Genevieve Sadler, a trainee teacher at St Peters Catholic School (Guildford, UK), used the averages prompt with a high-attaining year 8 class. She reports on how the inquiry developed: 

"The inquiry lesson was at the end of the averages topic and I wanted it to push the students out of their comfort zone. I started the lesson with independent work for the first 10 minutes. Students wrote everything they knew about the statement on the page before I gathered the information on the white board. I then allowed them to work in pairs, while getting them all to focus on other students' findings around every 10 minutes. 

"At the end I asked the students if they had enjoyed the lesson and 30 out of 34 raised their hands. I thoroughly enjoyed leading the inquiry."

After the lesson, Genevieve collated the student's contributions on a sheet (above).

Presentation

Before running the inquiry lesson, Genevieve had attended an Inquiry Maths workshop at St Mary's University (Twickenham, UK) during which she had worked on the prompt. 

On hearing about Inquiry Maths, the mathematics department at St Peters asked Genevieve to give a presentation at a departmental meeting. Genevieve contacted the website to say that the presentation was well-received and she was hopeful that some of the other teachers would have a go at an inquiry lesson.

May 2023

As the aim of the inquiry, the class decided to find sets of data that satisfied the six inequalities possible with mean, median and mode:

median > mean > mode

median > mode > mean

mean > median > mode

mean > mode > median

mode > median > mean

mode > mean > median

The activity covered the first three cards and Ollie attended to those students who required more of an explanation than had emerged in the question and observation phase. At the end of the lesson, students presented their sets of data. Usually a quiet student, Billy was excited to create his own mathematics and enthusiastic to show his data set to the class. He explained how {1  1  9  11  12} satisfied the condition that median > mean > mode. Then Ahmad used negative numbers {-12  -2  1  1  2  3  4  5  6} to create a set of date for which median > mode > mean.

Ollie reports on how the inquiry framed students' learning: "Only a few students in my year 7 group had come across the three different types of averages and range as a measure of spread before. This inquiry was particularly useful for two reasons. First, it helped frame students’ understanding of averages: I found it a powerful way to introduce averages as it heightened students’ interest in the topic. After discussing the prompt, I taught them averages in the way that I normally do, but I then referred back to the prompt, which gave students a sense of closure and deeper understanding of the topic. Second, starting with the prompt sparked students’ curiosity, which assisted with their motivation to carry out the work and gave them more agency in doing so."

Half the class practised finding the three averages from sets of numbers designed by the teacher to provide increasing challenge. Other students used five numbers to satisfy inequalities involving the averages. At the end of the lesson, students either explained how to, for example, find the median with an even set of numbers or fed back on the inequalities they had managed to satisfy.

Lesson 2

The starter came from a student’s question: “Is it possible to find five numbers where mean = mode = median?”. The class presented some solutions (when the numbers were not all the same) and then expressed the general case (under the teacher’s guidance) as n – x, n, n, n, n + x (where n and x are positive integers). The main inquiry continued with students looking to satisfy the two inequalities for which a set of numbers had not been found in the first lesson:

median > mode > mean 

mean > mode > median.

As the inquiry progressed, more students claimed the task was impossible using five positive integers. The teacher required those students to develop an explanation by thinking about the process of selecting five numbers. Others tried decimal and negative numbers or used more than five numbers to find a solution. The lesson ended again with students presenting their reasoning.

Lesson 3

In the third lesson, the teacher introduced the range into the inquiry from the initial question: "What about the range?". Through discussion, the class constructed an understanding that range was a measure of dispersion, rather than an average. The starter involved the class in thinking about how many permutations there are with four separate items. Using the resource sheet (24 inequalities), the students created sets of seven numbers that satisfied the inequalities. When students found a set, they came to the board and wrote the numbers next to the inequality.

The first lesson ended with one student presenting a set of data that satisfies the condition, although he used nine numbers instead of the seven agreed upon by the class:

31    32    53    53    56    57    58    59    60

median 56 > mode 53 > mean 51 > range 29

In the second lesson of the inquiry, students established four aims of their own:

• Find sets of four, five, six, and seven numbers for which mean = median = mode = range.

• Find sets of five numbers that satisfy inequalities involving permutations of mean, median, and mode only.

• Find sets of seven numbers that satisfy inequalities involving permutations of mean, median, mode, and range.

• Find a set of numbers that satisfy the same permutation of mean, median, mode and range as in the prompt, but with the inequality reversed to "less than".

After selecting an aim, the students explored in pairs or groups before explaining their findings to the rest of the class.