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.
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 Marianne’s grandmother give her?
So, we can see that it’s not a simple 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 perimeter—two measures of the length and two of the width—the 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!