# Algebra

• Comparison of syllabus content between HL and SL
• Discussion of ways to approach the teaching of the Algebra topic - HL and SL considered separately
• Connections between Algebra topic and other parts of the syllabus
• Thorough discussion on helpful ways to introduce and teach proof by mathematical induction (HL only)
• Potential ideas for student Explorations involving concepts in the Algebra topic
• Exercise sets (including answers) and student activities (including teacher's notes)
• Unit tests and quizzes (with solution keys)

#### Comparison of syllabus content between HL & SL - Algebra topic

There is a significant difference in syllabus content between the Algebra topic in Maths HL and the Algebra topic in SL. The number of recommended teaching hours for the Algebra topic in HL is 30 hours but only 9 hours in SL. That is the biggest difference for any of the six syllabus topics. This is primarily due to the fact that mathematical induction and complex numbers are in HL but not in SL. Take a look at the Algebra syllabus content for HL in the Syllabus Content sub-section of the Basics section on this site where you can see that most of the content in the Algebra topic for HL is not in the Algebra topic for SL.

Although the number of instructional hours recommended for the Algebra topic differs significantly between SL (9 hours) and HL (30 hours), there are several syllabus content items that are in both the SL Algebra topic and the HL Algebra topic. These items include:

♦ arithmetic sequences & series
♦ geometric sequences & series (including infinite geometric series)
♦ sigma notation
♦ working with relevant laws for exponents & logarithms
♦ expansion of binomials (a+b)n
♦ binomial theorem
♦ binomial coefficients / Pascal's triangle / combinations nCr

#### Maths SL

It is possible to include all three syllabus points of the Algebra syllabus content for SL (1.1,1.2 & 1.3) into one teaching unit. However, I prefer that the Algebra SL content be covered in two separate teaching units. I recommend that the syllabus sections 1.1 (arithmetic & geometric sequences/series) and 1.3 (expansion of binomials / binomial theorem) be taught in a Sequences & Series; Binomial Theorem unit, and that syllabus point 1.2 (exponents & logarithms) be covered in an Exponential & Logarithmic Functions unit. This separation does create a potential issue with where to teach some natural applications of geometric sequences such as compound interest. Given that a student should be familiar with the basics of exponential expressions (Prior Learning Topics) prior to entering the SL course, it is suitable to introduce a topic such as compound interest before a unit that covers exponents and logarithms. But, just a simple introduction - i.e. no problems asking how to determine the amount of time it takes for a quantity to increase to a certain amount. The topic of exponential growth (including compound interest) can be further explored in the unit on exponents and logarithms during which logarithms can be applied. [ note: see other section on What About Counting Principles? ]

#### Maths HL

I do not believe it is possible, nor advisable, to teach all of the Algebra syllabus content for HL in one teaching unit. It's not just the issue of quantity (recommended teaching hours are more than 3 times that for Algebra topic in SL), but also an issue of items which do not require being taught together - e.g. Proof by Mathematical Induction can easily be taught elsewhere. Different to what I suggested for Maths SL, I suggest teaching a Sequences & Series unit that covers the content in syllabus section 1.1 (arithmetic & geometric sequences/series) but does not cover any of the other syllabus sections in the Algebra topic. The remaining content in the Algebra topic - binomial expansions, binomial theorem, combinations & permutations, and proof by induction - can all be covered in a separate teaching unit (Counting Principles; Binomial Theorem; Induction). As with SL, it might be best not to cover questions on sequences & series that will require logarithms to solve. It is not completely out-of-bounds, but the unit covering exponential & logarithmic functions will be taught after sequences & series. Certainly HL students should know enough about logarithms prior to starting the course to apply them in the context of sequences & series - but I would advise waiting to pose these kinds of questions during the exponential & logarithmic functions unit. It also serves as a nice review of sequences & series at a later time in the course.

The phrase "counting principles" could have a variety of interpretations but with regard to syllabus content for Maths SL & HL it basically includes the multiplication rule for counting*, factorial notation, combinations and - for HL only - permutations. For SL and the Sequences & Series; Binomial Theorem unit, it is not necessary to spend a great deal of time on counting principles. SL students need to be familiar with factorial notation so that they can understand the formula for combinations (binomial coefficients). It is expected that SL students can compute binomial coefficients by using both the formula for and their GDC. This skill will be very useful with binomial probability distributions (syllabus section 5.8) later in the course. HL students need to spend more time on counting principles, but this should be covered in the Counting Principles; Binomial Theorem; Induction unit.

* multiplication rule for counting (or, fundamental counting principle): If there are m ways an event can occur followed by n ways a second event can occur, then there are a total of m×n ways that the two can occur - and the idea can be extended to more than two events.
The rule can be very useful in other areas - such as in probability with computing the size of a sample space.

Series question - illustrating difference between HL and SL

Here is an example of a question involving series (calculator allowed) that would be suitable for HL but not SL .

Solution for series question above:

## Selected Pages

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### sequences & series2 January 2019

Quick links:► downloadable teaching materials for sequences & series► syllabus content for the Algebra Topic: SL syllabus...
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### proof by induction (HL)28 March 2016

Proof by mathematical induction is only in the HL course (not in SL) - and is the only formal proof method in the HL syllabus....
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