Teachers work all year to help children build conceptual understanding, make connections, and see relationships - working their behinds off modeling and practicing those concepts and helping kids gain a deep understanding. Then comes the mastery piece - fact fluency. It is a ton of hard work!
But the word problems. Oh, the word problems! That's the piece that sometimes gets left by the wayside in the desperate attempt to drive home fact mastery.
It doesn’t seem too complicated at first glance because we have used story problems since Kindergarten to help students build a conceptual understanding of addition and subtraction.
But we can easily get stuck in the basic join and separate word problem set and not even realize that there are many more ways that problems can be presented.
When pushing through that quick word problem of the day, it’s easy to forget to increase the complexity of the problems we present.
Now as a fourth grade teacher, I can see how essential those early word problems are. If students didn’t learn to understand and solve complex addition and subtraction word problems in the early grades, they have a much more difficult time understanding multi step multiplication and division problems in the upper grades. They need that foundation on which to build the new concepts.
It is so important for kids to have daily opportunities to solve all different types of word problems. And not just any old problem we come up with on the fly, but varied and wide ranging.
When teaching first grade, I found myself forgetting to include all the types of word problem types. I tended to go with the familiar. So I incorporated daily word problem practice and I found it helpful to keep a chart handy so I could keep track of the problems we were working on and ensure that my students were getting exposure to all problem types.
Did you know there are four different types of word problems?
And 11 subsets? Take a look at this chart based on the CGI model.
Let's take a closer look at each type:
Join problems involve an action, with a set being added to an existing set.
Result Unknown is the type of addition problem our kids are so familiar with when they come to us: 3 + 2 = __
We know the parts and need to find the sum. Joe had 3 pencils. Meg gave him 2 more pencils. How many pencils does Joe have now?
Change Unknown problems ramp up the complexity and throw in the missing part: 3 + __ = 5
Joe had 3 pencils. Meg gave him some more and now Joe has 5 pencils. How many pencils did Meg give Joe?
Start Unknown problems are very similar: __ + 2 = 5
Joe had some pencils. Meg gave him 2 more and now Joe has 5 pencils. How many pencils did Joe have to start?
All of these problems involve joining two sets, but once the start and change become unknown, the problem requires a deeper understanding.
Separate problems also involve an action with a set being removed from an existing set.
Result Unknown again, is a familiar form of a subtraction problem: 5 - 3 = __
Joe had 5 pencils. He gave 3 of them to Meg. How many pencils does Joe have left?
Change Unknown provides for the missing part and an opportunity for deeper understanding of subtraction: 5 - __ = 2
Joe had 5 pencils. He gave some of them to Meg and now he has 2 pencils left. How many pencils did Joe give Meg?
Start Unknown is looking to find the whole: __ - 3 = 2
Joe had some pencils. He gave Meg three and now he has 2 pencils left. How many pencils did Joe have to start?
All subtraction problems, but presented in ways that allow for a more conceptual understanding of subtraction.
Part part whole problems seem much like join and separate problems, but they do not involve an action. The problem is really looking at relationships between the quantities. There are two types of part-part whole problems:
Whole Unknown 2 + 3= __ Joe had 2 green pencils and 3 yellow pencils. How many pencils did Joe have altogether?
Part Unknown 2 + __= 5 or 5 - 2 = __ Joe had 5 pencils. Two are green and the rest are yellow. How many yellow pencils does Joe have?
Compare problems also look at relationships between quantities. But here we are not looking at a set and its subsets, but the relationship between two distinct sets.
Difference Unknown compares two sets. 5 - 3 = __ or 3 + __ = 5
Joe had 5 pencils. Meg had 3 pencils. How many more pencils did Joe have than Meg?
Quantity Unknown 3 + 2 = __
Joe had 3 pencils. Meg had 2 more pencils than Joe. How many pencils did Meg have?
Reference Unknown 5 - 3 = __
Joe had 5 pencils. He had 3 more than Meg. How many pencils did Meg have?
Each type of problem requires a different level of thinking and problem solving.
If you do daily word problem practice and would like to ensure that you cover all problems types, or if you haven't incorporated a word problem of the day and would like to start, grab this freebie! It contains a copy of the problem type poster and some word problems with work mats to get you started!