Bar Model in Math – Definition with Examples
What is a bar model in math, types of bar models , bar model fractions, solved examples, practice problems, frequently asked questions.
A bar model is one such tool that helps us visualize the given math problem using different-sized rectangles or bars. The size of the bars are proportional to the numbers that it represents.
Let’s take up some bar model examples to understand them better.
In the given example, the green bar represents 8 apples, and the yellow bar represents 3 pears. And their combination represents the total number of fruits, that is, $8 + 3$.
We can represent other operations like subtraction, multiplication, and division using a bar model, too. We can also show a comparison of numbers using this model.
The different types of bar models include
Equal parts of a whole model, comparison model.
The part-part-whole model consists of two parts that together form a whole. This model can be used to show addition and subtraction.
The equal parts of a whole model consist of multiple equal parts that together form a whole. This model can be used to show multiplication and division.
The comparison model consists of two or more bars placed one below the other to compare the quantities.
Making Bar Models to Solve Word Problems
Bar model addition.
Sam has 5 red marbles and 4 green marbles. Find the total number of marbles with Sam.
In this example, we are given a number of red and green marbles, and we need to find the total number of marbles. This means that the parts are known and the whole is unknown. So, we add the parts to find the whole.
Bar Model Subtraction
Kate has 10 flowers, out of which 2 are roses and the rest are daisies. How many daisies does Kate have?
In this example, we are given a total number of flowers and the number of roses, and we need to find the number of daisies. This means the whole and one part are known, and the other part is unknown. So, we subtract the given part from the whole to find the other part using the subtraction bar model.
Multiplication Bar model
Beth has 4 bags of 8 candies each. How many candies does she have?
In this example, we have candies divided equally among 4 bags. We know the number of bags and the number of candies in each bag. So, we need to find the total number of candies.
According to the model, we know the number of equal parts and the size of each part. So, we multiply them to find the unknown whole.
Division Bar Model
Tim had 18 candies, which he distributed equally among 6 of his friends. How many candies did each friend get?
In this example, we know the total number of candies and the number of friends. That is, the whole and the number of equal parts is known. We need to find the number of candies each friend will get, that is, the size of each part is unknown. We divide the whole by the number of parts to get the answer.
Lily has 10 crayons. James has 2 more crayons than Lily. How many crayons does James have?
In this example, the number of crayons with Lily (Quantity A) and how many more crayons James has (difference) is known. We need to find the number of crayons with James (Quantity B).
Bar models can also be used to show fractions, as shown below:
Bar models are a great way to visualize math problems and understand operations like subtraction, addition, multiplication, and division. They can help us compare different amounts and understand parts of a whole.
They can make word problems much easier to solve. Let’s try it!
Example 1: Write an equation for the given bar model.
The model clearly shows that the parts are known, and the whole is unknown.
So the equation that represents this model is:
$8 + 12 =$ ? = Total Number of Candies
Example 2: Write an equation for the given bar model.
The model clearly shows that the total and one of the parts are known, while the second part is unknown.
$56 +$ girls $= 85$
$85 – 56 =$ ? $=$ girls
Example 3: What would be the bar model for this equation: $7 + 16 = 23$ ?
Solution : There are two parts, 7 and 16, that together make 23. So the model would be as follows:
Example 4: Grant has 40 fruit bars. Ken has 15 less fruit bars than Grant. How many fruit bars does Ken have?
Represent this situation using a bar model to find the answer.
We know Grant’s number of fruit bars and how many more bars he has than Ken. Clearly, this is a comparison model and can be represented as:
Number of fruit bars with Ken $=$ Number of fruit bars with Grant $– 15$
$= 40 – 15 = 25$ fruit bars
Example 5: Create a word problem for the given model.
Nina sells chocolate and vanilla ice cream cones. On a particular day, she sold 13 ice cream cones out of which 10 were vanilla. How many chocolate cones did she sell?
(Please note that there can be multiple answers to this problem.)
Bar Model in Math - Definition with Examples
Attend this quiz & Test your knowledge.
Which of the following equations correctly represents the given model?
Find the correct value for the missing number.
Alex has 84 berries. He wants to eat them over 7 days, eating the same number of berries daily. Use the bar model to find how many berries he should eat daily.
A book consists of 5 chapters. Each chapter has an equal number of pages. Katie has read the first chapter. How many pages has Katie read if the last three chapters are 90 pages altogether? Draw a bar model to find the answer.
Why do bar models have different-sized boxes?
Bar models have different-sized boxes because the boxes represent different values or quantities. The size of each part shows how much it is as a proportion of the whole.
How can we write an addition statement where the whole and one part in the bar model are known?
We can write it like this: $3 +$ ? $= 24$?
How do bar models help solve math problems?
Bar models help solve math problems because they show us a visual representation of the problem and help us understand the missing number. They also help us compare different numbers or quantities and see how they relate to each other.
Can we represent different forms of numbers such as fractions, percentages, and decimals using a bar model?
Yes, we can represent fractions, percentages, and decimals using bar models.
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The Ultimate Guide To The Bar Model: How To Teach It And Use It In KS1 And KS2
Teaching the bar model in maths and different bar modelling techniques in KS2 and KS1 is essential if you want pupils to do well in their reasoning and problem solving. Here’s your step by step guide to how to teach the bar model as part of maths mastery lessons from early years through KS1 SATs right up to KS2 SATs type questions.
First we will look at the four operations and a progression of bar model representations that can be applied across school in Key Stage 1 and Key Stage 2.
Then we will look at more complex bar model examples for KS2 SATs, including how to apply bar models to other concepts such as fraction and equations.
Finally, look out at the end of the article for some further bar model worksheets and free resources to get you started.
What is bar modelling in maths?
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What is a bar model ? In maths a bar model is a pictorial representation of a problem or concept where bars or boxes are used to represent the known and unknown quantities. Bar models are most often used to solve number problems with the four operations – addition and subtraction , multiplication and division.
In word problems, bar models help children decide which operations to use or visualise problems.
The bar model is central to maths mastery, the pictorial stage in the concrete pictorial abstract (CPA) approach to learning. Bar models will not, however, do the calculations for the pupil; they simply make it easier for pupils to work out which calculation must be done to solve the problem.
Bar modelling is the term used when you are teaching, learning or applying your bar models, and drawing out each bar to represent the known and unknown quantities. Encouraging a child to ‘bar model’ a problem can help them to understand conceptually what maths operation is required from the problem, and how each part combines to make the whole.
Here bar modelling shows us that if 2 rectangles out of 3 rectangles are green, then two thirds of 12 must be 8. An understanding of bar models is essential for teaching how to solve all sorts of word problems especially those using four operations (addition, subtraction, multiplication, division), fractions and algebra.
Bar modelling is much used in Singapore and Asian Maths textbooks and is an essential part of the mastery maths approach used by schools at all stages of the national curriculum.
By using the bar method to visualise problems, pupils are able to tackle any kind of number problem or complex word problem.
Because bar models only require pencils and paper, they are highly versatile and can come in very useful for tests, especially SATs Reasoning Papers .
However the use of bar models can begin much earlier, from showing number bonds to ten or partitioning numbers as part of your place value work.
Once a child is secure in their use of bar modelling for the four operations and can conceptualise its versatility, they can start to use it to visualise many other maths topics and problems, such as statistics and data handling .
‘The Singapore Maths Model’ is another name for the bar model method. Despite this name however the Singapore bar model (like most of the maths mastery approach) is based heavily on the work of Bruner, Dienes and Bishop about the best way to help children learn : teaching for mastery.
Bar models act as a ‘bridge’ between the concrete, pictorial and abstract (CPA in maths); once children are secure with using pictorial versions of their concrete materials, they can progress to using bars as visual representations.
Bars are a more abstract way of representing amounts, making the transition to using wholly abstract numbers significantly less difficult.
Also known as the ‘part part whole’ method or the comparison model, this kind of bar model uses rectangles to represent the known and unknown quantities as parts of a whole. This is an excellent method to help pupils represent the very common ‘missing number’ problems.
This can be done in two ways:
- As discrete parts to a whole – each unit in the problem has its own individual box, similar to using Numicon cubes.
- As continuous parts to a whole – units are grouped into one box for each amount in the problem e.g. in 26 + 52, 26 would have one long bar, not 26 smaller rectangles joined together.
When using the part-whole method, proportionality is key; all the bars must be roughly proportional to each other e.g. 6 should be about twice the length of 3. Often you’ll find this is referred to as the
Part-whole models are generally used to visually represent the four operations, fractions, measure, algebra and ratio, but can be applied to many more topics (as long as they’re relevant!).
Read more: What Is The Part Whole Model ?
[FREE] Let's Practise Bar Model Word Problems KS2
25 scaffolded bar model word problems on the four operations. Questions suitable for Year 3 to Year 6.
There are a few steps involved in drawing a bar model and using it to solve a problem:
- Read the question carefully
- Circle the important information
- Determine the variables: who? what?
- Make a plan for solving the problem: what operation needs to be used?
- Draw the unit bars based on the information
- Re-read the problem to make sure that the bar models match the information given
- Complete the calculation using the determined operation.
Bar models KS1
Pupils in Reception and Year 1 will routinely come across calculations such as 4+3.
Often, these calculations will be presented as word problems: Aliya has 4 oranges. Alfie has 3 oranges. How many oranges are there altogether? With addition, subtraction and multiplication, to help children fully understand later stages of bar modelling, it is crucial they begin with concrete representations.
There are 2 models that can be used to represent addition:
Once they are used to the format and able to represent word problems with models in this way themselves (assigning ‘labels’ verbally), the next stage is to replace the ‘real’ objects with objects that represent what is being discussed (in this case, we replace the ‘real’ oranges with button counters):
The next stage is to move away from the concrete to the pictorial. As with all the stages, when pupils are ready for the next stage is a judgement call that is best decided upon within your school.
A general rule of thumb would be that towards the end of Year 1 or start of Year 2, pupils should be able to understand and represent simple addition (and subtraction) word problems pictorially and assign written labels in a bar model.
The penultimate stage is to represent each object as part of a bar, in preparation for the final stage:
The final stage stops the 1:1 representation. Each quantity is represented approximately as a rectangular bar:
As mentioned before, it is a judgement call for your school to make, but if you want pupils to use the bar model to support them in end of Key Stage 1 SATs tests, they are going to need to have had a fair amount of experience of this final stage.
The same concrete to pictorial stages can be applied to subtraction. However, whereas with addition it is really down to the pupil’s preference as to which of the 2 bar representations to use, with subtraction the teacher can nudge to pupils to one or other.
The reason? One represents a ‘part-part-whole’ model, the other a ‘find the difference’ model. Each will be more suited to different word problems and different pupils. Let’s examine those at the final stage of bar modelling:
Austin has 18 lego bricks. He used 15 pieces to build a small car. How many pieces does he have left?
Calculation: 18 – 15 =
Find the difference
Austin has 18 lego bricks. Lionel has 3 lego bricks. How many more lego bricks does Austin have than Lionel?
Calculation: 18 – 3 =
Bar model multiplication starts with the same ‘real’ and ‘representative counters’ stages as addition and subtraction. Then moves to its final stage, drawing rectangular bars to represent each group:
Each box contains 5 cookies. Lionel buys 4 boxes. How many cookies does Lionel have?
Due to the complexity of division, it is recommended to remain grouping and sharing until the final stage of bar modelling is understood. Then word problems such as the 2 below can be introduced:
Grace has 27 lollies. She wants to share them into 9 party bags for her friends. How many lollies will go into each party bag?
Grace has 27 lollies for her party friends. She wants each friend to have 3 lollies. How many friends can she invite to her party?
Now that we have established a structure across school that allows for children to use bar models for KS1 SATs, we are now ready to teach pupils how to use the bar model for a deeper understanding of complex problems during Key Stage 2 and particularly in preparation for KS2 SATs.
The key question at any stage, at any age is what do we know? By training pupils to ask this when presented with word problems themselves, they quickly become independent at drawing bar models.
For example, in the problem: Egg boxes can hold 6 eggs. We need to fill 7 boxes. How many eggs will we need?
We know that there will be 7 egg boxes, so we know we can draw 7 rectangular bars. We know that each box holds 6 eggs, so we can write ‘6 eggs’ or ‘6’ in each of those 7 rectangular bar. We know we need to find the amount of eggs we have altogether. We can see we will need to use repeated addition or multiplication to solve the problem.
Bar models KS2
Bar modelling can be used to solve Year 6 word problems . Let’s ramp up the difficulty a little. In a sample KS2 SATs, pupils are asked:
A bag of 5 lemons costs £1. A bag of 4 oranges costs £1.80. How much more does one orange cost than one lemon?
Pupils could represent this problem in the below bar model, simply by asking and answering ‘what do we know?’ This can
From here it should be straightforward for the pupils to ‘see’ or visualise their next step. Namely, dividing £1.80 by 4 and £1 by 5. Some pupils will not need the bar model to represent the next stage, but if they do, they would calculate and then allocate the cost onto the model:
Then those pupils that needed this stage, should be able to see that to answer the question, they need to calculate 45p – 20p. With the answer of 25p.
Download more bar modelling questions: Let’s Practise Bar Models Four Operations
Here’s another example from the sample key stage 2 tests involving fractions and how it can be solved using a fraction bar model . On Saturday Lara read two fifths of her book. On Sunday, she read the other 90 pages to finish the book. How many pages are there in Lara’s book? If we create our bar model for what we know:
Pupils will then see that they can divide 90 by 3:
As fractions are ‘equal parts’ – a concept they should be familiar with from key stage 1 – they know that the other 2 fifths (Saturday’s reading) will be 30 pages each:
Then they can calculate 30 x 5 = 150
Download more bar modelling questions: Let’s Practise Bar Model Fractions
There are lots of other areas bar models can assist pupil’s understanding such as ratio, percentages and equations. In this final example, we look at how an equation can be demystified using the comparison model:
2a + 7 = a + 11
Let’s draw what we know in a comparison model, as we know both sides of the equation will equal the same total:
The bars showing 7 and 11 could have been a lot smaller or larger as we don’t know their relative value to ‘a’ at this stage. However, it is crucial that the ‘a’ appearing first in both bars is understood to be equal (even if it is only approximately equal when drawn freehand in the bar). This allows the pupil to ‘see’ that to work out the second ‘a’ in the top bar, they can calculate 11-7.
So if that ‘a’ is 4, then both the other ‘a’s will also be 4. So each side of the equation will total 15. The below model shows all sections completed. This is not necessary for the pupils to do, the representation is merely useful until they can see the steps necessary to calculate whatever they are faced with:
At the SATs level most maths problems require multiple steps to solve, and incorporate numerous mathematical concepts e.g. money, fractions, four operations etc.
While bar modelling can be used to represent all these steps at the same time, pupils may find it easier to identify each step and draw bars separately, forming the answer gradually.
While there are some multi-step word problems that cannot be solved using this method, bar modelling does make a significant difference in pupil’s ability to work through the SATs.
Now you’re persuaded (we hope!) that bar modelling in maths is going to revolutionise problem solving across your school from early years onwards, here are some other bar modelling resources to help you out.
- FREE Ultimate Guide to Bar Modelling (including a bar modelling ppt) which will give a structure that can be put in place across the whole school to enable teachers to teach pupils the bar model consistently.
- FREE KS2 Bar Model Worksheet with blank bar models to support children who need a bit more help on using bar models for word problems
- Lots of Bar Modelling Worksheets on our Maths Hub , including those that look at the Bar Model for Multiplication, Bar Model for Division, Bar Model for Subtraction etc
- Bar Model Training through the Bar Model CPD videos on our Maths Hub specifically about teaching bar models – Year 1 through to Year 6.
What are you waiting for? Get a staff meeting booked, and encourage your staff to calculate difficult problems using bar models to support. They’ll be sold instantly and will be racing to go out and teach bar models to their own KS1 and KS2 pupils.
If you don’t believe me, use this problem to grab their attention:
Hussain wins first prize for his spectacular cake version of the Eiffel Tower. He generously gives three fifths of his winnings to his children and spends a third of what he had left. He has £80 left. How much money did he win?
With or without bar models? Which is easier? My guess is your staff will be hooked!
More on bar modelling and maths mastery techniques
- Third Space Sample Lessons on Bar Modelling KS1 and KS2
- Benefits of following a mastery in maths approach
- Bar modelling applied to improper fractions
If you’re looking for more maths resources that help develop maths problem solving techniques, all the White Rose Maths lesson slides contain a mix of fluency, reasoning and problem solving work as well as following a structured CPA approach; and on the Third Space Maths Hub you’ll find resources like Rapid Reasoning (daily word problems), and plenty of worked examples.
Do you have pupils who need extra support in maths? Every week Third Space Learning’s maths specialist tutors support thousands of pupils across hundreds of schools with weekly online 1-to-1 lessons and maths interventions designed to address learning gaps and boost progress. Since 2013 we’ve helped over 150,000 primary and secondary school pupils become more confident, able mathematicians. Learn more or request a personalised quote for your school to speak to us about your school’s needs and how we can help.
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Bar Model in Math – Definition with Examples
Created: January 13, 2024
Last updated: January 13, 2024
Welcome to Brighterly , the platform dedicated to making learning a delightful and fruitful journey for every child. We believe in transforming complexity into simplicity, taking the abstract into the concrete, and making learning accessible for all. Today, we will dive into one of the most effective math strategies that embodies our philosophy – the Bar Model. Often an overlooked tool, the Bar Model is an excellent strategy used in mathematics to simplify and solve problems in a way that children can understand intuitively. With its origins from the renowned Singapore Math curriculum, the Bar Model uses bars or rectangles to represent numbers and their relationships. It’s like turning word problems into a picture that speaks a thousand numbers!
What is the Bar Model in Math?
The bar model in math is a visualization method that transforms abstract mathematical problems into visual, easily understandable situations. Used predominantly in primary education, it simplifies problems by using bars or rectangles to depict relationships between numbers or variables. This allows for an intuitive understanding of mathematical principles, especially beneficial when dealing with word problems, fractions, ratios, and proportions.
For instance, consider a word problem: John has 8 apples, and he eats 3. How many apples does John have left? With a bar model, we visualize the problem by creating a long bar representing the 8 apples John initially has. Then, we shade a shorter part of the bar to represent the 3 apples John eats. The remaining part of the bar shows the solution – the apples left with John.
Definition of the Bar Model
In simple terms, a bar model is a graphical depiction of a mathematical problem or concept, where bars or rectangles symbolize known and unknown quantities. Its objective is to break down complex problems into comprehensible units, aiding students to visualize the solution path.
Returning to our apple example, we understand that the bar model isn’t just a visualization tool. It’s a method to make the mathematical relationship between quantities more explicit. The lengths of the bars in the model represent the quantities involved in the problem – the whole bar (8 apples), and its parts (3 apples eaten, 5 apples remaining).
Use of the Bar Model in Mathematics
The bar model method, originating from Singapore Math, has seen global adoption due to its simplicity and effectiveness. It is a versatile tool applicable to a variety of mathematical problems involving part-whole relationships, comparison, and proportional reasoning. Beyond basic arithmetic, it can simplify understanding of fractions, percentages, ratios, and even algebraic expressions.
Let’s consider a proportion problem: In a class, the ratio of boys to girls is 2:3. If there are 20 boys, how many girls are there? By creating a bar model with 2 parts representing boys and 3 parts for girls, students can easily see that the number of girls in the class is 30.
Properties of the Bar Model in Math
The power of the bar model lies in its simplicity. The lengths of the bars signify the quantities involved in the problem, and their relationship conveys the mathematical link between these quantities. This property allows for the separation of known and unknown quantities, which aids in the formation of suitable equations for problem-solving.
Visual Representation of the Bar Model
The effectiveness of the bar model method is directly tied to its visual representation. In its simplest form, a bar model consists of two bars, each representing a part of a whole. Complex bar models may involve several bars of varying lengths, each standing for different quantities in a problem. The visual comparison between these bars aids in a deeper understanding of the relationships among the quantities involved.
Benefits of Using the Bar Model in Math
The bar model method offers several benefits. By presenting problems visually, it fosters understanding and reasoning. It promotes numerical literacy by reducing dependency on rote memorization. By aiding in the identification of the underlying structure of problems, it hones problem-solving skills.
Difference Between Bar Model and Other Mathematical Models
The significant distinction between the bar model and other mathematical models is its visual nature. Unlike other models that rely on symbols and equations, the bar model uses lengths of bars to denote quantities. This visualization process translates abstract concepts into tangible forms, thereby enhancing understanding and retention.
Equations and Problem Solving Using the Bar Model
The bar model simplifies the creation of equations and problem-solving. By visually presenting the problem, it aids in identifying the known and unknown quantities, guiding students towards the formation of correct equations.
Writing Equations Using the Bar Model
The process of writing an equation using the bar model involves three steps. First, identify the quantities from the problem and represent them as bars. Then, understand the relationship between the quantities and arrange the bars accordingly. Finally, replace the bars with numerical or algebraic expressions to form the equation.
Solving Mathematical Problems Using the Bar Model
To solve a problem using the bar model, the first step is to create a bar model that represents the problem. This is followed by formulating an equation based on the bar model. Lastly, the equation is solved to arrive at the solution.
Practice Problems on the Bar Model in Math
There are numerous practice problems available online and in textbooks that students can use to hone their problem-solving skills using the bar model. These problems cover a wide range of concepts, from basic arithmetic operations to more complex algebraic problems.
In a world where math can often seem intimidating, the Bar Model stands as a beacon, guiding young learners on their journey with numbers. At Brighterly, we strongly believe that visualization aids understanding, and this is precisely what the Bar Model provides. It transforms abstract mathematical concepts into tangible and understandable visuals. While the bar model may initially seem simple, it is a potent tool that can be adapted for a wide variety of mathematical problems – from basic arithmetic to more complex algebraic expressions. By practicing with the bar model, children can develop robust problem-solving skills, setting them up for success in their future mathematical endeavors. So let’s embrace the Bar Model, make math fun, and watch our young learners thrive!
Frequently Asked Questions on the Bar Model in Math
What kind of problems can be solved using the bar model.
The Bar Model can be used to solve a wide range of problems. It is particularly effective for problems involving part-whole relationships, comparison, and proportional reasoning. Additionally, it is a useful tool for understanding arithmetic operations, ratios, fractions, percentages, and even algebraic expressions.
Why is the Bar Model more effective than traditional methods?
The Bar Model offers a visual perspective to problem-solving, which can be more intuitive for many learners. Instead of abstract calculations, it represents problems in a concrete, visual manner. This allows learners to better understand the underlying structure of the problem, making it easier for them to devise a solution.
How can I help my child practice with the Bar Model?
At Brighterly, we offer a plethora of practice problems designed to enhance problem-solving skills using the Bar Model. These problems range from basic arithmetic operations to more complex algebraic problems. Guided practice, coupled with our interactive learning methods, can significantly improve your child’s proficiency with the Bar Model.
Is the Bar Model applicable only to primary grades?
While the Bar Model is often introduced in primary grades, it’s a versatile tool that can be applied at various educational levels. As mathematical problems become more complex, the bar model adapts, offering a visual representation of the problem that can simplify the process of finding a solution.
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What Is a Bar Model? How to Use This Maths Problem-Solving Method in Your Classroom
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The bar model is a powerful visual strategy for solving arithmetic problems, commonly used in Singapore Math and other mathematics curricula around the world. It helps students understand the relationship between numbers and the operations that connect them. But what exactly is a bar model, and how can it be effectively employed in the classroom to enhance students’ problem-solving skills?
A bar model is a pictorial representation of numerical situations or problems. It involves drawing bars or boxes to represent known and unknown quantities, making it easier to visualize complex arithmetic problems. The bars are drawn to scale to reflect the value of each amount, with longer bars representing larger amounts and shorter bars for smaller quantities.
To use this method in your class, introduce it step by step:
1. Start with simple addition and subtraction problems – Use physical objects at first (like blocks or counters) to help students see how numbers can be represented visually, then transition to drawing these representations as models.
2. Teach students to label their bar models – Each part of the bar or box should be labeled with the corresponding value or variable. For example, if students are working on a problem involving the total number of apples, each section of their model could be labeled with ‘apples’.
3. Progress to more complex problems – Once your students are comfortable with basic problems, introduce varied types such as multiplication, division, ratios, or even algebraic expressions and equations into bar modeling exercises.
4. Encourage creativity and flexibility – There is often more than one way to draw a model for a given problem. Encourage students to experiment with different models until they find one that helps them understand and solve the problem best.
5. Practice regularly – Include bar modeling in regular practice exercises so that students develop familiarity with this approach and gain confidence in using it.
6. Word Problems – Bar models thrive in breaking down word problems into digestible parts. Have students identify what they know, what they need to find out, and how the parts of the problem relate to each other before drawing their models.
In practice, here’s how you might use a bar model:
Suppose your objective is to solve a word problem like “John has 20 apples; he gives 5 to Mary. How many does he have left?” A bar would be drawn representing John’s 20 apples. Then a smaller section (representing the 5 apples he gives away) would be shaded or separated from the rest of this bar. The length of the unshaded part of the bar represents how many apples John has left.
The strength of teaching through bar models comes from its capacity to provide an external aid in understanding abstract concepts. Fluency with this technique allows students not only to answer textbook questions but also equips them with analytical tools that apply far beyond formal math problems—into everyday numeracy.
Integrating this tool into your math instruction encourages strategic thinking, deepens comprehension, and fosters an environment where mathematical discussion and exploration are part of everyday class activities.
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Bar modelling is an essential maths mastery strategy. A Singapore-style of maths model, bar modelling allows pupils to draw and visualize mathematical concepts to solve problems.
At a glance
- A versatile maths model strategy that can be used across a wide range of concepts and topics
- Gives pupils a powerful and adaptable strategy for solving increasingly difficult problems
- Allows pupils to understand on a conceptual level what occurs when using complex formulas (for example, algebra)
- Draws on the Concrete, Pictorial, Abstract approach
- Used extensively in Singapore-style maths mastery textbooks and workbooks
- Based on three pedagogical theorists — Jerome Bruner, Zoltan Dienes, and Alan J Bishop
Bar modelling and the CPA approach
The bar model method draws on the Concrete, Pictorial, Abstract (CPA) approach — an essential maths mastery concept. The process begins with pupils exploring problems via concrete objects. Pupils then progress to drawing pictorial diagrams, and then to abstract algorithms and notations (such as the +, -, x and / symbols).
The example below explains how bar modelling moves from concrete maths models to pictorial representations.
As shown, the bar method is primarily pictorial. Pupils will naturally develop from handling concrete objects, to drawing pictorial representations, to creating abstract rectangles to illustrate a problem. With time and practice, pupils will no longer need to draw individual boxes/units. Instead, they will label one long rectangle/bar with a number. At this stage, the bars will be somewhat proportional. So, in the example above, the purple bar representing 12 cookies is longer than the orange bar representing 8 cookies.
The lasting advantages of bar modelling
On one hand, the Singapore maths model method — bar modelling — provides pupils with a powerful tool for solving word problems. However, the lasting power of bar modelling is that once pupils master the approach, they can easily use bar models year after year across many maths topics. For example, bar modelling is an excellent technique (but not the only one!) for tackling ratio problems, volume problems, fractions, and more.
Importantly, bar modelling leads students down the path towards mathematical fluency and number sense. Maths models using concrete or pictorial rectangles allow pupils to understand complex formulas (for example, algebra) on an intuitive, conceptual level. Instead of simply following the steps of any given formula, students will possess a strong understanding of what is actually happening when applying or working with formulas.
The result? A stable, transferable, and solid mathematical framework for approaching abstract concepts. Combined with other [ Error: Link is empty → essential maths mastery strategies and concepts ← ] (https://mathsnoproblem.com/en/approach/number-bonds/), bar modelling sets students up for long-term maths success.
Find out how proven mastery strategies, world-class training, and step-by-step teacher guides can give you and your school a maths boost.
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The Bar Model
A representation used to expose mathematical structure
The bar model is used in teaching for mastery to help children to 'see' mathematical structure. It is not a method for solving problems, but a way of revealing the mathematical structure within a problem and gaining insight and clarity to help solve it. It supports the transformation of real-life problems into a mathematical form and can bridge the gap between concrete mathematical experiences and abstract representations. It should be preceded by and used in conjunction with a variety of representations, both concrete and pictorial, all of which contribute to children’s developing number sense. It can be used to represent problems involving the four operations, ratio and proportion. It is also useful for representing unknowns in a problem and as such can be a precursor to more symbolic algebra.
It is helpful to introduce children to the bar model as part of a sequence of learning so they can connect their understanding of the real world to this mathematical representation. In the NCETM Primary Professional Development Materials they are introduced in Year 1. The bar model can continue being used to expose mathematical structure throughout school maths and beyond. The NCETM Secondary Professional Development Materials contain a section on bar models for use in Key Stage 3 and beyond.
Addition and Subtraction
The bar model supports understanding of the relationship between addition and subtraction in that both can be seen within the one representation and viewed as different ways of looking at the same relationships.
This diagram encapsulates all of the following relationships;
a = b + c ; a = c + b ; a – b = c ; a – c = b
To prepare young children for the bar model it is a good idea to encourage them to line up objects in a linear arrangement when representing addition and subtraction problems.
Such arrangements will also help children to organise their counting. The physical objects can then be replaced, in time, with linking cubes and with a bar drawn next to it. The question can then be asked “what’s the same, what’s different?” to support the children in their reasoning and in making sense of the bar as an abstract representation of the physical objects. It is useful for children to work in pairs with one manipulating the cubes, while the other records by drawing the bars and then writing the number sentence underneath. The children can then swap roles.
Sam had 10 red marbles and 12 blue marbles. How many marbles did he have altogether?
In problems involving addition and subtraction there are three possible unknowns as illustrated below and given the value of two of them the third can be found.
The examples below illustrate a variety of ways that the bar might be used for addition and subtraction problems. A question mark is used to indicate the part that is unknown.
The model can be rearranged to demonstrate equivalence in a traditional layout
Pupils need to develop fluency in using this structure to represent addition and subtraction problems in a variety of contexts using the bar model. The model will help children to see that different problems share the same mathematical structure and can be visualised in the same way. Asking children to write their own problems, using the bar as the structure will help to consolidate this understanding.
Multiplication, Division, Fractions, and Ratio
All of these concepts involve proportional and multiplicative relationships and the bar model is particularly valuable for representing these types of problems and for making the connections between these concepts visible and accessible.
Notice how each section of the bars in the problem below has a value of 4 and not 1. This many-to-one correspondence, or unitising is important and occurs early, for example in the context of money, where one coin has a value of 2p for example. It is also a useful principle in the modelling of ratio problems.
Peter has 4 books Harry has five times as many books as Peter. How many books has Harry?
4 × 5 = 20 Harry has 20 books
When using the bar model for division it is the image of sharing rather than grouping which is highlighted in this representation.
Mr Smith had a piece of wood that measured 36 cm. He cut it into 6 equal pieces. How long was each piece?
36 ÷ 6 = 6 Each piece is 6 cm
Problems Involving Proportion
When modelling problems involving proportion it is useful to divide the bar into equal parts so that the proportional relationship and multiplicative structure are exposed.
Once the value of one part is labelled, the other parts can be identified as they are the same, for example in this KS2 SATS Question.
The bar model is valuable for all sorts of problems involving fractions. An initial step would be for pupils to appreciate the bar as a whole divided into equal pieces. The number of equal pieces that the bar is divided into is defined by the denominator. To represent thirds, I divide the bar into three equal pieces, to represent fifths I divide the bar into five equal pieces. A regular routine where pupils are required to find a fraction of a number by drawing and dividing a bar, using squared paper would be a valuable activity to embed both the procedure and the concept and develop fluency.
Find 1/5 of 30
The same image can be used to find 2/5 or 3/5 of 30 etc.
Finding the original cost of an item that has been reduced in a sale is one that pupils find particularly tricky. The ease at which such problems can be solved is demonstrated below:
A computer game is £24 in the sale. This is one quarter off its original price. How much did it cost before the sale?
The bar represents the original cost. It is divided into quarters to show the reduced cost of £24.
£24 ÷ 3 = £8, giving the value of three sections of the bar. The final section of the bar must also be £8, since it represents the same proportion as each of the other sections.
£8 × 4 = £32
The original cost of the computer game is £32
Problems involving percentages are solved in a similar way to those involving fractions. The key is to divide the whole into equal parts.
A computer game is reduced in a sale by 30%. Its reduced price is £77. How much was the original price?
Dividing the bar into ten equal pieces allows us to represent 30% and keep the other pieces the same size.
£77 ÷ 7 = £11
The original cost (the whole bar) is £11 × 10 = £110
Problems involving Ratio
The ratio problem outlined below is represented with double sided counters. These can be used as an alternative representation to drawing a bar. The advantage of this is that they can be moved around to represent a change in relationship.
Sam and Tom have football stickers in the ratio of 2 to 3. Altogether they have 25 stickers. If Sam gives half of his stickers to Tom, how many will Tom have?
Represent the ratio
Recognise that if together the counters have a value of 25 then one has a value of 5
Give half of Sam's stickers to Tom
Multiply 5 by 4 to give the total value of Tom's stickers
Notice how the many-to-one correspondence, as discussed above, allows this problem to be modelled efficiently. Children may at first use a one-to-one correspondence (i.e. each counter having a value of 1) and represent the problem using 25 counters, setting them out in a two to three ratio as illustrated below.
Children can then be supported in transferring their understanding to many-to-one correspondence as illustrated above (i.e. that every 5 counters in each column can be replaced with one counter worth 5). The flexibility of appreciating that the value of one counter can change depending on the context and the total quantity helps to develop the pupils’ algebraic reasoning.
Notice how the strategy explored above supports this KS2 SATS question
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Bar Model Method: In Problem Solving
This is the second unit of the special topic " The Bar Model Method ". This is a pedagogical strategy widely used in Singapore to help students solve word problems.
This unit Bar Models in Problem Solving , designed for students around Year 6, shows how to use the bar model method for handling multi-step problems involving four operations with whole numbers and fractions. The overall objective is to promote flexible thinking to solve non-routine tasks. Using a bar model provides a concrete approach for visualising the information in a mathematics problem and for organising it in a helpful way.
There are 8 lessons, which can be taught together or across the year.
Advice is given for using the unit with students who have not completed the first unit Introduction to Bar Model Method .
Read the Teachers' Guide (also included in the download) before using this resource.
Lesson 1: Part-whole Model - Division
In this lesson, students study multi-step problems which need a combination of whole number arithmetic operations for their solution. They make two different types of bar models for different division situations. They structure their problem solving by Polya’s four phases. Students then solve problems independently or in groups to consolidate their learning.
Lesson 2: Part-whole Model - Fractions
Students learn how to use the part-whole model to represent fractions in different real-world contexts. Solving the word problems is supported visually by the bar model and structured by Polya’s phases of problem solving. Students study examples and practise with further tasks.
Lesson 3: Part-whole Model - Fractions of Fractions
In this lesson, students encounter problems where the part-whole model is used to represent fractions, and fractions of those fractions. The bar models support an intuitive approach, building understanding of fractions and fraction calculations. Students study examples where either parts, parts of parts or the whole itself are to be found, and practise with similar tasks.
Lesson 4: Comparison Model - Whole Numbers
In this lesson, students use the comparison bar model to help solve multi-step word problems that include information about additive and multiplicative relationships between quantities. They study worked examples and practise with further tasks.
Lesson 5: Comparison Model - Fractions
In this lesson, students solve word problems that give information about fractions of different quantities that represent equal amounts. They use the comparison model for these problems, and solve them by identifying a common unit in the different quantities. Students see worked examples, then practise on several tasks.
Lesson 6: Stack Model – Whole Numbers
This lesson introduces the stack model. Students study problems where they compare two groups of quantities, with either a multiplicative or additive relationship given between individual items. Students work on two examples together, then practise independently or in pairs to consolidate learning.
Lesson 7: Change Model - Whole Numbers
In this lesson, students learn to use the change bar model, a variant on the comparison model. They construct visual representations of complex stories, usually involving multiple quantities that change over time, and use the representations to find a strategy for solving the problems. Students study examples, then practise the techniques on other tasks.
Mathematical Problem Solving - The Bar Model Method
A professional learning workbook on the key problem solving strategy used by global top performer, singapore, by scholastic staff , soo vei li , and liu yueh mei.
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Bar model method as a problem-solving heuristic: an investigation of two preservice teachers’ solution paths in problems involving ratio and percentage
- Original Article
- Published: 22 August 2022
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- Serife Sevinc ORCID: orcid.org/0000-0002-4561-9742 1 &
- Cheryl Lizano 2
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This study aimed to explore preservice elementary teachers’ use of a bar model as a heuristic for conceptualising relationships between quantities in situations involving ratio and percentages. As a part of a larger project, we focused on two preservice teachers, Maia and Jane, and investigated their solution paths in ratio and percentage problems and the role of the structural representations (i.e., bar model and percent line-bar model) in their solution paths. We analysed these focus participants’ written solutions and their think-aloud process during interviews using three coding phases with a constructivist grounded theory approach. We observed that the problem text fed the construction of the structural representations that served to make sense of the quantities and the relationships among them. Furthermore, the structural representations enhanced Maia and Jane’s mathematical insights about the problem situations. In addition, we observed that preservice teachers’ solution paths were neither linear nor similar, which indicates a mathematical richness in learning. In those solution paths, the model construction played a role in guiding the mathematical operations of constructing and operating on a unit rate in the ratio problem and identifying and operating on a referent whole unit in the percentage problem. We interpreted that variety and nonlinearity in solution paths suggested mathematical richness in learning, the educational implications of which were further discussed.
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Problem solver in this manuscript refers to an individual (at any age) involved in the process of solving a mathematical problem, irrespective of whether the problem is non-routine or procedural in nature. Therefore, we used the term in our review of the literature to refer to students engaged in problem-solving and throughout the manuscript to refer to the participants of the present study (i.e., the pre-service teachers) engaged in solving ratio and percentage problems.
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Mathematics and Science Education, Middle East Technical University, Ankara, Turkey
Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, USA
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Correspondence to Serife Sevinc .
All procedures followed in conducting the research reported on in the manuscript are consistent with the principles of research ethics outlined by the American Psychological Association. For ethical considerations, IRB approval was received from Indiana University Bloomington (IRB Protocol No. 1401241233). Based on the regulations of the IRB Approval, informed consent forms were received from the participants. Since the participants are adults, there was no need for parental consent forms. This research is our original work and has not been published or under revision elsewhere.
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Sevinc, S., Lizano, C. Bar model method as a problem-solving heuristic: an investigation of two preservice teachers’ solution paths in problems involving ratio and percentage. Math Ed Res J (2022). https://doi.org/10.1007/s13394-022-00427-9
Received : 08 March 2022
Revised : 15 June 2022
Accepted : 26 June 2022
Published : 22 August 2022
DOI : https://doi.org/10.1007/s13394-022-00427-9
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