# Averages inquiry

# The prompt

**Mathematical inquiry processes:*** *Generate examples and counter-examples; test particular cases; change the condition; reason. **Conceptual field of inquiry:*** *Averages and range; listing permutations.

The inquiry starts with students trying to understand the statement and, particularly, the constraint placed on the data set. Questions and comments that have arisen in the classroom are:

What do median, mode, mean, and range mean?

What is a set of data?

Why are there ‘greater than’ signs in the sentence?

What does it all mean?

How many numbers are there in a set of data?

Do the median, mode, mean, and range have to be in that order?

If the mean is higher than the mode, you have to start with the mean, then work out the mode.

It must be possible.

When we tried this, it worked for the median and mode, but went wrong on the mean.

You can’t do it because the range is too small.

The prompt leads students into trying to construct sets of data that fulfil a certain condition. The teacher might orchestrate a discussion on how many numbers are appropriate in the data set and guide students towards using five or seven to simplify the start of the inquiry. If students are unfamiliar with the terms in the prompt, or need to revise them, they have in the past chosen a **regulatory card** that gives them time to practice finding the median and mode and calculating the mean and range before exploring the statement itself. However, it is possible to develop fluency in these procedures *during* the inquiry.

In the search for a set of data described in the prompt, students begin to create sets that satisfy different inequalities. So, for example, students found this set that has mode > mean > range > median:

### 3 5 6 6 10 10 10

Thus, a period of exploration can lead to a question about how many permutations of median, mode, mean, and range exist. In one inquiry, a group of students found the 24 permutations and explained the result by using four ‘blocks’ based on the one they demonstrated to the class, which started with the median (see below).

Another class working collectively found that, along with the constraint in the prompt, there are four other permutations for which it is impossible to create a data set:

**Median > mode > range > mean**

**Median > range > mode > mean**

**Mean > mode > median > range**

**Mean > mode > range > median**

This final result, if confirmed by other classes, leads onto other questions: Why are the five permutations impossible to achieve? Do they have something in common? Do they remain impossible to create if we use more numbers in the data set?

# Counter-examples

## Classroom inquiry

These are the initial questions and observations of a year 9 (grade 8) class about the prompt. When invited to decide on the direction of the inquiry by selecting a **regulatory card**, the majority of students opted to *Find more examples*.

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.

**Sarah Keller**, a math teacher at Allen Middle School (Camp Hill, Pennsylvania), contacted the Inquiry Maths site with this set of 6 data values that satisfies the condition in the prompt.

### 10 28 28 30 32 34

**median (29) > mode (28) > mean (27) > range (24)**

Sarah explained how she was preparing to use the prompt in her classroom: "I am teaching my seventh grade mathematics students about measures of center and spread in a few weeks, so I was exploring some different inquiry methods of teaching these skills.

"I came across this prompt and before I decided to implement this in my classroom, I wanted to see if it was really impossible to create a set of data that satisfies the prompt! While I was not able to think of a set of five data values, it only took a few minutes to develop a set of six data values that worked. Now that I have the set I created, I can keep that in the back of my mind when my students are working to create their own sets of data to attempt to satisfy the prompt, and perhaps if they are really struggling, they can manipulate my data set to create a new one."

**James Handscombe** (Principal of Harris Westminster Sixth Form) has found a counter-example to the proposition in the prompt:

### 6 23 23 24 25 26 27

**median (24) > mode (23) > mean (22) > range (21)**

James is confident that a data set of **seven numbers **exists for each of the 24 permutations. One question that follows concerns the smallest set of data that satisfies all the permutations - is it 7, 6, or 5 numbers?

This picture was posted on social media by a middle school math **teacher** in Mississippi who reported that the prompt initiated deep thinking and good discussions. The class went on try to show the prompt was false by finding a data set that contradicted the contention.

# Inquiry as a frame for learning

**Ollie Rutherford**** **who is on the Teach First programme at Haverstock School (Camden, UK) used the alternative averages prompt to initiate the first inquiry with his year 7 class. The picture below shows the students' initial questions and observations.

After the initial phase of the inquiry, Ollie used the pack of six** ****regulatory cards** to give students the opportunity to direct the inquiry. He recorded their choices on the board:

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."

# Question-driven planning for inquiry

## Lesson 1

Year 7 students posed questions and made observations about the prompt (above). It was evident that many of them had met the concepts of median, mode and mean before. The teacher allowed students to select a **regulatory card** to decide how to proceed (below).

The inquiry was the tenth that the class had carried out and it had become routine for students to choose a combination of cards. Eight pairs selected *Practise a procedure* and others wanted to *Change the prompt *and *Decide on the aim of the inquiry*, so the teacher focused on the question "Can it be in a different order?"

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.

# Constructing an explanation

The inquiry can lead to some ideas that students find difficult to explain. Why is it that they cannot create a data set that fulfils the inequality in the prompt? In one discussion, the teacher and class tried to construct an explanation by focusing on one case. They used seven positive whole numbers in the data set. The class also tried to make as many of the median, mode, mean, and range as possible positive whole numbers. (See reasoning in the table.)

This did not constitute a proof of impossibility, even if we have exhausted all the possibilities in this one case. The class decided to inquire into whether changing the assumptions at the beginning - that is, only positive whole numbers in the data set, aiming at positive whole numbers for the median, mode, mean and range, using seven numbers, and starting with five as the median - led to a different outcome. Students chose one assumption to disregard. However, they could still not create a set of data to meet the conditions in the prompt. The class concluded that it is impossible.

# The prompt as a problem

Two months after the averages inquiry was posted on the site, NRICH published **Unequal Averages** (below) – a problem solving task for students based on the well-established “M, M, and M” problem. While the task is similar to the inquiry presented on this page, the differences reflect the distinction between problem solving and inquiry.

Problem solving has a pre-determined beginning (the question) and end (the solution). In the NRICH activity, the set of inequalities is defined and so is the size of the sets to be used (four, five, and six numbers). It is also clear that the problem setter knows the solutions, including those inequalities that cannot be satisfied. The problem solver is left with an 'open middle' in which to arrive at the required end-point from the given starting point.

In contrast, an inquiry has the potential to have an open beginning, middle, and end. This is more motivational for students, as well as giving them the opportunity to pose mathematical questions and determine appropriate solution sets.

The NRICH activity does, however, suggest one way to run a **structured inquiry**. The teacher could restrict the prompt to the three averages and limit the size of the set of numbers in the initial inquiry.