Category Archives: Investigations

Lessons where pupils have explored ideas and posed their own problems

How many rectangles can you see?

how many rectangles can you see?This excellent puzzle is based on “Rectangles within Rectangles” from the book “Mathematical Snacks” by Jon Millington. Highly recommended as a source of enriching mathematical thinking!

Problem solving is a key part of 1-1 tuition – because the student is not in competition with others, the tutor can provide as much or as little help as is needed and the rewards are potentially very great, in terms of enjoyment, confidence, and increased skill in mathematical approaches.

A very elegant piece of algebra

I set this problem today, during a lesson in which we had been looking at simultaneous equations with 2 unknowns:

A + B = 92

A + c = 57

B + C = 59

M came up with this answer:

clever algebra

 

POSTSCRIPT –

M then set me this problem:

A-B = 25

B+C = 151

A+C = 176

which turned out to have an unexpected twist in it….. Have fun!

Problem Solving Skills – How much is a tin of tomatoes?

Image from the Sainsbury's website, I hope they are pleased to get some free advertising :)

Ever since I went to school. the format for a “normal” Maths lesson has been the same:

  1. Teach a new method
  2. Show the class some examples
  3. Give them a set of similar problems to try.

Just at the moment I’m exploring an alternative approach

  1. Think about ONE interesting* problem to give the pupils
  2. Give it to them, and see what they come up with, being open to them experiencing a dose of “I can’t do it” and “I’m stuck”

The definition of *interesting is a big challenge of course, and will rest on how well you know your pupils. Today I picked food because it is an area of common interest between me and one of my pupils. I gave her a problem which I would classify as a “backwards” problem – I was asking her to discover the price of a tin of tomatoes, given some clues.

Click Here to see the printable Tomato Puzzle

Isosceles Triangle Numbers

9 is an isosceles triangle number
9 is an isosceles triangle number
16 is a square number
16 is a square number
The next person produced this answer, which means the number 16 went on the Isosceles Triangle list as well.
16 is an isosceles triangle number

A bit more experimentation, and we realised a couple of things – it was easy to play around with this investigation on 1cm squared paper, and it let us try find some MUCH bigger isosceles triangle numbers. We also noticed that the bottom row of counters has to contain an ODD number or the triangle won’t come to a point at the top.

We also found some bigger square numbers, and both lists got quite a bit longer.

Then, we noticed something about the lists…. try it. What do you notice? Why??

Isometric Drawing

challengeOne of my Year 6 groups tried some Isometric Drawing for the first time today. It’s perfectly possible for people to get hopelessly confused the first time they try this, but they all listened carefully, started simple (drawing one cube, then two) but it didn’t take long before they were digging into the box of Cuisinnaire Rode and setting themselves some really tough challenges. The most extreme one was this, six “ten” rods, piled up. Groans ensued and most of them decided it was impossible. However, some of them stuck at it and their drawings were brilliant!

 

 

 

 

Insometric drawing fourth answer Insometric drawing first answer Insometric drawing second answer

I noticed a couple of them had abandoned their pre-printed paper and were busy drawing soomething of their own on the class whiteboard:

Isometric drawing second challenge

The 90 times table.

I like Wednesdays. There are just seven pupils that I see twice a week, and they arrive, in a 3 and then a 4, on Wednesdays. When this “second lesson” was instigated, I tried just NOT planning anything, but to allow them to use it as a workshop to air their particular struggles with Maths. Today was unusual because there is a lot of illness in school at the moment so I just had Guy* and Becky*. On the way to the lesson….

Guy (jumping and landing a quarter turn around) We did 90 degrees today.

Oh yes, what happened?

We did (jumping some more) 90, 180, jump jump….

partial 90x table
partial 90x table

We had arrived in our teaching room by now, so I wrote up 90, 180 on the board. Thanks to Jan Poustie’s brilliant work on Dyscalculia**, I ALWAYS (and there’s not many things I’m strict about in lessons) write tables in a 3×3 grid. Poustie’s theory is that this makes them easier to memorise and recall, for kids who struggle to learn them. I’ve worked like that for 3 years now and there is more, so much more to it than that, but that’s for another blog post. So, anyway, up on the board goes 90, 180, and they supply 270, 360 and then get stuck. I asked them, did they stop there? Yes, that’s all the way round. But the table has 5 empty slots in it… what next? What does it look like?

Becky “It looks like the 9x table” .

90x and 9x table, in a 3x3 grid ref Jan Poustie
90x and 9x table, in a 3×3 grid ref Jan Poustie

OK you can trust that, let’s write up the 9x table (Lots of OOhs and Aahs favourite table because of the pattern in it, and because it also works so beautifully on your fingers). So up goes the 9x table with them providing all the numbers correctly. Back to the 90x table, it’s easy now cos we’ve got the 9x table as a crib…

So I volunteer to do the jumping bit if they chant…. 90, 180, 270, 360, 450, 540, 630, 720, 810, 900… now the room is gently spinning and I am facing away from the students. They seem to think this is hilarious, note to self, get a pupil to do this next time so I can be the one laughing…

But still this feeling that the angles above 360 don’t mean anything. They agree (me jumping more gently this time) that it makes more sense to say 90,180,270,360,90,180,270,360….

One of the best decisions I made this term was to keep a big box of STUFF at school, this is what enables me to teach these drop-of-a-hat lessons when the kids provide a topic. For me, the REALLY interesting sequence is 180,360,540,720… and later, when enough of the boys have got hooked on skateboarding, this will be a beloved sequence for them too, but they’re a bit young yet.

I draw an equilateral triangle on the board (we worked a lot with triangles last week, and I discovered that for all of them, this is THE triangle). They could tell me that all the sides would be the same length, but no idea about the size of one angle (90? 45? 25?) so we set up a little production line, drawing triangles, measuring the angles and adding them together. They knew (from last week) how HARD it is to draw an equilateral triangle by eye, so we settled on any triangle, as odd and different as you like. The numbers started to mount up. 196, 181, 180… for me, the tension was mounting. It is SOO tempting to correct and reject outliers, and this wasn’t looking promising. But holding my hair on, I reminded myself that a  small group is an ideal venue for measuring angles, cos protractors can get a bit of reputation for being tricky customers to use, and by Year 9 or 10, some kids have got themselves REALLY confused. Even if The Sequence didn’t emerge, they would gain a lot of confidence measuring angles.

How to measure an angle using a protractor

working out the angles in a triangleIt took me a lot of trial and error to work this one out. Kids really get confused over WHAT they are measuring, so I start by reminding them how they find the length of a line using a ruler. One end of the line sits under the zero, the other end of the line gives you the answer. I bring the ruler near the protractor and show them that the protractor is just a little round ruler – the degrees are almost 1mm wide. I draw the angle in, running the pencil around the edge of the protractor, and show them they are measuring the curve, and it is … degrees. Becky just needed reminding once that the centre of the protractor MUST be on the corner of the triangle, otherwise the answer is wrong. They drew the big curves on to the first few triangles. It looked a bit odd, and we discussed it, but later they dropped that and just measured the angles.

Fairly quickly we had 10 results and I asked for comments. Becky thought all the numbers were pretty close together, but lots of different ones. I confessed that the 196 at the beginning did look a bit odd, and checked the triangle – a mistake had been made and they were happy to change the result to 182.

So we had a variety of different results, I suggested we find a mean average (I could see it would work out how, but it seemed sensible to summarise the data). They thought they could do it… but after a crash revision of means, becky added them all, and got 1812, divided by 10 and got 181.2.

measuring the angles in a quadrilateral

This, frustratingly, drew a blank and our half hour was nearly up but I had mentioned that task 2 would the the same, with 4-sided shapes, and they begged an extra 5 minutes. We rapidly found 254, 366, 360 and 360, and since Guy had intended to draw a rectangle, he was adamant that his 360 was “right”. The mean average (worked out unassisted by Becky) was 360. Another blank.

So, I write them up near the 90x table on the board. Becky gets that lightbulb look on her face:

Well the “2” is a bit wrong on 182, but the 360 is in the table and so is 180…?

I allowed that we might make little mistakes and they got really excited about the idea that every other number in the table was appearing, the next one, for 5-sided shapes, could be 540, then 720 for 6-sided shapes….

35 minutes into a 30 minute session the time had REALLY come to throw them out of the door. After half term, we will draw and measure 5- and 6-sided shapes.

a big hint

* Names changed, always!!

** Jan Poustie, 1990, Mathematical Solutions – an introduction to Dyscalculia Part B How to teach children and adults who have Specific Learning Difficulties in mathematics  Available direct from the author’s website

Maths with spaghetti

Introduction – Where did the spaghetti idea come from?

I’m a sucker for quirky Mathematical instruments and a couple of weeks ago, I got out my weird protractors. Most of my sets of protractors are full circles but the strangest set have all different sizes of circular holes in them, as well as the normal degree markings around the edge.

I gave the Year 5 extension group a protractor each and asked them what they were. They were pretty hesitant except for one boy who said, very confidently, “It’s a spaghetti measurer!”.

This appealed to me for several reasons. One was the fact he had obviously been participating in life in the kitchen, asking questions and getting answers. Another was the fact that my Year 6 extension group has just finished their work on areas of shapes, and they found measuring the area of circles in square centimetres a big challenge. It would have been so much easier to measure the areas of circles in spaghetti sticks…

The Spaghetti Starter – Year 6 extension group

Each of  them had a mini whiteboard, silence, and 4 minutes, to come up with 3 or 4 ideas each for a question you could pose, using the spaghetti.

Possible spaghetti questions
Possible spaghetti questions

I’m transcribing them because they are a bit too small to make out from the photo… I have paraphrased slightly for clarity, and omitted duplicates:

  1. How many pieces of pasta are there?
  2. Are there the same number of pasta strands in another packet?
  3. What is the average weight of 1 pasta piece?
  4. Is the weight of each pasta strand the same?
  5. What is the length of one spaghetti string?
  6. The length of all pieces of spaghetti put together
  7. Is the length of each piece of pasta the same?
  8. What is the mode of the lengths of the pasta strands?
  9. How many calories in 1 piece?
  10. How much carbohydrate in one piece?

Would the spaghetti reach the length of the field?

The next Year 6 group came up with an even better question:

Would the spaghetti, laid end to end, reach the length of the school field?

This encompassed questions 1,5,6 and 7, and required us all to have an opinion before we even opened the packet. Really we hadn’t got a clue but we agreed on, it would probably reach from side to side, but not end to end. Looking back, I’m amazed we reached an answer within the allotted half hour, but we managed it. It went like this:

  • We estimated how many pieces were in the packet (just for fun really), then split the packet up between the 4 of us to speed up counting. We agreed not to eat it as we went… (hygiene!!)
  • We discussed that there were some broken ones (we left them out), and that the whole ones were pretty much the same length.
  • We measured the spaghetti with a ruler and they used a calculator to find the total length of the spaghetti, converting correctly to metres on the way.
  • Two of them dashed back to class to get a trundle wheel but discovered the cupboard was locked… meanwhile, the other pair and I decided to see if we could do giant 1m steps to measure the field, since measuring it properly with a single metre stick wasn’t really an option… which is why the three of us may have been spotted by several classes as we did huge steps across the whole length of the field…. then I defrosted inside while they all giant-stepped across the field to measure its width.
  • We reconvened and compared the spaghetti with the field size. Our results are a closely guarded secret because other groups may want to do the same work.

How many pieces of spaghetti are there?

The next Year 6 group, given the packet of spaghetti, decided to count the pieces of spaghetti. Momentarily I was a bit disappointed because it seemed a less rich task, mathematically, but a big part of what I want to achieve for these groups is to help them to trust their own mathematical processes, so I cooperated. Again, the group began by sharing out the spaghetti so we could all count some. We all stuck our sub-totals on the board, and I waved at the calculators but they weren’t interested, they decided to see if they could add the numbers mentally. And here comes the mathematical richness of the task – these individuals have their own mathematical goals (why do I keep forgetting this?). I don’t mean a goal a teacher has put on a computer for them, I mean a bit of Maths they keep returning to, and will continue to be fixated by, until they are good and ready to drop it. For this group, in this moment, it was mental addition. A couple got it exactly right, but Hattie was out by 2. We discussed why, and it transpired she had rounded them to the nearest 10, then added. Cate was eager to teach her something, and proceeded to show her, with the spaghetti, that if she moved a couple of bits of spaghetti around instead of rounding, she would get the right answer. Here’s what I mean, but with smaller numbers:

19 + 29 + 24
Take one from the stack of 24, and give it to the 19 stack:
20 + 29 + 23
Do the same, move one from 23 to 29:
20 + 30 + 22
Now add 20 + 30 + 20 = 70, and 70 + 2 = 72

Hattie got it of course, Cate teaches very well!

There were still 10 minutes left so they picked up my challenge of:

What does one piece of spaghetti weigh?

So, we had a total, they knew the packet weighed 500g, and they unanimously decided to divide (on the calculators) and correctly arrived at a good answer for the problem. I asked Cate to dictate the number onto the board. She reeled off the expected 10 digits from the scientific calculator. Then Hattie chipped in with ten more. Eh? What was going on? Oh, she said, just mouse right and you get the next bit.

This was the most surprising moment of the whole morning. I love these calculators. I’ve read the instruction book several times, but I didn’t realise there was more accuracy like that…. But even more surprising, it was Hattie who found it, Hattie who was close to tears the other day at the sheer terrible scariness of the decimal number world.

So up on the board we now had a 20-digit decimal number and about 120 seconds left. Noone could tell me what the number actually MEANT. So homework was to ask 3 adults what the number means, and write down what they say. I await next week’s lesson with great interest!

Just finished counting....
Just finished counting….

Problem Solving in Maths Lessons

I have been a member of the Association of Teachers of Mathematics  for years now, and very much enjoyed the 2011 conference – a brilliant opportunity for a Freelance Maths Tutor to do some networking and catch up with CPD. So I was very pleasantly surprised recently to find myself a member of the ATM Facebook Group, where more networking but at less cost, suddenly becomes a possibility. On February the 23rd Mike Ollerton is doing a session in Leicester entitled

“How can we develop a problem solving classroom culture?”.

I’d love to come but can’t, so I have had to content myself with holding this question in my mind for the past fortnight as I teach my Primary School groups.

It is SO tempting just to TEACH Maths, and it can be quite effective with some pupils, especially in the short term. I have  bunch of tricks I can serve up and they are simple, powerful, they go down well. But I have taken a break from that recently and given the kids some actual problems to solve… The results have not been quite as I expected…

How much does one Strawberry Pencil weigh?

I picked up a packet of 12 strange-looking sweets on the way into school recently, and put this question on the board at the start of each of the group sessions. The extension group, who decided to run this as a boys-v-girls challenge, sailed into a process with ease, using the “obvious” method, which was to read from the packaging the total weight of the sweets and divide by 12 using a bus stop division. It was so quick that they had time to work with another pack of sweets as well….

The interesting outcome for this group was they found three different answers. The division was 75/12, and I was offered 6.3, 6.6 and 6.25. We’ve done a lot of work recently on team work and collaboration, and so I offered this back to the group and asked them to resolve the differences. Pretty soon 6.6 was withdrawn (Oh, I made a mistake), but two remained for consideration. I was pretty chuffed when one of the girls (Kate) admitted to the group that she couldn’t do the division, and she knew her answer (6.3) was wrong. This took a lot of courage and I think was a complement to the level of trust she now feels in the group’s process.

She wrote her workings on the board and, predictably, several hands shot up accompanied by “Oooh”s and much enthusiasm. It would have been easy to get one of them (or several) to teach Kate how to do division when a remainder is not acceptable as an answer. However, I was skeptical whether she would get it, or remember, so I said to the group that we were ALL hugely tempted just to teach Kate, but I was going to NOT teach her, she was going to work it out herself. The atmosphere crackled – the hands reached higher and dislocated shoulders threatened… this was not at all what the audience wanted… one wannabe teacher appeared to be on the verge of exploding, so I offered her the chance to go and briefly run around outside. To the huge amusement of the group, she kangarooed theatrically up and down outside the window then returned…

Back to Kate…

I began to gently engage her with finding her own solution. I got her to rework the bits of the sum that she was confident with and identify the “unstuck” moment (which was the arrival at “remainder three”). She put “point three” into the answer area.. but looked worried. More hands reaching up. This time the exploding pupil was a boy but was sent out with instructions to kangaroo OUT OF SIGHT so the focus could remain on Kate. I continued to ask Kate questions about where she might put the three, where did she normally put remainders, just opening her to her own solution-finding strengths, and then the magic moment, the penny dropped and was missed by several of the group whose desire to teach had become almost unbearable – but Kate’s face as she inserted the required noughts, put the remainder 3 in the correct place and finished the sum correctly, was a picture of triumph. 100% agreement now on 6.25 being the answer.

We finished the lesson with a quick bout of the Inverse Function Dance to release the required steam.

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I think sequels are important. The long term effects of an experimental approach matter and it is important not to make judgements too quickly. So…

Ten days later

Kate assures me that she is still 100% confident with divisions that push on into the decimal world in this way. The group’s cohesion, focus and mutual trust continues to grow and they have spent two one-hour lessons now exploring “What is the largest (area) shape you can draw with a perimeter of 30cm”, they now believe it’s a circle but are very keen to spend next lesson getting to grips with HOW to best find the area of a circle – they are becoming dissatisfied with counting squares, and have found the investigation a genuine struggle in places, but seem to be experiencing a delight in their learning which is motivating them to go deeper.

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 A Quote..(from ATM)

The power to learn rests with the learner. Teaching has a subordinate role. The teacher has a duty to seek out ways to engage the power of the learner