# Bar Model Method: In Secondary Mathematics

ACMNA153; ACMNA173; ACMNA175; ACMNA187; ACMNA188; ACMNA123; ACMNA126; ACMNA152; ACMNA155; ACMNA157; ACMNA158;; ACMNA179;

This is the third unit on the special topic “The Bar Model Method”. This is a pedagogical strategy widely used in Singapore to help students solve word problems.

This unit, Bar Models in Secondary Mathematics, is designed for early secondary students. Six lessons show how the bar model method can assist in solving problems involving fractions, ratio, and percentages. Two lessons can be used as part of introductory algebra to make a transition from solving problems with arithmetic to solving them with algebra. The aim is to demonstrate how the bar model can be used as a flexible problem solving strategy in many topics.

The lessons can be taught as a group, but are probably best taught alongside the relevant topics.

A bridging lesson, Lesson X, is provided for students who have not previously used bar models.

### Lesson X: Preparation for Bar Models

This lesson provides a concise introduction to the bar model method. Students are familiarised with the two most commonly used bar models (Part-Whole and Comparison models), using selected worded problems as exemplars. Students then use the bar models as a scaffold to solve the problems.

### Lesson 1: Varieties of Bar Models (Whole Numbers)

The bar model is a tool for students to use when solving worded problems. In this lesson, students learn to use this tool to represent contextualised multi-step, multi-art worded problems, by studying worked examples and practising with further tasks. This lesson focuses on problems (involving whole numbers only) that utilise a variety of bar model types. The skills developed are useful across all mathematics topics, and especially when solving problems with algebra (optional in this lesson).

### Lesson 2: Varieties of Bar Models (Fractions)

In this lesson, students learn to use the bar model method as a tool for solving worded problems involving fractions (and fractions of fractions) by studying worked examples and practising with further tasks. Students represent contextualised multi-step, multi-part worded problems with three different types of bar models (part-whole, comparison, change). The skills developed are useful in many mathematics topics.

### Lesson 3: Algebra with Bar Models 1

In this lesson, students learn to use the bar model method to assist in forming and solving simple linear algebraic equations from word problems. This is done through whole class examples and individual or group practice. Students label unknown quantities using algebra, then set up equations, then solve them, with the bar model as a supporting visual representation.

### Lesson 4: Algebra with Bar Models 2

Students learn to use the bar model method as a scaffold to solve word problems algebraically. They set up bar models to represent problems, label quantities with algebraic variables and expressions, set up equations, and then solve them to find unknowns. Understanding the standard algebraic techniques is supported by the visual representation. All the problems involve changing relationships (change bar models) and the equations are simple linear equations.

### Lesson 5: Percentage of a Percentage

This lesson is for students who can find a percentage of a quantity (e.g. 15% of 28), and can express a proportion as a percent (e.g. 6 is 40% of 15). Students learn to solve word problems that involve finding a percentage of a percentage of a quantity (e.g. finding a discount of 10% on a price already discounted by 25%). Students represent the key components of the problems in bar models, which highlights which ‘whole’ they are using at each of step in these multi-step problems. The visual support of the bar models facilitates an organisation and understanding of problems, which can be used in many mathematical topics. Polya’s four steps of problem solving are used to structure the solution process.

### Lesson 6: Comparing Percentages

In this lesson, bar models are used to enable students to solve problems that involve percentages of different ‘wholes’. The lesson emphasises that a percent is always relative to the nominated ‘whole’ quantities. Worked examples are followed by group or individual practice. The examples involve equal quantities, expressed as percentages of different ‘wholes’. The comparison bar model supports finding a conversion factor between the percentages of the different ‘wholes’.

### Lesson 7: Bar Models and Ratios 1

Students first learn how to solve simple practical problems involving a single ratio, using the bar model method as a tool for visualisation. Students then learn how to solve more difficult problems involving two ratios, where the ‘units’ from one ratio need to be subdivided to make them compatible with the other ratio. The lesson ends with students classifying the different ratios involved in computer monitor resolutions.

### Lesson 8: Bar Models and Ratios 2

This lesson extends students’ exploration of bar models as a visualisation and organisation tool whilst solving ratio problems. Students work with two and three ratios of quantities, mainly both before and after an amount has either been transferred between the quantities or otherwise changed. The problem structure is slightly varied throughout the lesson, resulting in students encountering a variety of different approaches to solving problems involving ratios. Alternative solutions to the problems using fractions or algebra can be explored, connecting these topics with ratio. The bar model method is intended to be a flexible tool that students can take with them from this topic to support problem solving in all mathematics areas.

Last updated December 11 2018.