### AS Maths

*Welcome to the Tunnelworks AS Maths area.*

Here you'll find one lesson, complete with an online presentation, student worksheets and detailed lesson plans. The 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 able students.

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

**Content**

In this lesson students practice generating and transforming trigonometric graphs, including translations and stretches. Using information about tidal patterns in the River Thames, students develop a simplified chart for tides at London Bridge.

**CREST Gold Award**

Research project: students research the science that helps us understand why the number of sewage discharges into the River Thames has increased over time and the effect these have on river wildlife and the environment. They will discover what people think about the River Thames and create a communications activity to explain the science behind the project.

# Introducing the Thames Tideway Tunnel

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

# AS Maths Trigonometric graphs

# AS Maths Trigonometric graphs

**Screen 1:** What would the formula and graph for a stretch along the x-axis look like?

# AS Maths Trigonometric graphs

**Screen 2:** Some Thames Tideway Tunnel construction sites will be built in the foreshore of the River Thames. The barges that take materials to and from the site will rest on ‘campsheds’ at low tide. This requires accurate tidal data for each location. We can simulate this using trigonometric graphs.

# AS Maths Trigonometric graphs

**Screen 3:** Look at the diagram to see how water depth is calculated.

# AS Maths Trigonometric graphs

**Screen 4:** Calculate the time difference from Tower Bridge using distance data. Modify a basic function using a transformation that matches this delay. Use CWD, MLWS and MHWS to scale and translate. Write the function for each location, generate a graph and annotate with key information.

# A2 Maths Translations and stretches to complex functions

# A2 Maths Translations and stretches to complex functions

**Screen 1:** How can engineers calculate the volume of water the Tunnel may have to capture during different rainfall events? We can start by modelling the flow through each CSO. Click next to see the diagram.

# A2 Maths Translations and stretches to complex functions

**Screen 2:** How can engineers calculate the volume of water the Tunnel may have to capture during different rainfall events? We can start by modelling the flow through each CSO.

# A2 Maths Translations and stretches to complex functions

**Screen 3:** How could you transform this graph to show the curve of flow rate for different: Rainfall intensity? Rainfall duration?

# A2 Maths Translations and stretches to complex functions

**Screen 4:** How does rainfall of different intensities and durations affect the flow through the sewer? What transformations to the graph of this function could represent these?

# A2 Maths Translations and stretches to complex functions

**Screen 5:** This equation represents a real rainfall event that lies on the CSO’s overflow threshold of 0.6m³/s: **Intensity = 2mm / hour; Rain duration = 1 hour; Flow duration = 2 hours **

# A2 Maths Translations and stretches to complex functions

**Screen 6:** What pair of transformations will modify the basic equation (shown in the previous slide) and account for this minimum flow level?