Understand what is meant by UPPER and LOWER BOUNDS Objectives Understand what is meant by UPPER and LOWER BOUNDS Use these to calculate bounded answers Terms and Conditions: To the best of the producer's knowledge, the presentation’s academic content is accurate but errors and omissions may be present and Brain-Cells: E.Resources Ltd cannot be held responsible for these or any lack of success experienced by individuals or groups or other parties using this material. The presentation is intended as a support material for GCSE maths and is not a comprehensive pedagogy of all the requirements of the syllabus. The copyright proprietor has licensed the presentation for the purchaser’s personal use as a teaching and learning aid and forbids copying or reproduction in part or whole or distribution to other parties or the publication of the material on the internet or other media or the use in any school or college that has not purchased the presentation without the written permission of Brain-Cells: E.Resouces Ltd.

What is the height of each of these herbs to the nearest centimetre? 8 cm 8 cm

We have rounded this size up to 8 cm We have rounded this size down to 8 cm

Measuring to the nearest centimetre, we would round this plant height - which looks to be 7.6 cm - up to 8 cm

Measuring to the nearest centimetre, we would round this plant height - which looks to be just under 8.4 cm - down to 8 cm

We call the numbers 7.5 cm and 8.5 cm the bounds of 8 cm. To the nearest cm, a length of 8 cm is any length between 7.5 and 8.4999999… (8.5) cm. 8.5 cm is the upper bound We call the numbers 7.5 cm and 8.5 cm the bounds of 8 cm. 7.5 cm is the lower bound

1 mm is 0.1 cm so it will be plus and minus half of 0.1 To the nearest mm, a length of 2 cm is any length between 1.95 and 2.05 cm. 1 mm is 0.1 cm so it will be plus and minus half of 0.1 ± 0.05 2.05 cm is the upper bound 1.95 cm is the lower bound

Using the units required for the bounds, what would be the upper and lower bounds of these: Units for bounds Size To nearest L. Bound U. Bound 12 cm cm 54 cm mm 27 metres 46 litres 1/10 litres 11.5 cm 12.5 cm 53.95 cm 54.05 cm 26.995 m 27.005 m 45.95 litres 46.05 litres

What are the upper and lower bounds? Size To nearest Lower B. Upper B. 19 metres metre 33 cm cm 7 mm mm 67 cm 63 cm 72 cm 156 metres 81 metre 40 km

What are the upper and lower bounds? Size To nearest Lower B. Upper B. 19 metres metre 18.5 metres 19.5 metres 33 cm cm 32.5 cm 33.5 cm 7 mm mm 6.5 mm 7.5 mm 67 cm 66.95 cm 67.05 cm 63 cm 62.95 cm 63.05 cm 72 cm 71.95 cm 72.05 cm 156 metres 155.995 m 156.005 m 81 metre 80.995 m 81.005 m 40 km 39.9995 km 40.0005 km

Bounds and Linear Calculations

Five cuboids each have a height of 12 cm to the nearest mm Five cuboids each have a height of 12 cm to the nearest mm. If the cuboids are stacked as shown, what is the minimum and maximum height of the pile? Height 12 cm

To the nearest mm, the bounds of 12 cm are… Five cuboids each have a height of 12 cm to the nearest mm. If the cuboids are stacked as shown, what is the minimum and maximum height of the pile? To the nearest mm, the bounds of 12 cm are… Height 12 cm Lower Bound is 11.95 Upper Bound is 12.05 Minimum height 11.95 x 5 = 59.75 cm Maximum height 12.05 x 5 = 60.25 cm

To the nearest mm, the bounds of 5 cm are… To the nearest millimetre, the sides of this regular hexagon are 5 cm long. What are the upper and lower bounds of its perimeter? Lower Bound is 4.95 Upper Bound is 5.05 5 cm Lower bound of the perimeter is: 4.95 x 8 = 39.6 cm Upper bound of the perimeter is: 5.05 x 8 = 40.4 cm To the nearest mm, the bounds of 5 cm are…

Here are four for you to try…

If the dimensions are to the nearest mm If the dimensions are to the nearest mm. What are the upper and lower bounds for each shape’s perimeter? Regular hexagon 7 cm Isosceles triangle 12 cm LB 35.85 cm UB 36.15 cm 6.95 x 6 7.05 x 6 LB 41.7 cm UB 42.3 cm 11.95 x 3 12.05 x 3 Regular pentagon Rhombus 8.95 x 5 9.05 x 5 LB 44.75 cm UB 45.25 cm 9.95 x 5 10.05 x 5 LB 39.8 cm UB 40.2 cm 9 cm 10 cm

Bounds and Area Calculations

Find the bounds of its lengths Use these to find respective areas… If the lengths are to the nearest mm, what are the upper and lower bounds of this rectangle’s area? Upper B. 4.55 Find the bounds of its lengths 12.5 cm 4.5 cm Lower B. 4.45 Upper B. 12.55 Use these to find respective areas… Lower B. 12.45 Lower Bound Area 12.45 x 4.45 = 55.40 cm2 to 2 dp Upper Bound Area 12.55 x 4.55 = 57.10 cm2 to 2 dp

Here are four for you to try…

If the lengths are to the nearest mm, what are the upper and lower bounds of the shape’s areas? 15.2 cm 5.6 cm 12.7 cm 8.9 cm LB  84.08 cm2 to 2 dp UB  86.16 cm2 to 2 dp LB  55.98 cm2 to 2 dp UB  57.06 cm2 to 2 dp 10.3 cm 6.2 cm 2.7 cm Assume ∏ = 3.14 10.2 cm diameter LB  21.73 cm2 to 2 dp UB  22.83 cm2 to 2 dp LB  80.87 cm2 to 2 dp UB  82.47 cm2 to 2 dp

Bounds and Volume Calculations

Find the bounds of its lengths Use these to find respective volumes… If the lengths are to the nearest mm, what are the upper and lower bounds of this cuboid’s volume? Find the bounds of its lengths Use these to find respective volumes… UB 10.05 LB 9.95 UB 17.55 LB 17.45 10 cm 17.5 cm UB 9.25 LB 9.15 9.2 cm Lower Bound Volume 9.15 x 9.95 x 17.45 = 1588.69 cm3 to 2 dp Upper Bound Volume 9.25 x 10.05 x 17.55 = 1631.49 cm3 to 2 dp

Here are two for you to try…

Find the bounds of its lengths If the lengths are to the nearest mm, what are the upper and lower bounds of this cuboid’s volume? Find the bounds of its lengths Assume ∏ = 3.14 8.3 cm radius UB 8.35 LB 8.25 15.2 cm UB 15.25 LB 15.15 Lower Bound Volume 3.14 x 8.252 x 15.15 = 3237.80 cm3 to 2 dp Upper Bound Volume 3.14 x 8.352 x 15.25 = 3338.66 cm3 to 2 dp

Find the bounds of its lengths Use these to find respective volumes… If the lengths are to the nearest cm, what are the upper and lower bounds of this triangular prism’s volume? Find the bounds of its lengths 1.5 metres 1.7 metres 2.5 metres Use these to find respective volumes… UB 1.505 LB 1.495 UB 2.505 LB 2.495 UB 1.705 LB 1.695 Lower Bound Volume 1.695 x 1.495 ÷ 2 x 2.495 = 3.16 metres3 to 2 dp Upper Bound Volume 1.705 x 1.505 ÷ 2 x 2.505 = 3.21 metres3 to 2 dp

Bounds and Calculations

A joiner has lengths of timber that are 2 A joiner has lengths of timber that are 2.4 metre long to the nearest cm. It is possible to set a joiner’s circular saw so that it will consistently cut wood accurately to the nearest mm of the setting size. The saw’s blade is exactly 2 mm wide. If the joiner cuts 5 pieces on a setting of 36 cm long, what will be the lower bound length of the unused timber? 36 cm Unused length Do a sketch of the problem 5 pieces 36 cm to nearest mm from a length 2.4 metres long to the nearest cm We can assume blade cut is exactly 2 mm wide We want the upper bound of the unused length

A joiner has lengths of timber that are 2 A joiner has lengths of timber that are 2.4 metre long to the nearest cm. It is possible to set a joiner’s circular saw so that it will consistently cut wood accurately to the nearest mm of the setting size. The saw’s blade is exactly 2 mm wide. If the joiner cuts 5 pieces on a setting of 36 cm long, what will be the lower bound length of the unused timber? 36 cm Unused length Work in cm Find the bounds UB 36.05 LB 35.95 UB 240.5 LB 239.5 36 cm 240 cm Unused LB = 239.5 – 5 x 36.05 – 1  58.25 cm Blade 5 x 2 mm = 1 cm

Here is one for you to try…

Upper Bound Petrol consumption = Notice that to find upper bound consumption, the upper bound is divided by the lower bound Kathy drove 238 miles correct to the nearest mile. She used 23.7 litres of petrol correct to the nearest tenth of a litre. Workout the upper bounds for the petrol consumption for Kathy’s journey. Give your answer to 2 d.p. Miles travelled Litres of petrol used Petrol consumption = Find the bounds 238 miles UB 238.5 LB 237.5 23.7 litres UB 23.75 LB 23.65 238.5 23.65 Upper Bound Petrol consumption = 10.08 miles/litre