 I drew a picture... For my most recent birthday, my family hid 102¹ ducks around the house. Fortunately, we’re not talking about real ducks, rather tiny plastic yellow ducks. I’ve been finding them in strange spots ever since: in drawers, mugs, shoes, the ice tray… I’m never quite sure where the next one will unexpectedly emerge from². I enjoy a good joke and started displaying the ducks on the architrave above the kitchen window. And of course, I had to line them up in numerical order³. What has now emerged is a fun game where every time I find a duck, I have to try and guess where on the architrave it will sit. There’s something deeply satisfying about predicting where duck number 37 should sit… and occasionally being right. And then I realised I’d created a number line made out of ducks.  First: an admission... I didn't appreciate the power of number lines for a long time. I saw the problems students had with them and took the risk-averse response of going, well, let’s not use them at all (or only when I really must). How foolish I was! I didn’t want to use them because students would ALWAYS get confused about how they were used in addition and subtraction. If the question was 3+4=□, students would consistently end up with an answer of six⁴. And subtraction would always leave the kids prone to even more errors. But, none of those errors were on the kids; they were all on me and how I introduced them to the number lines. I was introducing number lines far too late, and pairing them with concepts students already found difficult. I should have introduced number lines earlier on, with concepts students were comfortable with, so that they could be used as the powerful tool they are to support understanding difficult concepts! When should we introduce number lines? There are a few key ideas we need students to be familiar with before introducing number lines. To understand the number line, students need to understand the five counting principles⁵. That is, students should know:
Once students understand these 5 principles, they are ready for number lines! Let’s begin with number tracks… Number tracks place the numbers in boxes. They are really useful for students as they bridge the gap from concrete to representational to abstract, which is a constant cycle throughout mathematics. We typically begin with tracks from 1–5, then 1–10, then 0–10 and 0–20. This progression naturally leads to a 0–100 number track, which becomes the hundred chart. My mistake early on was thinking that they would suffice. I now think number tracks are important, but not sufficient.  A variety of "important, but not sufficient" number tracks What’s the difference? A number line differs from a number track because the numbers sit along a continuous line instead of inside boxes. This is a crucial distinction that opens up the possibilities of number lines well beyond those of number tracks. This representation supports the idea of numbers as magnitude. And this might seem trivial in the early years, but is critical later on. Once students are familiar with a number track to 10, we can start introducing them to number lines to the same number. But we have to be careful, because they are not the same thing! We need to be very clear about the difference: we are no longer counting the boxes. Instead we are exploring the magnitude of numbers. It moves them away from just counting items (the numbers in the boxes) and gets students to begin considering the value of the number. For example, 4 on a number track looks like this:  While 4 on a 0-10 number line represents this:  My hot tip: begin with number lines that look very similar to number tracks. This ‘minimal variation’ helps students notice the key differences without being overwhelmed by a new format. Of course, at this stage many students benefit greatly from concrete representations, so finding a structured number line that matches the lengths of unifix⁶ is a really nice way to support this. Just make sure to teach students that the concept of ‘4’ is not just a little tick mark along a line: it’s the value it represents.  Yeah, I know that these are duplo blocks, not unifix... but it's what I have at home! So, why does it matter? Well, before long, we progress to addition and subtraction. And while number lines can be useful, they can also lead to confusion. The puzzled face of a child as they try to work out the jumps forwards or backwards was enough to drive me away from using them as a scaffold⁷. My mistake was that I was introducing both the concept of a number line and the operation (addition or subtraction) at the same time. If they weren’t confused about the number line, then they were confused about the addition/subtraction. Quite simply, there were too many new bits competing for attention… and students got lost along the way. If I’d introduced number lines earlier, with content students already felt confident with, then the number line itself would have been the only new idea. Then when we got to addition/subtraction, the new bit would have been how this is represented on the number line. And as fluent mathematicians, we use these little jumps and think our students will comprehend them. But even that is fraught with danger: I know how often students struggle to make the jumps accurately, often jumping over numbers or putting tiny jumps between the numbers. Now consider using a structured number line that matches your preferred blocks. Instead of showing ‘jumps’ (which are rather abstract), we can further support students by physically making the difference between the numbers.  Note how the blocks make the concept of '+4' tangible? Of course, we want our students to move to more abstract representations (i.e., jumps), but we need to support all our students. And if we lose or confuse our students along the way, the achievement gap widens, making our work as teachers catering to different needs even harder. Can kids draw their own number lines? Another mistake I made was asking students to draw their own number lines, without realising how difficult this is. Yeah, that’s right… I forgot how difficult it is to divide a line into 10 approximately equal lengths (especially for the kids who sometimes struggled to count to ten). This is why we should start with structured labelled number lines.  A structured 0-10 number line Spoiler alert: They don’t always have to start at 0. Introduce this idea early and we’ll be able to reap the rewards! Then we can move to semi-structured number lines⁸, which have a few numbers labelled.  A semi-structured number line And then we can move to empty number lines (which is just a line until the student starts adding to it). It’s important to remember that whenever we are introducing a more complex maths task, we should increase the structure of the number lines we’re using. The harder the maths, the stronger the initial scaffold. And number lines are great when we get into two-digit and three-digit maths (if we’ve taught the prerequisite skills).  Okay, so number lines are just good for addition & subtraction? Nope! Number lines help when we are measuring. Most⁹ scaled instruments use some kind of representation of a number line: rulers, beakers, thermometers… In fact, I argue that the best way to introduce students to analogue clocks is by using string to show them that the hours are a circular 0-12 number line, and the minutes are a circular 0-60 number line. Number lines help make multiplication and division clearer because they show the movement behind the maths. When students use equal jumps along a line, they can see the pattern of “lots of the same size jump” for multiplication, and how division breaks a length into equal parts. It turns repeated actions into a simple visual story, which helps kids understand what the operations really mean. Does this mean that we should always use number lines? No, because they are a scaffold. And the point of a scaffold is that we should remove it¹⁰. Great, so number lines are brilliant for early years maths… Actually, number lines become even more important as students move into more complex maths. Number lines are the most powerful way to introduce formal notation for fractions¹¹. I’ve adapted the routine below to help students understand two essential ideas:
Step 1: Show Whole Units Clearly Begin by drawing a number line with several whole units marked by tall lines. Point to each tall line and name it as a whole. You might say:
Step 2: Count the Equal Parts in Each Whole Underline the first whole unit and ask students to count the small tick marks within it. They should count aloud: 1, 2, 3, 4. Then ask: “How many parts are in this whole?”  Repeat for the second and third whole units. Students should see that each whole is divided into the same number of parts. This is the moment where the denominator finally becomes meaningful. Say “There are 4 parts in each whole unit. So the bottom number of the fraction is 4.” Step 3: Write the Denominator Once students confirm the number of equal parts, write that number as the denominator for every fraction on that number line. This reinforces that the denominator describes the structure of the whole: not the shaded part, not the number of wholes, and not the number of ticks counted so far. Repeat this process with number lines divided into other fractions such as thirds and halves. Step 4: Count From the Beginning to Find the Numerator Now shift to the top number of the fraction (the numerator). Point to a location on the number line and say: “To figure out the top number¹², we count from the beginning and stop right here.” Students count the tick marks from zero up to that point. If they stop at the third tick, the numerator is 3. If they stop at the sixth tick, the numerator is 6. If they stop at the twelfth tick, the numerator is 12.  This routine makes the numerator feel concrete: it’s simply how far along the number line you’ve travelled. Once students have grasped this pattern, this routine is easy to repeat with any fraction. What I particularly like about this introduction is how it immediately challenges the misconception that fractions are always less than one. And from fractions to decimals… From here, it becomes remarkably easy to use number lines to introduce decimals¹³. Of course, this only works if students are already familiar with number lines. This is why it’s crucial to lay the foundations¹⁴ early on! But wait; there’s more! Number lines are powerful across a huge range of mathematical ideas:
One last quack… What began as a short, whimsical ode to the power of number lines has somehow grown into over two thousand words exploring their importance. If we don’t teach students how to understand number lines early, we are doing them a massive disservice. Number lines give our students a way to organise ideas, see relationships and make sense of the abstract world of maths. And just like the growing parade of ducks¹⁵ in my kitchen, when students can confidently place numbers in order and understand the space between them, everything starts lining up beautifully. ¹They apparently bought a pack of 100, but when my daughter labelled each one discovered there were 102 in the pack
²At the time of writing, I’ve found 72, so only 30 left to find. ³I’m not a monster after all! ⁴By counting the numbers including 3 ⁵See Gelman & Gallistel’s The Child's Understanding of Number (1978) ⁶Or whichever type of stacking block is your personal favourite/your schools has invested in ⁷I was young, foolish & didn’t know better ⁸Check out chapter 4 in Siemon et al.’s Teaching Mathematics: Foundations to Middle Years (2021) ⁹I haven’t actually worked out whether more than 50% of scaled measuring instruments represent a number line, but a lot of them do ¹⁰Unless you’re the building in the centre of Castlemaine that seems to have a permanent scaffold around it ¹¹ a la Stein et al.’s Direct Instructions Mathematics 5th Edition (2018) ¹²I realise in my script I’m saying “top number” and not “numerator” (and likewise “bottom number” vs “denominator”). This is deliberate: I want to use the language that will give more students success with the mathematical content. Do I want students to be able to use the vocabulary of “denominator/numerator”? Absolutely! Is it critical that they learn it at this very moment? No. ¹³You’re welcome. ¹⁴Arguably, a good name for an educational consultancy… ¹⁵Surely 2000 words means that the ducks are now tax-deductible?
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I'm JamesI have been teaching for over a decade in Australia. I have worked as a classroom teacher, lead teacher, learning specialist, and principal. Archives
October 2026
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