Coordinate Geometry – Underground Maths

geometry-of-equationsExploring Algebra Review Questions from Underground Mathematics I came across some Coordinate Geometry questions I really like and yesterday spending a day with the very talented writing team and my fellow Underground Mathematics Champions we explored Straight Line Pairs, a question with much scope for exploration and possible methods of solution.
straight-line-pairs

The image above straight-line-pairs-1has been created from the Printable/supporting materials.

My Year 11s will be looking at Coordinate Geometry this week and I have some other questions I would like them to try. It is possible to create pdf files for a collection of questions, see Saving Favourite Resources, one of Underground Mathematics’ How To Videos. (See the tutorials page I have in the Underground Maths series of pages – a work in progress).

coordinate-geometry-resource-collection

You will find a whole collection of such questions if you look at Geometry of Equations. This includes many resources including Review questions. Note the Building Blocks resources. I think I’ll be using Underground Mathematics resources with ever younger students – Year 9 can try Lots of Lines! You will see from the the supporting materials that this has come from the brilliant Standards Unit (A10) collection. Students must sort the lines into six pairs, each pair matching one of the given descriptions.
lots-of-lines
Staying with the Building Blocks I do like Straight Lines where students must decide which of 17 equations are equations of a straight line.
Look at the list – a wonderful lesson in not jumping to conclusions here! Both my Year 9 and my Year 11 are going to be trying these this week!

ax^2+2hxy+by^2=0.

Straight Lines reminded me of Line Pairs, I feel an extension for Year 11 coming on!

 

Review Questions – Underground Mathematics

Underground Mathematics - QuadraticsI wrote earlier on the wonderful resources on Underground Mathematics from the University of Cambridge. Thinking about the new A level specifications  I believe this site will provide us with rich resources for these new specifications.

Each section includes Review Questions, look at Thinking about Algebra for example; scroll down the different resource types for the Review questions for this station.

Alternatively you can browse all the Review questions.

Review Questions - types
An excellent feature of Underground Mathematics is the excellent search facility; we could look at the Review questions by type. One can also search by line ( Number, Geometry, Algebra, Functions or Calcuus) and by Station.
Review Questions - search

See the example question below, for each review question you will find the question, a suggestion, the solution and sometimes suggestions for taking it further with for example GeoGebra resources.

Note the star by the title – if you choose to log on to the site (you don’t have to but it’s a very good idea!) you can save any favourite resources to your collection.

Review Question - Gemetry

I can never resist a quick illustration on Desmos! I think I’ll start an Underground Maths Desmos collection! Note the use of the hyperlink on Desmos to link to the question.
Select the image for the Desmos page.
Desmos illustration

I think we have a wonderful supply of excellent questions here to challenge our students and help them see connections between the various areas of the subject. These are ideal to use with A Level students; some are also useful for higher level GCSE students aiming at those top grades or Level 2 Further Maths students. Any student who wants to study Mathematics at university should certainly be using this site.

Looking for Resources

See also: Problems and Activities

Q3 Resources

Pondering a question on Twitter I realised that I always have a few sites I rely on where I know I can always find something. So I thought I’d pick a random example to illustrate.

A3 Constructions
So – constructions, for demonstrations I always use John Page’s Math Open Reference, his demonstrations are so clear and can be shown step by step; students can also be given the website so they can access them themselves. I found this many years ago when I wanted some demonstrations for constructions – a Google search returned it as the first entry!

bisect angle Math Open Ref

Math Open Ref – Bisecting an Angle

So obviously we need some questions / activities. Where to look – our textbooks are fine – plenty of questions there, but what else is available?

On Nrich, try a search by topic facility to find all the resources for a particular topic; searching on constructions there are several resources returned.

Nrich constructions

Nrich - triangle construction

Nrich – triangle construction


CIMT – I don’t think CIMT have ever failed me! One can actually do a Google search such as CIMT constructions to very quickly find resources. It is worth being familiar with the site so you know what is where; I would always check the Year 7, 8 and 9 material and also the GCSE course. In this case, the Year 9 resources include Unit 12 on Constructions and loci. As well as the text we have all the supplementary teacher resources. Note that for some Teacher Resources you will need the CIMT password.

I often find Nrich and CIMT more than sufficient and

I want to spend time planning my lesson and thinking about my students’ learning and how I’m going to help them understand and make it stick.

And how will I know what they know?

So of course quality resources are key but I don’t want to spend too much time looking for them if it stops me spending sufficient time on the above. I believe it is very worthwhile to have a few key sources so you can find something efficiently and quickly.

Having said that, since this post is on finding resources I’ll mention a few more!


The old Exemplification examples for Key Stage 3 have some very useful example. In this case use the Geometry and Measures document and do a search for constructions.


Teachit Maths though a subscription site offers its entire collection of activities as pdfs free. A search on constructions returns a small number of resources including a good card sort.

TeachIt Maths constructions


I’ll finish with Craig Barton’s and Simon Woodhead’s wonderful Diagnostic questions site. (Select this link for all posts on Diagnostic Questions, these include some instructions for use and other resources for rich questions.) Start typing construction into the search box and various choices will be returned.

Diagnostic Questions

See also: Problems and Activities

Introducing Trigonometry

Introducing Trig - NZmaths

This week I will be introducing Trigonometry to Year 9 (UK Key Stage 3, age 13-14). I always like to begin trigonometry with students actually measuring lengths in triangles, I believe they get more of a feel for the meaning of the ratios of the sides of a triangle if they have actually measured the length of the sides and calculated the ratios themselves.

I decided what I need is some accurate drawings of triangles of various dimensions that they could work on. It took a few seconds (the third entry in the search results for introducing trigonometry) to discover not only the drawings I wanted but a perfect recording sheet! From NZmaths (New Zealand Maths) – a site I have mentioned before for its excellent resources comes Introducing Trig. As well as the resources, teachers’ notes are provided. (Scroll to the end, past the teachers’ notes  for the resources)

This slideshow requires JavaScript.

So after an introduction including a reminder of Pythagoras they met last year, we’ll just need to be clear on the terms opposite, adjacent and hypotenuse, how to complete the recording sheet and the measuring can begin! This is a very able group of students and I suspect several of them to be telling me for example by the end of the lesson that sin 60° is the same as cos 30°.

Checking a few more links in the results of the search I see the excellent Math Open Reference site which I have referred to on several occasions. I also see that I am in very good company in my desire to get the students measuring themselves and using the nzmath resource – see Dan Pearcy’s post.

Something else I like to do when discussing trigonometry is to discuss all the possible types of problems that can come up because whether they are disguised as buildings / trees / ladders or whatever there are still only a limited number of problem types, eg find the angle given the opposite and hypotenuse. The students can work out how many problems there are.

 

Robocompass

Robocompass - simple demonstration

Robocompass – simple demonstration

Draw geometric constructions using the very attractive interface that is Robocompass.

demo commands

Robocompass is easy to use, simply type in commands. Select How to for a list of supported commands. Robocompass How To

Select the triangle image to go directly to the Robocompass file. Selecting play allows you to easily see each step. Note that you can select the page to rotate it in any direction.

Robocompass - play

Note that you can also look at individual steps,

demo 2

easily change colour or play speed:
demo 3

Reflection example

Robocompass – Reflection example

Experimenting with Robocompass made me realise that it can provide rather good demonstrations for transformations; see the above example (select the image for the file). Having set this up it is easy to change the line MN:

Reflection example 2

Or we could try a rotation.
rotation example

 

 

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examplesLearn more about Robocompass by following their blog and/or follow Robocompass on Twitter.

You can learn how to use Robocompass by studying examples; note the given examples or perhaps have a look at this on a Pythagoras proof.

Pythagoras