# Number & Algebra

'A guide for for teaching! Ideas and resources and lessons to look forward to'

*This page is good for the 'Number and Algebra' units from both the new and previous syllabi! There is also a sub-section for 'Financial Mathematics' which is wrapped up in this unit in the new syllabus but was separate in the old syllabus*

This topic is a curious mix of 'Mathematical skills', 'applications' and 'far reaching new concepts' and as such it can be a challenge to arrive at consistent style for teaching it. At first glance, many of the ideas look like revision of basic number skills like rounding, percentage error, classification of numbers and scientific notation. Each of these however offer an opportunity to explore the purpose of Mathematics as a modelling tool. SI units are what we use to achieve that modelling and add a very applied element to this topic. When this is all combined with the study of sequences and solving of algebraic equations the topic quickly upgrades itself to fascinating one with many challenges for teachers and students.

This page is a guide to the topic, ideas for classroom activity that encourage critical thinking, practise exercises and some of the best resources available on the Internet to help! The overview written below aims to help teachers think about the main objectives of the module and potential opportunities and issues there are with teaching it! Please follow links to the pages on Ideas and resources, exercises and the internet guide.

#### Ideas and resources

Classroom experiences are the most important thing that we as teachers are responsible for. This links to a page of activities and ideas for teaching this unit that are aimed to be both engaging and effective in encourage students to enjoy, discover and understand mathematics. The list is a brief outline of an activity that links to a page that gives more detail and the associated resources. This is an area that should develop regularly. Think about subscribing to the RSS feed to get notifications of updates.

#### Practice

This page has a variety of tasks designed for practise and revision. Practice questions are an important part consolidating students' understanding of a concept. They can be found from numerous sources and the following is not intended as a 'fix all' solution. Most teachers appreciate that questions gathered from numerous different sources make the best diet!

#### The Internet Guide

This page is intended to list and outline some of the best resources available online to support the teaching and learning of this module. These will mostly be videos, virtual manipulatives and self help sites. Where appropriate, there will be a short commentary of what they are and how they might best be employed! It is also intended that this selection grows over time and that users contribute.

## Overview

Few of the elements in this topic should be considered presumed knowledge and it is crucial to give each of them the required time to make sure they are understood. There are some key links to be made between geometric sequences and exponential functions, arithmetic sequences and linear (n^{th} term) functions. SI units links to indices and algebraic fractions. Scientific notation relies on understanding the core structure of the decimal system. Rounding and estimation becomes a truly fascinating topic when you realise that almost all of the numbers we use to describe the world around us are estimates certainly containing errors that can compound (perhaps even geometrically) to distort or limit our perception of truth. This is a great ToK link, particularly when challenging the notation that Mathematics is exact. Arithmetic sequences offer a great opportunity to tackle Mathematical proof with Maths Studies students. There are fewer more pertinent applications of Mathematics than the study of population growth or spreading of disease, both of which can in part be modelled by a geometric sequence. All round, this topic can sometimes feel like a hotch potch of ideas but each of them are fascinating and important to explore and will filter through the rest of the syllabus. It can be a useful topic to teach early in the course if not first!

### Key elements

- sets of numbers
- rounding, estimation and percentage error
- scientific notation
- SI units
- arithmetic sequences
- geometric sequences
- solving equations using the GDC

Please refer to the subject guide for full details

### Teaching time

This topic is given

### Sticking points

In time these sticking points will be illustrated with a video or screen cast....

#### Defining and remembering

Sets of numbers is an interesting topic and lends itself very well to exploration. It also links well with the Sets, Logic and Probability (SLP) topic. I find, however, that remembering the sets of numbers and their definitions proves consistently difficult for students. This begins to hint that there is a danger in trying to 'teach' these ideas. There is not always time to explore in this syllabus and often there will only be a handful of marks available on this topic, but if you consider its relevance to SLP then it may be worth getting students to make their own classifications of a group of numbers and try to discover the fundamental differences in these types of numbers for themselves. It can seem trivial but actually there are many conceptual misunderstandings about sets of numbers that can hinder progress elsewhere. The fun video below can be a good way to introduce the discussion with students.

#### Routines

Rounding to 3 significant figures is the default position for answering IB questions unless otherwise stated and being fluent with rounding is worth a lot of marks on a Maths Studies paper. Although rounding is an instinctive process, the formal teaching of rounding often undermines that instinct and creates a mental block that prevents students from achieving the fluency they need. As such, it should not be considered routine until it is!

Calculating a percentage error is a concept that relies entirely on a thorough understanding of percentages or a crystal clear process for applying a formula! The latter is faster but riskier, the former can be a big can of worms. The aim is just to point out here that what look like small syllabus objectives can easily expose frailties in students understanding. If some of these can be tackled on this course then great!

#### SI units

Again, this is an example of a topic that looks small but relies on a deeper understanding. Students are asked to consider units like algebraic variables and manipulate them accordingly. Its quite a shift to treat units like variables although they actually are not. To understand that m/s is the same as ms^{-1 }and that acceleration can be measured in ms^{-2} relies on understanding about negative indices. That 36km/h is the same as 10m/s is initially surprising and when this is equated roughly to the speed of 100m sprinter even more so. Asked to make this conversion themselves, many students will not be able to. If students are then faced with concepts like momentum and density that they use considerable less then the task becomes even more difficult. There is a big opportunity here to help students understand the sticks with which our world is measured but it is not easy.

#### Formulae

This information booklet for this topic contains a number of use formulae and I am not suggesting for a minute that students should not make good use of these. They do, however, pose some danger. For example, a student may incorrectly use the formula to calculate the 20th term of a given sequence when their actual understanding of how a sequence works would get them to the correct answer. I am suggesting that if students can rely on their understanding of these situations then they are less likely to make incorrect substitutions into formulae. A reliance on formulae can detract from this understanding. Equally students can go too far the other way and work out the 20th term by working out all of the terms in between without stopping to think about how they could carry out their calculations with less steps. It is a tricky balance and one worth considering.

### Teaching Opportunities

#### Exploration

As mentioned above, students can really explore some fundamentals of Mathematics in this topic. A simple question like 'Are there more rational or irrational numbers?' can spark a revealing exploration that helps define the two sets and consider where and when they occur with a nice little voyage into the wold of different sized infinities! Why are integers important? What would it be like without them?

What is the effect of compounding percentage errors? When is an error too big to be acceptable? What do we know that is exactly measured or defined?

#### Proof

Set the task and tell the story of how Gauss added up the first hundred integers so quickly! Arithmetic sequences provide an excellent opportunity to derive the formula for the sum of an arithmetic sequences for Maths Studies students. In doing so students get an experience of a mathematical proof that helps them to understand that all formulae are derived. They start as conjectures that then have to be proven. This is a good ToK moment examining the nature of mathematical knowledge.

#### Applications

- geometric sequences and population growth
- geometric sequences and investments
- SI units used to describe our world
- percentage errors and tolerance levels
- scientific notation and the universe

These and more ideas can help add relevance for students and as such help them to understand.

#### Links

This is the key point for this topic. Most of what is studied in this topic can and should be linked to other areas of the syllabus.

- arithmetic sequences and linear functions
- geometric sequences and exponential functions
- geometric sequences and compound interest for Financial Mathematics
- SI units for geometry and trigonometry
- rounding and approximation filters through the whole syllabus
- sets of numbers for Sets, Logic and Probability
- solving equations for functions and calculus and more

### Ideas and resources

This links to a page of activities and ideas for teaching this unit. The list is a brief outline of an idea; some are just ideas and others link to a page that gives more detail and some resources. This is an area that should develop regularly. Think about subscribing to the RSS feed to get notifications of updates.

## Selected Pages

### Visual Sequences Friday 23 September 2016

'Get out the multilink cubes and literally build an understanding of arithmetic sequences'Build a visual representation...

more

### The 6 million question Thursday 22 September 2016

Big numbers present us with a real challenge! How do we understand their size and the significance of what they represent?...

more

### N&A Teaching ideas Thursday 22 September 2016

Here you should find a wealth of resources that can be used by you and your students for teaching this topic. The resources...

more

### N & A Internet Guide Tuesday 24 July 2012

This page is intended to list and outline some of the best resources available online to support the teaching and learning...

more

### Financial Mathematics Monday 26 September 2016

This is one of the most obviously relevant parts of the course and from my experience, students show an appreciation for...

more

### Focus on - Geometric Sequences Monday 26 September 2016

The Geometric sequence is a fundamental bit of mathematics that is an inherent part of exponential growth and decay. Their...

more

# Number & Algebra

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