# The Power of Part/Whole Thinking

Part/whole thinking begins in Kindergarten when students are expected to learn the number combinations for the numbers through 10. Below you see a graphic representation for the combinations for 5. These are often called number bonds. This is a year-long process as students progress through the combinations for each number. As you can imagine, they don’t all progress at the same rate.

Knowing these combinations is critical in 1st and 2nd Grade as students are learning their basic facts and multi-digit addition and subtraction.

While part/whole thinking definitely leads to computational flexibility, it has some unexpected benefits as well.

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### Part/Whole Thinking and Word Problems

Now we know that students have trouble deciphering word problems, right? To complicate matters, students need to know all the different structures for math word problems. What’s the good news? Part/whole thinking can help students determine what operation to use to solve word problems.

Let’s look at this word problem:

Marianne had $20. Her grandmother gave her some money for her birthday. Now she has $28. How much money did Marianneโs grandmother give her?

So, we can see that it’s not a simpleย *result unknown* problem. By analyzing each number in the problem to determine if it is a part or the whole, we can use a simple part/whole diagram to represent the problem.

Be careful, however, about calling this a subtraction problem. You could certainly subtract 20 from 28 to find the money her grandmother gave her, but you could just as easily count up from 20 to 28. Allow for flexible strategies for solving problems.

### Part/Whole Thinking and Missing Part Problems

Consider the problem shown below.

I think you can see where the rigor comes from in this problem. Rather than knowing the length and width and being asked to find the perimeter, the perimeter and length are given and the width must be determined. In other words, this is a missing part problem.

Model drawing, which is based on part/whole thinking, can be used to represent this problem. Understanding that 22 is the whole, the model begins like this:

Assuming that students understand the basics of perimeterโtwo measures of the length and two of the widthโthe model can be extending like this:

Want more information on model drawing? Check out Char Forsten’s Step-by-Step Model Drawing: Solving Word Problems the Singapore Way.

These are just a few out-of-the-box ways that part/whole thinking can be used, and I hope it illustrates how powerful part/whole thinking can really be!

I’m wondering if the Number Combinations: Using Number Bonds to Develop Part/Whole Thinking only goes up to combinations of (5) or does it include combinations for 6, 7, 8, 9, and 10 also. I tried to send my question via the website and it wouldn’t allow me to type. Hopefully you will receive my question and reply before the sale is over.

Thanks!

Hey, Susanne! It has all the combinations from 3 through 10.

Love this thinking, and as a 3rd/4th grade teacher who uses area models (vs the perimeter image in this article) to talk about multiplication and division, I wonder about the vocabulary for multiplication / division is in this system of thinking (I am not very familiar with Singapore math, but curious!) I.e. instead of part/ whole, does it become factor/ multiple? What representations hold or need to change between the two sets of inverse operations? For example, how would you think about transitioning “number bonds” to multiplicative thinking? Thanks for any thoughts–I’ve been wondering a lot about how to connect my curriculum to my students’ prior learning, and managing any initial misconceptions.

With multiplication and division, the parts have specific meanings. One is the number of groups and the other is the number in each group. If you know the parts and are looking for the whole, it’s multiplication. If you know the whole and one of the parts, it’s division, although you can solve the problem by thinking about multiplication. For example, if the problem you’re solving is

Ms. Boucher had 24 roses. She put them equally into 4 vases. How many roses were in each vase?, we know the whole (24 roses) and one of the parts (the number of groups) and we’re looking for the other part (the number in each group). So we traditionally think of it as division, but I could certainly think 4 x 6 = 24, so there are 6 roses in each vase.