This unit supports students to learn and apply Pythagoras’ theorem and trigonometry in an engaging context which leads to the production of an authentic and useful outcome: a resource “Goodnight Stories for Builders and Architects to be…” for other classes to use. This frames Pythagoras and trigonometry in interesting and researched 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.
This unit presumes that students have previously encountered and used Pythagoras’ theorem. While there are revision activities relating to Pythagoras’ theorem included in Session 1, the teaching of Pythagoras not explicitly supported in this unit. Refer to the Pythagoras' Theorem or Gougu Rule or Pythagoras' Theorem units for more teaching activities relating to Pythagoras' theorem. You could plan to carry out additional teaching around Pythagoras’ theorem between Sessions 1 and 2.
English:
Processes and strategies
Purposes and audiences
Language features
Structure
The learning opportunities in this unit can be differentiated by providing or removing support to students, and by varying the task requirements. Ways to support students include:
The context for this unit can be adapted to suit the interests and experiences of your students. For example, you might adapt the purpose of this unit to be focused on the creation of a book of stories to be placed in the school library. You might also have students feature a local building in their stories.
Te reo Māori kupu such as ture a Pythagoras (theorem of Pythagorus), pākoki (trigonometry), and koki hāngai (right angle) could be introduced in this unit and used throughout other mathematical learning.
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.
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
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.
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.
Activity 3
Activity 4
Activity 5
By now students should be ready to take on the idea of the tan ratio relatively easily.
Activity 6
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
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
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:
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.
Session 7
Activity 1
Dear families and whānau,
We have been exploring trigonometry and applying to solve practical problems. Ask your child to share their learning with you.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/goodnight-stories at 8:27pm on the 26th February 2024