = 1440 and 48 x 6 = 288, which total 1728 apples. A quick check with 50 x 36 = 1800 shows the answer is reasonable.
6. Step 6, Your turn: estimate then work out 63 x 24. Round to 60 x 20 first, then use 63 x 20 and 63 x 4. Answer: estimate 1200, exact 1260 + 252 = 1512, close to the estimate but a bit higher because both numbers were rounded down.

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. Multiply by one and two digits and use estimation: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Division', you learn to divide up to four digits by one digit. 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 'Division' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: divide up to four digits by one digit. 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, Divide 756 by 3 using short division. 7 hundreds / 3 = 2 with 1 left. That 1 hundred joins 5 tens to make 15 tens; 15 / 3 = 5. Then 6 / 3 = 2. Answer 252.
2. Step 2, Check by multiplying: 252 x 3 = 756 (2 x 3 = 6, 5 x 3 = 15 write 5 carry 1, 2 x 3 + 1 = 7). The product equals the dividend, so 252 is right.
3. Step 3, Division with remainder: 1,475 / 6. 14 / 6 = 2 rem 2; 27 / 6 = 4 rem 3; 35 / 6 = 5 rem 5. So 245 remainder 5. Check: 245 x 6 = 1,470, plus 5 gives 1,475.
4. Step 4, Four digits: 3,648 / 4. The 3 thousands cannot be shared among 4, so join them with the hundreds: 36 hundreds / 4 = 9 hundreds. Then 4 tens / 4 = 1 ten and 8 ones / 4 = 2 ones. Result 912. Check: 912 x 4 = 3,648.
5. Step 5, Word problem: 1,284 oranges are packed in boxes of 6. 12 / 6 = 2; 8 / 6 = 1 rem 2; 24 / 6 = 4. So 214 boxes with nothing left. Check: 214 x 6 = 1,284.
6. Step 6, Your turn: divide 2,596 by 4 and 905 by 5. Answer: 2,596 / 4 = 649 since 649 x 4 = 2,596; and 905 / 5 = 181 since 181 x 5 = 905.

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. Divide up to four digits by one digit: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Factors and multiples', you learn to find factors, multiples and common multiples. 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 and multiples' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: find factors, multiples and common multiples. 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, Factors of 24: test numbers in pairs. 1 x 24, 2 x 12, 3 x 8, 4 x 6. Factors are 1, 2, 3, 4, 6, 8, 12, 24. A factor divides the number exactly.
2. Step 2, Multiples: the multiples of 6 are 6, 12, 18, 24, 30, 36. Multiples of 8 are 8, 16, 24, 32, 40. Multiples come from counting by the number, while factors go into it.
3. Step 3, Common multiples of 6 and 8: both lists contain 24, and the next common multiple is 48. The lowest common multiple (LCM) is 24. Write the first match you find.
4. Step 4, Common factors of 12 and 18: factors of 12 are 1, 2, 3, 4, 6, 12; factors of 18 are 1, 2, 3, 6, 9, 18. Common factors: 1, 2, 3, 6. Greatest is 6.
5. Step 5, Word problem: two buses leave a stop, one every 6 minutes and one every 8 minutes. They leave together at 9:00. They next leave together after 24 minutes, at 9:24.
6. Step 6, Your turn: list the common factors of 20 and 30, and the LCM of 4 and 10. Answer: common factors are 1, 2, 5, 10; LCM of 4 and 10 is 20.

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 factors, multiples and common multiples: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Fractions and decimals', you learn to compare, add and subtract fractions; link tenths and hundredths to decimals. 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 and decimals' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: compare, add and subtract fractions; link tenths and hundredths to 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 3/5 and 4/5: same denominator, so compare numerators. 4 > 3, so 4/5 > 3/5. For unlike denominators, 1/2 is greater than 1/4 because halves are bigger parts.
2. Step 2, Add: 3/8 + 2/8 = 5/8 since the denominators match. Subtract: 7/10 - 3/10 = 4/10, which simplifies to 2/5 (divide top and bottom by 2).
3. Step 3, Tenths to decimals: 3/10 = 0.3 because one tenth is 0.1. So 7/10 = 0.7. Hundredths: 45/100 = 0.45, with 4 tenths and 5 hundredths.
4. Step 4, Compare decimals: 0.6 and 0.45. Write 0.6 as 0.60. Now compare hundredths: 60 > 45, so 0.6 > 0.45. A longer decimal is not always bigger.
5. Step 5, Money link: 0.75 dirham is 75 fils, which is 75/100 = 3/4 of a dirham. Check: 3/4 of 100 fils = 75 fils. Fractions, decimals and money agree.
6. Step 6, Your turn: find 5/9 - 2/9 and write 3/100 and 6/10 as decimals. Answer: 3/9 = 1/3; and 0.03 and 0.6.

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. Compare, add and subtract fractions; link tenths and hundredths to decimals: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Measurement and perimeter', you learn to find perimeter and area of rectangles and convert units. 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 'Measurement and perimeter' 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 rectangles and convert units. 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, Perimeter of a rectangle 9 cm by 5 cm: add all four sides, 9 + 5 + 9 + 5 = 28 cm. Or use 2 x (9 + 5) = 2 x 14 = 28 cm. Perimeter is the distance around.
2. Step 2, Area of the same rectangle: length x width = 9 x 5 = 45 square cm. Area counts squares inside, so the unit is square cm (cm2), not cm.
3. Step 3, Find a missing side: a rectangle has perimeter 30 cm and length 10 cm. Half the perimeter is 15, and 15 - 10 = 5. So the width is 5 cm and area is 10 x 5 = 50 cm2.
4. Step 4, Convert units: 2.5 m = 250 cm since 1 m = 100 cm. A ribbon 3 m 40 cm long is 340 cm. For mass, 3 kg 200 g = 3,200 g, and 4,500 ml = 4 l 500 ml.
5. Step 5, Word problem: a garden is 12 m by 8 m. Fencing is its perimeter: 2 x (12 + 8) = 40 m. Grass area is 12 x 8 = 96 m2. Fence length and grass area use different units.
6. Step 6, Your turn: find the perimeter and area of a 15 m by 6 m field. Answer: perimeter 2 x 21 = 42 m; area 15 x 6 = 90 m2.

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 perimeter and area of rectangles and convert units: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Time and calendar', you learn to solve problems with time, timetables and the calendar. 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 'Time and calendar' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: solve problems with time, timetables and the calendar. 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 a timetable: bus leaves School at 7:45 a.m. and arrives at Market at 8:20 a.m. From 7:45 to 8:00 is 15 minutes, then 20 more. Journey time 35 minutes.
2. Step 2, 24-hour time: 3:15 p.m. is 15:15, found by adding 12 to 3. And 18:40 is 6:40 p.m. by subtracting 12. Midnight is 00:00, noon is 12:00.
3. Step 3, Calendar: October has 31 days, and if 5 October is a Monday then 12 and 19 are Mondays too. So 26 October is also a Monday, and 31 October is 5 days later, a Saturday.
4. Step 4, Days between: from 12 March to 4 April. Days left in March are 31 - 12 = 19, plus 4 days of April gives 23 days. Always count the month length you cross.
5. Step 5, Combine time: a film runs 1 hour 45 minutes and begins at 6:30 p.m. Add 1 hour to get 7:30, then 45 minutes to get 8:15 p.m. Check: 7:30 + 30 = 8:00, 15 more = 8:15.
6. Step 6, Your turn: a train leaves at 10:50 and arrives at 13:05. How long is the journey? Answer: 10 minutes to 11:00, then 2 hours to 13:00, plus 5 minutes; total 2 hours 15 minutes.

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 problems with time, timetables and the calendar: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Geometry', you learn to classify angles, triangles and quadrilaterals and find lines of symmetry. 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 'Geometry' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: classify angles, triangles and quadrilaterals 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, Classify angles by size. Acute is less than 90 degrees, such as 35. Right is exactly 90. Obtuse is between 90 and 180, such as 120. A straight angle is 180, and reflex is between 180 and 360, such as 250. Sort five angles by comparing with a paper corner.
2. Step 2, Classify triangles by sides. Equilateral has 3 equal sides and 3 angles of 60 degrees. Isosceles has 2 equal sides. Scalene has no equal sides. The three angles of any triangle add up to 180 degrees, so 60 + 60 + 60 works.
3. Step 3, Classify triangles by angles and find a missing one. A triangle with a right angle is right-angled. If all angles are under 90 it is acute, and if one is over 90 it is obtuse. With angles 90 and 35, the third is 180 - 125 = 55.
4. Step 4, Classify quadrilaterals. A square has 4 equal sides and 4 right angles. A rectangle has 4 right angles and equal opposite sides. A rhombus has 4 equal sides. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal neighbouring sides. Their angles add to 360 degrees.
5. Step 5, Find lines of symmetry by folding paper shapes. A rectangle has 2, a square has 4, an equilateral triangle has 3, an isosceles triangle has 1 and a circle has infinitely many. A parallelogram that is not a rectangle has none.
6. Step 6, Your turn: a triangle has angles 72 and 48 degrees. Find the third angle and name the triangle by its angles. Then give the lines of symmetry of a rhombus that is not a square. Answer: 180 - 120 = 60 degrees, acute triangle, and 2 lines of symmetry.

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. Classify angles, triangles and quadrilaterals and find lines of symmetry: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Data handling', you learn to read bar graphs and line graphs and find the mode. 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 handling' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: read bar graphs and line graphs and find the mode. 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 a bar graph: bars show pupils per sport. Football 12, cricket 9, tennis 5, swimming 9. The scale is 1 per square, so read the top of each bar against the axis.
2. Step 2, Find the mode: the value that occurs most often. In 12, 9, 5, 9 the number 9 appears twice, so the mode is 9. In the graph, cricket and swimming share it.
3. Step 3, Line graph: temperatures at 6, 8, 10, 12 o'clock were 18, 21, 26, 29 degrees. The line rises each time. From 8 to 10 the rise is 26 - 21 = 5 degrees.
4. Step 4, Find the greatest rise: between 6 and 8 the rise is 3, 8 to 10 is 5, 10 to 12 is 3. The steepest part of the line is 8 to 10, with a rise of 5.
5. Step 5, Total and difference from bars: total pupils = 12 + 9 + 5 + 9 = 35. Football minus tennis is 12 - 5 = 7. State what the numbers mean in a sentence.
6. Step 6, Your turn: find the mode of 4, 7, 7, 3, 7, 4 and the rise from 21 to 29 degrees. Answer: mode is 7, which appears three times; rise is 8 degrees.

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 bar graphs and line graphs and find the mode: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Patterns and money', you learn to continue number patterns and solve money 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 'Patterns and money' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: continue number patterns and solve money 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, Number pattern: 7, 14, 21, 28, __. The rule is add 7, so the next terms are 35 and 42. These are the multiples of 7. Check: 6 x 7 = 42.
2. Step 2, Pattern with a changing step: 2, 3, 5, 8, 12. Differences are 1, 2, 3, 4, so the next difference is 5 and the next term is 17. Always check the gaps.
3. Step 3, Money problem: three books cost 12.50, 8.75 and 15.25 dirhams. Total: 12.50 + 8.75 = 21.25, plus 15.25 = 36.50 dirhams. Align the decimal points in columns.
4. Step 4, Change: pay with 50 dirhams. 50.00 - 36.50 = 13.50. Check by counting up: 36.50 to 37 is 0.50, to 50 is 13, so 13.50. Change is 13.50 dirhams.
5. Step 5, Unit price: 4 pens cost 14 dirhams, so one pen costs 14 / 4 = 3.50 dirhams. Then 9 pens cost 9 x 3.50 = 31.50 dirhams. Check: 9 x 3 = 27 and 9 x 0.50 = 4.50, total 31.50.
6. Step 6, Your turn: continue 5, 12, 19, 26 and find the cost of 6 notebooks at 2.75 each. Answer: next 33, 40; and 6 x 2.75 = 16.50 dirhams.

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. Continue number patterns and solve money problems: 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 04 Mathematics is at this level. Its units include Large numbers; Addition and subtraction; Multiplication; Division 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.