Going Deeper with Visual Models: Part-Whole Multiplication/Division

Oct 8, 2020

Have you tried visual models yet? Ready for more? Let’s bring in the rest of the Part-Whole family of word problems in today’s blog!

Watch this first!

Prerequisite:

If you haven’t watched our Visual Model Basics video yet, stop – right now! – and go watch it! It will explain the thorough and effective step-by-step process for creating visual models that we use, and it will help you understand why we use it to help students comprehend word problems.

This group of word problems earned the family name – Part-Whole Problems – because we’re taking a piece and subtracting from it, or adding to it, or trying to find the missing part. After students have a grasp on part-whole addition, subtraction, and multi-step problems, we also want them to understand part-whole multiplication and part-whole division, which sometimes look a little different than our regular part-whole operations.

In addition and subtraction, we use a non-proportional bar to represent an amount. If it’s a larger number, like 54, it would be a larger bar. For a smaller number like 24, it would be a smaller bar.  When we move into part-whole multiplication and part-whole division, we are still using a non-proportional bar, but we’re chopping it up into units.

Part-Whole Multiplication

Here’s an example problem: I saw 5 spiders, each with 8 legs. How many legs were there all together? Instead of making a large bar, we’re going to look at the intricacies of the individual pieces of information that are in the problem. 

Since I have 5 spiders, I might show 5 units (spiders) and inside each unit, I would put 8 (legs). At this point, in part-whole multiplication and division, we start to label our bar diagram a little bit differently. The total of that bar will represent the total number of legs, which we’ll write to the left of the starting point. The question mark at the end of that bar shows that we’re looking for the total number of legs. We’re talking about individual spiders that are being counted in the problem, so, across the top of the bar, we label it to show that. 

Part-Whole Division

It’s the same idea if we were to think about maybe a child having money that they might go to the candy store and they have $10 and they want to buy candy that is, maybe 50 cents each. How many pieces of candy could they buy with the $10?

We’re now talking about that whole entire unit bar equaling the money the child had, and I want to break that up into 50 cent pieces to see how many total pieces they could get out of the $10. 

A Visual Shortcut

As the problems get larger, you  might find yourself thinking I have to draw how many boxes?? Professor Barble gives us the answer for both multiplication and division types of problems. He can help us understand when things are quantifiable for drawing units on the bar, or when its just too many.

Imagine if I saw 48 spiders, instead of 5. That would be way too many to draw! Instead, I can use an ellipses ( “…”) to represent space between the first and last unit. We can do the same thing if we were dividing something large as well. If we had 426 apples and we wanted to put them into bags of 20, it would be too many to draw! But we can use the ellipses to still give us the visual, of dividing the bar into units, without using all our time drawing boxes!

Study the anchor posters (included in the Visual Models for Word Problems bundles!) from Professor Barble to see more about how and when to use the ellipses. 

When students have a solid grasp on the part-whole family of problems – the first set when they can execute problems with a mixture of part-whole addition, subtraction and multi-step and then the second set when they’re adding in multiplication and division – then they really have a handle on a large portion of the types of problems they’ll encounter in elementary school. 

Danger, Danger!: Multi-Step Problems 

One of the areas I’ve seen students really start to get tripped up in as they do part-whole word problems is when the two “sets” of operations cross paths in a multi-step problem. 

Let’s look at this problem: If I have $250 and I wanted to buy new ski equipment. I bought skis for $100 and gear for $75. How much money do I have left?

That’s a part-whole multi-step problem that students might find pretty easy since they’re just adding and then subtracting. Some students might also find problems that combine multiplication and division to be easy. But when we cross operations, and have to complete two or three steps in a problem, students tend to revert to guess/check tactics and appealing to you for help (“Do I just add? Do I just multiply?”)

However, if we’ve taken our time to lay in the foundation of our step-by-step visual model or reading comprehension process will help students slow down and really figure out what the problem is actually asking.

We’re getting ready to go on a field trip and we have 152 students signed up to go. Ten teachers are going and we also have 8 parents/chaperones. We want to load the vehicles, and each can only fit 10. How many vehicles do we need?

This kind of problem is clearly going to involve higher order thinking. You can’t just quickly add or subtract to find an answer. Students really have to draw out that bar model to help them understand what’s going on!

We start by drawing a unit bar that will equal the total number of people going on the field trip. That bar shows a section for students, parents and teachers. Then, the information from that part of the problem is needed to solve the next portion of the problem, which is how many vehicles we’ll need. Again, I’ll draw a bar to represent the total number of people going, but this time, I need to divide it into groups of 10. Now this is starting to look more like a part-whole division problem! 

Drawing a multi-step bar model helps students see that they have to do a little bit of transferring in this process. They take that bar which shows the answer they found in the first part of the problem and re-represent it in another bar below so they can go on to the next part of the problem.

Multi-step problems really just involve a little bit more grit!

When students can become efficient in this step-by-step process for solving word problems with visual models, It’s going to help them when they start to go in to the idea of comparison, which is what we’re going to talk more about next week!

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