Grade 02: Mathematics

In this unit, 'Place value to 100', you learn to show a number as tens and ones and compare 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 'Place value to 100' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: show a number as tens and ones and compare 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, Break 63 into tens and ones: 6 tens and 3 ones, which is 60 + 3. Draw 6 rods of ten and 3 unit cubes. The 6 in the tens place is worth 60, not 6.
2. Step 2, Compare 47 and 74: both use digits 4 and 7. Look at the tens first. 7 tens is more than 4 tens, so 74 > 47. Only look at the ones if the tens match.
3. Step 3, Compare with equal tens: 58 and 52. Tens are both 5, so compare ones: 8 is more than 2. So 58 > 52. Write it as 52 < 58 and read it two ways.
4. Step 4, Order numbers: 39, 93, 29, 91. Tens: 3, 9, 2, 9. Smallest tens is 29, then 39. The two with 9 tens are compared by ones: 91 before 93. Order is 29, 39, 91, 93.
5. Step 5, Build a number: with digit cards 5, 8 and 2, make the largest two-digit number using two cards. Put the biggest digit in tens: 85. The smallest is 25. Check, 85 is above 25.
6. Step 6, Your turn: write 76 as tens and ones, and compare 68 and 86. Answer: 76 is 7 tens and 6 ones, 70 + 6. And 68 < 86 because 6 tens is less than 8 tens.

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. Show a number as tens and ones and compare numbers: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Addition to 100', you learn to add two-digit numbers with and without carrying. 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 to 100' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: add two-digit numbers with and without carrying. 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 without carrying: 34 + 25. Ones: 4 + 5 = 9. Tens: 3 + 2 = 5. So the answer is 59. Line up the ones under ones and tens under tens before adding.
2. Step 2, Add with carrying: 47 + 28. Ones: 7 + 8 = 15, write 5 and carry 1 ten. Tens: 4 + 2 = 6, plus the carried 1 is 7. The answer is 75.
3. Step 3, Check with rounding: 47 is about 50 and 28 is about 30, so the sum should be near 80. Our answer 75 is close to 80, so it is sensible. A far-off answer means check again.
4. Step 4, Word problem: a shop sold 56 mangoes in the morning and 38 in the evening. 56 + 38: ones 6 + 8 = 14, write 4 carry 1; tens 5 + 3 + 1 = 9. Total 94 mangoes.
5. Step 5, Add by partitioning: 56 + 38 = 56 + 30 + 8 = 86 + 8 = 94. Same answer as the column method, so both agree. Using two methods catches slips in the carry.
6. Step 6, Your turn: find 68 + 25 and check by rounding. Answer: ones 8 + 5 = 13, write 3 carry 1; tens 6 + 2 + 1 = 9. So 93. Rounded 70 + 30 = 100, near 93.

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 two-digit numbers with and without carrying: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Subtraction to 100', you learn to subtract two-digit numbers with and without borrowing. 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 'Subtraction to 100' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: subtract two-digit numbers with and without borrowing. 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, Subtract without borrowing: 79 - 34. Ones: 9 - 4 = 5. Tens: 7 - 3 = 4. The answer is 45. Write the bigger number on top and line up the places.
2. Step 2, Subtract with borrowing: 52 - 27. Ones: 2 is less than 7, so borrow a ten from the 5. The 5 tens become 4 tens and the 2 ones become 12. Then 12 - 7 = 5 and 4 - 2 = 2. Answer: 25.
3. Step 3, Check by adding: 25 + 27 = 52, since 5 + 7 = 12, write 2 carry 1, and 2 + 2 + 1 = 5. This matches the number we started with, so 25 is correct.
4. Step 4, Word problem: a library had 83 books and lent out 46. 83 - 46: borrow, so ones 13 - 6 = 7 and tens 7 - 4 = 3. Answer: 37 books are left.
5. Step 5, Check with counting up: from 46 to 50 is 4, then 50 to 83 is 33, so 4 + 33 = 37. This second method agrees with 37 and proves the borrowing was done right.
6. Step 6, Your turn: find 71 - 38 and check by adding. Answer: ones 11 - 8 = 3, tens 6 - 3 = 3, so 33. Check: 33 + 38 = 71.

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. Subtract two-digit numbers with and without borrowing: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Multiplication as equal groups', you learn to use groups, arrays and repeated adding; the tables of 2, 5 and 10. 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 as equal groups' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: use groups, arrays and repeated adding; the tables of 2, 5 and 10. 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, Make equal groups: draw 3 plates with 4 apples on each plate. Count all the apples by adding the groups, 4 + 4 + 4 = 12. We say 3 groups of 4 make 12. Write it as 3 x 4 = 12 and read it as three times four equals twelve.
2. Step 2, Build an array, which is rows and columns of equal size. Put 5 counters in each of 2 rows. Add by rows, 5 + 5 = 10, so 2 x 5 = 10. Turn the array and see 5 columns of 2, so 5 x 2 = 10 too. The order does not change the answer.
3. Step 3, Learn the table of 2 by skip counting in twos: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. Each answer is a double, so 6 x 2 = 12 because 6 + 6 = 12. Touch a finger for each jump and say the count aloud, then say the whole table without looking.
4. Step 4, Learn the table of 5 by counting in fives: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. Every answer ends in 5 or 0. To find 7 x 5, count 7 jumps of 5 and reach 35. A clock helps, since each number on the face is 5 minutes.
5. Step 5, Learn the table of 10 by counting in tens: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. Four groups of ten is 4 x 10 = 40, so the answer is the number with a 0 after it. Use ten-rods or ten coins to build 4 tens and count them.
6. Step 6, Your turn: there are 6 bags with 5 marbles in each bag. How many marbles in all? Then find 8 x 2. Answer: 6 x 5 = 30 marbles, by counting 5, 10, 15, 20, 25, 30, and 8 x 2 = 16, because 8 + 8 = 16.

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 groups, arrays and repeated adding; the tables of 2, 5 and 10: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Sharing and grouping', you learn to share equally and link sharing to multiplication. 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 'Sharing and grouping' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: share equally and link sharing to multiplication. 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, Share equally: 12 sweets among 3 children. Deal one at a time: each child gets 1, then 2, then 3, then 4. Each gets 4 sweets and none are left. So 12 shared by 3 is 4.
2. Step 2, Group instead: 12 sweets in bags of 3. Circle groups of 3: you can make 4 bags. Sharing 12 into 3 groups and making groups of 3 both give 4.
3. Step 3, Link to multiplication: 4 bags of 3 sweets is 4 x 3 = 12. Reverse it: 12 divided by 3 = 4. The multiplication fact and the sharing fact are about the same numbers.
4. Step 4, Check unequal sharing: 10 apples among 3 children gives 3 each with 1 left over, because 3 x 3 = 9. Equal sharing needs the same number for everyone, so one apple remains.
5. Step 5, Word problem: 20 pencils shared equally among 5 pupils. Count in fives: 5, 10, 15, 20 is four jumps, so each gets 4. Check: 5 x 4 = 20.
6. Step 6, Your turn: share 18 stickers among 3 friends, and write the multiplication check. Answer: each gets 6, because 3 x 6 = 18.

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. Share equally and link sharing to multiplication: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Fractions: halves and quarters', you learn to name and show halves, thirds and quarters of shapes. 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: halves and quarters' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: name and show halves, thirds and quarters of 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, Half of a shape: fold a square paper so the two parts match exactly. Each part is one half, written 1/2. Two halves make a whole, and the parts must be equal in size.
2. Step 2, Quarters: fold the square again so there are 4 equal parts. Each part is one quarter, 1/4. Colour 1 part and say one quarter is shaded; 3 parts shaded is three quarters.
3. Step 3, Thirds: a rectangle cut into 3 equal strips gives thirds. Shade 1 strip and write 1/3. If the strips are different sizes then they are not thirds, so measure them.
4. Step 4, Half of a number: half of 12 cookies means sharing into 2 equal groups, so 12 shared by 2 is 6. A quarter of 12 is 3 cookies since 4 groups of 3 make 12.
5. Step 5, Spot a wrong split: a circle cut into 2 pieces, one big and one small, is not in halves. Both parts need the same size, so the split fails and is not 1/2.
6. Step 6, Your turn: what is a quarter of 8 sweets, and how many equal parts make thirds? Answer: a quarter of 8 is 2, since 4 groups of 2 make 8; thirds have 3 equal parts.

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. Name and show halves, thirds and quarters of shapes: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Length, mass and capacity', you learn to measure with units and choose the right tool. 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 'Length, mass and capacity' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: measure with units and choose the right tool. 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 length: line a ruler's 0 mark with the left end of a pencil. The right end reaches 15, so the pencil is 15 cm long. Always start at 0, not at the edge of the ruler.
2. Step 2, Choose the tool: use a ruler for a pencil, a tape measure for a desk, and a metre stick for a door. Pick a tool that fits the size, so a ruler for small things.
3. Step 3, Mass: a bag of sugar on a kitchen scale reads 2 kg. A bag of rice reads 5 kg. The rice is 3 kg heavier since 5 - 2 = 3. Scales measure mass in grams or kilograms.
4. Step 4, Capacity: a jug holds 1 litre and a cup holds a quarter of a litre. Pouring 4 cups fills the jug, because 4 x 1/4 = 1. Choose litres for large amounts of liquid.
5. Step 5, Compare lengths: a rope is 90 cm and a stick is 65 cm. The rope is longer by 90 - 65 = 25 cm. Check by adding: 65 + 25 = 90.
6. Step 6, Your turn: which tool measures a classroom floor, and how much longer is a 120 cm ribbon than an 85 cm one? Answer: a tape measure; 35 cm longer, since 120 - 85 = 35.

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 with units and choose the right tool: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Time and money', you learn to read time to five minutes and add small amounts of money. 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 money' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: read time to five minutes and add small amounts of money. 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 the clock to 5 minutes: the minute hand on 3 means 15 minutes past, because each number is 5 minutes. With the hour hand just past 4, the time is quarter past 4, or 4:15.
2. Step 2, Count in fives: minute hand on 7 means 7 x 5 = 35 minutes. Count 5, 10, 15, 20, 25, 30, 35. With the hour hand between 6 and 7 the time is 6:35, which is 25 minutes to 7.
3. Step 3, Add money: a pen costs 15 dirhams and a notebook costs 28 dirhams. Add: 5 + 8 = 13, write 3 carry 1; 1 + 2 + 1 = 4. Total is 43 dirhams.
4. Step 4, Change: pay 50 dirhams for the 43 dirham items. Count up from 43 to 45 is 2, then to 50 is 5, so 7 dirhams change. Check: 43 + 7 = 50.
5. Step 5, Time later: it is 3:25 and a film starts in 10 minutes. Count on 10 minutes from 25 to get 35. The film starts at 3:35. Check on a clock face: minute hand moves from 5 to 7.
6. Step 6, Your turn: say the time when the minute hand is on 9 and the hour hand is between 8 and 9, and add 24 + 19 dirhams. Answer: 8:45, quarter to 9; and 24 + 19 = 43 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. Read time to five minutes and add small amounts of money: at least 4 of 5 attempts correct, said or written, with one real example from home or school.

In this unit, 'Shapes and data', you learn to name flat and solid shapes and read simple picture 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 'Shapes and data' before using symbols.
• Say the rule in your own words, then write one worked line, naming each step.
• Use it: name flat and solid shapes and read simple picture 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, Name flat shapes: a circle is round with no corners, a triangle has 3 sides and 3 corners, a square has 4 equal sides, and a rectangle has 4 sides with two long and two short. Trace a coin, a book and a sandwich slice to find them.
2. Step 2, Name solid shapes: a ball is a sphere, a dice is a cube, a cereal box is a cuboid, a tin is a cylinder, and a party hat is a cone. Solid shapes can be held. Pick up a tin and roll it, then try to roll a box.
3. Step 3, Count the parts of a cube: it has 6 flat faces, 12 edges and 8 corners. Each face is a square. Touch each face of a dice and count aloud. A flat shape lives inside a solid, such as the circle on top of a tin.
4. Step 4, Read a picture graph. Each smiley stands for 1 child's favourite fruit. Apple has 4 smileys, banana has 6 and orange has 3. Put a finger on the first row, say apple, and count the pictures to get 4.
5. Step 5, Answer questions from the graph. Which fruit is most popular? Banana, with 6. How many more chose apple than orange? 4 take away 3 leaves 1 more. How many children in all? Add 4 + 6 + 3, which makes 13.
6. Step 6, Your turn: a graph shows dog 5, cat 3 and fish 2 pictures. How many more children chose dog than cat? How many children in all? Name the solid shape of a can. Answer: 2 more chose dog, 10 children in all, and a can is a cylinder.

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. Name flat and solid shapes and read simple picture graphs: 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.

This course: Grade 02 Mathematics is at this level. Its units include Place value to 100; Addition to 100; Subtraction to 100; Multiplication as equal groups and more.

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.

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.