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Mathematics Department Workshops
Topic: Angle Properties and Reasoning
Overview
This workshop explores:
• the importance of giving a context to geometry, in this case the use of parallel lines and triangles in bridge
design;
• the significance of being able to see equalities between angles before becoming too concerned about the
formal language that fits the description;
• familiarity with formal language;
• the use of a practical “demonstration”, with associated reasoning, that establishes that the angles in a triangle
sum to 180° as a preliminary to using a card sort as a formal proof
Objectives related to this topic are:
• identify parallel and perpendicular lines; know the sum of angles at a point and on a straight line; recognise
vertically opposite angles (Year 7)
• identify alternate angles and corresponding angles; understand a proof that the exterior angle of a triangle is
equal to the sum of the two interior opposite angles (Year 8)
• solve problems using properties of angles and of parallel and intersecting lines, justifying inferences and
explaining reasoning with diagrams and text (Year 9)
• distinguish between conventions, definitions and derived properties (Year 9)
• distinguish between practical demonstration and proof in a geometrical context (Year 10)
Materials required
•
Resource Sheets HT2.APR.1 to HT2.APR.7
•
Data projector and internet access
•
Scissors
Web links
•
To pursue the ideas of bridge design:
http://www.matsuo-bridge.co.jp/english/bridges/index.shtm
•
This site gives details on balsa wood bridges and spaghetti bridges:
http://abcdpittsburgh.org/kids/kids.htm
•
This site allows you to generate word-searches about the angles associated with parallel lines (and supplies
the answers):
http://argyll.epsb.ca/jreed/math7/strand3/3203.htm
Suggested activities
Activity 1: Getting Started
Provide your team with resource sheet HT2.APR.1 and ask them to mark the first four circles with 1 point on the
circumference, 2 points on the circumference, 3 points on the circumference and 4 points on the circumference.
The task is – for each circle in turn – to join every point to every other point with a straight line, and count the
number of resulting regions. They should end up with the sequence 1, 2, 4, 8 – and most learners would agree that
there is a straightforward pattern here. Now predict and test for the fifth case. The answer will be 16. But when
you challenge your team (or learners) to find 32 regions on a sixth diagram they will not be able to do it (though it
is fun to watch them try!) 30 or 31 are both possible, depending on the regularity of the positioning of six points.
Ask your team if this is helpful in distinguishing between practical demonstration and proof. Discuss how you
would use this with learners.
www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics
teaching
Ask your team to write down a demonstration that the sum of the angles in a triangle is 180°, and a proof that the
sum of the angles in a triangle is 180°.
Activity 2: Setting the scene
The set of slides HT2.APR.2 show various bridges to emphasise the importance of the geometry of triangles and
parallel lines in engineering. As a department you might like to discuss how learners whether this sort of context
can motivate in mathematics, which may require general “scene setting” rather than detailed scientific explanation.
You might like to discuss the level of detail needed in this particular case.
The slides suggest that triangles and parallel lines are important in bridge design. If you want to pursue these ideas
look at the “Hooks for Learning” web sites.
If the photographs are used in the classroom you might like to set learners the challenge: “How can we guarantee
that the relevant struts in the bridge are parallel?” (See HT2.APR.7 for the relevance of the question to
mathematical proof).
Activity 3: Sample Learning Activities
As a team use the following resource sheets to explore some learning activities, which are designed to show
progression through a series of lessons.
Resource Sheet HT2.APR.3: Colourful angles
Learners often find it difficult to “mentally manoeuvre” their way around a diagram. This activity helps learners to
master this ability. If you use this during a departmental workshop you might explore some interesting sequences
with your team.
Resource Sheet HT2.APR.4: Word search
In the previous activity learners used their own descriptions of the angles but for formal proofs the correct terms
need to be mastered. The word search is intended to help learners to become familiar with the formal terms. You
can generate as many new word searches as you wish at this web site:
http://argyll.epsb.ca/jreed/math7/strand3/3203.htm.
Resource Sheet HT2.APR.5: Proving a point
In this activity there is a demonstration that the angles of a triangle add up to 1800 rather than a traditional
geometric proof.
The activity is written as if it was being used with a group of learners and in this context can be used to introduce
them to a way of learning maths by justifying arguments using logical reasoning.
Resource Sheet HT2.APR.6: Formality
This activity introduces learners to the process of formal proof. There is a suggested visual aid to help learners see
the first stage of the proof, which associates what has been learned about the angles of parallel lines and the angles
of a triangle, but the main vehicle used is a card sort activity.
Activity 4: Reflection
Resource sheet HT2.APR.7 suggests the importance of a corollary and the formal proof of the sum of angles in a
triangle. Challenge your team to find as many different proofs (and demonstrations) that the sum of the angles in a
triangle is 180°. Which ones is it worth keeping a record of and using with learners?
Embedding in practice
Hooks for Learning
• The importance of using parallel lines and triangles in bridge design is demonstrated at this web site, where
different designs are illustrated. For each bridge design there is a set of statistics for the “longest bridge” of
www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics
teaching
that type. http://www.matsuo-bridge.co.jp/english/bridges/index.shtm
•
A more practical approach may be to construct some model bridges. This site mentions balsa wood, toothpicks and spaghetti as possible materials. It also mentions some target weights to aim for if you want to test
the bridge to destruction. http://abcdpittsburgh.org/kids/kids.htm
Action points
At the end of the session, spend time recording some actions.
What do you need to do:
• Next day?
• Next week?
• Next year?
Further reading
•
This web link has an animation to show the angles associated with vertical lines. It also shows corresponding
angles, and alternate angles but uses terms like transversal so there may be preliminary work to complete
before using the site.
http://www.ies.co.jp/math/products/geo1/applets/kakuhei/kakuhei.html
•
This web site introduces the terminology of transversal, corresponding etc. but there is also a section for
angles at a point and angles on a line that can be measured using a virtual protractor. This could have many
uses for learners who find using a protractor confusing.
http://www.mathsisfun.com/geometry/parallel-lines.html
•
This is a site using interactive geometry, but it needs an activity to make use of it. This might involve a
discussion about how close to each other do angles have to come before they are classified as “equal”.
http://www.saltire.com/applets/parallel/parallel.htm
www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics
teaching