Whilst this unit is presented as a sequence of 7 sessions, more sessions than this may be required. It is also expected that any session may extend beyond one teaching period. 

Session 1

This session introduces the topic over the course of two to three lessons.

Activity 1

Play a game of classroom Pictionary.

1. Provide everyone with a small card (about 3cm x 6cm) and have students write on it their name and the name of something relevant, related to buildings and monuments (e.g. New Zealand landmarks, buildings in our school/town/city). Students need to keep whatever they draw on their card a secret.
2. Collect in the cards.
3. Organise students into at least three groups to play the game.
4. Explain that the cards are going to be used to play Pictionary.
5. Play a practice round using the cards that they have made to see how it works.
6. Rules of the game: 
   • A drawer comes up from a group and has 60 seconds to draw the building while members of other teams try and guess what it is.
   • The drawing group does not get to guess.
   • The drawing group can use their devices to search the building in question if they don’t know what it is.
   • If another group guesses correctly both teams get a point.
   • If no-one guesses correctly then the drawing team lose a point.

Activity 2

Use a stations activity (Copymaster 1) to help students pick up key ideas and get them engaged in the topic. Note that research-based stations will require students to have access to a device. As students go around each station it is intended they write up and/or glue their answers in their workbook. 

Allow about 10 minutes per station. Some stations finish a bit quicker, others take a bit longer so some management of students is required. Stations such as the Pythagoras puzzles and the triangle measurements could be doubled up on to allow groups to go to if they have finished a station early.

The stations could be changed to reflect the prior experience of students and/or to assess the level of students at the beginning of the unit. Consider what additional teaching and modelling will be needed to ensure your students can successfully participate in the learning at each station.

Session 2

This session is likely to take several teaching lessons to complete.

The learning of trigonometry is developed around similar triangles. The activities are developed from ideas in “Towards Better Trigonometry Teaching (Equals Network, 1989)” and are presented here with the permission of the authors of the original material. Students work together in groups and then groups pool their ideas for whole class use.  From this the need for and use of sin, cos, and tan are developed. They are not developed into an algorithm (e.g. SOHCAHTOA) until a later session.

The English activities focus around research and creative writing. The silent reading at the beginning of each lesson is now excerpts from the “Goodnight Stories for…”  and “...Dare to be Different” books, particularly stories of mathematicians. You might consider providing students with e-books, audiobooks, or copies of other, relevant texts that are more suitable to your students literacy needs, if necessary. The point of this silent reading is to give students a feel for what a “goodnight story is”.  There is also the opportunity to play some more “Buildings and Monuments Pictionary” to start generating ideas about what buildings or monuments students might like to research. Once each student has chosen a building, they begin their research.  A set of prompting questions for students who need it is provided.

Activity 1

1. Teach students how to label adjacent, opposite and hypotenuse sides of a right-angled triangle. Use Copymaster 2 as whole-class questions and/or as independent tasks.

Activity 2

Use the triangles resource pack (Copymaster 3). Ideally, you will have printed these onto light coloured card - a different colour for each type of triangle. E.g. 20° triangles on yellow card, 30° on red, etc. Each type of triangle makes up a set.

Having a pre-prepared table on the whiteboard to summarise results is also useful. A PowerPoint file is provided to help with this.

1. Explain that the purpose of the activity is to do some exploring of right-angled triangles and the relationships between their sides. The work is divided up so each group can contribute to the findings.
    
2. Provide each group of students with one set of triangles from Copymaster 3 and a recording sheet (Slide 3) for their work.
    
3. Have the students in each group measure the opposite side and the hypotenuse of each triangle, record them and calculate the ratio. Complete the sheet by working out the average of the ratios.

As groups come close to finishing, there is the opportunity to ask them what they are seeing and the implications. E.g. all 30° triangles, no matter their size have a ratio of 0.5.

Once groups have reported to the class, there should be a table of ratios for each of the triangle types. There is now the opportunity to consolidate these results and use them to estimate the solution for a right-angled triangle problem.

4. Explain to the class that you have fine-tuned their results and show a table of the correct ratios. Students can now use these to solve problems involving calculating the opposite side. These could be sourced from a text book or teacher-generated.

Activity 3

1. Students can either complete the measuring activity again for the adjacent side or you can introduce “results from another class who measured the adjacent side”.
    
2. Students can now use this table of results to solve problems involving the calculating the adjacent side. These could be sourced from a text book or teacher-generated.

Activity 4

1. Introduce the problem of triangles that do not match our table. For example, triangles with an angle of 23°.  Students could then make an estimate of the solution.
2. Draw on this idea to introduce the sine and cosine functions on calculators. For the purposes of later activities you may find it best to use the idea of hypotenuse x ratio to calculate the missing side.

Activity 5

By now students should be ready to take on the idea of the tan ratio relatively easily. 

1. Provide students with an exercise that requires them to use using the adjacent side to calculate the opposite side. These could be sourced from a text book or teacher-generated.
2. Review and summarise the learning from the previous activities – to find a missing side, choose the trig function and then calculate hypotenuse (or adjacent) multiplied by the ratio.

Activity 6

1. Introduce a problem that requires the hypotenuse to be calculated. Have students  in groups to solve this and/or use "guess and check" to get the correct answer.  This can be used as a teaching opportunity that to calculate the hypotenuse (or adjacent) you use side ÷ ratio.
    
2. Provide time for students to practise this using sourced from a text book or teacher-generated examples.

Activity 7

This activity could be used for all students or used as an extension activity.

In groups, have students try to come up with a set of steps, or algorithm that other students can use to solve a trigonometry problem. You could introduce them to the word SOHCAHTOA to help them. This provides an opportunity to summarise the learning from the activities in Session 2 with an algorithm or overall strategy for solving trigonometry problems.

Session 3

Activity 1

1. Introduce students to clinometers as a way of measuring angles of elevation and declination. Point out that for any object, they can measure their distance from it, and the angle to the top of it. With a diagram on the board, it becomes pretty clear that this is a problem they can solve.
    
2. Provide time for students to complete their own investigation, following the process outlined below:

• Choose a building/tree/object around the school
• Draw a diagram to show the measurements you will take
• Go out and take the measurements
• Draw a right-angled triangle to help you solve the problem
• Solve the problem

Depending on the depth of your students' thinking, some may just do the task, others will lie down to measure the angle from the ground, and others might choose to take into account the height of the person taking the measurement.  An extension to the activity could be to work out the difference in heights between two buildings. Some basic examples are provided on Copymaster 4.

Session 4

Activity 1

1. Introduce a problem with a triangle of 30° and the opposite side. Have students think about and discuss what the length of the hypotenuse might be. A variety of methods might come up for solving the problem but hopefully someone will suggest using the tables from Session 2. If not, suggest it.
    
2. Have students do a few problems before you ask: what happens if it is not a ratio from the table? 
    
3. Put up an example and support students to estimate from the tables.  You now have the opportunity to show students a set of Eton tables if they are available and get them to find a more precise answer by hunting down the ratio in the tables. Enjoy some of your more engaged student thinking this is pretty cool and asking for a set of the tables!
    
4. You can then explain that sin^-1 etc. on a calculator will give the exact ratio too. They can explore and use this further. 

Activity 2

Students are now equipped to solve these problems. Depending on the time you have left in your unit, either they can develop their own methods or you can add a method directly to the algorithm developed/used at the end of Session 2. The aim is for students to solve problems involving calculating angles in right-angled triangles.

Session 5

Activity 1

Provide time for students to carry out sufficient research, write their goodnight stories, and write a problem based on their building. The final task is to write a textbook style problem to go with the story they have written. When complete, this will make a mathematics resource that another class could use.

Using a local example of a building or monument, you can explain what students are going to do, and how to raise their level of thinking. SOLO is used as it matches up with levels of achievement in mathematics in NCEA. Levels of achievement are below:

• Demonstrate a problem that is incomplete and therefore cannot be solved, for example, does not have enough information. (Uni-structural/NA)
• Adjust the problem so it can be solved to get the height of the building but because the numbers are just made up, the height is not the actual height of the building. (Multi-structural/A)
• Get students to problem solve for you to adjust the angle or distance or both to get the height to come out right.  Now adjust the problem to represent this and the solution is now in context. (Relational/M)
• Now demonstrate or just suggest how insight can be shown.  One way is to make sure the problem works in context.  Is it possible to stand where you are suggesting in the problem? Students can use the distance measurer in google maps to check this or to find a realistic place to stand.  Another way might be to make it a multi-step problem.  For instance, imagine you are looking out from another building – this would require angles of both elevation and declination.  (Extended abstract/E)

Provide time for your students to research, write, and draw their Pythagoras and trigonometry problems.

Session 6

Prior preparation:

Prior to session 6, it is helpful if the format of the book is already set up.  This allows student work to be dragged and dropped into the book as teachers receive it. With a little planning, and the time if you have it, it is possible to do this with the final pieces of work and to print out the book for each student without a gap between Session 6 and Session 7.

1. Have students complete their Pythagoras’ Theorem or Trigonometry problem and creative writing and submit this for collation into a book. There may be students at many different levels of completion and there is the opportunity to extend students who finish early. Some six-bit problems are provided for this (Copymaster 5). Students work on these problems collaboratively. Once they have successfully tried and solved them, they can then have a go at writing a six-bit problem using their own building.

Session 7

Activity 1

1. Provide time for students to read both their own published work and the stories of their friends and classmates.
2. Provide time for students to try solving the problems set by other students.