This unit supports students to learn and apply Pythagoras’ theorem and trigonometry in a context that enhances their engagement and produces an authentic and useful outcome. Students to create a resource “Goodnight Stories for Builders and Architects to be…” for other classes to use that puts Pythagoras and trigonometry into interesting and researched contexts.
- Know some of the history behind Pythagoras’ theorem.
- Apply Pythagoras theorem in mathematical and real world contexts.
- Discover what sin, cos and tan are.
- Apply trigonometry ratios in mathematical and real world contexts.
At its heart, the idea of this unit is that students discover and create the mathematics that becomes Pythagoras’ theorem and trigonometry.
The main mathematics within the unit is using Pythagoras’ theorem to find any side of a right-angled triangle and using ideas about similar triangles to solve problems involving unknown sides and angles of right-angled triangles.
As presented, the unit presumes that students have encountered and used Pythagoras’ theorem previously. While there are revision activities relating to Pythagoras’ theorem in the stations activity in Session 1, the teaching of Pythagoras not explicitly supported in this unit. For more teaching activities relating to Pythagoras' theorem you may want to use the Pythagoras' Theorem or Gougu Rule or Pythagoras' Theorem units. If you wanted to include Pythagoras’ theorem within the unit, between Session 1 and Session 2 would be an appropriate place to slot it in.
Associated Achievement Objectives:
Processes and strategies
- Integrate sources of information, processes, and strategies purposefully and confidently to identify, form, and express increasingly sophisticated ideas.
Purposes and audiences
- Show an understanding of how to shape texts for different audiences and purposes.
- Select and use a range of language features appropriately, showing an understanding of their effects.
- Organise texts, using a range of appropriate, effective structures.
- Clinometers or clinometer apps.
- Access to devices for research and word processing.
- Copies of:
- Goodnight Stories for Rebel Girls, by Francesca Cavallo and Elena Favilli
- Goodnight Stories for Rebel Girls 2, by Francesca Cavallo and Elena Favilli
- Stories for Boys Who Dare to be Different, by Ben Brooks and Quinton Wintor
- Stories for Kids Who Dare to be Different, by Ben Brooks and Quinton Wintor
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. Every lesson starts with silent reading.
This session introduces the topic. It is intended that it would take two to three lessons to complete.
- Set the scene for the unit and start to connect the mathematics and English to the context.
- Learn about what Pythagoras’ theorem and trigonometry are and where they are used.
- Become familiar with the ideas and language that will be used later in the unit.
Play a game of classroom “Buildings and Monuments Pictionary”.
- Everyone gets a small card (about 3cm x 6cm) and writes on it their name and a favourite and/or famous building. It can be anything, but keep it a secret!
- Collect in the cards.
- Organise students into at least three groups to play the game.
- Explain that the cards are going to be used to play Pictionary.
- Play a practice round using the cards that they have made to see how it works. If this first round is too tricky or fails, give students the opportunity to think of more suitable things to put on their cards.
- 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.
A stations activity (Copymaster1) to help students pick up key ideas and get them engaged in the topic.
Note that research-based stations will need a device for students to do the research. As students go around each station it is intended they write 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.
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 such as 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. This starts 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.
- Develop an understanding of trigonometry ratios.
- Apply knowledge of trigonometry ratios to solve problems involving right-angled triangles, including:
- Finding the adjacent or opposite side using sin or cos
- Finding the hypotenuse
- Using tan
- Finding an angle using two sides.
Students are taught how to label adjacent, opposite and hypotenuse sides of a right-angled triangle.
A practice sheet is provided (Copymaster 2).
This activity uses the triangles resource pack (Copymaster 3). It works best if the triangles are on 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.
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.
Each group of students gets one set of triangles from Copymaster 3 and a recording sheet (Slide 3) for their work.
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 are getting 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.
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.
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”.
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.
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.
This idea can be used to introduce the sine and cosine functions on their calculator.
For the purposes of later activities you may find it best to use the idea of hypotenuse x ratio to calculate the missing side.
By now students should be ready to take on the idea of the tan ratio relatively easily. They can practise using an exercise that involves using the adjacent side to calculate the opposite side. These could be sourced from a text book or teacher-generated.
Sum up 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.
Introduce a problem that requires the hypotenuse to be calculated. Students in groups can problem solve and/or 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.
Students then practice this using examples sourced from a text book or teacher-generated.
This activity could be used for all students or used as an extension activity.
In groups, students try to come up with a set of steps, or algorithm that other students can use to solve a trig 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.
- Apply trigonometry to solve a practical problem.
- Students engage in collaborative problem solving
Introduce students to clinometers as a way of measuring angles of elevation and declination.
As part of the conversation, 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.
Students now complete their own investigation:
- 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 thinking of your students, 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.
- Calculate angles given two sides of a right-angled triangle.
Introduce a problem with a triangle of 30° and the opposite side. Students then 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, you could suggest it.
Students could do a few problems before you ask what happens if it is not a ratio from the table. Put up an example and students can 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!
You can then explain that sin-1 etc on a calculator will give the exact ratio too and get them to use this.
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.
Students practice solving problems involving calculating angles in right-angled triangles.
- Students write a Pythagoras or trigonometry problem for their researched building.
By now, students should also have been researching and writing their goodnight stories and will know at least enough about their building to write a problem involving based on their building. The final mathematics 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/school 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. (Unistructural/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)
Students can now go and research, write, and draw their Pythagoras and trigonometry problems.
- Students complete and submit their work from both subject areas
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.
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.
- Students read their book and celebrate.
For silent reading hand out a book to each student to read. Students can read both their own published work and the stories of their friends and classmates.
Students can try solving the problems set by other students.