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Why Do Students Struggle in Geometry?
Tell your students that all the information they need to provide for a proof will be in these axioms.
Working backwards The student should ask themselves what do I have to prove last to complete my proof, e.g. to prove lines are parallel, either alternate angles are equal; corresponding angles are equal or cointerior angles are supplementary. If a student does not know where to start, then teach him/her to work backwards. Teachers should demonstrate this often. Different proofs Geometry gives the teacher the opportunity to show students that there is often a variety of ways to solve a problem. Encourage your students to share different proofs. You might even like to create a contest where groups of students try to find the most proofs for a particular problem to reinforce this idea. Constructions Adding a simple line to a diagram can mean the difference between finding a proof or not. Thinking about what you might have to prove last to finish your proof will lead you to the type of construction you need to do that. Always pencil in the construction initially and use the diagram to go through your proof. Then, ink in your construction and add any axioms that come from the construction that assist your proof. Assumptions Euclidean Geometry is based on three assumptions. Explain how as you introduce geometry to your classes. Define exactly what is meant by an assumption. Axioms When you begin Euclidean Geometry, show how the assumptions and the terminology fit together to create/prove the first axiom, i.e. the assumption, "Adjacent angles forming a straight line are supplementary" is used to prove, "Vertically opposite angles created by straight lines are equal in size". In lower high school classes, when geometry is the topic, I will always give a revision lesson where I develop each of the axioms (to be revised) beginning from the assumptions to remind students how they all fit together. It also shows that geometry is a continuum. Model proofs As with all parts of Mathematics, it is important for the teacher to model regularly, proofs for students. Some teachers actually give students a 'model' to follow. It might be divided into these headings: given; to prove; constructions; and proof. Justification for each step would also be included as well as a detailed diagram. Further hints for the student Look for the obvious or the simple ideas/axioms to use to start. Don't assume you need a complicated idea to begin. Work down the page. It is easier to check and reduces errors. Remember that angles are named because of their (a) size; (b) shape and (c) position. Often they come in pairs. Even an obvious fact must be justified. Geometry is fun to teach. Often students find it difficult to see its relevance but the mere fact that it teaches logical thinking makes it an important learning experience for them. Geometry is the study of two and three dimensional shapes, lines, and planes. They cover all relevant topics and make Geometry an easy subject to grasp. Geometry deals with different shapes and the measurement of their size, volume and area. These are measurements that we all make in our minds, on a daily basis. http://www.kiwibox.com/mice87storm/blog/entry/108519507/tipsinstudyinggeometry/?pPage=0 , http://www.go2album.com/pg/groups/1793887/tipstohandlegmatgeometry/ , http://www.sayjack.com/learn/english/vocabulary/73470/ 