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.

]]>The schools that I have worked in have all used DIRT marking which I agree with but the question has always arisen of what actually is DIRT marking? For me, I see it as something the students should respond to and it should show progress.I don’t think this means that it requires lots of writing from me, in fact, it can be quite the opposite. I found some excellent DIRT marking feedback sheets on Miss B’s resources which you can find here https://www.missbsresources.com/teaching-and-learning/dirt-resources

I took inspiration from these but decided to adapt these slightly for my own purposes. Some examples of completed DIRT marking are below:

The column’s replicate questions from the homework activity so that if the students have not fully understood the quetstion in the homework then the DIRT marking sheet gives them an opportunity to show that they can understand the topic after receving feedback on the homework. Some students are asked to complete only one column whereas some students are asked to complete two columns. The extension question goes beyond the homework questions and could be an exam style question or a reasoning question of some kind to encourage the students to apply their knowledge.

These sheets have been very easy to use. I have them ready to hand out to the students on the lesson that I am marking their books, I get the students to stick the sheets into their books and then hand their books in open on the page with the DIRT sheet on. It then doesn’t take an age to look at their homework, correct mistakes and then tick the relevant boxes on the sheets. The students on the following lesson receive 10-15 minutes to respond to my feedback so there is no reason why the sheet should not be completed. This then shows the students are responding to my feedback and should use the notes on their homework to complete the sheet. If it turns out that all of the students have not done a particularly question very well, then at the start of the next lesson I will go through the question with the whole class and ask the students in a different coloured pen to write down the feedback on their homework.

Personally, I think this style of marking fits with the DIRT (dedicated, improvement, response time) ethos and as a teacher, makes me feel as if my marking is worth the time as the students do have to respond to it insetad of either not reading it or simply glancing at it and moving on. I would recommend giving sheets like these a trial.

As mentioned earlier, you can find lots of DIRT marking sheets at https://www.missbsresources.com/teaching-and-learning/dirt-resources

]]>At my first Maths Conf in Manchester I met some other maths teachers the night before at the pre conference drinks, I then spent the Maths conf 15 with these people and followed each other on twitter. It was great to see that they were also attended Maths Conf 17 and so yesterday I met back up with these teachers and spend the Maths conf with them again. The people at the conference are really friendly so don’t be afraid of going on your own; like me you too could also make friends with people at the conference. What was lovely about yesterday was that the day kicked off with maths speed dating so if you had come to the conference alone this would be a great way to meet a few people and exchange some excellent teaching ideas.

**Speed Dating**

We were asked to find another teacher in the room, sit down with them and in 2 minutes talk about a resource that we use in the classroom. Once the two minutes was up the other person had 2 minutes to explain their resource. Once both partners had explained their resources you found another person in the room to repeat the process with. Through this activity I sat down with 5 other teachers and was given 5 different ideas. Some ideas shared was how one of the teachers broke down factorising quadratics into several different sections and focussed questions on each of these areas in turn; a game entitiled dangerous 7s in which 2 dice are used. Each student starts standing up, the two dice are rolled and each student recueves that many points (unless a 7 is rolled). Students can then bank these points by sitting down or can remain standing and continue to collect points. If a 7 is rolled howoever and some students are still standing then they lose all points for that round. A great game about probability and one I am sure the students enjoy. Another idea was to present a problem solving question (this one was an image of a trapezum in top of circles) with vital information missing. The students in silence then have to write down questions about what information they think they need to know. The teacher can then decide if the question needs to be answered and the idea is that the vital information is soon given via the students questioning. I was told by the teacher who shared this that the first time they did this the students didn’t come up with that many questions but on the second time round they did ask a lot more questions.

**Session 1 – Literacy in Mathematics by Jo Locke**

The first session I attended was literacy in mathematics. The session kicked off with us being given mathematical dingbats that we needed to solve. These were a great way to introduce key words or to keep key words that students have already learnt at the forefront (not to mention a bit of fun). Jo did say that she uses these as starters and the students do enjoy them. If you want to use them in your lessons simply google ‘mathematical dingbats’ and there will be plenty out there to use.

Jo then introduced us to five golden rules of literacy in Maths found at http://www.resourceaholic.com/2015/09/gems40.html.

The five rules are:

1) Know your words

2) Mind what you write

3) Mind what you say

4) Be discerning

5) Make corrections

It is so important that we model good literacy in our lessons and encourage use of correct terminology by ensuring that we use it as well. A good example from Jo was that if a he answer is -3 and the student says ‘ the answer is minus 3’ instead of correcting in a negative way we can easily say ‘that is correct, the answer is negative 3’.

To help students become familiar with exam words Jo showed us some fantastic posters by MrsHnumeracy:

There are lots of resources on MrsHnumeracy, I definitely recommend that you spend some time taking a look.

Jo told us that towards the exam each week or each lesson she has the ‘word of the week’ by missrodders at the side of the board to keep the students familiar with the key words. Missrodders has produced approx 60 words which can be found at https://ideasfortheclassroom.wordpress.com/2017/07/15/classroom-display-maths-word-of-the-week/.

If you want some starters that focus on key words then transum have plenty. They may not be the easiest to navigate from the website but if you google transum mathanagrams or transum voweless then they come up straight away. The questions on transum are still free but the answers are not so you can still use their resources. These activities have 5 levels of difficulty so can easily be differentiated for classes.

I am assuming you have heard of elf on the shelf? Well, what about the mathematical version?

That’s right, this is pi on the tie. There are LOADS of these that you can access at https://padlet.com/tessmaths1/elfonashelf. Some of them will have to be used with 6th formers due to the key word but there are plenty you can use with all key stages. These are great fun and a fantastic way to once again get the students thinking about key words.

Some other excellent resources that Jo recommended and uses with her students are word wheels (tes.com/teaching-resources/word-wheels-for-maths), mathematical versions of 4 pictures 1 word, teachit maths scrabble tiles and wordsearches. Mathematical key word wordsearches can certainly have a place in the classroom. Jo recommended creating a wordsearch with the key words of the topic and then using this as a starter. Instead of the students crossing out the words they should highlight them and this can be stuck into their books. Doing this now gives the students the correct spellings of the key words and they should refer back to this throughout the topic of the correct spellings.

The final resource that Jo recommended was using Frayer models for key words:

These are a great way to get students to really think about the word and have a true understanding of the concept. Another way to do this would be to give the students the definition, chararcteristics, examples and non examples but hold back the key word and ask the student to state what the key word is. The other way to use this is to give the students the key word and then get them to complete the sections of the Frayer model.

Overall, session one was an excellent session with lots of excellent ideas and a lot to try out in lessons.

**Session 2 – GCSE Mathematics: Reflections from the second Summer by Andrew Taylor
**

The second session I attended was focussed on the AQA papers from summer 2018.

There was a slight shift in entries for tiers with 45% entered for foundation and 55% for higher which was a -2% from the previous year for foundation and a +2% from the previous year for higher. Overall, it was thought that teachers had made good decisions on entry of tier.

Below is box plots showing results from both foundation and higher and the different papers:

You can see that in every paper (except F3) at least one student did achieve full marks in the paper and at least one student achieved 0 marks in each paper. The higher box plots would suggest that students found paper 1 the most difficult of the papers.

Andrew then went on to look at a comparison between some of the common questions. One question that particulalrly stood out was:

This question was not answered too well on either tier. This could have been due to the nature of the question e.g. a negative fraction and the concept of range using fractions. Andrew highlighted that students do need to be prepared for novelty questions such as these. As a teacher it does make me question how often when I teach fractions do I use negative fractions both during questions and for answers. Also, how often do I use fractions when getting students to calculate averages and range.

Below is some more information on common questions:

Andrew also went through comparisons on questions between those students who achieved a grade 5 and a 3 and then those that achieved a grade 9 and a 7.

From foundation:

As you can see, 40.1% of students who achieved a 3 answered this question correctly and 83.9% of those achieving a 5 answered this question correctyl. This question sparked some comments in the room. There were some questions about what was the examiners actually wanting to test with this question; was they trying to test ratio skills or division skills. Andrew responded that in this question it was both. Had one part been an integer then this question would probably have appeared earlier on in the paper but due to one part not being an in integer it did appear later in the paper.

The focus of this question is the part (b), is your answer sensible? There was conversation around whether had this question been asked the opposite way around would it have had an impact in results? Was it possible that some students couldn’t access this question due to the fact that they had used a calculator in part 1 therefore could not understand why their answer would not be sensbile. Had they been asked to approximate and then check on a calculator, would it have made more sense to students?

From higher:

Most of these questions from the higher paper were towards the end of the paper therefore it is not unexpected that these questions would produce a large difference between the grade 9 and 7 students as the questions towards the end of the paper would be expected to be more grade 9 focussed.

This sessions provided some useful insights into the summer exams and was again worthwhile attending.

**Lunch – Tweet up**

Durng lunch there is an opportunity to attend a ‘tweet up’ session if you wish. This is an opportunity to put faces to twitter handles. There was some great activities to get involved with and all of them are designed to be social. I got involved with creating a pringle circle, which I am proud to say, I completed (just). It’s nice to know that over lunch there is somewhere to go to network with other teachers (again, if you are on your own there is no need to think you will spend the day on your own. There are lots of opportunities to meet people and chat). Below is the finished result of my efforts:

**Session 3 – The wonderful world of maths revision by Julia Smith**

One of the most memorable things for me from this session was how Julia described revision – revision is simply re-visioning the learning. It is seeing what we have seen before in a different way.

Julia certainly agrees with many other teachers out there that the best way to revise maths is simply to do maths. She encourages her students to aim to revise for between 15 and 30 minutes per day.

Some excellent resources that she uses with her classes are:

corbett maths 5 a day

onmaths (online platform that gives instant feedback)

Mr Chadburn maths – a little bit of maths everyday

M4ths challenges

Access maths

and many, many more. For some more resources you can access a padlet that Julia made here: https://padlet.com/tessmaths1/revision

One thing that I thought was a brilliant idea was that Julia created a padlet with her students for their revision. She sat with her class and discussed what resources they like to use and would use for their revision and put them altogether in one place. By involving the students in this process the hope is they will actually use the resources. Her fantastic padlet can be seen here: https://padlet.com/tessmaths1/FRA.

A big topic of conversation in my school recently has been around revision and do students actually know how to revise? Do we actually go through what revision is and how the students can do it? I have always mentioned revision to my year 11s and been through they should be hydrated and not revise for more than 30 minutes at a time but maybe I never went into enough detail. Below are some good points about revision:

Julia suggested having these conversations in lessons with the students and also encouraging students who revise in similar ways to team up. Any revision that students can do is positive therefore they could revise on the bus with a mate on the way home. Doing this everyday for 20 minutes would soon add up to several maths questions completed.

Another resource that was recommended was CGP guidance 21 killer revision tips which can be found at https://www.cgpbooks.co.uk/interactive_tips_exam.

During this session Julia made a point that some teachers may or may not agree with: it is not our job to like a method, it is our job to help students re-vision concepts. If a student has failed maths after 11 years of studying the subject then it is our job as the teacher to help them see it in a different way. This could be by using fraction tables for adding fractions, using alternative methods to grid and column for multiplication such as chinese method and instead of using bus stop method for division writing the division as a fraction and simplifying the fraction down. If the answer isn’t an integer then this method should still help students get to a times table that they are more familiar with.

One of the final points suggested was that if we can enable students to be confident with the four operations, FDP, ratio and scaling then we should be putting them into good standing for achieving a grade 4.

**Session 4 – Exploring creative thinking and problem solving by Drew Foster **

This final session was more of a hands on session in which we were presented with problem solving questions and given some time to think about them. I really agreed with some of the points that Drew made: Maths is about resillaince. Problem solving is about building resilliance. We need to allow students time to explore concepts and give them time to solve problems instead of presenting a problem and then moving on 15 minutes later. No one put a time limit on solving Fermat’s last theorem.

A website that Drew recommended was youcubed.org. I have used this website before but mainly looking at the growth mindset section. I have some of the growth mindset posters in my classroom. Youcubed.org have some excellent problem solving questions that you could use in lessons.

Drew also referred to Alcuin of York who wrote a book of mathematical problems. The problems and answers from this book can be found at http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Alcuin_book.html.

As Drew pointed out, the joy of problem solving is the actual solving of the problem of our own accord and not attempting it for 10 minutes, giving up and then asking for the answer. Drew doesn’t always reveal answers (unless you DM him). I have used a similar tact in my classroom. I set problem solving questions as extension to my starters and keep the same question as the extension until someone in the room solves the problem. One of my students eventually went home one night spent the evening solving the problem and presented the answer to me the next day so that another question could be asked. Problem solving is about encouraging curiostiy and as always, building resilliance.

Some of the question posed in the session are below:

**Final Thoughts **

Mathsconf17 certainly lived up to my high expectations. After how much I gained from mathsconf15 I had certainly bigged up mathsconf17 in my head. I was expectant of another excellent day of CPD and networking and the day certainly delivered. I highly recommend attending mathsconf, it is genuinely some of the best CPD that I have received. Do not worry about going on your own as everyone is genuinely really friendly and LaSalle do a great job of ensuring there are plenty of opportunities to network during the day. I am so pleased that I decided to attend mathsconf15 as it has opened up a whole new world of resources and ideas. What are you waiting for? Look at the dates for the next few maths confs and get yourself booked on; you will be pleased that you did.

]]>I fell in love with using the inquiry maths prompts last year after using them for the first time. I have now used them several times with different classes and enjoy using them. The value that the students get from these lessons is fantastic and I love that it makes them think for themselves and encourages them to ask questions.

The inquiry I used this wek was ‘The Surface area of a cuboid can never be less than its volume’.

This inquiry was used with my new year 7 class so it is their first time using inquiry with me. I am no longer nervous about using inquiry for the first time as I know that the students will gain something from the lesson and that I can structure it to help guide the students.

The first thing I did was place the prompt on the board along with the question stems that are provided on the website and told the students in pairs to use the question stems to think about the prompt and come up with statements/questions. After about 5 minutes I brought the class together and took in their thoughts, Below is what they came up with:

I had already pre-empted that my students might need a refresher on surface area and volume and so I already had examples ready to go on my lesson slides. I went through an example of volume and an example of surface area and made the students copy this into their books. From there, I put the prompt back on the board and told the students that they needed to investigate. I have previously used inquiry flowcharts (again from the website) to help students form their inquiry and these have been succesful and so I used them again this lesson. I made it clear that it didn’t matter if they did not have example of things that work or do not work as this may just be part of the inquiry.

The students worked in pairs and were given time to start drawing cuboids and finding their SA and volumes. One of the questions that one of my students asked was brilliant ‘Miss, you know how a few lessons ago you said that a square is a special type of rectangle, does that mean that a cube is a special type of cuboid? Can we use cubes?’. This question shows that my student was making a connection with prior learning. It also showed that they were thinking about properties of 2D and 3D shapes and that they truly were inquiring about the prompt.

Some of my students got excited when they realised this was true and started calling me over during the lesson when they were using the cube that has lenghts of 6cm. A cube of 6cm produces the same SA and volume and the students were asking whether this was an example for or against the prompt and so we started to disucss the wording ‘it can never be less’. Is the SA less than the volume? No, it is the same. Correct, therefore, does it show that the prompt it wrong? No, because they wer equal. It’s great to be able to have these conversations and to really get to the root of mathematical language such as less than and the difference between statements such as less than and less than and equal to.

Towards the end of the lesson I asked the class about what examples they had found and what were they thinking. One of my students put their hand up to say they hadn’t yet found an example that was against the prompt but they were just about to use a cube of 1cm lengths. This gave me an opportunity in front of the class to go through this thought and ask the students what the SA and volume was. It’s also great to show the students that we can have thoughts, try them out and them not always work. This would be how mathematicians would discover things.

Some of my students then shared some examples that did not work and did show that it is possible to have a SA that is less than the volume. Some of these examples are below:

Another excellent part of inquiry maths is the encouragement for students to make links and spot patterns. As you can clearly see from this bottom picture, the student has started to conjecture that a cube that has length above 6cm will always produce a smaller SA than the volume. This is now something that this student can investigate further.

This lesson was a great way to once again expose my students to thinking skills and starting to make links and conjecture. The students all appeared to be engaged and at the same time were practising the skills of calculating the SA and volume of cubes and cuboids.

If you have not yet used an inquiry lesson, I do recommend that you give one a go. Head to http://www.inquirymaths.co.uk/ and look through the prompts and read the musings of other teachers. I am so pleased that I decided to given inquiry a go and now I try my best to incoorporate an inquiry lesson into my teaching when I can.

If you want to use the recources that I used during this lesson they are here:

Power point: inquiry the surface area of a cuboid can never be greeater than its volume

Inquiry student sheet: inquiry the surface area of a cuboid can never be less than its volume

]]>A new colleague of mine shared an exit ticket and I was talking to them about it. I decided to give an exit ticket a try and get out of my plenary ‘rut’ sort of speak.

I decided a great opportunity was with my year 9 students when we had been looking at how to factorise more complex expressions. I planned my lesson and activities and so had my examples and questions in mind when I was creating my exit ticket. I also remembered a great piece of advice from Mr Barton in his book ‘How I wish I had taught maths’ which is that the plenary can potentially be the 5 minutes of our lesson that students really remember. He suggested that a plenary may not be a time to introduce a problem solving type question or an exam question but instead a question that will actually build the students confidence and allow them to leave the room feeling that they have achieved something and remember that final question that they could answer. With this in the back of my mind I chose my question carefully for the exit ticket.

I handed out the exit tickets at the end of the lesson and asked the students to bring them to my desk before they left. I know some teachers chose to check them once the students have gone but I decided to check the tickets whilst the student was there and had conversations with students if necessary and asked them to make any corrections there and then. Of course you could just ask for a pile on your desk and then check them in your own time ready for the next lesson.

Some examples of students exit tickets are below:

The top left exit ticket is an example of a very small mistake that a few of my students made which was that they had placed a plus sign between the terms instead of a negative because the second term is negative. This mistake was very quickly rectified by me simply asking ‘why is your second term in the bracket positive?’. The top left exit ticket was again an example that I saw and showed that this student had understood the concept of factorising but had not selected the highest common factor. Again, I just had to point out that the terms in the bracket had a common factor greater than 1 therefore this was not fully factorised. The bottom example is one of a student who had struggled more and therefore a more detailed conversation was needed and I had to help guide the student more.

I definitely think that this exit ticket brought a great insight into what my students had truly understood. It also helped me decide upon what to use for my next starter. It had become apparent from the exit tickets that some students had struggled with identifying the highest common factor of numbers, some had struggled with the highest common factor being a letter that is not linear and some had made mistakes with the second term being negative. My next starter addressed these issues and was designed to help students rectify these areas. In the same lesson that I used this starter I handed back the exit tickets and asked my students to stick them in their books.

Overall, I certainly feel the exit ticket was a valuable plenary. The ticket itself did not take long for me to make. I already had the template and then printed 4 to a page and just had to print and cut them out. The information I was able to glean from them in a matter of minutes was excellent and certainly allowed me to really understand which of my students understood the lesson and which had struggled. It also gave my students the opportunity to decide how confident they felt. I would recommend using an exit ticket for a plenary.

The template which I used can be found here: factorising an expression exit ticket

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What I like about this LO is that chances are, all of my class are going to be able to do this. It means that I get to think of a different way to introduce practising this skill.

Can they be equal by nrich is an excellent activity to do for this.

The first part of the problem is below:

After going through one basic example of how to find the area and perimeter of a rectangle and how I wanted them to set the work out, I introduced them to this problem.

As I was walking around the room it was really interesting to see what rectangles the students were trying and the conversations they were having. . Some students without prompting soon realised that the width needed to be 2.5 for this problem to produce a rectangle with an area of 25 and a perimeter of 25. Some of my other students said they thought they were becoming close with a width of 2 and a width of 3 but then wasn’t quite sure what to do. Some of my students asked if it was possible to use a decimal for a width to which I simply replied ‘Is a decimal a number?’ and ‘did the question specifically state that you needed to use an integer?’. When these students were confident that they could use a decimal they were soon on their way. This is my first reason for thinking this task is brilliant. Quite a few of my students when conducting investigations often stick to integers and have a hard time with using other types of numbers such as fraction, decimals or as they get to KS4 surds etc. I think students need to realise that there is a whole range of numbers out there not just integers and start to realise that fractions, decimals, surds etc are all just numbers. A classic example is when students solve equations and tell me that they got a fraction for an answer and is this OK? To which my reply simple is, is a fraction a number? After a moment of thought they normally go, well yes. I want my students from early on to realise that these things are numbers and to feel comfortable working with them.

Once the first part of this problem was completed, students moved onto the second part of this problem which is below:

Once again students got to work trying out different rectangles and trying to find the answers. At the end of the allocated time for the task I asked the students to share their findings. One answer given was a 6 by 3 rectangle which all of my students accepted and said yes they agreed this was indeed an answer. Another one of my students gave the answer a 4 by 4 square. This however caused many comments to be made around the class ‘You can’t use that, that is a square’, ‘the question specifically states a rectangle, that answer surely can’t be allowed’. Here comes the second reason why this activity has great value. Once the comments had been made I challenged one of the students: ‘What is the definition of a rectangle?’ It has 4 sides. So I continued to press on, is that the only defining feature of a rectangle? We soon got to that a rectangle has 4 sides, 4 right angles, 2 sets of parallel sides that are the same length. From there I drew a square on the board and showed that a square is in fact a special type of rectangle therefore the answer was indeed valid. This is a brilliant way to challenge the misconception that so many year 7s posses.

Overall, I think this is an excellent activity to undertake in lesson. It’s a great way to introduce problem solving into lesson and to take an area of maths that a lot of students can already do and make them think about it in a different way and encourage team work. It also encourages students to realise that unless a question stipulates that an integer must be used then they can use an array of numbers and hopefully build their confidence with these numbers. Finally, it stimulates good conversation about the properties of squares and rectangles and challenges that common misconception that year 7s can posses, that a square is not a rectangle. I do truly believe that this activity has a lot of value and I certainly encourage teachers to use this activity.

I would encourage teachers to visit the nrich website when planning lessons and look through the excellent resources that are on that website: https://nrich.maths.org/

]]>If you are about to teach a lesson on square and cube numbers then look no further than the following two resources:

square numbers: 10 questions:

And cube number problems:

The lesson to be delieverd to my year 8s was for students to be able to work with powers. Due to my class being year 8 I knew that they will have met square numbers many times before and I certainly knew that I had taught cube numbers to my year 7s last year. In which case this lesson became more of a recap on prior knowledge and consolidating my students knowledge on square and cube numbers. I quickly decided that this would be the perfect opportunity for my students to once again embark on problem solving. What better way to get my students thinking about square and cube numbers than making them think about them in a different way and also help them enahnce their problem solving skills.

After a recap on square and cube numbers and checking that my students had the correct numbers, I placed a selection of the questions on the board. I chose to use questions 3, 4 and 6 from square numbers: 10 questions and questions 1 and 3 from cube number problems. It was a great way to also recap other number types such as prime numbers and once again reinforce previous key words that we had met such as consecutive. What I also liked about the cube number problems was the link to algebra and some of the questions it caused my students to ask such as can the value of a be the same as the value for b? Does the value of a in the top row have to be the same as the value for a in the bottom row? Important questions that as a teacher I can sometimes take for granted.

It was a pleasure to listen to the conversations my students were having and wonderful to see them engaged with the activity. This is such a brilliant way to get students consolidating their knowledge of square and cube numbers whilst also incorporating that all important problem solving. I highly recommend using these questions the next time you are teaching square or cube numbers.

]]>I always try to start my new school year as I mean to go on; in this case it is using problem solving where possible.

This week with my year 8 group we were looking at Highest Common Factor. Factors were covered last year and so they do have a concept of factors. To try and get them thinking about factors and also start to introduce common factors I used Gabriel’s problem. I used this as a starter activity so I did not specify that we were focusing on factors and HCF at this point. I gave the class about 10 – 15 minutes to attempt this task.

Essentially, the students are presented with a 3 by 3 grid in which they have to place the numbers 1 to 9 so that each row and each column multiply to the product shown. Each grid grows increasingly more difficult but each contains one number that could be a great starting point e.g. 21. I placed the first grid (the green grid) on the board with instructions and let my students attempt the problem. Some students were struggling with this and others were getting on with the task. Some students appeared reluctant to write anything down in case they were wrong. One of my classroom rules is that its Ok to be wrong, its OK for exercise books to have mistakes in as this is how we learn. Students need to learn to be resilient and to not give up because they find something hard. Once some students had started completing the first grid i presented the next few grids to them. Those who were still struggling i went and gave some hints to e.g. which number can only be made in one way? (the answer was 21 so 1 x 7 x 3). Then i would ask why can the number 7 not go in the same row as the number 40? (because it does not go into 40. Ok so its not a factor) etc. From here more of my students were able to start thinking for themselves more and soon my whole class completed the first grid. Some of my students were even able to complete the second grid.

As a class we then had a discussion about what were the students thinking about as they were completing this task. The students said they were thinking about what numbers fitted into the products. From there I asked them what was the mathematical word for this and they correctly identified factors. I asked them if there were certain numbers that could only go in certain places and they identified that 7 could only go in the column with 21 and the row with 378 etc. From here we identified what common factors were and then I was able to lead them to the idea of a highest common factor. Once we reached this point I introduced the LO and key words and used the Mr Barton idea of me as the teacher doing a whole class example and then the students having a go through your turn.

What I really enjoy about using problem solving in my lessons is the value that the tasks bring. Problem solving encourages students to think for themselves and be less reliant on me to lead the activity. It encourages team work and students talking about their thoughts and what they are doing, It encourages them to give tasks a go even if they find it difficult and usually along the way they get something wrong. This then helps them build resilience (if as a teacher I handle this carefully) and understand that its ok to be wrong as this is how we learn. Also, I like to think that its fun for the students.

Gabriel’s problem can be found here: https://nrich.maths.org/11750

The power point I used with my class is here: HCF

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Be prepared for a few lessons where things may feel somewhat chaotic but the truth is there is so much value in this activity.

When I stop and think about how statistics is generally taught, I realised that I don’t really introduce the idea of larger data sets until A-Level. Our students are used to us presenting data in perhaps already grouped forms or with raw data that is actually quite simple i.e. a row of numbers that need averages finding. We don’t really give them data to analyse that could resemble something closer to real life; enter lives of presidents.

To see what I mean, go now to: https://nrich.maths.org/11007

The students are simply presented with facts about the presidents of the united states (see below)

The students are asked to create their own research question and then set about answering their question. What a brilliant task for them to do! This is where you need to be prepared for the little bit of ‘chaos’. Now you may be thinking that this is a process that our students should be used to from science, and I would agree, but I also know that some students find it hard to make that link. So straight away this process allows for a link to be made between similar processes in science: make a hypothesis, go about testing it, make a conclusion.

What is interesting is that some students find it difficult to create a question. Nrich have themselves suggested lines of inquiry and so I always put these on the board as a possible option.

Some of my students choose one of these questions to research where as others create their own lines of inquiry.

I then set about visiting the pairs of students asking what it is they are researching and asking how they are going to go about this. I am also on hand to answer questions they may have e.g. should I draw a scatter graph for relationship? Can you remind me how to calculate what degrees I need for a pie chart? etc

Apart from that, I allow the students some freedom to go about their research and give them an opportunity to experience what it feels like to be presented with a sea of data and make some sense out of it.

Some great conversations come out of doing this project. One group was looking into the age of deaths of the presidents and decided to group the presidents based on the year of inauguration. Below is their grouped frequency table:

You can see that the average age of death goes down between being inaugurated in 1860 to 1900 and so I asked them why they thought this might be. They told me that 3 of the 4 assassinations had happened to the presidents in this group and so this was clearly affecting their data. We then discussed what skew meant and that data can only highlight that something has happened but you need to look elsewhere to find why (cause and effect). We then also discussed that it could be possible to state that there were anomalies at this time and that they could take these anomalies out and re-evaluate the data to see if there actually is a genuine dip at this point or it these 3 presidents were having a major affect. From there a discussion about which averages are the best to use in different situations is possible. I know this group had used the mean and so I am hoping that next lesson they may re-evaluate this as a whole and take other averages to compare. If not, that is a direction I can suggest to them.

This activity is certainly adaptable to all year groups based on what knowledge of data handling they have. I have done this activitiy with years 8 to year 10s and just encouraged them to use different representations of data e.g. with year 10 they drew box plots and cumulative frequency diagrams with my current year 8s they are producing pie charts, scatter graphs and frequency polygons. There is so much that can be done with this data and the fact that the task self differentiates is brilliant. I highly recommend using this next time you are doing a statistics topic and allowing the students to get a sense (all be it a small one) of what it is to be presented with larger data and research a question.

Examples of my year 8s work is below:

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The lesson before had been on writing numbers as the product of prime factors and this lesson was on HCF/LCM. I saw that inquiry maths had an inquiry that could be used ‘The product of two numbers is equal to the product of their highest common factor and lowest common multiple’.

I introduced the lesson by placing the prompt on the board and showing the students the question stems that they could use to help them create questions/statements. I gave the students 3 minutes to discuss their ideas with a person near them and then took in their thoughts. The picture below shows what they came up with:

Once we had done this, I then went through an example of how to find the highest common factor and lowest common multiple of a pair of numbers using prime factor decomposition (this was to link this to the prevouis learning). I then went through one example of what the prompt was suggesting due to some of the questions the students asked were ‘what does this statement mean?’ (referring to the prompt).

Due to this being the first inquiry I have done with this class, I used the guided sheets to help them form their inquiry. Below is an example of this:

The guided sheets can be found on inquiry maths and I do recommend using them to help the students know how to set out their work. I made it clear to my students that they had to show working out in their books and then put their thoughts on the sheet. This inquiry was certainly a great way for students to practise the methods for HCF and LCM.

Not only was it great for practise but the inquiry also brought out some good questions and conversations. One pair of students decided to use the numbers 60 and 79. As they worked through the question they noticed that there wasn’t a number to be placed in the middle of the venn diagram. They then concluded that this could mean that there is no HCF. From there, we discussed was it posible for the HCF to be 0? They realised that this in fact was not true and that the HCF must indeed be 1. They also made the connection it was because 79 was prime and the other number they were using was less than this hence why the HCF was 1.

I personally think this was a useful conversation to have and I started to think about is this an example I would use normally? I don’t recall a time when I have specifically done an example using prime factor decomposition where the HCF is 1 meaning that technically no number would belong in the intersection.

Another excellent thing about inquiry maths is that it encourages students to notice patterns and write down their thoughts. I actually think the statement below is excellent:

I like the fact that the student felt confident enough to write down that they were wrong. I think it is powerful for students to truly grasp that being wrong is perfectly acceptable and is actually a part of learning. That we all get things wrong but this is how we develop and move on. I hope doing these inquiries will continue to help my students search for connections and be able to admit and accept when they are wrong and see this as a normal part of learning. What an excllent learning environment to hone and to encourage. One in where all thoughts and ideas are welcome and one in which it is OK to be wrong.

Find this inquiry and many more at: http://www.inquirymaths.co.uk/

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