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:
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!
Do any of your students get in a mess with “double borrowing”? There’s a very simple solution which builds on their existing knowledge of subtraction with borrowing….
Easy subtraction – one “layer” of borrowing
There is a very simple solution to this, encourage the student to look at the number differently. Borrow ONCE from the “200” on the top, which becomes 199.
Borrow ONCE from the “200” in the top number – now this sum is ready to do, and there’s far less that can go wrong.
Try it on a student near you, and use the reply box to tell readers what happened next!
Chinese multiplication has been explained many times in many places on the Internet. This is a quick recap of the way I do it….
The kids I’ve taught, especially the more able ones, really like this way of multiplying numbers because it’s SOOOO easy to build up to very large numbers.
Within 20 minutes, a group of ambitious mathematicians has commandeered the class whiteboard and tried to do an ENORMOUS sum like 185936296722 x 15436796 and got an answer. This gives the teacher a problem. How can the sum be checked? Calculators and EXCEL will round the answer to only 10 or so significant figures, which is pretty hopeless for checking the work.
Here is a link to an EXCEL spreadsheet that will do these HUGE sums so you can check pupils’ (or your own) work.
The extra challenge that these interpid mathematicians give themselves, of course, is how to add together huge long lists of numbers. Here’s an example of one of the additions in the sum mentioned above:
One of the additions needed to do the sum…
The student has to add 4,1,7,5,2,1,2,5,7,1,4,2,5,1,2 and 0. It’s tough to add all that without errors, so encourage them to look for TENS, and cross them out, “carrying” them into the next column…
They could make ten from the 4,1 and 5, then another from 7,2 and 1, and anotherfrom 1,4 and 5. Cross them out neatly and there’s not really much more to add! The nice thing is you can tackly any column you like, in any order, which is great for mathematicians who don’t know their right from their left! (except of course the TENS have to move left!).
Most of the numbers made TENS! Each ten has been done in a distinct colour for clarity.
About half the adding done now…
Once this TEN-hunting is complete, the final pass is to add up any digits that are left.
Most of the adding done now…
Completed! The answer!
Finally, some thoughts about the process of learning Chinese Multiplication:
Decimal points in Chinese Multiplication
These instructions work on a CASIO fx-83GT PLUS, which is widely used in schools and will, if you learn to drive it with confidence, do a tremendous number of different sorts of Maths.
If you want the calculator to give all its answers in standard form, follow this sequence:
Red ring round the standard form button
To type in a number that is in standard form, for example 4.5 x 10^12, use the button with x10x on it (it’s in the middle of the bottom row of buttons, and I will use square brackets to mean button) so the keystrokes you need, are 4.5 [x10x]12
A game for 2 players. The winner is the player with most points at the end.
Starter (optional)
Introduction:
Each turn:
What Maths is learned:
Mathematical Language Covered
You will need:
Introduction:
Creating the Puzzles
Give out 2 pieces of card each. Create your choice of sequence, write it on the coloured side of the card. On the back write F(X)= and the function you used.
Discuss boundries – numbers under 10. (I should have set more – no decimals and no division yet. But it did make life interesting!
As they produce the cards, check the function – if it isn’t in simplest form, then simplify it with them and write that version underneath. Point out that their friends won’t be able to guess any more complicated version of the rule. eg if the rule they used was F(X)= 2X+3X+1 then the friends will guess 5X+1
If you think any of them are quite difficult, give them one or two stars. This will let the pupils self-differentiate.
Solving the puzzles
Put the cards down SEQUENCE SIDE UP and the pupils choose which one they want to do. Try to guess the F(X), key your guess onto the calculator, can you make that exact sequence? Once you have made it, turn over and check the rule on the card. Then initial the front of the card to note that you have done it.
Plenary:
What was easy? What was more challenging? Do we need another lesson on this (My group decided they found the minus ones harder so we pencilled in a lesson on handling negative numbers.
You will need:
Slow Rounds:
Fast Rounds:
Super Fast Round:
Bonus level thinking:
One 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!
I noticed a couple of them had abandoned their pre-printed paper and were busy drawing soomething of their own on the class whiteboard:
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
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 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.
It 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.
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.
* 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
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…
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
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:
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:
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:
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….
Bellropespider 