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Bar modeling is officially called the Model Method in Singapore, and is more commonly known as "tape diagrams" and "strip diagrams" in the U.S. It is a visual problem-solving technique that uses rectangular bars to represent quantities and the relationships between them in a word problem. Dr. Kho Tek Hong and his team developed it in Singapore in 1983, originally for 4th graders preparing for algebra, but it proved to be so effective that instruction of it began in first grade. It's widely considered the signature feature of Singapore's math curriculum.
Watch "Singapore math methodology: bar models (aka tape or strip diagrams) in action"
Why bar modeling exists
Bar modeling is one of three core methods behind Singapore math, alongside the spiral curriculum and the CPA approach. (See What Is Singapore Math? for the full framework.) Bar modeling sits inside the Concrete-Pictorial-Abstract (CPA) approach, acting as the visual bridge between physical manipulatives and abstract equations. (For the full breakdown, see What Is the CPA Approach?) Without a bar model, most curricula ask kids to jump straight from a word problem to an equation. This is a leap that many children struggle with because the words and the math don't automatically connect to each other in a child's head.
A bar model forces a student to slow down and represent what the problem is actually saying, sentence by sentence, before ever touching an equation. It makes the thinking process visible and visual.
According to Dr. Kho, bar modeling serves four specific purposes.
- Helping students plan out their problem-solving steps
- Strengthening conceptual understanding of ideas like fractions, ratios, and percentages
- Building algebraic thinking before formal algebra is introduced
- Giving students a way into challenging problems they might otherwise not attempt
The two basic types of bar models
Part-whole models
Part-whole models represent situations where a whole is made up of separate parts. That's the foundation for teaching number bonds, addition, and subtraction in early grades. In upper elementary, the same structure extends to multiplication, division, fractions, ratios, and percentages.
Addition, finding the whole
Allison has 3 cookies. Her brother Alan has 4 cookies. How many cookies do they have altogether?
A student draws one bar for Allison's 3 cookies and a longer bar underneath for Alan's 4 cookies (since 4 is bigger than 3), then places the question mark spanning both bars, because the problem is asking for the total. The equation follows the model, 3 + 4 = 7.
Subtraction, finding a part
Jacob has $10. He used $2 to buy a pen. How much money does he have left?
A student draws one bar for Jacob's $10, then splits it into two parts, a labeled $2 part for the pen and a question mark for what's left. Since the whole is $10 and one part is $2, the equation is 10 − 2 = 8. Jacob has $8 left.
A fraction of the whole
Ben spent a quarter of his money on a toy car. If the toy car costs $10, how much money did Ben have originally?
A student draws a bar split into 4 equal units, since Ben spent 1 out of 4 equal parts of his money. That 1 unit is labeled $10, the cost of the toy car. If 1 unit is $10, the whole (4 units) is 4 × 10 = 40. Ben had $40 originally.
Equal groups (division)
There are 100 chairs in 4 classrooms. A teacher wants to have an equal number of chairs in each classroom. How many chairs should be in each classroom?
A student draws one bar for the 100 chairs, split into 4 equal parts, one for each classroom. Since all 4 parts are equal, the equation is 100 ÷ 4 = 25. Each classroom should have 25 chairs.
Comparison models
Comparison models show the relationship, specifically the difference, between two quantities.
Addition comparison (more than)
Ben has 5 books. Jacob has 10 more books than Ben. How many books does Jacob have?
A student draws a bar for Ben's 5 books, then a longer bar for Jacob directly underneath, matching Ben's length plus an extra part worth 10 for how many more books Jacob has. The equation is 5 + 10 = 15. Jacob has 15 books.
Subtraction comparison (the difference)
Allison has 3 cookies. Her brother, Alan, has 20 cookies. How many more cookies does Alan have than Allison?
A student draws Allison's bar shorter (3) and Alan's bar longer (20), stacked directly underneath. The gap between the two bars is the difference the problem asks for. Reading it as a part-whole model, Alan's 20 is the whole, Allison's 3 is one part, and the gap is the missing part. So the operation is subtraction, 20 − 3 = 17. Alan has 17 more cookies than Allison.
Multiplication comparison, finding the larger amount
Jacob has 5 times as many books as Ben. If Ben has 5 books, how many books does Jacob have?
A student draws 1 bar for Ben's 5 books, then 5 equal bars for Jacob, since Jacob has 5 times as many. If 1 unit is 5, Jacob's 5 units equal 5 × 5 = 25. Jacob has 25 books.
Multiplication comparison, finding the smaller amount
Allison has 40 jellybeans. She has 5 times as much jellybeans as pieces of chocolate. How many pieces of chocolate does Allison have?
A student draws 1 bar for Allison's chocolate and 5 equal bars for her jellybeans, since she has 5 times as many jellybeans as pieces of chocolate. The 5 bars together equal 40, so 1 unit, the chocolate, is 40 ÷ 5 = 8. Allison has 8 pieces of chocolate.
Comparison bar models also quietly solve a classic word-problem trap. Kids often treat "how many more" and "how many fewer" as needing opposite operations, when the model shows both questions are asking for the exact same gap between two bars.
How bar modeling builds algebra readiness, without x or y
This is the part most people miss. Take this problem. A bookstore sold 3 times as many fiction books as non-fiction books. If it sold 480 books total, how many were fiction?
A student taught algebra would set non-fiction as x. That gives 3x + x = 480, x = 120, fiction = 360. Many students trip on that first step, because deciding what to call x is challenging.
A Singapore math student solves the same problem with bars, no variable required. Draw 1 bar for non-fiction, 3 equal bars for fiction. That's 4 equal bars totaling 480 books, so 1 bar = 120, and fiction (3 bars) = 360.
Compare the two solutions and the steps are identical. The bar-model student is doing algebra, just without the formal notation. When that same student encounters x and y for the first time in a few years, the leap isn't nearly as abstract, because they've been reasoning with "1 unit" the same way for years already. The bar is the variable. The student just hasn't named it yet. This kind of multi-step reasoning is exactly what shows up in Singapore's TIMSS and PISA scores. (See Why Singapore Ranks #1 in Math.)
Bar modeling and fractions
The same visual logic extends to fraction problems that trip up a lot of older students. Try this one. 2/3 of a class is wearing jeans. 1/5 of those kids are also wearing a jacket. What fraction of the class is wearing both?
Many students see the two fractions and just multiply or add on instinct, without reading carefully. A bar model forces a sentence-by-sentence read. Draw a bar for the whole class split into 3 parts, shade 2 for "wearing jeans." Then draw a second bar the same length as just the shaded section, split into 5 parts, shade 1 for "also wearing a jacket." Once both bars convert to a common number of equal pieces, the answer (2/15) falls out of the picture directly, no memorized fraction-multiplication rule required, and no chance of skipping straight to the wrong operation.
Why bar modeling is harder to teach than it looks
Bar models are the crown jewel of Singapore's math curriculum, and it's a secret weapon nobody else has quite mastered yet. Most kids can draw a bar model on command for an easy problem, one that looks like the worked example right above it. Fewer can look at an unfamiliar, multi-step problem and know how to apply it, deciding on their own what the bars should represent and how they connect. That's the actual skill, and it's the one most instruction never gets to. I've taught Singapore math for close to a decade, and I've met very few students who could do that second part without being shown how first.
Frequently Asked Questions
What is a bar model in math? A rectangular diagram used to represent the quantities and relationships in a word problem, developed in Singapore in 1983 by Dr. Kho Tek Hong and his team. It prolongs the pictorial phase of the Concrete-Pictorial-Abstract approach and functions as a visual bridge to algebraic reasoning.
Is a bar model the same thing as a tape diagram? Yes. "Tape diagram" and "strip diagram" are the terms many U.S. curricula use for the same tool. "Bar model" or "Model Method" is the original Singapore terminology.
What grade do kids start learning bar modeling? Singapore originally introduced it at 4th grade, but it proved so effective it was pushed down to first grade, where it's now typically introduced.
Is bar modeling the same as algebra? Bar modeling is the visual foundation to algebra because it builds the exact reasoning structure algebra requires, which is why students who learn bar modeling early often find formal algebra far less abstract when they get there.
Do I need special tools to teach bar modeling at home? No, pencil and paper is enough to get started. For a faster and more accurate way to draw bar models, Wenxi uses and recommends this bar model ruler that she used as a student in Singapore.
Why do kids still have to draw a bar model if they already know the answer? Bar modeling isn't primarily about solving for the answer on simple problems. It's building the visual and proportional reasoning that makes harder problems solvable later. A problem like "6 trays of 8 cookies each, how many cookies total?" gets modeled as 6 equal parts labeled 8, with the reasoning "if 1 unit is 8, what are 6 units?" That unit-based language is the exact vocabulary students will need for ratios, fractions, and percentages down the line. Skipping the model on easy problems skips the rehearsal that makes the hard ones possible.
Why do kids who know bar modeling still get stuck on hard problems? Because drawing a bar model for a familiar problem and knowing how to apply one to an unfamiliar problem are different skills. Most instruction only teaches the first.
Written by Wenxi Lee, founder of Singapore Math Circle. Wenxi grew up in Singapore, learning bar modeling from first grade as part of the first generation of Singaporeans taught the method from the start of elementary school. She holds a PhD in math education from the University of Illinois at Chicago.