# Proportion

These resources are designed to encourage students to see the bigger picture of proportional reasoning and cover:

- conversion between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts
- the understanding that X is inversely proportional to Y is equivalent to X is proportional to 1/Y
- the construction and interpretation of equations that describe direct and inverse proportion
- the interpretation of the gradient of a straight line graph as a rate of change and to be able to recognise and interpret graphs that illustrate direct and inverse proportion

### Key National Strategy Resources for GCSE Mathematics

Proportional thinking is essential in the solution of many problems. This may not be immediately apparent and may not be accessible through a single technique such as the unitary method. The underlying ideas need to be developed systematically.

This is a website with a selection of CPD resources including teaching ideas from the Secondary National Strategy, chosen to support planning, teaching and assessment. It includes a section on proportional reasoning which starts by reviewing why developing proportional reasoning is so important and establishes links between different strands of mathematics. The resources include a proportional reasoning summary table that can be used to gain an overview of the teaching and learning support available in the four related sections which are fractions, ratio, scaling, proportional sets. The resources include comprehensive teacher notes and related resources.

Each of the four sections includes aspects of proportional reasoning which pupils find hard to understand and which are often overlooked in text books. Here is the structure of each section:

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an overview of concepts and relationships

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an outline of associated language and notation, supported by teacher notes for short mental activities

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suggestions for making connections through supporting images, along with teacher notes and resources.

### Developing Proportional Reasoning N6

In this resource students are required to reflect on the reasoning they currently use when solving proportion problems, examine proportion problems, appreciate their multiplicative structure and create their own variants of proportion problems. Proportional reasoning is notoriously difficult for many students and many have difficulty in recognising the multiplicative structures that underlie proportion problems. Instead, they use addition methods or informal methods using doubling, halving and adding. This session aims to expose, and build on, this prior learning.

Students are given four direct proportion problems to solve, taken from different areas of the mathematics curriculum. They then compare their methods for solving these with methods produced by other students. This leads to a discussion that compares the use of more primitive informal methods that use adding, doubling and halving with the use of more sophisticated methods that use multiplication.

This resource is aimed more at building understanding with students who may previously have struggled with this concept.

These materials exemplify the ideas and approaches adopted in the Standards Unit pilot. To get best value from them use them in association with the guidance and other materials in the resource. http://stem.org.uk/rxzr

### Hot Under the Collar

In this task, students consider two different methods for converting temperatures from degrees Celsius to degrees Fahrenheit. Students have to select a way of comparing the two methods, explore the effects of varying temperature, make accurate calculations and devise a method of deciding when the approximate method gives a value that is too high. They are then required to interpret their tables and graphs to help solve the problem and relate their findings to the original context.

Teacher guidance includes concepts which may be discussed with students, examples of probing questions which may be useful and assumptions that students need to make when completing the task.

The assessment guidance consists of a progression table detailing how students may improve in each of the key processes. The resource concludes with a number of examples of students' work, together with comments, probing questions and feedback.

### Interpreting Distance-Time Graphs with a Computer A5

In this resource, students interpret linear and non-linear distance-time graphs using the computer programme Traffic. This program provides a simple yet powerful way of helping learners to visualise distance–time graphs from first principles. The program generates situations involving traffic moving up and down a straight section of road. It then allows the user to take ‘photographs’ of this situation at one-second intervals, places these side-by-side, and then gradually transforms this sequence of pictures into a distance–time graph. In this way, direct correspondences between speeds and gradients are obtained.

Learners are asked to describe situations, and draw and interpret distance–time graphs. Later, examples are offered that involve cars travelling at non-uniform speeds

These materials exemplify the ideas and approaches adopted in the Standards Unit pilot. To get best value from them use them in association with the guidance and other materials in the resource. http://stem.org.uk/rxzr

### Interpreting Distance – Time Graphs A6 *suitable for home teaching*

In this resource students interpret and construct distance–time graphs; relating speeds to gradients of the graphs and accelerations to changes in these speeds. Students have often constructed distance–time graphs before. However, experience shows that many still interpret them as if they are pictures of situations rather than abstract representations. In addition, they also find it difficult to interpret the significance of the gradients of these graphs. In this session, students begin by discussing a question that is designed to reveal common misconceptions about distance–time graphs. They then work in pairs and threes to match descriptions, graphs and tables. As they do this, they will interpret their meaning and begin to link the representations together.

These materials exemplify the ideas and approaches adopted in the Standards Unit pilot. To get best value from them use them in association with the guidance and other materials in the resource. http://stem.org.uk/rxzr

### Some Simple Functions

This book addresses the topic of functions. The content of the book is intended primarily for teachers, for whom, it was believed, it would provide a valuable background which would deepen their understanding of mathematics and give them insight into mathematical thinking.

Teachers will find that the functional approach links aspects of mathematics which, on the surface, seem disconnected and compartmentalised. There are a range of examples both abstract and linked to real life. It is quite rigorous and would probably suit teachers planning for students aiming at the higher levels.

Chapter 2 covers direct proportion and chapter 5 covers inverse proportion.

### Ratio, Proportion and Rates of Change

This sub-collection of resources from the Virtual Text Book collection contains three resources designed for use on an interactive whiteboard to aid the teaching and learning of a variety of topics in ‘Ratio, Proportion and Rates of Change’. Each resource is an interactive Excel worksheet to encourage teacher-student interaction. The topics covered are:

• Direct Proportion

• Interest: Simple and Compound

• Inverse Proportion