The Vectors Topic in both SL and HL contains quite a bit of content that is linked to other areas of mathematics (algebra, geometry, trigonometry and kinematics) and certainly to other subjects - especially physics. Practical applications of vectors arise in many situations. There are not only many ways to apply vectors for modeling various phenomena - but also certain problems encountered earlier in a students' mathematical career can be answered more efficiently by using vectors. Thus, the teaching of vectors and their applications can benefit greatly by making connections with other topics - especially ones involving practical applications.
Syllabus content - SL & HL
The key difference to recognize between the two courses is that there is a great deal more syllabus material on three-dimensional vectors in HL than there is in SL. The fact that the HL syllabus includes both the scalar product (dot product) and the vector product (cross product) whereas the SL syllabus has only the scalar product is a clear sign that SL students do not do as much with three-dimensional vectors as HL students. SL students are required to know how to work with three-dimensional vectors and the vector equation of a line in three dimenstions (also 2D, of course). Along with the vector product and its properties, syllabus content which is exclusive to HL includes: properties of the scalar product and its properties, and equations of a plane (3 different forms and finding intersection between a line and plane and between two planes).
One way to begin the study of vectors (even if students have encountered vectors in a previous mathematics or physics course) is to actually avoid using the word 'vectors', and use some other term or phrase which better fits what a vector actually does. It's the concept of a vector (what information a vector provides) that is most important to get across to students when first starting to teach the Vectors Topic.
♦ Two initial activities ♦
Movement Instructions: This student activity attempts to give an engaging and informative introduction to two-dimensional vectors referring to them as 'movement instructions' rather than 'vectors'. Take a look at this activity (there are six separate sections - each with its own set of instructions) and try it with your students at the very beginning of the Vectors Topic. This student activity is suitable for both SL and HL. download file: ACT_4-1-15v1_SLHL_movement_instructions
Another activity that I like doing with SL and HL students at the start of their formal study of vectors is answering a question that they should be able to answer using some trigonometry. The question is simple and short: find the measure of the acute angle between two diagonals of a cube. In fact, I prefer to have students do this question as a class 'warm-up' exercise before I've even announced that we're starting a unit on vectors. I simply give them an exercise with no instructions on how to approach it - i.e. an open-ended question. Unless someone in the class has studied vectors before the expectation is that all students should answer the question using the cosine rule - so, the exercise also serves as good review for an important item in the Trigonometry Topic. Then what really makes the exercise useful is to re-visit the question at the end of studying vectors (or at least after a thorough study of the scalar product). Answering this question again using knowledge of vectors should give students an appreciation of how using vectors to solve geometric problems (and exam questions) can be very practical and efficient.
[ ANSWER: angle is approximately 70.529 degrees ]
Test covering most of the content for SL vectors. Contains 7 questions (total of 57 marks) - 5 questions with no GDC, and 2 questions with GDC allowed. Answers included on last page ( be sure to detach last page before giving to students ! )
Test for SL vectors. Contains 8 questions (60 marks) - 4 questions with no GDC, and 4 questions with GDC allowed. Worked solutions (solution key) available below.
Set of worked solutions for Test2 SL vectors above.
Test covering most of the content for HL vectors. Contains 6 questions (total of 50 marks) - 3 questions with no GDC, and 3 questions with GDC allowed. No space is provided for students to show working - so separate answer paper will be needed. Answers included on last page ( be sure to detach last page before giving to students ! ). Solution Key available below.
Full worked solutions for Test1_HL_vectors above
Test (longer than Test1 above) covering majority of content for HL vectors. Contains 11 questions (total of 80 marks) - 5 questions with no GDC and 6 questions with GDC allowed. Space is provided below each question for students to show working. Answers provided on last page (so be sure to detach last page before giving to students). Solution Key below.
Full worked solutions for Test2_HL_vectors above
Unit test on vectors with 9 questions - total of 70 marks. 4 questions with no GDC (Part1), and 5 questions with GDC (Part 2). Worked solutions available below.
Full worked solutions for Test3_HL_vectors above
Set of six exam-style questions on vectors suitable for SL and HL; answers included