This week with my year 7 group I have been looking at averages; mean, median, mode and range.
The previous lesson we went through median, mode and range and then for this lesson we were looking specifically at the mean. The students had done well with the median, mode and range and also expressed that this was something they had done before. Taking this information on board, I wanted to use a problem solving activity for the mean as my students will have come across this before as well and my lesson will be more of a re-cap.
Looking on nrich I found a really nice activity for averages: M, M and M. (https://nrich.maths.org/mmandm)
There are several sets of five positive whole numbers with the following properties:
- Mean = 4
- Median = 3
- Mode = 3
Can you find all the different sets of five positive whole numbers that satisfy these conditions?
Can you explain how you know you’ve found them all?
After doing a starter recapping on last lesson and completing a few whole class examples I placed the problem on the board.
I asked one of the students to give me 5 numbers and I did a whole class example of how I wanted the work setting out. I then told the students they could work in pairs and simply let them go. I gave the students 30 minutes to explore this question. After about 3 minutes one pair put their hand up and told me they had completed the task. I then redirected them to the question and said ‘ have you found all of the numbers?’. They then realised this question had many different solutions. Some students asked if once they had used 2 3 3 3 9 could they then use 9 3 3 3 2? This is an important point to highlight, that this is the exact same number conbination in a different order therefore cannot be used. I also re-iterated that for a median, you need to put the numbers in size order therefore it once again is the exact same solution.
I walked around the room checking their examples and answering any questions and listening to their thoughts. Once the students found a few examples some of them started to produce a system.
One of my students said to me’ Miss, I have started to use a system. I have realised that I need at least 2 threes for this to be a mode and that one of these threes need to be in the middle. From there, If i take this example 2 3 3 3 9, all I have to do is add 1 onto one of the values and then subtract 1 from one of the other numbers e.g. 1 3 3 3 10′.
It was brilliant that this activity encouraged the students to think systematically and to spot patterns.
This activity was also great for students thinking about the mode in a different way, some students at the start trialled numbers such as 2 2 3 3 10. This stimulated conversation about what the mode was. It was an oppotunity to dicuss the difference between having two modes and having no mode.
Towards the end of the task, students began to ask me if they could use zero or not. I replied with ‘why do you thinl zero could not be used?’. Some of the students said they didn’t know why they thought this and proceeded to use zero, some of them replied with well zero isn’t technically a number, its a place holder. Both of these replies are indeed valid and once again great to see the students questioning. Nrich do have combinations with zero on their website therefore I did allow zero to be used.
At the end of the allocated time I collated the student’s answers and placed them on the whiteboard. Some student’s shared their systematic approach and we dicussed the parameters of this approach. Using this system, would we get a finite amount of solutions? I always think that Nrich are brilliant for key words and encouraging students to really think about their meaning, Students may not have thought about what a finite amount of solutions really means and why they are produced.
Overall, I think this activity was worthwhile using and th students certainly engaged well. I think it’s important to allow the students the time to work together and investigate, and to give them the freedom to work systemtically. It can be daunting giving students 30 minutes to work on a task but for the students to have real time to think and understand the task, I felt it was important to give them a longer period of time. My students are becoming used to having more open ended taks and longer to work on them as I am working them into my teaching more often. I especially think that with year 7 it is a perfect opportunity to implement these tasks, specifically when covering topics that they will have already seen and be familiar with. I highly recommend using M, M and M by Nrich.