Generalising with Areas of Circles

So, at #MathsMeetGlyn last weekend Don Steward briefly put this picture up on the screen but then decided to talk about something else as time was pressing.

That’s a bit like a “Wet Paint” sign.  You have to touch it just to see if it really is wet…

So I had a little play with these. I couldn’t find anything about it on his blog but of course there is lots of other lovely stuff there.

A few different questions you could pose here:

• What is the ratio of red area to blue area?
• What is the red area as a fraction of the blue area?
• What is the red area as a fraction of the whole circle?
• What is the blue area as a fraction of the whole circle?

And then of course, can we generalise?  What about when there are n small circles.  What happens as n gets bigger? Why? Can we find a general expression for the area?

Have a go.  It actually turns out to be quite simple but depending on which question you start with, you can get into a lot of practice with ratio and dividing fractions.

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Let’s fix marking

I’ve been reading a few things recently about marking and thought I would consolidate these into how I am going to tweak things next term in my classes.

Marking is often cited as one of the biggest causes of workload pressure and thus dissatisfaction with teaching.  Probing deeper, I think it is the time taken and the monotony of marking that is the problem, not the actual activity itself.  Most teachers would agree that personalised feedback is a crucial aspect of learning. I often feel a stronger personal connection with my students on marking their work.  And I think they usually like to see comments in their book.  But how many times have you spent maybe 2 hours wading through a class set of books and at the end asked yourself the question, “Was that really worth it?” or “What else could I have done in that time?”

In my experience, there are two types of marking:

1. Marking tests and assignments

2. Marking books

It is impossible to avoid the first one.  The department head or KS coordinator sets these, they have to be done at the same time by all classes and once they are marked the data has to be entered on a spreadsheet usually within a week or so.  At least that has been my experience. I find this work usually gets bumped to the weekend.  It’s fairly mundane, but takes longer if tired, so it usually ends up being a Saturday or Sunday morning job.  The outcome of this effort is useful as it gives us insight into individual student’s progress and can (although often doesn’t) highlight specific topics which need to be retaught across the whole class. I try to be ruthlessly efficient when doing this type of marking: no comments, just marks, but it can be helpful to scribble down some notes as you go on how to plan the next lesson.  I’ve written before about ideas for how this can be done in Giving the Tests Back.  It’s the type of marking you get quicker at with practice, but it is still a fairly significant workload. In my view we shouldn’t be doing more than 3 per year per class and they should be spread out evenly.  I’d be interested to hear comments on that number by the way!

Final point on tests – how effective are you at enforcing exam conditions in a classroom of 30+ students? Especially if you want to use the time to get on with something else.  Eyes wander, you might read them all the riot act at the start and that might have some impact, but make sure you take all those test results with a pinch of salt.

So, on to book marking.  The fact that marking tests and assignments is non-optional and book marking is, let’s say, less non-optional, means that book marking invariably gets squeezed.  A typical school or department policy might state that books should be marked every two weeks. In reality that often doesn’t happen and for good reason – there are better things to do!  Personally I wouldn’t go so far as Giving feedback the ‘Michaela’ way which espouses simply browsing books frequently and making separate notes, so not putting any mark on a child’s work.  It’s an interesting post (written from the perspective of an English teacher but equally valid for Maths) but I feel that pupils expect something in their book from their teacher.  Personally, I have no interest in providing evidence to Ofsted or anyone else (if you want to provide useful feedback on my teaching that will help me improve, I’ll listen, but if you don’t trust me as a professional to do what I need to do, then I’ll go do this somewhere else!). But, rightly or wrongly the students I teach have come to expect teachers to mark their books and it damages my relationship with them if it doesn’t happen.

A desire I have often heard expressed is to establish a “marking dialogue” in books. The student writes a response next to each of your scribblings. So, as a teacher, you try to phrase a question to elicit a response. e.g. “What have you forgotten to do here?”  But the risk here is that the question is too hard and the response is “I don’t know” – pointless. Alternatively, you may trigger a valid question back from the student, but that question is likely to go unanswered.  Are you really going to go back over all of the previous comments next time you take the books in? It gets somewhat unwieldy and falls squarely into the bucket of “I could be doing something better with this time!”

So I felt enlightened when I saw this recently on @reflectivemaths blog:

Instead of writing the same thing in 10 exercise books, you have a key. I mean, it’s not rocket science, but I like the use of symbols.  There are 2 brilliant reasons why this works:

1, You are marking with purpose.  Obviously it saves you time writing, but more than that, you can probably anticipate what the comments will be before you start marking based on the work you did in class.  If you have a go at writing these down first you are looking to use them and I can just see that process being more purposeful and quicker. I would want to include specific positive comments too, e.g. “You have mastered cancelling down fractions”.  I would also use this for additional questions for students to have a go at.

2, It provides a structure for student engagement with your marking. There is nothing worse than spending two hours marking a set of books to then watch students spent 30 seconds looking at it followed by the conclusion, “meh”. For our own sanity (!), we should be spending 10 minutes of the following lesson making sure that students are properly engaging with the feedback and, more importantly, self-reflecting on their learning. So you put your key on the board and they copy down the comment next to the symbol.  I would also ask them to write a statement saying whether or not they agree. Realistically, I’m never going to look at that statement but the point is that it has got them to engage.

It is completely flexible and specific to the work you have been doing.  And you can always add new symbols as you go, there are plenty to chose from!

So, that’s my plan for next term.  What’s yours?

Decimal-Crazed Lunatics

Love this. My students do this all the time. I love the “door” analogy!

Math with Bad Drawings

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Stumped by a Foundation GCSE Maths question

I have just returned from an inspiring morning at #mathsmeetglyn organised by @mathsjem watching Don Steward give a fantastic, brain-stretching whistle-stop tour of some of the great problems on his site.

The theme was “Generalising” – he started off by saying that his current mission is to get some generalising into every single lesson because without it, well, you’re not doing real maths.

I filled 10 pages of notes, some of which I’ll come back to, but wanted to share how he took this Edexcel Exemplar Foundation GCSE question and “generalised” it.

Now, first up, I don’t mind admitting that after working at it for about 3-4 minutes I was completely stumped.  Don had added “You can’t cut the tiles” to the question which was essential in my view, but didn’t actually help me. I was completely fixated on the fact that all tiles have to be in the same orientation.  They don’t.

I think it’s a pretty poor question, probably mainly because I couldn’t do it. But there is a serious point around what questions like this are really testing. If there is a simple “trick” you need to get, is that fair?

A debate to be had there, I’m sure, but more interesting was what Don did with it next.

What other areas can you fit 40 x 30cm “carpet tiles” into and how many do you need? Start with:

120cm x 60cm
120cm x 70cm
120cm x 80cm
120cm x 90cm

Do you need to go any further?  Can you write a general statement from this? i.e. can you prove that all multiples of 10 will work if the width is 120cm?  What other widths does this work for and why? And then, of course, what happens if you try different sized carpet tiles.

It feels a bit like one of those Maths GCSE coursework questions that were set in the days before I was a teacher.  But I really like the idea of taking what is a pretty bad question and turning into some interesting maths.

Solving Linear Equations

This is a pretty old flash-based resource hosted on STEM.org.uk (requires a log-in but it’s free).  It is still my favourite tool for practising solving linear equations. I’m worried that I haven’t found anything better yet and that as Flash become less and less supported, I’m not going to be able to use it much longer.  Let me know if you know of others!

I use it to really reinforce the idea of “doing the same to both sides”.  It can be used both in whole class explanations / discussions and also with students using it themselves.  The process of “getting it wrong” is really productive as they can see what would happen if they did that.  I also like the fact that there is no “undo” button. They need to work out that “undo” just means do the inverse operation so it reinforces that idea too.

 

 

Some great problem-solving tasks

FMSP isn’t all about A level Further Maths.  They have some great GCSE resources too.

This page contains some superb resources for problem solving and group work with some very helpful discussion prompts and worked solutions for each question.

These look like nice resources for introducing group work, as each member of the group has “their” cards but they work collaboratively to complete the task.

Also, this post from Number Loving has some good suggestions.

Reflections from Teaching: Fraction Talks

Some great low threshold, high ceiling ideas here. I could see a whole lesson talking about fractions and getting students to create their own. Would work well for mixed attainment groups.

Also, this is a great digital manipulative for lots of things fractions, decimals and percentages.

 

For the Love of Maths

In this post, I want to describe my experience with the Fraction Talks Activity I originally borrowed from Nat Banting’s site. He has since become a curator to www.fractiontalks.com, a website devoted to various templates teachers may use – if you haven’t seen it yet, it is very worthwhile to check out!

After pondering about the original blog post for a few minutes, I was struck with just how versatile these templates are at connecting certain mathematical aspects together, and providing a visual representation of certain operations/ideas. The fraction talk template that I used when I worked with two classes of Grade7/8 was the following:

Concept of Fraction

Perhaps one of the most basic uses of the template is to connect it to the general concept of a fraction: a shaded portion of a whole, in which the whole has been divided into equal parts. To do this, we…

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Vertical Number Lines in Books

I made this a while ago for supporting Year 7 students on directed number (i.e. positive and negative numbers).  I think there is something more intuitive about a vertical number line – if you are adding you go up, subtracting you go down.  Having said this, I have always had a horizontal number line across the top of my board!

If the number line is stuck on the inside back cover of the exercise book, it is always there whatever page the child is working on. It can then be folded safely away whether using large or small format books.

Doing some work in primary this week, I realised that the same idea could be useful for supporting younger children learning the essential skills of counting back and counting on when doing addition and subtraction of positive numbers.  So I have made another version just with positive numbers.

The end points of these lines are arbitrary of course.  I have deliberately gone for something a bit random to start a discussion, “Sir, why does it stop at 44?”.  But if that offends your preference for order in life, then feel free to adjust it on the spreadsheet that I used to create the pdfs in the first place.

Vertical number line to stick in books.xls

Vertical number line to stick in books zero to 44.pdf

Vertical number line to stick in books minus22 to plus22.pdf

 

Things I love about Pi

Pi day on Monday…

As a pdf: img069

Multiplication methods – introducing formal long multiplication

Today I was with a group and we were trying to formalise long multiplication methods.  They were fairly secure on 2 digit by 1 digit using formal method (not grid/box method).  i.e. they were happy doing this.

So after a while, I tried to introduce them to 2 digit by 2 digit.  It was quite a leap, so I’ve been thinking about the microsteps in between.

Before moving on, however, I really like some of these ideas on Don Steward’s Median (still my favourite maths website!)  They make for good extension work as they go deeper and involve problem solving before moving on the “next” thing.

2-digit x 1-digit to 2-digit x 2-digit

It’s a big leap, so I’m thinking about the microsteps.

1. Multiples of 10.  We can start with a discussion about 12 x 20.  Some might well be able to do that mentally, and writing out their explanations should eventually lead to:
   We can discuss the practice of “putting a zero in the ones column” which then enables us to just focus on the number in the tens column and multiply that by the number above.  Here is a worksheet of these that I created using Math-Aids.
   multi_digit_power_ten
2. I’m assuming that the idea of partitioning a number into its tens and ones is pretty secure by now, so I might then look at this.
   
   Here is a worksheet that I have created to practise these. It’s a spreadsheet that you can change if you want to alter the questions.  If you’d prefer just simple pdfs to print, here are the questions and these are the solutions.
3. So, now we are ready to combine it all together and introduce the efficient method. Effectively doing the same calculations but with less repetition in the writing. Some suggested language to help solidify the steps:
   • always start with the ones – this is what we use the first row for
   • put in a zero when you are ready to multiply the tens – the answer goes on the second row
   • keep your columns in order, think about the place value of each number you write down
   • use the final row to do your column addition

If you are looking for some worksheets, here are some which I like because they are on squared paper.  The early ones have the hint of putting the zero in, the latter ones don’t. If you don’t like those, a quick Google images search on long multiplication worksheet will soon get you the one you want.

I’m sure there are lots of nuances I’ve skipped over here, but if this triggers any thoughts about how you would teach it, please leave a comment below.