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Mathematics Teaching Self-evaluation Tools

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Here you can see a summary of the areas in which you are confident and those in which you are less confident; there are some ideas and suggestions which may help you in your professional learning.

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properties and transformations of shapes?(show/hide all)

What this might look like in the classroom

Task
Find an image of a Mondrian painting or construct a similar image of your own
Pupil's work in pairs sitting back to back. One pupil has a copy of the image which the other cannot see and describes the picture in mathematical language so that the second pupil can draw his/her version from the description.

Task
Choose the shape of a shadow; e.g. a square. What 3D shapes could cast such a shadow?

Taking this mathematics further

Find out about the 17 wallpaper patterns.

Transformations have a range of applications in other fields, particularly in chemistry and crystallography. Optics in physics also relies on reflection in mirrors to produce a focus point. Biological systems often use symmetry to retain balance and efficiency.

At a higher level transformations can be described by matrices and in advanced mathematics symmetries are often used to teach group theory.

Making connections

Learners will have worked with basic transformations from an early point in their mathematical education. They should be able to carry out reflections, rotations and translations, and be able to describe them.

Understanding basic properties and transformation shapes develops into a formal understanding of similarity and congruence.

Question
Rotate the triangle with vertices (1,0) (3,−1) and (2,1) through 90 degrees anticlockwise about (0,0).

Answer

Taking this mathematics further

Encourage learners to look for examples of rotation outside of the classroom. There are plenty of opportunities for them to notice them. Particular examples of note are:

Wallpaper patterns; of which there are seventeen different groups relying on basic transformations − including rotation.

Frieze patterns; which are defined as 2D designs that are repetitive in one direction only. There are seven different groups, again relying on basic transformations, and car tyre treads almost always correspond to one of these.

Rangoli patterns: traditional Indian geometrical patterns based on reflection, rotation and translation. Often displayed during the Hindu festival of lights: Diwali.

Ancient tiles offer endless possibilities to consider the transformation of shapes.

Making connections

Rotation is one of the four basic transformations, the others being reflection, translation and enlargement. Translations, reflections and rotations result in congruent shapes; i.e. they have the same shape and size, but the position varies. Enlargement of a shape results in a similar shapes; mathematically similar shapes have the same shape, but a different size.

In rotation, the anticlockwise direction is taken as the positive direction. If a direction of rotation is not specified, then a rotation of 90^{o} (for example) would be 90^{o}
Learners should be able to carry out combinations of transformations.

Question
Use dynamic software to enlarge the quadrilateral with vertices (0,

1

3

), (

1

3

, 0), (2,0) and (1.5, 1) by a scale factor 3 with centre of enlargement (0,0). Use the software facility to experiment with moving the position of the centre of enlargement / scale factor.

Answer

Taking this mathematics further

Encourage learners to look for examples of enlargement outside of the classroom. There are plenty of opportunities for them to notice them. Particular examples of note are:

Printing

Photographs

‘A’ paper sizes

Projectors

Making connections

The concept of enlargement is first addressed by carrying out enlargements of simple shapes given a scale factor. This is developed by considering the use of a specific centre of enlargement.

Learners should be aware that changing the position of the centre of enlargement changes the position of the enlargement. Changing the scale factor changes the size of the enlargement. Exploring this with dynamic software provides a powerful visual aid.

The dynamic software approach is particularly good for extending into fractional and negative scale factors.

Investigation
A series of rotations about the same centre could return the shape to its original position depending on the angle of rotation.
Think about what combinations of other transformations will ultimately have no effect on the shape; e.g.
Enlarge scale factor 2 centre (0,0)
Reflect in the line y = x
Enlarge scale factor 0.5 centre (0,0)
Reflect in y = x
Does the order of the transformations matter?

Taking this mathematics further

Transformations have a range of applications in other fields, particularly in chemistry and crystallography. Optics in physics also relies on reflection in mirrors to produce a focus point. Biological systems often use symmetry to retain balance and efficiency.

At a higher level transformations can be described by matrices and in advanced mathematics symmetries are often used to teach group theory.

Making connections

Once learners have understood the effect of individual transformations then the next stage is to combine these transformations. Questions may be set which require the learner to deduce from the resulting diagram what single transformation has been used to effect this change.

Question
Imagine that, like Alice in Wonderland, you had a drink which would make you smaller and cake that would make you bigger.
You have a drink which makes you one tenth of your original height but you remain exactly the same shape. How many times smaller is the palm of your hand?

Answer
The palm of your hand (area) will be 10^{2} times smaller

Problem
Investigate pizza sizes and prices. A pizza size is its diameter. What is the (multiplicative) difference between the amount of topping on a 9 inch pizza and a 12 inch pizza?

Answer
The length scale factor of the two pizzas is

12

9

so the area scale factor is ^{
(
12
9
)2} =

144

81

. This is nearly double: the larger pizza is likely to be far better value for money!

Taking this mathematics further

Use the concept to explain why giants / King Kong / … cannot exist:

If a gorilla became 5 times bigger (length scale factor) then its mass would increase by a factor of 5^{3} = 125 (volume scale factor). However, for the enlarged gorilla to support its own weight, it needs its bones to be strong enough. The force of its weight acts through the 2−dimensional cross−section of its bones, and the area scale factor is 5^{2} = 25. So its enormously increased mass is transferred through its comparatively much weaker bones − it cannot support its own weight.

Making connections

Learners will need to be confident at finding the scale factor from a shape and its enlargement, including cases where the numbers do not provide a ‘neat’ answer (such as the pizza example).

An understanding of proportionality will also aid learners.

You might like to consider a diagram to support this:

Question
What single vector would take you back to your starting point?

Answer (

-1

-7

)

Taking this mathematics further

Velocity is a vector as it has magnitude and direction; speed is scalar. Learners will meet this idea in GCSE physics

Forces, because they have both magnitude and direction, can be represented by vectors. By combining vectors we can work out the resultant if a number of forces act on the same object.

Making connections

Learners are likely to have first encountered vectors when describing translations.

There is no substitute for using a diagram to support this process as it makes it very clear what is happening − just ensure that learners are careful to interpret the direction of movement correctly.

Adding vectors should not be confused with adding fractions!

the exterior angle of a triangle is equal to the sum of the two interior opposite angles, the sum of the angles in a triangle is 180Â°, and the sum of the exterior angles of any polygon is 360Â°?(show/hide all)

What this might look like in the classroom

Activity 1: Match the cards to their correct place on the grid:

Solution 1:

Some of the cards go in more than one place, and misconceptions are addressed. Make sure that learners justify their answers!

Question 2 A regular polygon has interior angles of 160°. How many sides has it got?

Answer 2 Since it is a regular shape every interior angle is the same.
We know that interior angle + exterior angle = 180° so each exterior will be 180° − 160° = 20°.
As all the exterior angles of 20° add up to 360° there must be 18 sides.

Taking this mathematics further

This type of geometry is known as Euclidean geometry; that is geometry based on the definitions and axioms stated in Euclid’s ‘Elements’. Find out more about Euclid and this pivotal piece of work which is considered more widespread than any other book in the Western world (except the Bible). In what situations might it not be appropriate to use the axioms set out in Euclidean geometry?

Find out about the role of Pythagoras in the development of mathematical proof.

Making connections

Learners are likely to have experienced the ‘rip off the corners of a triangle’ approach to demonstrate the angle sum is 180°, and a formal proof of this might well be the first geometrical proof that they come across. It is important that the notion of proof is explored, and this is a good opportunity. However, simple algebraic proofs may have been encountered before and it is worth pointing out that formal proof is one of the most important underlying concepts in mathematics.

An unjumbling approach is often particularly effective when encouraging learners to develop their own proofs. In this, learners are presented with the correct steps in a formal proof, but in the wrong order. They have to place the steps in the right order.

the opposite angles of a parallelogram are equal?(show/hide all)

What this might look like in the classroom

Activity 1
Place the following cards in the correct order to produce a correct logical proof that the opposite angles of a parallelogram are equal

So opposite angles in any parallelogram are equal

Label each angle that is corresponding to ‘B’ with an ‘N’

Therefore A = M = P = D

Extend each of the four sides in both directions

Label each angle that is corresponding to ‘M’ with a ‘P’

Since vertically opposite angles are equal, D = P

Consider a parallelogram with angles ‘A’, ‘B’, ‘C’ and ‘D’

And B = N = Q = C

Label each angle that is corresponding to ‘A’ with an ‘M’

And C = Q

Label each angle that is corresponding to ‘N’ with an ‘Q’

Solution 1
Is given on page 2 of KS4 additional file 7

11. So opposite angles in any parallelogram are equal

5. Label each angle that is corresponding to ‘B’ with an ‘N’

9. Therefore A = M = P = D

2. Extend each of the four sides in both directions

4. Label each angle that is corresponding to ‘M’ with a ‘P’

7. Since vertically opposite angles are equal, D = P

1. Consider a parallelogram with angles ‘A’, ‘B’, ‘C’ and ‘D’

10. And B = N = Q = C

3. Label each angle that is corresponding to ‘A’ with an ‘M’

8. And C = Q

6. Label each angle that is corresponding to ‘N’ with an ‘Q’

Taking this mathematics further

Can a parallelogram have 4 acute angles?
Ask to your learners to prove to you why this cannot be so.

Making connections

Knowledge of angle facts involving parallel lines is required, and an understanding of the difference between demonstration and proof can be explored through this activity.

the conditions for congruent triangles and can prove that the base angles of an isosceles triangle are equal?(show/hide all)

What this might look like in the classroom

Question 1 PQR is an isosceles triangle with PR = PQ.
A is a point on PR and B is a point on PQ such that PA = PB.

Are triangles PQA and PRB congruent?

If X is the point of intersection of AQ and BR, find another pair of congruent triangles in this diagram.

Answer 1

Yes. Angle P is common to both triangles, PA = PB is a given fact and PR = PQ since PQR is an isosceles triangle. Therefore the triangles are congruent using the SAS condition.

Triangles BXQ and AXR

Taking this mathematics further

Do you think that the following is an acceptable proof that the base angles in an isosceles triangle are equal?

Triangle ABC is isosceles, so AC=AB.

angle BAC = angle CAB

Triangle BAC is congruent to triangle CAB (using the SAS condition)

Therefore, angle B = angle C

Making connections

Learners will first encounter the idea of congruence when exploring reflections, rotations and translations

Once the idea of conditions for congruent triangles has been established, learners will progress to conditions for similar triangles. What similar triangles can you find within the diagram described in ‘What this might look like in the classroom?’

how to use a dynamic geometry computer program to model geometric constructions?(show/hide all)

What this might look like in the classroom

Activity 1 Use dynamic geometry software (e.g. Geometer’s Sketchpad, Cabri Geometer) to construct each of the following:

The perpendicular bisector of a given line segment

The angle bisector of a given angle

A line through a point F, on a line segment XY, that is perpendicular to XY

A triangle with sides 6 cm, 7 cm and 8 cm.

The point equidistant from any three given points; A, B and C (think perpendicular bisectors!)

Activity 2
Two intersecting circles that have the same radius. Construct the chord that is common to both circles and the line joining the two centres. What do you notice about these two lines?

Taking this mathematics further

Research examples of geometrical constructions in the real world that can be modelled by these methods.

Rose windows of some cathedrals in particular, are wonderful examples that can be constructed mathematically using just the standard constructions of perpendicular bisector, angle bisector and so on.

Making connections

All of these constructions can (and should) be done using pencil and paper methods, but the process of using dynamic geometry software to do so can really embed the understanding. Make sure that learners explore the constructions by, for example, dragging some of the points around and observing the impact on the constructed diagram

This software also allows all learners access to the mathematics; some may find constructing by hand more challenging, and the use of ICT will help.

how to use a dynamic geometry computer program to investigate geometrical properties?(show/hide all)

What this might look like in the classroom

Activity 1
Use dynamic geometry software (e.g. Geometer’s Sketchpad, Cabri Geometer) to construct each of the following circle theorems:

The angle in a semicircle is a right angle

The angle subtended at the centre of a circle is twice the angle subtended at the circumference

The radius of a circle is perpendicular to the tangent at that point

Angles in the same segment are equal

Opposite angles in a cyclic quadrilateral sum to 180^{o}

The alternate segment theorem

Taking this mathematics further

Find out about:

Napoleon’s Theorem

Aubel’s Theorem

Thebault’s Theorem

Use dynamic geometry software to model these and explore the effect of dragging chosen points around the screen.

Making connections

All of these constructions can (and should) be done using pencil and paper methods, but the process of using dynamic geometry software to do so can really embed the understanding. Make sure that learners explore the constructions by, for example, dragging some of the points around and observing the impact on the constructed diagram

This software also allows all learners access to the mathematics; some may find constructing by hand more challenging, and the use of ICT will help.

Pythagorasâ€™ theorem and its application to solving mathematical problems?(show/hide all)

What this might look like in the classroom

Question

If a = 11 cm and c = 6 cm find the area of the triangle.

Answer Area of a triangle =

1

2

x b x c
To find b:
11^{2} = b^{2} + 6^{2}
121 = b^{2} + 36
85 = b^{2}
9.219544457… = b
So area =

1

2

x 9.219544457… x 6 = 27.65863337… = 27.7 cm^{2} (3 s.f.)

Taking this mathematics further

How many different proofs of Pythagoras’ Theorem can you find? Which of them would be accessible to your learners?

The conjecture that the sum of the squares of the shorter sides equals the square of the hypotenuse (in a right−angled triangle) was well−known before the time of Pythagoras. Indeed, the 3, 4, 5 relationship was used to measure right angles accurately. But Pythagoras was the first person to actually prove it was true for all right−angled triangles. In doing so he simultaneously promoted the notion of mathematical proof as important. It is for this reason that the theorem is named after him.

Research the mathematician Pythagoras. Other than his famous theorem, what other contributions did he make (consider music, fractions, proof, …)?

Making connections

Learners need to be clear that Pythagoras' theorem only applies to right angled triangles, and understand the meaning of the hypotenuse − it is not always the sloping side!

Learners need to know when and how to apply Pythagoras' theorem, being careful to identify whether the hypotenuse, or whether one of the shorter sides is required. It helps if learners always set up an equation, such as 11^{2} = b^{2} + 6^{2}, and see it as an equation solving problem.

Ensure that learners retain the accuracy of any solutions found, and only round at the end of a calculation.

Pythagoras' theorem can be used to calculate the length of a line segment expressed by the coordinates of its endpoints.

Some learners will progress to using Pythagoras' theorem to solve problems in three dimensions.

Pythagorasâ€™ theorem to solve problems in three dimensions?(show/hide all)

What this might look like in the classroom

Problem 1 Find the exact length of the longest diagonal that can be made by joining vertices of the cuboid ABCDEFGH.

Solution 1
The longest possible diagonal is the hypotenuse of the right−angled triangle FDG

First we need the length DG, which is the hypotenuse of the triangle CDG:
DG^{2} = 5^{2} + 2^{2}
DG^{2} = 29
DG = √29
Now, FD^{2} = DG^{2} + FG^{2}
FD^{2} = 29 + 3^{2}
FD^{2} = 38
FD = √38
The exact solution was required so there is no need to evaluate √29 or √38. In fact, it was easier not to evaluate √29 anyway.

Taking this mathematics further

How can you use Pythagoras' theorem to show that
sin²θ + cos²θ = 1

What are the other ‘Pythagorean identities’?

Making connections

Learners need to be confident with 2D situations before tacking questions in three dimensions.
Any difficulties may not be with the use of Pythagoras’ theorem, but with visualising the shapes themselves.

Increasingly learners will find that Pythagorean and trigonometric approaches are needed in combination to solve problems.

The cosine rule can be seen as Pythagoras’ Theorem ‘with an adjustment’. In fact, the cosine rule works in right−angled triangles since cos90^{o} = 0.
a^{2} = b^{2} + c^{2} − 2bc cosA

similar triangles and other similar plane figures?(show/hide all)

What this might look like in the classroom

Question
Two triangles ABC and EFG are similar.
AB = 4 cm, EF = 6 cm and angle ABC = 35°
If EG = 8 cm, find

the length AC

the angle EFG

Answer Construct a ‘proportional reasoning’ table from the information provided

4

6

8

so AC = 8 ÷

6

4

= 5.33 cm (to 3 s.f.)

The angle is unchanged by enlargement so angle EFG is 35°

Taking this mathematics further

Making connections

An understanding of similarity starts through an exploration of enlargement.

If the triangle or other shape is orientated differently this may lead to difficulties. Fractional scale factors will also be more challenging.

Interpreting the information and presenting known facts within a ‘proportional reasoning table’, such as the one in ‘What this might look like in the classroom’ is a powerful way to solve these problems.

trigonometrical relationships to solve problems in three dimensions?(show/hide all)

What this might look like in the classroom

Question

Find the size of angle α

Answer

DB is the diagonal of the base. By Pythagoras’ theorem:
DB^{2} = 3^{2} + 5^{2}
DB2 = 34
DB = √34

Now consider the triangle ABD: AB is the ‘opposite’ side, and DB is the ‘adjacent’ side:

Taking this mathematics further

Research the history of trigonometry. What are the origins of the three trigonometrical functions sine, cosine and tangent. Why are they called ‘circular functions’?

Making connections

KS4 learners should have knowledge of the trigonometric functions, sine cosine and tangent. To solve problems in three dimensions no new skills are required, but there is a need to offer learners the opportunity to visualise or view the shape, perhaps with the use of dynamic geometry software.

Examples and questions;
This power point shows how to apply trigonometry and Pythagoras' theorem to real life 3 D problems including the pyramids of Giza

National Stategies- Geometry

Framework for mathematics – learning objectives and supplementary examples

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