### Maths KS4

*Welcome to the Tunnelworks KS4 Maths area.*

Here you’ll find three ‘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.

After that, you can ‘dig deeper’ with follow-up ideas for more lessons for more able students.

Finally, a CREST silver project is outlined, if you wish to connect your work to this well regarded science award scheme.

**Essentials**

Explore mean, mode and median using test data for a Tunnel Boring Machine to plan an inspection schedule. Use the geometry of circles and cylinders to calculate how much concrete will be needed to make the Tunnel segments. Explore how to model some simple real-life situations using graphs of the form y = mx + c.

**Dig Deeper**

Dig Deeper ideas explore grouped frequency tables and cumulative frequency and extend segments and sectors into 3D to consider the volume of water stored in the Tunnel. Can students model the total time to load and unload multiple barges, allowing for docking and undocking times and to load each barge in turn?

**CREST Silver Award**

**Research projects:** research and design a fan system to maintain a constant airflow through a pipe or build a prototype or proof of concept demonstration for an autonomous Tunnel Inspector.

# Introducing the Thames Tideway Tunnel

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

# Maths KS4 Lesson 1: Mean, mode and median

# Maths KS4 Lesson 1: Mean, mode and median

**Screen 1.1:** Battersea is one location where the Tunnel Boring Machine (TBM) will be lowered down a large shaft, to bore through sand and clay deep under London and the River Thames. Click next to see an image of a TBM at work.

# Maths KS4 Lesson 1: Mean, mode and median

**Screen 1.2:** The TBM has a rotating cutter head with rows of cutting disks. These grind away the rock as the head rotates. The rock is crushed and removed by conveyor. The TBM moves by pushing against the sides of the Tunnel with thrust cylinders.

# Maths KS4 Lesson 1: Mean, mode and median

**Screen 2:** Play the video to hear Nasreen discuss how she uses data to plan machine maintenance.

# Maths KS4 Lesson 1: Mean, mode and median

TBMs remove the rock they grind away by mixing it with water and pumping it through a long pipe, or by crushing the rock and carrying it away on a long conveyor belt.

**Screen 3.1:** This is a TBM conveyor. It must work constantly in difficult conditions. Sometimes rock particles may interfere with the smooth working of the conveyor. How can the engineers predict how often this might happen?

# Maths KS4 Lesson 1: Mean, mode and median

This raw data shows the number of days between conveyor shut-downs during the manufacturer’s testing of a TBM:

**5, 10, 11, 6, 7, 9, 12, 14, 10, 15, 8, 13, 12, 11, 7, 8, 10, 10, 9, 11, 9, 10**

The engineers need to know the average time before the conveyor may experience a problem so they can develop a planned maintenance schedule. Is this time the:

a) Mode b) Mean c) Median

**Screen 3.2:** Engineers on the Thames Tideway Tunnel often have to make sense of raw data.

# Maths KS4 Lesson 1: Mean, mode and median

To make sense of the raw data, the engineers organise it into a frequency table.

**5, 10, 11, 6, 7, 9, 12, 14, 10, 15, 8, 13, 12, 11, 7, 8, 10, 10, 9, 11, 9, 10**

What’s the next step to calculate the mean for this data?

a) 1st row total x 2nd row total

b) Calculate value. x frequency for each column

c) Find the most common

**Screen 3.3:** Look at the newly organised data and calculate the mean.

# Maths KS4 Lesson 1: Mean, mode and median

**Screen 3.4:** Engineers use data to calculate the time between any unplanned maintenance.

# Maths KS4 Lesson 1: Mean, mode and median

**Screen 3.5:** The manufacturer uses this data to create planned maintenance schedules for the TBM and conveyor system, so there are minimal delays in getting the Tunnel operational and keeping sewage out of the River Thames.

# Maths KS4 Lesson 1: Mean, mode and median

**Screen 4:** Did you get the answers right? Listen to Nasreen to find out.

# Maths KS4 Lesson 2: Geometry of circles and cylinders

# Maths KS4 Lesson 2: Geometry of circles and cylinders

**Screen 1:** Meet Alex, a Construction Planning Engineer and find out how concrete segments are used to line the Tunnel as the TBM excavates.

# Maths KS4 Lesson 2: Geometry of circles and cylinders

**Screen 2.1:** The Engineers need to calculate how many concrete segments will be needed to line a section of the Tunnel.

# Maths KS4 Lesson 2: Geometry of circles and cylinders

**Screen 2.2:** The Engineers need to calculate the volume of concrete needed for the segments that make up the Tunnel.

# Maths KS4 Lesson 2: Geometry of circles and cylinders

**Screen 3.1:** Click next to see how segments are transported to construct the Tunnel.

# Maths KS4 Lesson 2: Geometry of circles and cylinders

**Screen 3.2:** This is how concrete segments create the Tunnel.

# Maths KS4 Lesson 2: Geometry of circles and cylinders

**Screen 4:** Now you’ve completed the worksheet, do you have the correct answers? Listen to Alex to find out.

# Maths KS4 Lesson 3: Straight line graphs

# Maths KS4 Lesson 3: Straight line graphs

**Screen 1:** Watch the video to learn about the work you will be doing in this lesson.

# Maths KS4 Lesson 3: Straight line graphs

**Screen 2a:** Straight line graphs have the form **y = mx + c**. m is the **gradient**: the **rate of change**. It is a measure of how much y changes with each value of x. c is the **intercept**: the **value of y when x = 0**.

# Maths KS4 Lesson 3: Straight line graphs

**Screen 2b:** Straight line graphs have the form **y = mx + c**. m is the **gradient**: the **rate of change**. It is a measure of how much y changes with each value of x. c is the **intercept**: the **value of y when x = 0**.

# Maths KS4 Lesson 3: Straight line graphs

**Screen 3:** Barges must dock at each Tideway foreshore site. Because the River Thames is tidal in some parts, the barges must not sink too low as they are loaded with spoil.

# Maths KS4 Lesson 3: Straight line graphs

**Screen 4:** Different barges take different times to load. This depends on how fast the barge can tie or untie from the dock, and how long it takes to load each tonne of spoil into the barge. These times vary depending on the barge design.

# Maths KS4 Lesson 3: Straight line graphs

**Screen 5:** Worksheet answers

# Maths KS4 Lesson 3: Straight line graphs

**Screen 6:** Worksheet answers

# Maths KS4 Lesson 3: Straight line graphs

**Screen 7:** Worksheet answers

# Maths KS4 Lesson 3: Straight line graphs

**Screen 8:** Worksheet answers