Example Problem Pairs

Example Problem Pair: Circumference A circle has diameter of 8cm. What is its circumference in terms of 𝜋? A circle has diameter of 16cm. What is its circumference in terms of 𝜋? A circle has diameter of 1.6cm. What is its circumference in terms of 𝜋?

Example Problem Pair: Circumference from Radius A circle has radius of 8cm. What is its circumference in terms of 𝜋? A circle has radius of 4cm. What is its circumference in terms of 𝜋? A circle has radius of 1.6cm. What is its circumference in terms of 𝜋?

Higher Only: Example Problem Pair: Circumference from Radius The shape below is made up of a rectangle and a semi circle. What is its perimeter? The shape below is made up of a rectangle and a semi circle. What is its perimeter? 3cm Higher Tier Only 6cm

Example Problem Pair: Diameter from Circumference A circle has circumference of 90cm. What is its diameter? A circle has circumference of 30cm. What is its diameter? A circle has circumference of 1.8m. What is its diameter in cm?

Example Problem Pair: Revolutions A wheel has radius of 80cm. How far does it travel in one revolution? A wheel has a radius of 40cm, how far does it travel in one revolution? A wheel has a radius of 40cm. How many full revolutions will it make in 500m?

Example Problem Pair: Area from Radius A circle has radius of 8cm. a) Estimate its area. b) Calculate its area in terms of 𝜋 c) Calculate its area to 1 decimal place A circle has radius of 4cm. Estimate area: Area in terms of 𝜋 Area to 1 d.p. A circle has radius of 1.4cm. Estimate area: Area in terms of 𝜋 Area to 1 d.p.

Higher Problem Pair: Dimensions from Area A circle has area of 48cm2. a) Estimate its radius. b) Calculate its radius. c) Calculate its diameter. A circle has area of 125cm2. a) Estimate its radius. b) Calculate its radius. c) Calculate its diameter.

Example Problem Pair: Area of Part Circles Calculate the area of this semi circle with radius 10cm. Calculate the area of this semi circle with radius 12cm. Calculate the area of this semi circle with diameter 10cm.

Example Problem Pair: Area of Part Circles Calculate the area of this quarter circle with radius 10cm. Calculate the area of this quarter circle with radius 12cm. Calculate the area of this shape with radius 10cm.

Example Problem Pair: Shaded Areas (i) Calculate the shaded area. The length of the square is 8cm. Calculate the shaded area. The length of the square is 6cm.

Example Problem Pair: Shaded Areas (ii) Calculate the shaded area. 8cm 8cm 20cm 14cm Calculate the shaded area. 8cm 20cm

Example Problem Pair: Shaded Areas (iii) Calculate the shaded area. The radius of the white circle is 2m. Calculate the shaded area. The radius of the white circle is 6m. 6 Not sure if this is necessary - ONM

Example Pair: Total surface area 8 cm 4 cm 3 cm 6 cm 10 cm 5 cm

Example Pair: Volume from Areas

Example: Pair Volume of Prism (Cubes) 2 cm 2 cm 3 cm 4 cm 2 cm 2 cm Use the questioning from the previous slide to assist.

Example Pair: Calculate the volume 6 cm 10 cm 3 cm 5 cm 4cm 8 cm

Example Pairs: Volume of Cylinder (i) 10cm 2cm 12 cm 12 cm 2cm 4 cm

Example Pairs: Volume of Cylinder (ii) 10cm 2cm 12 cm 12 cm 4cm 2 cm

Area of a triangle 20m 10m 20m

Model: Area of a triangle 9cm 6cm 3cm 6m

Model: Area of a triangle 12cm 24cm 16cm 10cm 8cm 5cm

Example Problem Pair – F/P/S Elevation

Example Problem Pair – F/P/S Elevation

Example Problem Pair: Decimal Place Rounding Round to 1 decimal place: 0 . 0 8 0 . 1 8 0 . 1 5 0 . 1 4 8 . 1 4 8 . 1 5 1 8 . 2 5 Round to 1 decimal place: 0 . 0 4 0 . 3 4 0 . 3 9 0 . 3 5 7 . 3 5 7 . 3 2 1 7 . 3 9

Example Problem Pair: Significant Figure Rounding Round to 1 significant figure: 4 3 3 2 5 4 6 3 2 . 5 4 8 3 . 2 5 4 2 . 3 2 5 4 . 4 3 2 5 0 . 4 8 3 2 5 0 . 0 4 3 3 5 Round to 1 significant figure: 5 4 3 0 5 4 5 0 2 . 5 3 6 1 . 2 3 2 8 . 0 2 1 4 . 3 0 2 5 0 . 4 9 3 0 1 0 . 0 5 2 1 8

Example Problem Pair: The 9 and 99 Effect Round to 1 significant figure: 4 9 3 2 5 9 9 3 2 . 5 8 9 3 . 2 5 9 9 . 3 2 5 1 . 9 3 2 5 0 . 9 9 3 2 5 0 . 0 3 9 0 5 Round to 1 significant figure: 3 9 3 2 9 2 . 5 9. 9 3 9 9 . 9 0 . 2 9 2 0 . 0 9 9 1 4

Example Problem Pair: Speed Distance Time Calculate the speed of a train travelling 300km in 3 hours Calculate the speed of a coach travelling 300km in 6 hours Calculate the speed of a plane travelling 300km in 30minutes Bar model approach

Example Problem Pair: Speed Distance Time Calculate the speed of a train travelling 300km in 3 hours Calculate the speed of a coach travelling 300km in 6 hours Calculate the speed of a plane travelling 300km in 30minutes Non- bar model

Example Problem Pair: Speed Distance Time A cyclist is travelling at an average speed of 40km/h. How far will they go if they cycle for 5 hours? A cyclist is travelling at an average speed of 20km/h. How far will they go if they cycle for 5 hours? A cyclist is travelling at an average speed of 20km/h. How far will they go if they cycle for 30 minutes?

Example Problem Pair: Speed Distance Time A person runs at an average speed of 10km/h for 3 hours. How far will they run? A person runs at an average speed of 12km/h for 3 hours. How far will they run? A person runs at an average speed of 12km/h for half an hour. How far will they run?

Example Problem Pair: Distance Time Graph Non- bar model

Intelligent Practice

Example Problem Pair: Distance Time Graph Non- bar model

Example Problem Pair: Average Speed Non- bar model

Intelligent Practice 2

Example Your Turn Divide 30 in the ratio 4 : 1

Example Your Turn Divide 300 in the ratio 7 : 3 Alternatively Your Turn 2 could have divide 30 in the ratio 17:3

Example Your Turn Divide 12 in the ratio 2 1 2 : 1 1 2 Higher only

Intelligent Introduction – 10 minutes: Divide 20 in the ratio 3:2 Divide 20 in the ratio 2: 3 Divide 20 in the ratio 4 : 6 Divide 40 in the ratio 4 : 6 Divide 40 in the ratio 4 : 1 Divide 40 in the ratio 7 : 1 Divide 20 in the ratio 7 : 1 Divide 20 in the ratio 4:3:1 Divide 30 in the ratio 4:3:1 Divide 40 in the ratio 3 1 4 : 4 3 4 Divide 4 in the ratio 3 1 4 : 4 3 4 Divide 3 1 4 in the ratio 3:7 Craig Barton’s Drill

Example Problem Pairs. Bar model problems Between them, Abi and Barry have collected 40 cubes. Abby has three times as many cubes as Barry. How many does Barry have? Between them, Abi and Barry have collected 80 cubes. Abby has four times as many cubes as Barry. How many does Barry have? 40 Between them, Abi and Barry have collected 40 cubes. Abby has four times as many cubes as Barry. How many does Barry have? Barry Abi BAR MODEL RE-ARRANGED BY GXT

Example Problem Pairs. Bar model problems Between them, Abi and Barry have collected 4000 cubes. Abby has four times as many cubes as Barry. How many does each person have? Between them, Abi and Barry have collected 2000 cubes. Abby has four times as many cubes as Barry. How many does each person have? BAR MODEL RE-ARRANGED BY GXT

Examples: Andrew and Belle share some cubes in the ratio 5:3 Andrew received 10 more than Belle. How many cubes did they have altogether? Andrew and Belle share some cubes in the ratio 5:3 Andrew received 30 more than Belle. How many cubes did they have altogether?

Examples: Andrew and Belle share some cubes in the ratio 8:3 Andrew received 10 more than Belle. How many cubes did they have altogether? Andrew and Belle share some cubes in the ratio 8:3 Andrew received 30 more than Belle. How many cubes did they have altogether?

Intelligent Practice 1. Andrew and Belle share some money in the ratio 16 : 10 Andrew received £12 more than Belle. How much money does each person get? 5. Andrew and Belle share some money in the ratio 6:1 Andrew received £4 more than Belle. How much money does each person get? 2. Andrew and Belle share some money in the ratio 8 : 5 Andrew received £12 more than Belle. How much money does each person get? 6. Andrew and Belle share some money in the ratio 1:3 Belle received £2 more than Andrew. How much money does each person get? 4. Andrew and Belle share some money in the ratio 8 : 3 Andrew received £12 more than Belle. How much money does each person get? 7. Andrew and Belle share some money in the ratio 8:3 Belle received £6 less than Andrew. How much money did they have altogether? 3. Andrew and Belle share some money in the ratio 18 : 3 Andrew received £12 more than Belle. How much money does each person get? 8. Andrew and Belle share some money in the ratio 8:3 Belle received £3 less than Andrew. How much money did they have altogether?

Example: Find the perimeter and area of this shape Paired Example: Find the perimeter and area of this shape 10cm 12cm

Example You try The cost of entry to a theme park is decreased by 40% using a voucher. After using a voucher it costs £60. How much did it cost before? The cost of entry to a theme park is decreased by 20% using a voucher. After using a voucher it costs £60. How much did it cost before? The cost of entry to a theme park is decreased by 40% using a voucher. After using a voucher it costs £120. How much did it cost before?

Example You try The cost of a collectors item increases by 40% over time. The item is now worth £120. How much was it worth originally? The cost of a train ticket increases by 4%. The ticket now costs £120. How much did it cost originally? The cost of a train ticket increases by 4%. The ticket now costs £60. How much did it cost originally?

Intelligent Practice The cost of entry into a theme park is reduced by 20% using a voucher. After using the voucher, the cost of entry is £40. How much was it before? The cost of entry into a theme park is reduced by 40% using a voucher. After the using the voucher, the cost of entry is £40. How much was it before? The cost of entry into a theme park is reduced by 40% using a voucher. After the using the voucher, the cost of entry is £80. How much was it before? The cost of entry into a theme park is reduced by 4% using a voucher. After the using the voucher, the cost of entry is £80. How much was it before? The cost of entry into a theme park is increased by 4%. After the increase, the cost of entry is £80. How much was it before? The cost of entry into a theme park is increased by 40%. After the increase, the cost of entry is £80. How much was it before? The cost of entry into a theme park is increased by £4. After the increase, the cost of entry is £80. What percentage increase is this? The cost of entry into a theme park is decreased by £4. After the decrease, the cost of entry is £80. What percentage decrease is this?

A voucher takes 15% of a restaurant bill. Same Surface, Different Depth A voucher takes 15% of a restaurant bill. A voucher takes 15% of a restaurant bill. After using the voucher the bill was £45. How much was the bill before using the voucher? Before using the voucher the bill was £45. How much will I pay? A voucher takes 15% of a restaurant bill. A voucher takes 15% of a restaurant bill. What fraction of the total bill will I pay? Give your answer in its simplest form. After using the voucher I split the bill with my three friends. What fraction of the total bill will I pay?

Example Problem Pair Your Turn a° 310° 1 a° 320° 60° a° Lower

Example Problem Pair Your Turn a° 50° a° 230° 20° 260° 60° 20° a° 1 Lower 20°

Example Problem Pair Your Turn 1 a° 200° a° 220° 120° a° Lower

Example Problem Pair Your Turn 100° a° 100° 50° a° 110° 70° 120° a° 80° a° Lower 50° 110°

Example Problem Pair 130° 110° 30°

Example Problem Pair 10° 100° 168°

Example Problem Pair 125° 115° 45°

Example Problem Pair 42° b 42° a 72° 62° 54° c 68°

Example Problem Pair: Vertically Opposite Angles 110° 120° 35°

Amy has 3 counters Barney has 7 counters. What is the mean amount of counters? Amy has 5 cubes. Barney has 7 cubes. What is the mean amount of cubes.

Amy has 13 cubes Barney has 7 cubes. What is the mean amount of counters? Amy has 15 counters Barney has 7 counters. What is the mean amount of counters?

Amy has £100 Barney has £180. What is the mean amount of money? Amy has £130 Barney has £170. What is the mean amount of money?

Amy has 3 counters Barney has 16 counters. What is the mean amount of counters? Amy has 22 counters Barney has 3 counters. What is the mean amount of counters?

Amy has £12 Barney has £1. What is the mean amount of money? Amy has £3 Barney has £12. What is the mean amount of money?

Intelligent Practice: The mean of two numbers Find the mean of 8 cubes and 14 cubes 13cm and 7cm 11m and 4m £3 and £4 £400 and £200 £7.50 and £6.50 3m and 14m 17 stickers and 6 stickers £6.50 and £8.50 10 cubes and 1400 cubes

Example Problem Pairs: The mean of a set of numbers Amy has 1 counter. Barney has 7 counters. Charlie has 10 counters. What is the mean amount of counters? Amy has 1 cube. Barney has 7 cubes. Charlie has 4 cubes. What is the mean amount of cubes>

Example Problem Pairs: The mean of a set of numbers Amy has 1 counter. Barney has 7 counters. Charlie has 7 counters. What is the mean amount of counters? Amy has 2 cubes. Barney has 8 cubes. Charlie has 8 cubes. What is the mean amount of cubes?

Example Problem Pairs: The mean of a set of numbers Amy has 1 counters Barney has 7 counters. Charlie has 10 counters. Dave has 2 counters. What is the mean amount of counters? Amy has 2 cubes. Barney has 7 cubes. Charlie has 6 cubes. Dave has 9 counters. What is the mean amount of cubes>

Example Problem Pairs: The mean of a set of numbers Amy has 1 counters Barney has 7 counters. Charlie has 10 counters. Dave has 2 counters. Edgar has 10 counters. What is the mean amount of counters? Amy has £2. Barney has £7. Charlie has £6. Dave has £9. Edgar has £1. What is the mean amount of money?

Intelligent Practice: The mean of a set of numbers Find the mean of 6 cubes and 10 cubes. 6 counters, 10 counters and 2 counters. 6 cubes, 10 cubes, 2 cubes and 2 cubes. £3, £4 and £8 £3, £4, £8, and £9 £3, £4, £8, and £5 £300, £400 and £800 £10.30, £10.80 and £8.90 £6.50, £7.50, £2.50 and £5.50 £1.50, 75p, £1.20 and 80p

Example Number Frequency CF 0 < N ≤ 5 26 5 < N ≤ 10 100 8 40 < N ≤ 60 4 60 < N ≤ 100 2 Number Frequency Midpoint (X) FX CF 0 < N ≤ 2 4 2 < N ≤ 6 10 14 6 < N ≤ 10 18 10 < N ≤ 20 2 20 Total Modal class Median value Median Class Interval Modal class Number Frequency CF 0 < N ≤ 5 26 5 < N ≤ 10 100 10 < N ≤ 20 2 20 < N ≤ 50 20 Median value Median Class Interval Modal class Median value Median Class Interval

Intelligent Practice: Find the median class interval CF 0 < N ≤ 2 1 2 < N ≤ 6 5 6 6 < N ≤ 10 12 10 < N ≤ 20 11 23 F CF 0 < N ≤ 2 2 2 < N ≤ 6 10 12 6 < N ≤ 10 24 10 < N ≤ 20 22 46 F CF 0 < N ≤ 2 2 2 < N ≤ 6 10 12 6 < N ≤ 10 24 10 < N ≤ 14 34 14 < N ≤ 20 46 F CF 0 < N ≤ 2 2 2 < N ≤ 6 8 10 6 < N ≤ 10 14 24 F CF 0 < N ≤ 2 14 2 < N ≤ 6 8 22 6 < N ≤ 10 2 24 F CF 0 < N ≤ 2 120 2 < N ≤ 6 20 140 6 < N ≤ 10 100 240

Example Your Turn Your Turn Time taken (t seconds) Frequency Class Width Frequency Density 10 < t ≤ 50 5 50 < t ≤ 60 4 60 < t ≤ 70 8 70 < t ≤ 90 27 90 < t ≤ 130 24 Time taken (t seconds) Frequency Class Width Frequency Density 10 < t ≤ 30 5 30 < t ≤ 35 4 35 < t ≤ 40 8 40 < t ≤ 50 27 50 < t ≤ 70 24 Your Turn Time taken (t seconds) Frequency Class Width Frequency Density 10 < t ≤ 50 10 50 < t ≤ 60 2 60 < t ≤ 70 40 70 < t ≤ 90 9 90 < t ≤ 130 3

Standard Form 6.708 x 103 4.708 x 102 5.6 x 103 6 x 103 4.302 x 103 1.5672 x 102 4.6 x 103 8 x 105

Standard Form 6.708 x 10-3 6.5 x 10-2 7x10-1 4.302 x 10-2 1.3 x 10-3 7x10-1

Writing in Standard Form 6000 320 625.2 400 45000 4354.3

Standard Form 0.8 0.064 0.04305 0.4 0.052 0.0608

Words to Standard Form What is 7 million in standard form? Write 3 million in standard form. Write 40 million in standard form. Write a quarter of a million in standard form? What is 7 million in standard form? Write 60 million in standard form. What is half a million in standard form?

Compensating to ‘fix’ standard form Remind pupils that indices are added when we multiply. Point out that 14.4 × 108 is not in standard form and discuss how it can be converted into the correct form. 510 x 105 181 x 103 5100 x 105

Compensating to ‘fix’ standard form Remind pupils that indices are added when we multiply. Point out that 14.4 × 108 is not in standard form and discuss how it can be converted into the correct form. 0.015 x 105 0.0523 x 103

Multiplying with standard form What is (2 × 105) x (7 × 103)? What is (2 × 105) x (8 × 103)? What is (2 × 106) x (8 × 105)? Remind pupils that indices are added when we multiply. Point out that 14.4 × 108 is not in standard form and discuss how it can be converted into the correct form. What is (3 × 106) x (8 × 105)?

Dividing with standard form What is (16 × 109 )÷ (4 × 107 )? What is (16 × 109 )÷ (2 × 107 )? What is (16 × 109 )÷ (8 × 107 )? Remind pupils that indices are subtracted when we divide. Discuss how 0.25 × 10-13 can be converted into the correct form. What is (16 × 107 )÷ (8 × 109 )?

What is (8 × 109 )÷ (16 × 107 )? What is (4 × 109 )÷ (16 × 107 )? What is (2 × 109 )÷ (16 × 107 )?

Finding the median and IQR from a list 8 , 4, 13, 12, 14, 20, 11, 7, 10, 12, 13, 15, 16, 20 7, 10, 12, 13, 15, 16, 20, 25 17, 20, 42, 53, 55, 86, 200, 250 3, 4, 5, 6, 6, 8, 9, 9, 9, 13, 17, 50, 10 , 7, 13, 12, 15, 20, 16, 20, 22, 30,