start of Maths HL yr.1
Below are some activities, assessments, materials and ideas that I often find useful at the start of teaching the 1st year of Maths Higher Level. I've presented them in a timeline (week 1,2, 3 etc) of sorts, but there is no necessity to use them in the order they are listed here.
Although it might be a bit on the harsh side, I give my first year Maths HL students an assessment on the very first day of class. See my Start-of-Course Assessment page for a copy of the assessment (and solution key). Students should be able to complete it in less than one hour. I use this primarily as a diagnostic tool for helping me identify students who are not sufficiently prepared for Math HL.
Activity #1 - no technology
Rather than trying to tell students what the Maths HL course is all about - either by talking (too much) the first day of class, or handing out some kind of 'course information' document - I prefer to have the students work together on an activity in which they can experience what I consider a couple of the most important aspects of the course: technical skills and problem solving skills. In my mind, technical skills, mostly have to do with techniques in algebraic manipulation and remembering basic mathematical facts and formulas. Problem solving skills are self-explanatory - but it includes the ability to think critically and creatively, and to break a problems down into smaller, more manageable, tasks. I feel very strongly that for a student to be successful in a rigorous and demanding mathematics course such as IB Maths HL it requires a combination of fluency and confidence in the application of specific techniques and logical thinking.
There is no one perfect activity - and what works well with one group of students may not work as well with a different group of students. However, here is an activity that I have found to be an effective 'starter' for the first day of class (or at least during the first week). I want something that will set a good tone for the rest of the course - and to emphasize the need for strong technical skills combined with problem solving skills.
I call the activity 4 Spheres. Here are the instructions:
Three spheres each with a radius of one unit are placed on a table such that each sphere is touching the other two. A fourth sphere of the same size is stacked on top of the three spheres in an efficient manner such that each sphere is touching the other three. What is the (shortest) distance from the table to the top of the fourth sphere (top sphere)? Give the distance exactly. Do not use a calculator.
Setting up a model of the four spheres stacked in a pyramid structure can help students visualize the problem. As the photo shows above, I find that four inflated beach balls work well. It helps to build a triangular enclosure (mine is made out of cardboard) to keep the three spheres on the bottom layer from rolling around. I also add a large flat piece of cardboard below the bottom layer and balance another on top (see photo at right) and have one of the students come up with the fact that the question in this activity is essentially asking for the distance between these two planes (represented by pieces of cardboard). A photo of a solution written on my whiteboard is shown below.
I have students attempt to answer the question working in small groups of 2 or 3. Being the first week of class, some of the students may not know each other - so this helps students become familiar with each other. Remind students that they are not to use a calculator.
You may have a different problem that would work well as a 'starter' activity. I like a question that combines algebra, geometry and that requires students to do a bit of problem solving, drawing diagrams and remembering basic facts - but also is not too difficult so that at least some students will answer it correctly. Be sure to emphasize that an exact answer is required.
After having taught secondary mathematics for over 30 years, it seems clear to me that there is a strong correlation between a student's fundamental algebra skills and the capability of that student to be successful in a challenging mathematics course such as Maths HL. A high competence in algebraic manipulation is certainly no guarantee that a student will do well in Maths HL, but without being fluent and confident in a range of algebra techniques a student will definitely struggle. Hence, I always have my Maths HL students complete a couple 'Algebra Refresher' assignments followed by an Algebra Review Quiz during the first two weeks of class. This aids greatly in identifying any gaps or weaknesses that some students may have in their algebra skills - and simultaneously communicating to students that they must be confident in a range of algebraic techniques. Any deficiencies in such skills must be addressed at the start of the course. If a student is not able to demonstrate a suitable proficiency in fundamental algebra skills within the first few weeks of Maths HL then it is best to advise the student to change to the Maths SL course.
Further Early Assessments
Quiz_1_HL_alg_refresher_v1 [worked solutions included]
The first assessment that I give my first year Maths HL students is a 7-question Algebra Refresher Quiz. This is given to students soon after then have completed the two 'Algebra Refresher' exercise sets (see paragraph above) and after they've had opportunities to ask questions in class on either of these assignments; and also you've probably provided some extra examples in class to review some of the basic algebra skills covered in the first two assignments.
As a follow-up to the Algebra Refresher Quiz, I often administer an Algebra Review Test to my first-year HL students. It gives me further very helpful information about the level of algebra and thinking skills for each HL student. You can access this test and a solution key containing fully worked solutions below.
Test with 9 questions for assessing HL students' algebra skills early in the HL course. Worked solutions available below.
Worked solutions for the HL Algebra Review - 1 Test above
Tests on 'fundamentals'
During the first couple of weeks in my Maths HL1 classes, I will spend some time reviewing a variety of skills and concepts (mostly algebra, but not exclusively) that I would expect a student starting Maths HL to be familiar with. I often give a short test (9 questions) that I've called Test 1 Fundamentals. It certainly is no where close to covering all prior learning topics, but it provides me (and the students) with helpful feedback on the level of their preparation. After reviewing this first test with students, I will often follow it up by giving them a Test 2 Fundamentals (perhaps in week 3, if not enough time in week 2 - that is longer and more rigorous than Test 1. See the downloable teaching materials on my Prior Learning Topics page for these two tests and answer/solution keys.
Activity #2 - technology allowed
In contrast to the first activity where students were not allowed to use any technology, I like to follow it with a similarly challenging prooblem where I allow students to use any technology they wish. Even access to the internet is OK since it is my experience that they will not easily find a solution by doing an internet search - and will most likely just waste a lot of time if they try. I call the problem semicircle chord cut v1. [Note: A more challenging semicircle dissection problem, v2, is presented in my July 30 blog entry ]
I usually have students work together in groups of two on this problem. Although most students find it more challenging that the 4 Spheres problem in Activity #1, it is possible for students to solve the problem by using technology (it asks for an approximate answer accurate to 3 significant figures). A student could use their GDC to solve the problem, or some mathematics software such as Geogebra or Autograph, could be applied very nicely to solve it. Of course, most students are either not familiar with such maths software or not sufficiently fluent with software tools like these. But, this is one reason why I like to give this problem early in the school year because it gives me an opportunity to demonstrate Geogebra (my choice) to students. See my Geogebra applet that can be used to find the solution to this problem: semicircle chord cut v1 (applet) . I will soon be uploading a worked solution with notes for this problem.