What Is a Part Whole Model? Explained For Teachers, Parents And Kids

Here you can find out what the part-whole model is, why it is called the ‘part-whole model’, and how you can help children to understand the part-whole model as part of their math learning at home.

The part-whole model is a way of representing the relationship between a ‘whole’ and its component parts. The visual representation used early on when students learn about the part-whole model is known as a number bond (model template shown below), but a bar model and bar modeling can also be used to show the part-whole relationship of numbers.

Read more: What is a bar model

The act of splitting a number into component parts is known as partitioning. The part-whole model is a useful model for students to learn and utilize as it will assist with addition and subtraction later on in their mathematical journey. A fraction bar model and bar model multiplication and division will also be useful.

What is the part-whole model?

The part-whole model is a pictorial representation that shows the relationship between a whole and its parts. While usually constructed with just two parts, a whole number can be partitioned into as many component parts as a person may choose.

For example, 6 can be partitioned into 2 parts (4 & 2), 3 parts (3, 2 & 1) or even 7 parts (0.5, 0.5, 1, 1, 1, 1, 1).

The greater a student’s understanding of the part-whole diagram, the greater their understanding of the number system and their sense of numbers. With a greater sense of numbers, the more the students are likely to manipulate numbers for addition and subtraction problem solving and rely less on an algorithm.

Algorithms may provide them with little understanding of what is actually happening mathematically. An early introduction to gaining this number sense is through the part-whole model.

Part whole model examples

In the part-whole model below, you can see that the whole is 5 and one of the parts is 2. Before learners use abstract numbers, it is important to use objects that can represent cardinality (the amount in a group) as this will be easier for young children to manipulate.

With part-whole models, it is important to remember that the parts combine to make the whole. The parts and the whole should not be combined to create a new whole.

Rather, the point of the part-whole model is to look at what the other missing part would be when combined with the other part that makes the whole. To write this model as a number sentence it would be 2 + __ = 5 or 5 = 2 + __.

A clear use for the part-whole model, as demonstrated above, is for students to become familiar with their number bonds with certain numbers. The CCSS and TEKS for mathematics required students to become fluent in their number bonds to numbers within 20 by the end of first grade.

While part-whole models will not be the only way that learners become automatic at such procedures, it is one tool that will help them.

Part-whole reasoning is being able to understand that whole numbers can be partitioned into any number of component parts. This will then allow the learners to manipulate the order in which numbers may be added or subtracted from each other (what is called the associative property of addition) while still getting to the correct answer.

For example, solving 17 + 25 can be done utilizing the part-whole model in the following way:

A common use of the part-whole model is to partition numbers in accordance with the place value of each digit. This has been demonstrated above. The digit 2 in the tens place has been partitioned to represent 20, and the digit 1 in the tens place has been partitioned to represent 10. These can then be added together to get 30.

The digit 7 which represents the ones can be partitioned into 5 and 2 as this will make adding these to the ones from 25 easier as many learners know the number bond that 5 + 5 is equal to 10. This can be added to the 40 before adding the final 2 to get the answer 42.

While in the initial stages of learning, this can take some time and lead some people to wonder whether it would be better to use a more conventional method, such as the standard algorithm for addition or a number line.

With time, students will become automatic at the process and there will be no need to write out the models. They will be able to solve such questions more quickly than a person could write out the expression in the standard algorithm.

What is part-part-whole?

Depending on the scheme of work used by the schools to deliver their math curriculum, some may refer to ‘part-part-whole’. The part-part-whole model is merely a synonym for the part-whole model as discussed above. Personally, I prefer the use of part-whole as it describes the relationship that the model attempts to demonstrate.

Part-part-whole narrows students thinking in that there can only be two parts and a whole. This can limit students’ understanding that a whole can be partitioned into a number of parts.

When do children learn about the part-whole model in school?

The CCSS and TEKS do not specify that students must learn this particular model. Nonetheless, many lessons do include the part-whole model. It is usually introduced in kindergarten and continues to be used throughout elementary school.

From my experience, it is used less in upper elementary levels as it is assumed that students can partition numbers mentally. However, this is not always the case and students should feel free to continue using it to support this important skill.

It can also be applicable in complex multiplication and division problems, as well as multi-step word problems, making it a useful tool for this age group.

How does the part-whole model relate to other areas of math?

As a model, it can be linked to other areas of mathematics where subtraction or addition is required. It is a good stepping stone to building strong mental methods of addition and subtraction as it opens up opportunities for learners to see the associative property of addition.

The part-whole model also relates to mental math where partitioning a number may be useful, such as measure, shape, and addition and subtraction with whole numbers and decimals.

Part-whole models and number bonds can also be used to make mental math easier by applying the distributive property of multiplication over addition when multiplying multi-digit numbers. This practice in upper elementary levels will make factoring much easier in later years when variables are used.

This same method can be used to make mental math easier when dividing larger numbers by creating a number bond of the dividend into smaller parts that can be divided mentally.

How does the part-whole model link to real life?

The purpose of these models is to scaffold students’ understanding of numbers. It is therefore unnecessary for learners to actually use these models in the real world.

However, the learning and understanding of numbers that students will gain from using the models early on in their mathematics journey will be used when computing in the real world.

For example, when solving $5 + $7, one could partition 7 into 5 and 2, add the two 5s together to make 10, and then add the 2 to get $12. All that is possible through understanding the part-whole relationship of numbers.

Part-whole model worked examples

1. Partition this number into tens and ones.

We can see that the whole is 89 and the question has asked us to partition this into tens and ones. Those will be the two parts. The 8 in 89 represents the tens, and 8 tens are 80, so that is one part.

The 9 represents 9 ones which can be written as ‘9’. The missing number is 9.

2. Find the whole number.

We can see that the parts of the whole are 50 and 3, so the whole must be the combined total of those numbers. 50 + 3 is 53 so that is my whole.

3. Complete the partition.

The whole is 45 and one part of the whole is 15. We need to find out what the remaining part is. By subtracting the known part from the whole, we find the remaining part. 45 minus 15 is 30. So the remaining part is 30.

Part whole model questions

1. Partition 57 into tens and ones.

(Answer: 50 and 7)

2. Find the whole.

(Answer: 87)

3. Complete the partition.

(Answer: 50)

4. Complete the partition.

(Answer: 40)

Wondering about how to explain other key math vocabulary to your children? Check out our Math Dictionary For Kids And Parents, or try these other terms related to part-whole model:

• Concrete Math Resources
• What Are Number Bonds?
• What Is Partitioning?
• What Is Place Value?
• What Is The Sum?

Read more:

• Number Bonds
• Concrete Pictorial Abstract

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The content in this article was originally written by primary school lead teacher Neil Almond and has since been revised and adapted for US schools by elementary math teacher Jaclyn Wassell.