### Maths KS3

*Welcome to the Tunnelworks KS3 Maths area.*

Here you'll find four 'essentials' lessons, complete with interactive online content, student worksheets and detailed lesson plans. Each lesson challenges students to solve a realistic engineering problem to help design, build or operate the Thames Tideway Tunnel. You can dig deeper with follow-up ideas for more able students.

In lessons one and two, explore right-angled triangles and Pythagoras' theorem using simplified engineering drawings, and use co-ordinates to create a simple map of the proposed Thames Tideway Tunnel route.

In lessons three and four, use algebra to calculate how fast a Tunnel Boring Machine (TBM) can bore through the ground, and use percentages and ratios to explore how rainfall across London drains into four Combined Sewer Overflows (CSOs).

**CREST Bronze Awards**

Two projects are also outlined, if you wish to connect your work to this well regarded science award scheme.

**Research project one:** students can investigate how the weather and our water use are changing in London and how they create the need for an upgrade to London's sewer system.
Investigative project: create two maps of your school grounds and buildings in as much detail as you can, using GPS for one map and a trundle wheel and compass for the other.

**Research project two:** students are set the challenge that their school is expanding and new buildings need to be built on the other side of a busy road.
Investigative project: plan a tunnel to connect the two sites and produce a report. They will consider different challenges such as the geology of the ground, lighting, materials and equipment to be used.

# Introducing the Thames Tideway Tunnel

Watch the video to find out why we need to clean up the River Thames.

# Maths KS3 Lesson 1: Right Angles and Pythagoras

# Maths KS3 Lesson 1: Right Angles and Pythagoras

**Screen 1.1:** This map shows the Combined Sewer Overflow (CSO) sites that we will connect to the Thames Tideway Tunnel to control sewerage discharge into the river. Click on the next arrow to see a foreshore site.

# Maths KS3 Lesson 1: Right Angles and Pythagoras

**Screen 1.2:** This is a typical 'foreshore' site. During construction a 'cofferdam' will reclaim some of the foreshore land for a temporary period, to allow construction to take place. Click on the next arrow to view a typical CSO.

# Maths KS3 Lesson 1: Right Angles and Pythagoras

**Screen 1.3:** Under the ground, we will build a drop shaft over 40m deep. This will link the existing sewer to the new Main Tunnel, deep under the River Thames.

# Maths KS3 Lesson 1: Right Angles and Pythagoras

**Screen 2:** Play the video to hear Suliaman discuss the role of a typical CSO and find out your challenge.

# Maths KS3 Lesson 1: Right Angles and Pythagoras

Click the next arrow to

show how Pythagoras'

Theorem helps us

find the lengths

of each side.

**Screen 3.1:** Find out how Pythagoras' Theorem helps us calculate the lengths of a right angled triangle.

# Maths KS3 Lesson 1: Right Angles and Pythagoras

**Screen 3.2:** Find out how Pythagoras' Theorem helps us calculate the lengths of a right angled triangle.

# Maths KS3 Lesson 1: Right Angles and Pythagoras

**Pythagoras' Theorem can help work out:**

How long the pipe should be

How far it should reach at ground level, or

The ‘drop’ between one end and the other.

If a culvert is 50m long and drops 4m from end to end, what horizontal distance does it cover? Click the answer you think is correct.

a) 49.8m b) 49.99m c) 48.9m d) 50.6m

**Screen 3.3:** By using Pythagoras' Theorem, Engineers can determine the length, depth and 'drop' of pipes.

# Maths KS3 Lesson 1: Right Angles and Pythagoras

Click the next arrow to show the equation.

If this Tunnel section is 7.5km long on the surface and must have a gradient of 1 in 2000, how far will B be below A, in metres? Click the answer you think is correct.

a) 37.5m b) 3.75m c) 12.5m d) 0.375m

**Screen 3.4:** Pythagoras' Theorem also helps calculate the gradient, or slope, of a pipe or tunnel section.

# Maths KS3 Lesson 1: Right Angles and Pythagoras

If this Tunnel section is 7.5km long on the surface and must have a gradient of 1 in 2000, how far will B be below A, in metres? Click the answer you think is correct.

a) 37.5m b) 3.75m c) 12.5m d) 0.375m

**Screen 3.5:** Pythagoras' Theorem also helps calculate the gradient, or slope, of a pipe or tunnel section.

# Maths KS3 Lesson 1: Right Angles and Pythagoras

**Screen 4:** Play the video to find out the answers.

# Maths KS3 Lesson 2: Co-ordinates

# Maths KS3 Lesson 2: Co-ordinates

**Screen 1:** Play the video to meet Lucy, a Planning Assistant on the Thames Tideway Tunnel project.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 2.1: **Engineers can use co-ordinates to locate site positions on a map. To do this, they need to use a grid. Click the next arrow to see how we find the point's coordinates.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 2.2:** Engineers can use co-ordinates to locate site positions on a map. To do this, they need to use a grid. Click the next arrow again to see how we find the point's coordinates.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 2.3:** Engineers can use co-ordinates to locate site positions on a map. To do this, they need to use a grid. Click the next arrow again to see how we find the point's coordinates.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 2.4:** Engineers can use co-ordinates to locate site positions on a map. To do this, they need to use a grid.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 2.5:** Watch carefully to see how a 1km x 1km square of Great Britain can be identified using a four figure grid reference.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 2.6:** Watch carefully to see how a 1km x 1km square of Great Britain can be identified using a four figure grid reference.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 2.7:** Watch carefully to see how a 1km x 1km square of Great Britain can be identified using a four figure grid reference.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 2.8:** Watch carefully to see how a 1km x 1km square of Great Britain can be identified using a four figure grid reference.

# Maths KS3 Lesson 2: Co-ordinates

The Blackfriars Bridge Combined Sewage Outlet is at Grid reference TQ 316 807.

How accurate are these co-ordinates? Are they to the nearest:

a) 1000m b) 10m c) 1m d) 100m

**Screen 2.9:** Using what you have learnt from the previous two screens how accurate are these coordinates?

# Maths KS3 Lesson 2: Co-ordinates

Pythagoras can help work out the distance between two locations on a co-ordinate grid:

How far apart are these two locations? Each grid square represents 100m.

Click next to see the right-angled triangle that connects them.

a) 720m
b) 360m
c) 1000m
d) 7.2km
**Screen 2.10:** Pythagoras' Theorem can help Engineers work out distances too.

# Maths KS3 Lesson 2: Co-ordinates

Pythagoras can help work out the distance between two locations on a co-ordinate grid:

How far apart are these two locations? Each grid square represents 100m.

a) 720m
b) 360m
c) 1000m
d) 7.2km
**Screen 2.11:** Pythagoras' Theorem can help Engineers work out distances too.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 3.1:** When you’ve finished plotting each location on your worksheet, click next to confirm which construction site goes where.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 3.2:** When you’ve finished plotting each location on your worksheet, click next to confirm which construction site goes where.

# Maths KS3 Lesson 2: Co-ordinates

**Screen 4:** Watch the video to find out more about the importance of maps for the Thames Tideway Tunnel project.

# Maths KS3 Lesson 3: Algebra

# Maths KS3 Lesson 3: Algebra

**Screen 1:** Play the video to hear Mohammed talk about the TBM and find out your challenge.

# Maths KS3 Lesson 3: Algebra

**Screen 2:** The tunnel needs to pass through soils made of clay, sand and chalk. The TBM’s progress will vary depending on each soil’s hardness, stability and the ease with which the spoil (waste soil) can be removed.

# Maths KS3 Lesson 3: Algebra

**Screen 3:** The TBM can travel a distance d in each shift. How can you create a formula for d that takes into account the maximum speed of the TBM, the length of the shift, any stoppage time, and the different speed through sand, clay or chalk soils?

# Maths KS3 Lesson 3: Algebra

**Screen 4:** Did you get the formula right? Listen to Mohammed to find out.

# Maths KS3 Lesson 4: Percentages and ratios

# Maths KS3 Lesson 4: Percentages and ratios

**Screen 1:** Play the video to hear Prunella discuss the role of the CSOs and find out your challenges.

# Maths KS3 Lesson 4: Percentages and ratios

**Screen 2:** As heavy rain falls over London it runs into drains along each street. These drains form a catchment network that feeds into London’s sewer system. However if there is too much rain the system overflows and CSOs discharge a mixture of rainwater and untreated sewage into the River Thames.

# Maths KS3 Lesson 4: Percentages and ratios

**Screen 3:** With the new Tunnel, these overflows will be intercepted and diverted into the Tunnel and then pumped out for treatment, via the Lee Tunnel at Beckton sewage treatment works.

# Maths KS3 Lesson 4: Percentages and ratios

**Screen 4:** We can model how rainfall flows through each CSO and into the Tunnel. What can this table tell you about the effect of rainfall on water flow into the Tunnel?

# Maths KS3 Lesson 4: Percentages and ratios

**Screen 5:** Did you get the answers right? Listen to Prunella to find out.