Two-Way Tables and Frequency Trees

Under exam conditions, students normally achieve most, if not all, of the marks when asked to complete a two-way table or a frequency tree. Both feel procedural and safe. Students trust them.

That trust is exactly what I want to use. If I put a topic students find hard, fractions, percentages, ratio or simplifying expressions, inside a structure they already believe they can do, they carry that confidence into the harder part of the question. When the calculation gets tough, the diagram or the table gives them something solid to hold onto. That is where resilience comes from.

Building Resilience Through a Familiar Structure

A grade 4 candidate rarely fails a two-way table question because they cannot read a table. They fail because somewhere inside it sits a fraction, a percentage or a ratio, and that is the part that unravels them.

So I stopped teaching those topics in isolation and started hiding them inside frequency trees and two-way tables instead. The structure does the organising. The student only has to focus on one calculation at a time, in a place they already know it belongs. Nothing about the topic changes, but the student’s relationship with it does.

How I Run This Lesson

I put one question on the board at a time. Students get two minutes of silent thinking time. They can jot ideas on a mini-whiteboard or in their exercise book, but they work alone. No talking, no comparing, just their own first attempt.

After two minutes, they turn to the student next to them and compare approaches. While they talk, I walk the room and listen. I do not feed back yet. I want to hear how they are reasoning before I shape it.

Once everyone has a final answer backed up with clear working, I bring the class together and go through it. The answer does not need to be correct. It needs to be thought out and backed up with working I can follow. That distinction is important. It values their reasoning, effort and resilience rather than the final answer. This is what it takes to teach problem solving in mathematics.

Some students finish early. Others are still waiting for the rest of the class to catch up. Rather than let them sit idle, I hand out the worksheet available to members, so they can move on to the next question independently while everyone else finishes.

Frequency Trees

How to draw and interpret frequency trees

To work out the number of girls and boys in the first branch of the frequency tree, students need to share the sample by a ratio.

To work out the second branches, students calculate a fraction of an amount.

Finally, when the tree is complete, students can work out the probability.

This is a non-calculator question, so students cannot lean on a calculator to find the fraction and the percentage.

For the first branch, students find seven tenths of 600 to work out how many people travel by car.

For the second branch, they calculate 85% of that amount to find how many of the drivers travel ten miles or more.

Once the tree is complete, students subtract to find the fraction who travel by car for less than ten miles, then write it in its simplest form.

Two-Way Tables

How to complete and interpret a two-way table.

This is a calculator question with links to:

• proportional reasoning
• sharing an amount to a ratio
• fractions of an amount
• percentages of an amount
• writing one number as a percentage of another.

This calculator question links to:

• reading values directly from a two-way table
• finding a missing value from a row or column total
• writing one number as a percentage of another.

These four questions are a sample of the full lesson. The full lesson includes five questions and each with a fully worked solution and a student handout.

As a member, you can download the complete lesson: all five problem solving questions, fully worked solutions and a printable student handout, ready to hand out the moment students need to work independently. Get the full Two-Way Tables and Frequency Trees pack here.