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How do I teach math word problems online? (Or: how to use bar models and tape diagrams on a shared whiteboard so the student can see what the problem is asking)

Teaching math word problems online works when the student can draw the problem on a shared whiteboard — not just talk through it — before they solve it. In practice that means using a bar model (also called a tape diagram) as the pictorial bridge between the English sentence and the arithmetic: place a rectangle for each known quantity, stretch each one to its proportional length, label the parts, mark the unknown with a "?" and an arrow, and only then write the equation next to it. On Koala Go, the Rectangle shape on Pro is the bar-model primitive (it resizes freely in both dimensions), the Math Symbols palette writes the equation on the same canvas, and — for younger students working with small whole-number quantities — the coloured Number Rods are the discrete-length variant. Fraction Bars on Platinum handle fraction word problems. The whole point is the same as it is in person with a pencil and paper: the student stops guessing at operations and starts seeing what the problem is asking, because the picture makes the structure of the problem visible.

Why math word problems are the hardest thing to teach on a webcam

Every math tutor who's worked online with a student stuck on word problems knows the shape of the failure: the student can do the arithmetic when it's presented as "27 + 15 = ?" and freezes when the same numbers arrive inside a sentence — "Alex has 27 stickers. She gets 15 more for her birthday. How many stickers does she have now?" The instinct is that the child "isn't a word-problem person", or "just needs more practice", or "hasn't read carefully enough". Occasionally that's true. Much more often the block is in translation: the student can compute, but they can't figure out which operation to use because the problem's structure isn't visible to them. In an in-person lesson you'd reach for a pencil, sketch two boxes side by side, put "27" in one and "15" in the other, draw an arrow with a question mark, and the "oh — I add them" moment lands in ten seconds. On a video call with a screen share, that same sketch happens on your side of the screen and the student watches. They see the answer without doing the work to see the structure themselves.

Three problems compound to make this specifically hard remotely, and the online workflow has to answer all three:

  1. Word problems are read differently from prose. A word problem is not a paragraph you read for meaning; it's a set of relationships you have to extract from a paragraph. Skilled solvers routinely re-read a word problem three times — once for what's happening, once for the numbers, once for the question — and mark it up as they go. On paper, the student annotates the sheet: underlining the numbers, circling the question, crossing out red herrings. On a webcam, if the problem is inside a PDF the student opens themselves and you can't see, none of that mark-up is happening on your surface. You're teaching to a black box.
  2. The structure of a word problem is spatial. "Alex has twice as many stickers as Maya" is a comparison between two quantities of different sizes. That relationship is naturally visual — two rectangles side by side, one twice as long as the other. Text can't carry that meaning as directly as a picture can. If your whiteboard forces you to type the relationship as a sentence, you're translating twice — from the problem's English to your typed English, and from your typed English to the student's mental model. Bar models cut out the middle translation.
  3. Word problems trigger the strongest math anxiety. More students freeze on word problems than on any other kind of math task. The freeze isn't about the math; it's about a paragraph of English that they know they're going to be judged on. Online, the freeze is worse — the student is in their bedroom with a webcam pointed at them and no easy way to hide "I don't know where to start". The workflow has to make it safe to begin before anyone gets to a correct answer. A bar model with the first bar wrong is still a start. A blank whiteboard is not.

An online word-problem workflow that works has an answer to each of the three: it puts the problem on a shared surface both people can annotate; it makes the structure visible through a bar model rather than through a sentence; and it gives the student a first move (place one bar, stretch it, label it) that isn't the same thing as committing to the final answer. The rest of this page is that workflow.

The move: bar models (tape diagrams) on the shared whiteboard

The bar model is the single technique that carries most word-problem teaching from about grade 2 through pre-algebra. It came out of Singapore's Primary Mathematics curriculum in the 1980s and now shows up under different names in Beast Academy (Art of Problem Solving), Eureka Math / EngageNY ("tape diagrams"), Maths — No Problem! (UK), Math in Focus (US), and most other mainstream problem-solving curricula. The technique is deceptively simple: for every quantity in the problem, draw a rectangle whose length is proportional to that quantity; for every relationship between quantities, put the rectangles next to each other so the relationship is visible; mark the unknown with a "?"; then read off the equation from the picture. It's the pictorial step of the concrete-pictorial-abstract sequence — students who can do the picture almost always can do the equation, and students who can't do the picture usually haven't understood the problem.

Koala Go's whiteboard ships the bar-model primitives inside the Whiteboard Library. What matters for word problems, before any workflow talk:

  • The Rectangle shape (Pro tier) is the bar-model workhorse. On the Pro plan (currently $25.99/month monthly, $21.99/month annual) you get a rectangle you can drop on the canvas and stretch to any width and any height independently — that's the "flexible" resize behaviour, contrasted with the square (which is proportional-only). So you can place one rectangle, drag its right edge until it looks the right length for "27 stickers", place a second rectangle underneath, stretch it to about 15/27 of the first's length, and the child can see the ratio in the picture before you write a number down. Because the rectangle is a first-class whiteboard object, both you AND the student can grab, resize, and move it. Which matters — the student doing the placement is where the understanding lives; a demonstration is not the same thing.
  • Number Rods (Pro tier) are the small-whole-number variant. When the quantities in the problem are small (typically ≤ 10) and the child is still building the concept, the 10 pre-sized coloured rods — values 1 through 10 — are often more useful than free rectangles, because the sizes are already exact and the student can count units directly. "Sam has 3 apples, Ana has twice as many" is one 3-rod above two 3-rods (or one 3-rod above a 6-rod), and the picture answers the question without any arithmetic. Rods are what you reach for in grades 2 through 4 with numbers that fit; rectangles are what you reach for beyond that.
  • Fraction Bars (Platinum tier) handle fraction word problems. "3/4 of the class went on the trip; 12 children stayed behind — how many went?" is a fraction-of-a-whole problem, and the model is the same shape as a whole-number bar model but the fractional pieces are placed one at a time. On Platinum ($49.99/month monthly, $39.99/month annual) you get individual 1/n bar segments for denominators 1/2 through 1/12 plus 1/16 — the same set the fractions spoke walks through in depth. See how do I teach fractions online? for the fraction-manipulative side; this spoke describes how those pieces show up specifically inside a bar-model word problem.
  • Shape Labels (Platinum tier) name each bar with a rescaling label. A "boys" label at the top of one bar, a "girls" label at the top of the second, and a "?" label above the difference — all four corner-anchored so they move and rescale with the bar when you stretch it. On Pro, the workaround is Sticky notes placed next to each bar (which drift if you resize the bar afterwards); most working tutors on Pro live with the sticky-note workaround for a while and upgrade to Platinum when they start doing multi-part bar models routinely.
  • Straight lines and arrows (Pro tier) are the "curly-brace" and "unknown" marks. A dashed line under a section of the bar with a "?" sticky above it marks the unknown quantity; an arrow from one bar to another with a "= 3 × ?" sticky above it marks a multiplicative relationship. Both are Pro-tier line primitives on the same canvas — the lines themselves don't take Shape Labels, so the annotation next to them is a sticky note or a text box. These small marks are what turn a bar-model diagram into a language you can read off — without them, two rectangles are just two rectangles.
  • The Math Symbols palette writes the equation on the same canvas. Once the picture is finished, the equation goes right next to it. The base Symbols set (Pro) covers the four operations, equals, and the basic "less than" / "greater than" comparisons — everything a pre-algebraic word problem needs. Advanced symbols (² ³ π ± infinity, ≤ ≥ ≠ ≈) sit on Platinum and are what you reach for once the natural next step becomes "let x = ...", which is where this page hands off to how do I teach algebra online?

Because all of these primitives live on the same shared whiteboard, the picture and the equation live in the same room — not in a slide-plus-Doc split that adds a cognitive tax every time the student's eyes move. That single property is most of the difference between "let me share my screen and we'll go through a problem" and "let's draw this together and figure out what it's asking".

The five bar-model shapes that carry most word problems

Working online math tutors typically converge on five bar-model shapes that between them handle most of the word problems in elementary and middle-school curricula. Learning to recognise which shape a problem needs is the highest-leverage move a tutor can teach — the shape names below are the ones used across Singapore's Model Method, Beast Academy, and most Eureka-adjacent curricula, so they'll be familiar if the student is already working in one of those systems.

  1. Part-whole (single-quantity addition and subtraction). One bar divided into two labelled parts; the whole is either known or the unknown. "Alex has 27 stickers, then gets 15 more" is one bar of length 27, a second joined to it of length 15, and a "?" over the total. "Maya had 40 marbles, gave some away, and has 24 left" is one bar of length 40, split into a known part (24) and an unknown part ("?"). This is the earliest bar-model shape and the one to start with in grade 2 or 3 — students who own this can usually generalise the others in a few lessons.
  2. Comparison (two quantities of different sizes). Two bars side by side; the difference between them is the answer, or the ratio between them is the answer. "Alex has 27 stickers. Maya has 12 more than Alex. How many does Maya have?" is one bar of length 27 with a second bar underneath that extends 12 beyond it, both labelled, the "?" over Maya's total. "Sam has twice as many pencils as Jamie" is one bar under a bar of twice the length. The comparison shape unlocks most of the elementary "how many more" / "how many fewer" / "how many times as many" problems.
  3. Before-and-after (change over time). Two rows of bars, one showing the state before and one showing the state after. "Jamie had 60 baseball cards. She gave 1/3 of them to her sister. How many does she have left?" — top row: one bar of length 60, split into three equal parts (three thirds); bottom row: two parts remaining, one part labelled "?". The before-and-after shape is what makes "some were removed" and "then more arrived" problems traceable. Younger students often need to draw both rows before the shift makes sense.
  4. Fraction-of-whole (fractional bar-model). One bar representing the whole, subdivided into fractional parts, with either the whole known and a fractional part unknown or vice versa. "3/4 of the class went on the trip; 12 stayed behind" is one bar of length "24 children" (student figures this out) split into four quarters, one quarter labelled 12 and three quarters labelled "?". The fraction bar-model is where the fraction spoke and this spoke intersect — the drag-and-drop 1/n fraction bars on Platinum are the tightest version, though a Pro tutor can achieve the same result by hand-drawing a rectangle and using sticky-note fractional labels. See how do I teach fractions online? for the fraction manipulative deep-dive.
  5. Ratio (two related bars with proportional units). Two bars whose lengths are in a stated ratio. "In a bag of 24 sweets, the ratio of red to yellow is 3:5. How many are red?" is one bar of length 24 split into 8 equal units, 3 shaded as red, 5 shaded as yellow. Ratio is the last of the five and usually shows up in grades 5 through 8; it's the bar-model shape that most directly bridges into algebra.

The fastest way to build a student's bar-model fluency is not to teach all five at once — it's to run one shape for two or three weeks until the student can draw it without prompting, then introduce the next shape when the student meets a problem the previous shape doesn't fit. Part-whole first, then comparison, then before-and-after, then fraction-of-whole, then ratio, is the order most curricula use in some form.

A working lesson: from the word problem to the answer, on the shared canvas

What the workflow looks like in a real 30 to 45-minute session. This is not a script — adapt to the student and the problem — but the shape is broadly what most working online math tutors run once they've been doing bar-modelling with a student for a few weeks.

  1. Put the problem where you can both mark it up (2 min). If the parent sent a worksheet, upload the PDF onto the whiteboard on Pro — it becomes a page both you and the student can annotate with the pen, sticky notes, and shapes. If the problem is from a textbook or a photo, drop it in as an image. If the problem is one you're inventing on the fly, type it in a sticky note at the top. What matters is that the problem lives on the same canvas as the bar model you're about to build — not in a link the student opens elsewhere.
  2. Read the problem twice, out loud, together (2 min). First read, both of you read it silently at your own pace. Second read, the student reads it out loud. This is the reading-strategy step and it matters more than tutors typically credit — students who freeze on word problems often haven't finished reading them; forcing a slow spoken read anchors the language. As the student reads, underline the numbers with the pen tool and circle the question.
  3. Name what's known and what's unknown (2 min). Ask the student: "What are the quantities in this problem?" and "What's the question asking us to find?" Take their answers verbatim onto a sticky note ("Alex: 27 stickers. Maya: ? stickers. Difference: 15 stickers."). This is the moment you catch a misread — if the student names quantities that aren't in the problem, or names the wrong unknown, stop and re-read. Fixing the misread here saves the next fifteen minutes.
  4. Choose the bar-model shape (2 min). Ask: "Does this problem have one quantity, or two? If two, is one bigger and we care about the difference (comparison), or do we care about the total (part-whole)? Or is the same quantity changing over time (before-and-after)?" For the first few weeks of bar-modelling, walk the student through this choice; after a few weeks, they'll start naming the shape themselves. The bar-model shape is the abstraction the student is really learning; the arithmetic is downstream.
  5. Draw the picture together (5 min). Place the rectangles or rods on the canvas. Stretch the first to roughly the right length; place the second and stretch it to a proportional length. Add the labels — Shape Labels on Platinum stick to the bar corners; sticky notes on Pro sit next to each bar. Add the "?" over the unknown and an arrow to the quantity that answers it. The student should be doing at least half of this — hand them the pen or the shape tool and let them place the second bar. If they place it wrong (too short, too long, in the wrong place), leave it and ask "does that look right compared to the first bar?" — the correction is the learning.
  6. Read the equation off the picture (3 min). Once the diagram is complete, the equation is usually visible in the shape. Two bars totaling a "?" means addition — write "27 + 15 = ?" on the canvas using the Math Symbols palette. Two bars where the difference is the "?" means subtraction. Ask the student to write the equation next to the picture (not you). This is the pictorial-to-abstract step of the CPA sequence, and it's the step that most naturally transfers to problems the student will meet without you.
  7. Solve the arithmetic (2 min). Almost always the shortest step — the student has already done the hard work of translating. If the arithmetic itself is a struggle (times-tables not automatic, addition-with-carrying wobbly), you've surfaced a separate gap; note it and route into the multiplication spoke or the number-sense spoke in a later lesson.
  8. Check by reading back to the problem (2 min). "So the answer is 42. Does that make sense — Maya has 42 stickers, which is 15 more than Alex's 27?" Yes / no + why. Answer-checking against the story is often skipped and always worth doing.
  9. Repeat with a variation (10-15 min). Same shape, different numbers. Same numbers, different question. Fresh problem of the same shape. The bar model is a template the student is grooving; three or four problems of the same shape in one session is what turns "we did that with the tutor" into "I know how to do these".

A single 30-minute session usually gets through 3-5 problems of the same bar-model shape at this pace. That's the target — not "how many problems can we get through" but "did the student's bar model for the third problem look better than the first". Fluency comes from repetition of the shape, not from a broad tour of shapes.

Where practice sites fit: the cobrowser for Beast Academy, Khan Academy, IXL, NRICH

Between lessons, the student needs word-problem practice they can actually complete without the tutor sitting next to them. Sending a worksheet home works, but the highest-leverage move is to open a curated problem-set inside the lesson using the cobrowser — a shared browser window where both you and the student can click, type, and interact with the same page — and finish the session with a few problems the student attempts semi-independently while you watch. Well-shaped problem sets for bar-model practice, in rough order of what most tutors reach for:

  • Beast Academy (Art of Problem Solving) — problem-solving-heavy curriculum from grades 2 through 5, with word problems in every chapter designed to be attacked with a bar model. Their problem sets are especially good for the "novel problem that doesn't fit the exact template" case.
  • Khan Academy — free, broad coverage, hints available. Their word-problem practice is calibrated for typical grade-band expectations and works well as spaced-practice homework. Search the site for the specific concept ("multi-step word problems 4th grade"; "fractions word problems").
  • IXL — subscription, but very deep problem-set library across every US state's curriculum. Good for parents who already have an IXL subscription. IXL's adaptive difficulty is a mixed blessing for bar-model practice — the student can slide into problem types the bar model doesn't fit; steer them back to a specific topic strand rather than free-choice work.
  • NRICH (Cambridge) — free, world-class problem collection. Older-student oriented (upper elementary and up). Good for the student who's fluent with the standard shapes and wants harder problems.
  • Bedtime Math and Illustrative Mathematics tasks — additional free sources for specific shapes.

Open the practice site in the cobrowser rather than as a link the student opens themselves. The cobrowser gives you actual shared control (not screen sharing) — you see exactly what the student clicks, why they got it wrong, and can jump in without them switching windows. On Koala Free the cobrowser is capped at 10 minutes per session with a 20-minute cool-down; on Pro it's unlimited. See what is a cobrowser? for the mechanic in depth and how do I give homework to online tutoring students? for the between-lesson practice cadence — the bar-model habit is one of the highest-ROI homework asks in elementary math.

Common failure modes and what to do about them

Six patterns that show up regularly with word-problem tutoring online, and the workflow adjustment for each.

  • The student draws the bar model correctly and still writes the wrong equation. The picture-to-equation gap. Usually the fix is one more round of "what does the picture say?" — literally reading each bar out loud and pointing at it: "This bar is 27. This bar is 15. The two together are what?" Sometimes the student can talk their way to the answer even though they can't write the equation; that's still progress. Write it for them once, ask them to write it themselves the next problem, and let the habit build.
  • The student can't decide which bar-model shape to use. Common in weeks 2-4. Fix: for a few sessions, tell them the shape ("This one's comparison — two bars of different lengths") rather than asking them to name it. The naming skill lags the drawing skill by a few weeks; that's fine. Once they've drawn twenty part-whole models and twenty comparison models, they'll start recognising the shape from the language.
  • The student stretches every bar to the same length regardless of the numbers. A common short-cut. Fix: place the "1" rod next to their bar and ask "how many of these fit inside your bar?" — the numerical mismatch becomes visible. For older students, add tick marks to the bar (labelled units) so the length constraint is explicit.
  • The student gets the bar-model but freezes when the problem has more than one step. Multi-step word problems are their own harder skill. Fix: draw the bar model in two stages ("first, let's answer the sub-question this problem needs before we can get to the real question"), and use the before-and-after shape (top row = state at step one; bottom row = state at step two). The two-row layout makes multi-step visible.
  • The student's block is in reading the problem, not in solving it. If the student reads "twice as many" as "two", or reads "combined" as an operation they don't recognise, or can't decode a word in the problem at all, you may have surfaced a reading rather than a math gap. Slow down; read the problem out loud with them; look up the confusing words together. If the pattern is persistent — the student reads the numbers fluently but can't parse the sentences — cross-link to how do I tutor a student with dyslexia online? for the reading-side workflow and consider suggesting the family talk to a reading specialist. Reading fluency is upstream of word-problem fluency.
  • The student's block is persistent and structural — even simple bar models don't stick after weeks of practice. If magnitude comparison stays unreliable, number-symbol mapping is confused, or the bar model keeps getting drawn without any grasp of what it represents, you may be seeing a pattern that could suggest a math learning difference. Read how do I tutor a student with dyscalculia online? for the SEN-specific workflow, and — separately — suggest the family talk to an educational psychologist or the school's SEN coordinator for actual assessment. A tutor's observation of a pattern is not a diagnosis; a qualified specialist diagnoses.

When to hand off to algebra: the "let x = " moment

Bar-modelling scales up remarkably well — Singapore's curriculum runs bar models through pre-algebra, and there are respectable versions of quadratic and simultaneous-equation bar models. But at some point in middle school, the natural next step becomes "let x = the number of pencils" and the picture stops being the primary tool. Signs the student is ready for the algebra hand-off:

  • The student is drawing the bar model in their head and getting to the equation without drawing.
  • Problems introduce quantities that need multiple operations combined ("If she buys 4 pencils and 3 notebooks and spends $23, and each notebook costs $2 more than a pencil…").
  • Problems introduce a variable in more than one place ("A rectangle has length twice its width; the perimeter is 30cm…").
  • The student is comfortable naming unknowns with letters rather than "?" boxes.

At that point, the workflow shifts: the bar model becomes a supporting scaffold ("let's draw the picture first, then write x = the unknown quantity"), and the algebra takes over as the primary technique. The algebra spoke covers this in depth — see how do I teach algebra online? for the balance-scale equation-solving workflow, the area-model for expanding products, and the Desmos cobrowser pairing for graphing. Bar-modelling is what gets a student ready for algebra; algebra is what they need once they arrive.

Honest limits

Four limits worth naming plainly rather than papering over.

  • Koala's whiteboard has no automatic bar-model generator. The tutor draws each rectangle by hand and stretches it to the right proportion. There are curriculum-specific tools (Thinking Blocks, some Beast Academy in-book apps) that let you type "27 and 15" and produce a bar model automatically; Koala doesn't. In practice most working tutors prefer hand-drawing because the manual step is where the student's participation lives — but if you want a click-to-generate diagram, the whiteboard isn't that.
  • Shape Labels are Platinum-only. If you're on Koala Pro ($25.99/month monthly, $21.99/month annual) and want your labels to rescale with the bar, you'll need to upgrade to Platinum ($49.99/month monthly, $39.99/month annual). On Pro, sticky notes next to each bar work as the label workaround — they don't drift much if you finalise the bar dimensions first and place the stickies after, and multi-part bar models can absolutely be run on Pro.
  • iPad is rougher than desktop for the drag-and-stretch gesture. The bar-model workflow depends on precise resizing of small rectangles, which is easier with a mouse than with a fingertip on an iPad. Students on iPads can still do bar models, but they often benefit from a stylus or from a slightly larger placement. On a desktop or laptop with a mouse, the gesture is smooth and quick.
  • The Playground and the 3D environment aren't the teaching surface here. Word problems live on the shared whiteboard; the Playground is where younger students take a 3-minute movement break between problems if focus is flagging. See what is a gamified virtual classroom? for the engagement layer around a math lesson without confusing it for the teaching surface.

Two implementation notes: (a) if you'd like to record a strong bar-model session for your own notes or to share a short clip with the family (a parent watching their child correctly draw a comparison bar for a problem they'd been stuck on is worth an hour of parent-conference explanation), see how do I record online tutoring lessons? — including the family-permission step; (b) if the student is in mainland China, the whiteboard, shape library, and Math Tools all work through our Hong Kong proxy, but cobrowser reliability into US-hosted practice sites can vary — see the China-specific answer for the workaround pattern.

Where word-problem teaching fits in the wider math cluster

Bar-modelling is one of several skills a working online math tutor teaches. The neighbouring pages that cover an adjacent question:

If you'd like a walk-through of the bar-model surfaces for the specific age band, curriculum, or word-problem type you're teaching, open Koala Go at classroom.teachwithkoala.com and try one of the shape types above on the whiteboard, or write to koala@teachwithkoala.com with a sentence about the student and the problem set you're working with, and we'll give you our honest read on the fit — including the parts of your specific curriculum (Beast Academy, Singapore Math, Eureka Math, or something else) where the bar-model on a shared whiteboard is the right primary tool and the parts where a subject-specific practice site in the cobrowser is doing more of the work.

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