Thursday, 14 May 2015


A scale, from the Latin word scala, a ladder
We've just finished a series of seven lessons on scaling with the Year 4 classes (8 and 9 year olds). It's not been a very prominent part of any curriculum I've worked with, but for a long time it's seemed to me such a ubiquitous bit of maths, that I was keen that our children have a handle on it. The immediate prompt that made me finally plan the lessons was a comment by Paula  Beardell Krieg on an interesting post by Tracy Johnston Zager:
"What I am wondering about these days is when are children developmentally ready to understand the concept of ratios and relationships? Personally, I love having been able to memorize all my “math facts,” but I thought, really really believed that doing calculations was doing math. If I had understood that math was more about the discovery and description of relationships it would have made a difference. I think that this is where the Wall might be: when students need to shift from “getting an answer” to understanding relationships. The calculations mind-set is so deeply hard-wired into the brain that this shift never happens.
So I am wondering, from your work with young children, when do you think you can start talking about math as if it’s tool for discovery?"
Paula puts relationships and ratios together; you could include proportion and scaling. You could include fractions or percentages. It's all part of one concept. I chose scaling as the word we would use, as it seems like the most everyday way of talking about it. We talk about scale models, the scale of a map, scaling up a recipe or picture. I dropped in  the words "ratio" and "proportion" occasionally, but not in any way I wanted the kids to remember. It was about the experiences primarily.

Reflecting on the lessons made me think a little more about what scaling is. It's not that easy to encapsulate what it's essentially about, and I'd appreciate other people's views on this. Thinking through what's common to all the activities, it seems it's a kind of "I'm taking you with me" thing. Change one thing, other things have to change too.

It's much easier for me to think about in particular examples. The longer the car journey is, the bigger other things get. You need a bigger bag of travel sweets for instance.

So, how well did these lessons do? (Click on the numbers in circles for more detail on each activity.)

Scaling up shapes on grid paper. Here, what goes together is the horizontal and the vertical scaling. Some of the children at first scaled one way but not both. But in the end they all got it. It's a really satisfying lesson for the children this, because they're creating their own shape, and the challenge level is just right.

Of course, there's something really interesting going on with the area within these shapes - there are square numbers involved - but though we touched on this, the thrust of the lesson was just to be able to create the shapes.

Julie, did something good with it in the other Year 4 class: she got the kids to add numbers to label all the lengths. This makes the numerical patterns much more obvious, and I'd do that next time.

There was triangle grid paper and some of the children wanted to use that. I went with it because I wanted to see how they'd scale up the three dimensions. But it's such a wonderful thing being able to draw 3D things on flat paper that some of them got a bit carried away with the first shape, and couldn't really scale it up easily!

If I had the same children again in later years, I would start to look at the areas and the volumes involved, how if you graph them, you don't get a straight line. This kind of non-linear scaling is really important in science, engineering and industry, not to mention maths itself!

Using the pattern blocks is a nice easy activity, and kids a few years younger could do this, making sure there were twice the number of triangles as squares. Because it was easy I could use it to link this scaling to its graphical representation as a diagonal line.

This incidentally led to a nice exchange of ideas on Twitter. John Golden, who'd originally suggested pattern blocks, came up with a brilliant ratio chart. This is something older children could investigate, as well as his suggestion of more complicated mixtures.

 Folding and halving A-size paper is something younger children could do. It could also go off in the directions of fractions and even the idea of infinite series. (I didn't mention in the post how we looked at other rectangles that don't scale up this way - and got into an interesting discussion about squares in particular.) We touched on the fact that the A-sizes have a 1:√2 ratio, but this is something older year groups could explore more.

The 1:100 and 1:500 scale models of the A380 came along at just the right time. Children have so much experience with scale when it comes to models. And as we were looking at maps of runways for our work on headings, it was natural to ask, if this is the plane scaled down, how much would the runway scaled down be? We could have done more on this if time allowed, and scales in maps is such an obvious place to deal with scaling.

Cuisenaire rods gave us the chance to pass by lesson 1 again. I think younger children could do this too. What's special about this approach is that it makes the square numbers really apparent, so it would help older kids too. You can actually stand blocks on their end in the trays, so you could scale up in three dimensions too, and think about cube numbers.

Kids should cook regularly anyway, but this time we especially focused on the scaling up the recipe. (I had to contrive it a bit, because the recipe was right to start with!) The online activity was good too. It showed me that scaling from a recipe for 2 to a recipe for 3 was very challenging for them. That's where to go next year and beyond. This activity also has the advantage that you can eat it!

The Zoolander question from Robert Kaplinksy was, in hindsight, quite challenging for this age group, maybe too challenging. The class got the humour of Zoolander not understanding a scale model though!

Strangely perhaps, I'm not satisfied with all this. And I don't even know why. Is it that scaling is such a slippery concept? Is it that the work we did didn't involve much calculation? Is it just that we need to be doing things like this every year? Maybe it's this... I don't know.

I'd be interested in your ideas about this. What, for you, is the essence of scaling? Do you think it's worth devoting time to? Are there other ways of approaching it you would choose? Let me know!


  1. I love all the lessons, and am curious about your dissatisfaction. I wonder if the kids connected one representation to another? Would they be able to say what each lesson had to do with the others? Perhaps take a page out of Kristin Gray's book and have the kids write about it? I'd be curious to hear what they took away.

    1. Yes, it's probably about the kids being able to group all these activities together. When we add things, whatever context it's in, we can use a nice simple + sign. There's no simple flag for scaling to fly under, perhaps because it's about *two* or more numbers having the same thing done to them.

      I think what you say about writing about it is very to the point. Perhaps it's the evidence of clear meta-thinking that I'm missing, that I don't really know the children have made the link.

      I don't see them again until Monday (the French love their days off!) but perhaps it won't be too late...

  2. Simon, since you first mentioned Dienes' theory of multiple embodiments to me I have been learning, reading, observing and writing with that idea in mind. The more I watch what math teachers do (or don't do) the more convinced I am that exploring a math idea in multiple ways and in many different contexts is SO important for conceptual development of the idea. I think you have a fantastically strong based here from which to build next year. I think adding the calculations in there is a good idea, and also, some body-based explorations. Here's an activity I did based on the book Ten Times Better. The book is nice, but the fun stuff is at the back. There's potential for a narrative and/or poetry exploration of scale as well!

    1. Malke, thank you. That's a great blog post. I can see how the Ten Times Better type facts could give a good opportunity for lots of calculations, as well as going off in all sorts of other directions. Getting the tape measure out and seeing what these things mean in real terms... I can see the kids running with that. They could also make choices with everyday objects. "If my pencil was a hundred times bigger, it would stretch from here to the whiteboard."

    2. Bringing the body in is also one of those multiple embodiments -- Dienes even has a book on body-based activities! But, even w/o that, there is a visceral experience lying in wait when you change the scale of scaling, so to speak, to include moving/body-scale activities in the world.

  3. I love everything about these lessons. I'm especially taken by Julie's addition of numbers to describe the side lengths, and also the use of triangle grid paper for 3-D shapes, which I have never seen before. And even if the Zoolander lesson was too challenging, the fact that they understood what was funny about the bit made showing it totally worth it.
    But now I'm going to finally ask you what's been on my mind since I first came across your work: What curriculum do you follow? And where can I find it? Because it isn't like anything I've ever seen, and I want in!

  4. Thanks Joe. One of the things that makes working with Julie great is that not only is she prepared to run with my ideas with her class, but she often improves on them!

    As for curriculum,.. long and complicated story.

    I've taught in inner London for most of my career. When the National Curriculum came in in England in 1988, I'd already been teaching for a while, in the Wild West days when schools, and often teachers, had to come up with their own ideas.

    The new National Curriculum was enough of a skeleton framework that schools and teachers could come up with either dull and depressing lessons or active and affirming ones (and I know I've come up with a fair number of the former too - there's something in us, that reverts to how we got taught at school ourselves, isn't there). I think having a curriculum, that also had some "give" in it was definitely a good thing - helping schools to structure their continuity and progression.

    1988 National Curriculum for mathematics

    Wikipedia gives another part of the political agenda which wasn't so good: "The purpose of the National Curriculum was to standardise the content taught across schools to enable assessment, which in turn enabled the compilation of league tables detailing the assessment statistics for each school. These league tables, together with the provision to parents of some degree of choice in assignment of the school for their child (also legislated in the same act) were intended to encourage a 'free market' by allowing parents to choose schools based on their measured ability to teach the National Curriculum." The emphasis shifted away from encouraging professionalism towards testing, league tables, inspection. I talk more about the problems with this in this post:

    Things got tighter in 2003 with the National Numeracy Strategy and the Numeracy Hour. Julie and I still have as a kind of starting point its now obsolete planning structure: Year 4 Numeracy Strategy blocks

    The structure was OK (except that in my opinion all curricula I know have too much emphasis on arithmetic - that's why they called it Numeracy rather than Mathematics, to really make the point); But the hard thing for me was The Numeracy Hour, which was structured rigidly, 20 minutes starter, 30 minutes main lesson, 10 minutes plenary, with lots of approved methodology delivered by advisers in all the local authorities. There were lots of good methods, but I also found it interfered with the way I knew how to teach.

    Now, last year there's a new national curriculum in England, but my school, an international school in France with a minority of English pupils, is looking to find a more international curriculum anyway, so we haven't followed it, this new NC. And so, for the time being... we have more freedom than ever.

    (The school has a calculation policy, which shows how we should progress from year to year in the way we teach calculation,; we also have what are called Numeracy Passports, which are where the kids, at their own level, learn number facts, times tables, etc, we check whether they know them and when they know a bunch they get a "passport", a kind of certificate effectively, awarded in assembly.)

    That's an already long, but very broad-brush answer to your question.

    And now there's the #MTBoS, and, for instance, I drop in on people in NJ for lots of my best ideas!

  5. Hi Simon, you know I've been following your projects with great interest.

    Your explorations have emboldened me to talk more about scale and proportion in the book making classes that I teach. As I've mentioned to you, scaling is embedded into book making skills. I work with all ages, but the work I am
    dong with 6 and 7 year old grabs my attention.I've noticed that what students
    learn at a young age can stay with them in a big way, hence my wondering
    about how early they can start thinking of math as more than a series of math

    Yesterday I worked with a group of first graders, and began our bookmaking
    journey by folding and slicing an 18" x 24"(about A2) paper until it became
    the size of a typical book. The paper was large ( completely covered their desk). I know that it doesn't occur to them that if they use a small piece of paper, the size of regular copy paper, that they can independently make a much smaller version of the project that I've brought to them. With a nod to theatrics, at the end of class I plucked a piece of copy paper from the recycle bin and showed them that they could use this to make a smaller, scaled down version of the origami pamphlet that they had made with large paper. Students were absolutely delighted with this. But what struck me was that they
    unmistakabley knew that the smaller paper would fold it's way down into a
    smaller book. There's no question that at 6 years old they are already
    understanding the concept of scaling. My original question about this, then,
    is changing, not to "when can we start talking to children about scaling" to
    "why aren't we talking to children about scaling?" It seems that at each age
    level there is age-appropriate language that can used to discuss
    relationships between things, and it would be wonderful to make this a part
    of the students' math journey, right from the beginning.

    I think I understand what you mean about not being satisfied with your
    seven scaling lessons. As brilliant, rich, creative and profound as your
    grouping is, there is one more lesson that I think goes with these, one that
    would link them together in an elegant way and create a bridge between the
    hands-on and the math language that the will soon be encountering. Have you
    considered introducing them to a line on the co-ordinate plane? You don't
    have to get into the definition of the equation-of-a-line or discuss the
    different quadrants, but if you make a line that starts at the orgin, then
    passes through a point where y=2 and x=3, then extend that line, you can show
    them that this is a way to illustrate scaling, as now every point on that
    line will be a way to express a 2:3 relationship. And then do this for
    currency, say measure the euro against the dollar,(today it's 1 euro: 1.14
    dollars) and they can see how, as they move up the line, they can scale any
    number of Euros into dollars. I'm sure your students can do this, esp since
    they've already scaled the runway for the A380! Then let me know how it goes.

    By the way, this morning Jo Morgan posted a link to Scaling the Universe. Have you seen it? Good timing! It's a perfect
    piece of work.

    Happy scaling!

  6. Thank you Paula for such an interesting and helpful reply! I make those booklets with my kids from time to time!

    Yes, I agree - "why aren't we talking to children about scaling?" When you think that so many toys are scale models. Things aren't right when a Duplo figure breaks in on a Lego figure party!

    I like what you think about the line on the Cartesian plane as the central reference metaphor. Maybe because I already have that place to go, it gives a kind of connectedness that, I feel that the kids haven't got yet with this.

    As I wrote somewhere else, As a kid I got to know various bits of London. Often I got to them by Tube (= Underground, Metro, Subway). Starting from my home in Paddington, I popped up in South Kensington or Tower Hill or Baker Street. They seemed like very separate places, unconnected, with their own climates and moods. Only later when I moved about overground a lot more did the islands of familiarity join up into one London.

    So, yes lots of experience of that line. We did do the line with ➋ and also saw that our A-size papers all kind of lined up. I should have mentioned that at this point we also looked at ProportioNiall, and saw how Niall is too wide or too narrow everywhere except on the blue line of proper scaling.

    In my own learning, I hadn't quite taken in how all shapes scale in terms of area until relatively recently - see this post here and this post here, and I did maths up to age 18, with a little bit at university too! So, somehow among all the trees I'd been sing I hadn't seen the forest.

    I probably won't have much time to fit in any more scaling with this class. we're going on to function machines, then revisiting addition and subtraction on number lines and in columns, then I'm hoping to get in some Geogebra and a few other things - more than I can fit. But in the future...

  7. Hello! This post was recommended for MTBoS 2015: a collection of people's favorite blog posts of the year. We would like to publish an edited volume of the posts and use the money raised toward a scholarship for TMC. Please let us know by responding via email to whether or not you grant us permission to include your post. Thank you, Tina and Lani.