A maths game using areas of rectangles

All my Primary-age pupils played this game this week. I originally created it three years ago but this week I made some pre-marked hundred squares just to save a bit of paper – by printing them smaller than 1cm squared, you can fit 6 on a page.

Screen Shot 2016-03-03 at 11.37.34You will need:

The winner will be the person who scores most small squares.

Do you remember how many small squares are in the 10×10 game board? That’s the same as saying “what’s the area in square centimetres”?

When it’s your turn, throw the 2 dice. I’m going to show you my moves in a particular game:

I could have drawn my rectangle anywhere, but it helps later if you try to keep things in the corners…..

I’ve drawn in two rectangles now and I must decide where to put this next throw of 1×5. Where would you put it?

Some of you preferred to count up every single square to decide the score after your turn, some of you counted them in lines (say, counting a 4×5 rectangle as 5…10….15…20 ) and some of you just said “4×5, that’s 20”. I think that depends how good your memory for tables is.

Here’s my board when I hit my first problem…

I can’t find anywhere to put this 5×6 throw, so it’s my first of three strikes…

I was lucky enough to throw a series of small numbers next, and build up my scores a bit…

But this throw was my second strike…

And this one was my third. I’m out, so I add up my score. What did I get? Here’s my opponent’s board. Who won?

Is there an easier way to work out your score?

What is the highest possible score for one rectangle?

Oh, and, what was the jam jar for? I got a bit fed up with some of you rolling the dice SO hard they rolled onto the floor, but one of you suggested we roll them INSIDE THE JAM JAR! Great idea! Well Done! A quick roll and it’s easy to see your score through the glass. Shake gently though….

PS for a whole class game, the teacher uses 2 big foam dice, and the winners are pupils who can place the rectangles in a logical way.

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….