I have been a fan of ratio tables for a number of years and we have recently decided to use them consistently across the department. It has been nice to see that they have been (deservedly) celebrated a lot recently, in particular at the latest Le Salle Maths Conference, so I thought I would offer my experiences.
What is a ratio table?
A ratio table is a way of representing two or more things which have a multiplicative relationship. I first came across them around seven years ago when I was part of a multiplicative reasoning pilot scheme run by the NCETM. There, we were shown a horizontal table linking the costs of different lengths of ribbon.
Horizontal or Vertical?
It was relatively recently that we discussed using ratio tables as a consistent method throughout the faculty. We listed the different topics that could be represented using ratio tables (more on that later) and its effectiveness. One of the biggest discussions was about whether using a horizontal table would be best overall, as students are used to showing their calculations vertically when dealing with percentages and equations etc. Although, a horizontal table is more useful when calculating with recipes, as they tend to be set out that way. In addition, a horizontal table shows clearer links between fractions and percentages. The picture below shows an example of this
We decided that a vertical table would be best overall, given that students are more familiar with working down the page. Here are some examples of the vertical ratio table in action:
I have made an adapted version of the ribbons resource to reflect this.
Using a ratio table
When using a ratio table I insist on two things. The first is to ensure that the table has clear headings, so that there is an understanding of what each value represents. Included within the heading is the units, so that there isn’t a need to repeat the units on each line of the table.
The second expectation is the inclusion of arrows to show the intended calculation, to mitigate against making mistakes and ensure that working out is evident.
Exploring with Ratio Tables
I like to spend lots of time just practising using a ratio table to perform different calculations. In the Using Ribbons task (see below), you may have noticed that I prompt the students to find as many different ways as possible to solve the problem.
This shows that there is more than one way to skin a cat – students are invited to find their own way to a solution, rather than follow a prescribed model. All of the ideas are shared so that there are opportunities for everyone to explore the different connections. Of course, we would like students to work efficiently as well as effectively, so there is lots of discussion about which may be the best way.
In doing this, students also get to see that when using a ratio table, not only can you multiply and divide, but you can also add and subtract rows of the table in order to find the values of other amounts.
Consistency is Valuable
Ratio tables can be used in many areas of Maths, including:
– Percentage (amounts, increase/decrease, reverse, express as a percentage, percentage change)
– Compound measures (speed, density, pressure)
– Proportion (direct, inverse, currency, best buy, similarity)
– Circumference (see examples above)
If students can use a ratio table in all of these topics (and even more!), there are fewer methods to have to remember and they will quickly become even more fluent in using the table to perform calculations. Furthermore, by using this method consistently, it can help students make connections between topics as they see the same processes being used throughout. It can also help students make even deeper connections within a topic, and a perfect example of this is in calculations involving speed, distance and time. Not only does it eliminate the need for awful formula triangles (don’t get me started), but it also can help students understand more that speed represents the relationship between distance travelled and time taken.
Although I am an advocate of ratio tables, I know that nothing is perfect. One of the biggest issues I have always had is that students have previously been taught to use a different method, so it has been counter-productive to undo prior learning. However, now that we have decided as a faculty to use a consistent method, this should (hopefully) be less of an issue. The next step I would like to take is to work with other faculties, in particular the Science department, to make it a whole school approach (and bin off the formula triangles for good).