Monday, 23 March 2015


I used to be the worksheet king. Making them clear, uncluttered, as simple as possible. They're all, hundreds of them, on the system at work. Here: I've found one from February 2005:

That must have been Year 6, ten, eleven year olds, I was teaching that year.

But these days I seem to be using them less and less. And the Abacus textbooks, with lots of sums and things in,
I've hardly touched them at all this year.

What is happening?

Partly it's that I'm more and more wanting the kids to be creative in their maths work, and neither the worksheets, nor the textbook seem to give enough space for this. With creative work, there's usually a stimulus at the beginning of the lesson (after we've done a bit of number circle or estimation or some starter or other) and then there are constraints (often the manipulatives we use are part of this) and a requirement that I give by speaking to the class, maybe an example. Then off you go...

I'm not against either worksheets or textbooks. It's just the way I've been moving. You can see why perhaps in my earlier posts. (You can see some of our work on the Year 4 blog.) And, besides, as well as my own ideas I've been getting so many from the great people in the Math-Twitter-Blog-o-sphere #mtbos that there's really not enough time to do half of the things I want to do.

So I was interested to read a post by Andrew Gael on different kinds of worksheets that approach the same task. I immediately wanted to do a worksheet-free lesson based on the same premise. But I also wanted to see what the worksheet would do for us. I chose the one I liked best, and passed it out without too much explanation to my Year 4s (8 and 9 year olds). I had to translate a little. I don't call it graph paper; I call it squared paper. But most of them could see the idea, though I think some them were a bit fazed by getting a worksheet with written instructions out of the blue like that. A few asked for a bit of supplementary explanation.

Quite a few found it straightforward, like this:
For one it was just so too easy, he went all one-dimensional:
Some got the side lengths wrong:
And one got a bit confused:
Aside from anything else, you can see it's a worthwhile activity just to draw a grid of a certain size (just as it's a worthwhile activity for young kids to create their own number line) and about half the class need a bit more practice at this. I think about a third of the class could do with repeating this activity and getting it right. So it's been informative. All of them could do with annotating their grid with a bit of explanation. Checking wouldn't be a bad thing either! Tomorrow.

What did we do next? More of an open-middle kind of thing. They cut rectangles form squared paper. Wrote how many squares on one side and their name on the other. Then they cut bites out.

We looked at them a bit together, and worked out the area of the original rectangles of some of them. We'll return to them, and the worksheet, tomorrow.

Anyway, what do you think? I've changed Andrew's question to fit my case. What are the advantages of the worksheet here? (Should I be using them more?) And how about the task afterwards? I'm interested in your thoughts.

The next day I gave everyone their work back to check, and change if need be. I also gave one of these orange sheets (with the orange taken out to make space for writing) to everyone.
We'd been talking about 16 X 5 type questions - and will talk more.
Most people seem to have got the idea; though I think I'll throw a few "draw me a rectangle with 45 squares on it"-type questions when we all have our whiteboards for a lesson starter.

Thank you, all you brilliant commenters! I've certainly got one or two things a lot clearer in my head about worksheets good and bad, and more besides.


  1. I think the concrete extension here is wonderful. I also like the combination, because I'm not sure you could have gotten to the middle without the intro. Maybe next step is thinking about practical applications. When would you need to figure this out in the real world. How many ideas can the class come up with?

    1. I'll ask them Lisa. In class we have carpet tiles on the floor and square ceiling tiles too. Most houses round here have square tiles on the floor too! I imagine they've got a lot of experience of this. We'll see...

  2. I'm wondering what the definition of "worksheet" is? Because I'm not thinking that's the word I would use to describe what you did here, or what Andy Gael does either. What you guys are doing is interesting and informative for you and the kids. My daughter's worksheets that I keep tweeting about angrily? Not so much. ;)

    1. Yes, I've seen those worksheets. Like
      The French like this kind of thing a lot too.
      What makes people come up with these strange solipsisms? Is there an insecurity, it all has to be written down, even if actually it will need quite a lot of oral explanation to make sense of it all? (I think I once felt a bit like this: reading and writing was the safe, controlled place, the secure limits of the A4 sheet of paper. Now I'm more comfortable with talk and doing things, which I always hoped I would be eventually.) Is it something about accountability, evidencing "teaching" to managers?

      I agree, Andrew's ones are good-worksheet, not bad-worksheet. And that's why I was happy to use one with my class.

  3. Hi Simon,
    It's pretty clear that you want to make the students work the prominent part of the worksheet. I remember my children bringing home worksheets that were so overwhelmingly pre-filled with the publishers images that my child's work looked sort of like an afterthought. What kind of message does that give the child? That their contribution of the least importance. It was so easy to just throw those papers right into the trash.

    The worksheet you showed leaves a good bit of space for the students to work out their response. Even though they are working within a space that you define, there is plenty of room for the personality of the student to come through. It looks like a fine balance to me.

    Thanks for asking for a comment about the task afterwords, because this is what I have been thinking about. There's something I want to comment on: it may be a bit off topic, but I think it's something to think about.

    I see this happening in elementary school. The worksheet states that the graph paper needs 56 units, but what you are asking for is really a grid. Counting the squares makes it a grid. But a grid and a graph, though they look identical close up, are really based on two completely different sets of rules. The grid defines squares. The graph defines points and relationships. I think that this distinction often blurred in elementary school, and this can set students up for confusion when algebra comes around. Students have to be nimble to switch to the idea of now there's a line for an independent
    variable and another for a dependent variable. The idea of a graph being a grid is really hard to shake after thinking of them as being interchangeable for so long.

    That said, if you drop the graph terminology that the worksheet references, I love the activity. It gets them looking at area in a concrete way. They can't possibly figure out the exact area of what's left after cutting out irregular shapes, but they can get really close. And, hey, maybe you can let them know, that sometime down the road in math, they'll learn calculus, which will teach them how to find the area of irregular shapes. Maybe, then, they'll be a the edge of their seats, waiting to sign up for that course!

    1. That's a good point about the graph / grid terminology; thank you for that Paula.

      I also like what you say about the bad kind of overstuffed worksheet:

      "What kind of message does that give the child? That their contribution of the least importance."

      The worksheet as a set of signs about how the teacher or establishment, or system even, regards learning.

  4. Agree with Tracy, although I admit I'm sort of confused by the task. (I'd be the one who needs the supplementary instruction!) Were the kids supposed to first identify which of the three graph paper images was 56 sq units? Then try to recreate what it would look like had the big coin (or whatever it was) not been covering the middle section? I do like the idea of making an attempt to draw the grid. Did they have access to rulers but not all elected to use them?
    I like the rectangles with the bites cut out. You could put them back together like puzzles to check after kids had drawn what the originals looked like. Maybe to make that consistent with the first part, the original worksheet should show the different graph papers with pieces torn out instead of covered up. Like, "The dog ate my graph paper!"

    1. Thanks Joe. I enjoyed your Building a Better Worksheet post:
      Take things away, ask more open questions, ask for more. It's great that there is this conversation, this reflection.

      They do have rulers in the middle of the table, but that I left up to them. I didn't suggest them.

      In the event, today I used the orange images, with the orange photoshopped out so they could write, and just got them to show what are the original rectangle would have been and why. Perhaps I'll scan one or two tomorrow and add to the post. Been a busy day. I think, like you say, the physical cut-out rectangles could be taken further. We hadn't kept the bites. But to make them out of bigger, stiffer card, maybe even drawn rather than on squared paper. Use them more after their creation; maybe display on either side of the window to the hallway...

  5. This is a really cool reflection! I definitely don't think you need to use MORE worksheets. The work you do with your students is something that continually inspires me!

    I think worksheets should be used to help students organize the problem solving process and for teachers to use as assessment tools. My post wasn't meant to be about the use of worksheets. It was (hopefully) about different ways to look at a similar task based on the needs of your student population.

    I'll have to update my post to include the final concrete, hands-on task I forgot to include! I'll update it tomorrow and send it to you.

    1. I know I went off at my own tangent from your post; I really hope you didn't mind at all. I think the comparison between the sheets is valuable too!
      And it was, as I said, definitely an instructive piece of assessment. I could have easily got stuck into say the scissors task and not known that some children couldn't reconstruct the rectangle with a pencil.

  6. It's not so much of a worksheet as it une page de l'exploration.

    There's nothing that implies that students have to solve it a particular way and I like that a lot! I think what's important here is that you've provided the possibility for multiple solutions which will only promote deep conversations when the opportunity to share presents itself.

    To open it up a bit more, maybe you could ask the question "I have a square with 56 units, what dimensions would give me the largest perimeter?" I'm assuming most students wouldn't immediately go to 27x1 and they'd start at 7x8.

    Merci de me tirer dans le mix ici Simon. The French-Canadian school boy in me couldn't resist. You make us all better and thanks for pushing me to find new ways to explore with my students.

    1. je m'excuse...I meant a rectangle with 56 units, not a square:-)

    2. I like that... une fiche d'exploration... I think I'll call it a Nemo.

    3. thanks so much for this lovely little bilingue tangent. I shall have a secret smile on my face now when I'm creating "les fiches d'exploration" for my French second language students, and will encourage them to "just keep swimming". The idea of leaving space for thinking/exploring/showing work is key, I think.

    4. I'm glad I'm not the only one who knows they're called Nemos!

  7. I didn't understand (didn't read carefully enough) that you were asking the students to find the area of the original rectangle before the bites were taken out. What a great worksheet! Here, the students have to insert themselves into the paper, to fill up the empty spaces, to solve the area problem. I'm not exactly sure why this feels so much more dynamic than giving them an full rectangle, but it does feel dynamic. It's fun, it's unexpected, and it tells them they can figure out something by gleaning "enough" information from something that seems incomplete.
    I've discovered that certain things that I teach to students are too hard to explain. When that happens I try to give them just enough information and let them figure out the rest themselves. Works like a charm. Your rectangles with bites reminds me of this kind of teaching.

    1. Thanks Paula. When I got them to do their own bitten rectangles, they were pretty good at pushing the limits of how much information was just enough. It's a nice idea isn't it? We've been making lots of KenKen puzzles recently, and there too the kids have had to think about whether they've given sufficient information for a solution. I'm not sure exactly where this fits into a maths, or any, curriculum, the art of clue-giving, but I feel like it's a really useful skill for students as well as teachers!