Transforming and constructing
Term 2 starting in week 7 :: Estimated time: 3 weeks
- Perform and describe line symmetry and reflection (review)
- Perform and describe rotation/rotational symmetry (review)
- Perform and describe translations of shapes (review)
- Perform and describe enlargements of shapes (review)
- Perform and describe negative enlargements of shapes (review)
- Identify transformations of shapes (review)
- Perform standard constructions using ruler and protractor or ruler and compasses (review)
- Solve loci problems
For higher-attaining pupils:
- Perform and describe a series of transformations of shapes (review)
- Identify invariant points and lines
- Understand and use trigonometrical graphs
- Sketch and identify translations of the graph of a given function
- Sketch and identify reflections of the graph of a given function
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Activities for pupils
Here are some related resources in alphabetical order. Some may only be appropriate for high-attaining learners while others will be useful for those in need of support. Click anywhere in the grey area to access the resource.
Rotation Visual Aid A Geogebra manipulative that shows the effect of rotating an object with a moveable centre of rotation.
Transformations Video A demonstration of the four basic transformations: reflection, translation, rotation and enlargement.
Mixed Transformations A worksheet containing mixed transformations questions and a grid for drawing the solutions on.
Which Transformation? Identify which simple transformations these diagrams represent.
Vector Maze Use vectors to navigate through a maze by the shortest distance.
Graph Paper Flexible graph paper which can be printed or projected onto a white board as an effective visual aid.
Vector Connections Video Learn how coordinates can be linked with vectors and how these vectors can help you decide if three points are collinear.
Vector Connectors Exercises about vectors and coordinates; using one to find the other.
Alpha Twist Develop your skills and understanding of rotation in this fast-paced challenge.
Tridots Find all the triangles that can be drawn by joining dots on a 3 by 3 grid of dots.
Straight Line Graphs Video After drawing a straight line graph learn about its equation in the form y = mx + c.
Transformations Draw transformations online and have them instantly checked. Includes reflections, translations, rotations and enlargements.
Vector Cops Help the cops catch the robbers by finding the vectors that will end the chase.
Constructions Construct the diagrams from the given information then check your accuracy.
Sheep Herding Arrange the sheep in the field according to the instructions. An introduction to loci.
Vectors An online exercise on addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic representations of vectors.
Scale Factors Video The scale factor, area factor and volume factor of similar shapes are quite different.
Similar Shapes Questions about the scale factors of lengths, areas and volumes of similar shapes.
Perpendicular Bisector Video A step-by-step guide showing how to construct the perpendicular bisector of a line segment.
Negative Enlargement A video from MathsWatch about Enlargement by a Negative Scale Factor.
Ruler and Compass Constructions Video A demonstration of standard ruler and compass constructions.
Blow Up Click on all the points that could be the centre of enlargement of the shape if the image does not go off the grid.
Loci Land Make scale drawings of real life situations using ruler, compasses and pencil to answer questions about loci.
Loci Construction Pairs Match the description of the loci to the diagram of the construction.
Straight Line Graphs 10 straight line graph challenges for use with computer graph plotting software or a graphical display calculator.
Graph Match Match the equations with the images of the corresponding graphs. A drag-and-drop activity.
Graph Plotter An online tool to draw, display and investigate graphs of many different kinds.
Transformations of Functions A visual aid showing how various transformations affect the graph of a function.
Here are some exam-style questions on this topic:
- "Shape A can be transformed to shape B by a reflection in the y-axis followed by a
translation \( {c \choose d} \)
Find the value of \(c\) and the value of \(d\).
" ... more - "The shape A is drawn on the coordinate grid as shown below." ... more
- "OABC is a parallelogram with O as origin. The position vector of A is \(a\) and the position vector of C is \(c\)." ... more
- "The diagram shows a red trapezium drawn on a grid." ... more
- "(a) Shape \(A\) is translated to shape \(B\) using the vector \( \begin{pmatrix}m\\n\\ \end{pmatrix}\). What are the values of \(m\) and \(n\)?" ... more
- "ABCD is a quadrilateral. The points E, F, G and H are the midpoints of the sides of this quadrilateral." ... more
- "The graph of the curve A with equation \(y=f(x)\) is transformed to give the graph of the curve B with equation \(y=5-f(x)\)." ... more
- "Construct a locus of points that are the same distance from points A and B." ... more
- "Construct a quadrilateral ABCD in which:" ... more
- "Plot the following points in order then join them up in order to make an irregular hexagon." ... more
- "In the parallelogram OABC two of the sides can be represented by vectors \(a\) and \(c\)." ... more
- "Describe fully the single transformation that maps trapezium A onto trapezium B." ... more
- "The graph of the curve with equation y = \(f(x)\) is shown on the grid below." ... more
- "The diagram, drawn to scale, shows a right-angled triangle ABC." ... more
- "The graph of the following equation is drawn and then reflected in the x-axis" ... more
- "\(f\) and \(g\) are two functions such that \(g(x)=3f(x+2)+7\)." ... more
Click on a topic below for suggested lesson Starters, resources and activities from Transum.
- Construction In a way this topic is quite different to all of the other topics in school mathematics. It requires a practical skill as well as the understanding of the geometrical concepts. It also requires a sharp pencil, a sturdy ruler and a decent pair of compasses. Younger children should practise using the drawing instruments to make patterns. They will then progress to constructing accurate diagrams, plans and maps. Older pupils are taught to derive and use the standard ruler and compass constructions for the perpendicular bisector of a line segment, the perpendicular to a given line from a given point and the bisector of a given angle.
- Enlargements When areas and volumes are enlarged the results are far from intuitive. Doubling the dimensions of a rectangle produces a similar shape with four times the volume! Doubling the dimensions of a cuboid produces a similar shape with eight times the volume! The activities provided are intended to give pupils experiences of dealing with enlargements so that they better understand the concept and are able to produce diagrams, make models and answer questions on this subject. Once positive scale factors have been mastered the notion of fractional and negative scale factors await discovery!
- Graphs This topic includes algebraic and statistical graphs including bar charts, line graphs, scatter graphs and pie charts. A graph is a diagram which represents a relationship between two or more sets of numbers or categories. The data items are shown as points positioned relative to axes indicating their values. Pupils are typically first introduced to simple bar charts and learn to interpret their meaning and to draw their own. More sophisticated statistical graphs are introduced as the pupil's mathematical understanding develops. Pupils also learn about coordinates as a pre-requisite for understanding algebraic graphs. They then progress to straight line graphs before learning to work with curves, gradients, intercepts, regions and, for older pupils, calculus.
- Loci A locus (plural: loci) is a set of points whose location satisfies or is determined by one or more specified conditions. This topic links to Geometry and Shape and provides opportunity for accurate drawing and 'people maths' where pupils can position themselves in the classroom according to certain conditions.
- Transformations A transformation in mathematics is an operation performed on a shape (or points) which changes the view of that shape (or points). This topic includes four transformations namely reflection, translation, rotations and enlargement. A reflection can best be described as the mirror image of a shape in a given line (which acts as the mirror). After reflection the shape remains the same size but the orientation is the mirror image of the original. The transformation known as a translation can be thought of as a movement or shift in position. The size and orientation of the shape remains the same but the position on the plane changes. A rotation can be described as turning. This transformation is defined by the angle of turning and the centre of rotation (the point which does not move during the turning). Finally enlargement is the term we use when a shape increases in size but maintains the same shape. The shape after enlargement is defines as being similar to the shape before enlargement. His use of the word similar has a precise mathematical meaning. All of the angles in the enlarged shape are the same as the angles in the original shape and the lengths of the sides are in the same proportion. An enlargement is defines by the scale factor of the enlargement and the centre of enlargement. We use the term enlargement even if the shape becomes smaller (a scale factor between minus one and one). A negative scale factor will produce an enlarged mirror image of the original shape.
- Vectors Vectors usually first make an appearance when pupils learn about transformations. A translation is best described with a vector written as a two by one matrix. Following that pupils learn how vectors can be used to prove geometric relationships in simple line diagrams. Ultimately vectors are studied as a major topic for A Level and International Baccalaureate courses where vectors in three dimensions are included.You can think of a vector as what is needed to 'carry' the point A to the point B. The Latin word vector means ‘carrier’ and was first used by 18th century astronomers investigating planet rotation around the Sun.
Lesson Starters
Here are some suggestions for whole-class, projectable resources which can be used at the beginnings of each lesson in this block.
1st Lesson
Mirror Maths
The bottom half of some symmetrical calculations are shown above. Can you work out the answers?
2nd Lesson
Wrapping Paper
Find the order of rotational symmetry of the repeating pattern.
3rd Lesson
Rotational Symmetry
Draw a pattern with rotational symmetry of order 6 but no line symmetry.
4th Lesson
Goat Grazing
Find the loci of the goat's position as it eats the grass while tethered to the rope.
5th Lesson
Hexagon
On a full page in the back of your exercise book draw a perfectly regular hexagon.
6th Lesson
Bridge Crossing
The classic puzzle of finding a route which crosses each bridge once.
7th Lesson
Countdown
How close can you get to the target by making a calculation from the five numbers given?
8th Lesson
One Ninth
Investigate three fractions which add together to give one ninth.
9th Lesson
Refreshing Revision
It is called Refreshing Revision because every time you refresh the page you get different revision questions.
Some of the Starters above are to reinforce concepts learnt, others are to introduce new ideas while others are on unrelated topics designed for retrieval practice or and opportunity to develop problem-solving skills.
