Congruent Triangles and Similar Shapes
Students are required to identify and construct congruent triangles. Students are also required to construct similar shapes by enlarging a given shape by a scale factor. Further work could include requiring students to enlarge a shape by a scale factor from a fixed point.
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Links and Resources
This booklet from National Strategies describes teaching approaches that can be used to develop mental mathematics abilities beyond level five. Originally written for key stage 3 the resources include some interesting activities and probing questions that can be used to check and challenge existing knowledge. It includes some of the aspects of geometry that have been identified as having implications in terms of understanding of geometry.
This booklet includes enlargement and similarity. Enlargements and similarity are applications of ratio and proportion. Making this link plays an important part in helping pupils to see the ‘big picture’. Mental methods that pupils develop to solve ratio and proportion problems can be extended to their work in geometry. Classroom experience suggests that pupils’ understanding of enlargements and similarity can be developed through a range of practical and mental activities.
Students work in groups and, intuitively, sort sets of photos to identify those which have been enlarged from an original image. They then justify their decisions and share their explanations with the rest of the class. Photo sets three and four may cause some disagreement, which should motivate students to find a rigorous approach to determine which ones are true enlargements. The resources include teacher’s notes. The discussions could be guided to include a discussion of fractional indices. Geogebra could be used to extend to enlargements with a negative scale factor.
This video is ideal as a starter activty to make students think. The first part of the video deals with constructing triangles. The second problem uses similar triangles.
The problem posed is how to find the length of the pier. The video should be paused regularly to discuss different options. The video releases clues as to how the problem may be solved, allowing students to suggest their own solutions. There follows a well-explained solution to the problem using similar triangles.
In this video, the challenge is how to measure something that is too high to measure. In this case, the challenge is how to measure the height of a central column in Durham Cathedral. The video shows how this can be done by correctly positioning a 30cm ruler. Students are challenged to find the height of the column given three of the measurements in similar right-angled triangles.
This resource has a number of activities appropriate to this topic. Shapes that can grow requires students to find scale factors by calculating how many times one shape fits into another. Areas of similar shapes is a little more advanced but ideal for extension work. Students are presented with a trapezium that has been enlarged by different scale factors. Students are required to investigate what happens to the areas of these shapes. Similar triangles presents students with a number of triangles and requires them to be grouped into sets of similar ones.
Activity 5 explores fractional scale factors. Activity 12 explores negative scale factors. Both sets could be used as part of the same lesson to make the connection clear to learners. The language used for negative scale factors may need some explaining.
Activity 7 investigates further the concepts of congruence and similarity in relation to area in similar figures. Activity 11 formalises the learning from the previous example, includes circles and real contexts. Activity 13 Similar Solids takes this one step further by extending the investigation to look at the relationship between the volumes of similar solids.
The text, Similarity begins by asking students to explore the concept of enlargement, identifying which shapes are enlargements of others and finding the corresponding scale factors. The second section introduces the concept of similar shapes with explanations, examples and exercises requiring students to calculate missing lengths from the given diagrams. The third section explores the connection between the line ratio, the area ratio and the volume ratio of similar shapes. The final section explores the use of similarity in the context of maps and scale models.
The activity sheet has three appropriate activities. Similar shapes requires students to which of the given shapes are similar. How far away uses similar traingles to calculate the width of a river. Paper sizes asks students to explore similarity of different paper sizes.
This resource contains two instant maths ideas. The first suggestion supplies students with the four conditions of which one is required to be satisfied for two triangles to be congruent. Students are asked to consider why these are the only conditions that guarantee congruence.
The second task asks students to find shapes which can be cut into two or more congruent pieces, each of which is similar to the original shape. It is suggested that students start by considering triangles before moving on to shapes with more sides. An example of a quadrilateral which exhibits the required property is given as an example.
Section 7 of this resource, Similar quadrilaterals, starting on page 42 of the pdf, begins by asking if two triangles with the same sized angles are always mathematically similar, and does this property also apply to rectangles. Further exploration considers the ratio of the length and the width of rectangles and the geometric properties of similar rectangles. Task number 8 requires students to consider two hypotheses and asks students to attempt to find a counter example to disprove the hypothesis. Further work considers similar rhombi, parallelograms, kites and trapezia.
The opposite corners of a trapezium are joined, creating four triangles within the trapezium. The areas of the triangles formed using the parallel sides as is given. The task is to calculate the areas of the other two triangles.
Working out the general result involves using ideas of similarity, area scale factor, length scale factor, multiplying out brackets and working with square roots. The final answer is a very elegant relationship between the triangle areas