The ratio table

In RME, the ratio table encourages learners to find their own route to solving proportional reasoning problems.

Its flexibility and wide range of applicability makes the ratio table a key model in RME classrooms. As students’ confidence grows in using this model, they will discover that many contexts can be represented in a ratio table. Speeds, rates, best buys, and density all appear in Section B of Seeing It Differently (N2), where the ratio table is first introduced. In the example above, a student has used the ratio table to calculate the cost of 33 eggs, without relying on an algorithm for long multiplication.

On this page we introduce you to RME’s approach to the ratio table by explaining:

• how the ratio table emerges from particular contexts and models,
• why the ratio table is a powerful and flexible tool for advancing understanding of number and proportional reasoning,
• and how an RME approach to the ratio table relates to other versions you may have encountered.

The ratio table is an emergent model.

Recipes provide a great natural context for generating a ratio table. Section B of Seeing It Differently (N2) begins with a “Junior Masterchef” school competition, challenging students to scale up several recipes to feed more people. Because ingredients are normally listed down the page in a recipe, this context already lends itself to jotting down an extra column of figures.

In the video clip below, Sue Hough and Kate O’Brien discuss how this context helps students to embrace the power of the ratio table. They also explore how teachers can ask questions that allow students to engage in proportional reasoning, while still making sense of the numbers. For more ideas about this context, see our teacher strategy pages  Go back to the context and “Say what you see”.

Recipes naturally generate rather complex ratio tables with many rows. As Kate and Sue demonstrate, the more columns students fill in together, the more possibilities they will see. Their collective ability to move freely between additive and multiplicative relations will be a very useful strategy across a range of topics.

The ratio table is an extension of the bar model.

The ratio table is closely related to the bar model. In the opening scenario of Section B of For Every One (PR2), students learn about this connection through the context of waist measurements from a young boy and his father.

As Sue explains in the short video below, the ratio table emerges from this example as a bar model “without a scale”. When we extend the number bar to values that no longer fit on the page, we need a new model for thinking: the ratio table steps in to fill this role. Sue also offers some ideas about how to support students to see these connections.

The ratio table allows students to reason for themselves.

Consider this problem: A photocopier takes 12 seconds to produce 20 copies. How long will it take at the same speed to produce 190 copies?

This task can be tricky. The numbers are hard to work with and knowing what to divide or what to multiply is not obvious for learners. Rote learned rules can fail them. Supporting students to use a ratio table representation of the problem enables them to make sense of what the question is asking.

Here is a possible route to solving the photocopier problem with the ratio table:

This learner has scaled up from 20 to 200 copies, then divided 20 by 2 to find the time taken to produce 10 copies. Subtracting 10 copies (in 6 sec) from 200 copies (in 120 sec), they quickly conclude that it will take 114 seconds to produce 190 copies.

Other learners may use repeated doubling — moving from 20 copies to 40, 80, and 160. Eventually, they will need to combine values to reach the time taken for 190. What’s important, however, is that students can write in as many or as few interim values as they need. The pathway to the answer is not fixed by an algorithm.

When first approaching a new problem scenario, understanding what you are doing (and why) is more important than reaching the destination value as quickly as possible. Students will develop their own short-cuts in due course.  When using the RME ratio table, we encourage students to label the rows, reminding them of the context and the meaning of what they are doing as they add in more numbers. Annotations, shown in blue above, are also a helpful way for learners (and their teachers) to keep track of their thinking.

The flexibility of the ratio table enables students to join up the curriculum.

The ratio table in RME can be applied to many topics where quantities are related in proportion, unifying the curriculum for students. Instead of needing to memorise a number of different methods for different topics, they learn to apply the same model. In doing so, they see links across many topic areas. In this clip, Sue discusses the impact of working with a ratio table on students’ (and teachers) engagement with mathematics:

RME’s approach to the ratio table is different.

In this final clip, Sue and Kate look at the RME ratio table alongside the contracted version which you may be familiar with. They think through some of the assets of these two styles of ratio table, and discuss the importance of annotation as a way to keep track of student ideas and strategies.