Crack the Code : Understanding Word Problem Types
Addition and Subtraction Word Problems
Students often rely on "key words" to solve word problems but very often this can become confusing. For example, students often think that the word give always means to subtract, but what if the problem says, "Bill has 5 apples, Jim gives him 3 how many does Bill have now?" This is a very simplified example but it illustrates how relying only on key words can be problematic. That is why teaching word problem types based on the Cognitively Guided Instruction research is so helpful. It teaches students to understand what is actually happening in the problem and provides categories and a common language in which to talk about the different types of problems. New Jersey's Learning Standards for Mathematics (NJLS-M) list these as the types of word problems with which students should be familiar.
In class, we always start with Join and Separate problems. These are the two types that involve an action or change. I begin by teaching students to identify what that looks like. Then we discuss the difference between joining and separating and look at lots of examples together.
Once students are comfortable with the idea of change, I introduce the Join and Separate template, which models the Start–Change–Result relationship. Using this template, students practice identifying which information is given and what we are solving for. From there, we explore problems with the unknown in all three positions: start, change, and result. Students are much more comfortable and used to solving result unknown types, and will often be confused when the unknown position changes. That is why understanding the relationship between the positions is so important. It is important to really take time and give each problem type and unknown position plenty of practice before moving on, because students will be using these throughout the grades, the only difference is that instead of small numbers they may be using fractions or decimals.
Join Problems
These involve bringing amounts together.
Example: Sarah has 4 toy cars. Her friend gives her 3 more. How many does she have now?
The three parts are:
Start (Sarah has 4)
Change (friend gives 3)
Result (how many she has now)
Students must practice with the unknown in different positions:
Result unknown: 4 + 3 = ?
Change unknown: 4 + ? = 7
Start unknown: ? + 3 = 7
Separate Problems
These involve taking some away or separating.
Example: David had 9 balloons. 4 floated away. How many does he have left?
Again, the three parts are:
Start (9)
Change (4 floated away)
Result (5 left)
With unknowns in all positions:
Result unknown: 9 – 4 = ?
Change unknown: 9 – ? = 5
Start unknown: ? – 4 = 5
After students have worked with join and separate problems, we move onto the problems that do not have action: Part–Part–Whole and Comparison. I usually start with Part–Part–Whole because it is very close to join/separate types of problems, but without the action.
Part Part Whole Problems
Once students identify that there is no action, we focus on identifying which number is the whole and which numbers are the parts in the problem, as well as which information is given. The students have a Part–Part–Whole template they use to fill in information and solve. We again practice with unknowns in different positions.
Example:
Whole unknown: There are 7 blue marbles and 5 red marbles. How many marbles are there in all?
Part unknown: There are 12 marbles in all. 7 are blue and the rest are red. How many are red?
Other part unknown: There are 12 marbles in all. 5 are red and the rest are blue. How many are blue?
Comparison Problems
The final type of addition and subtraction problem we work on is the Comparison Problem. These also do not have an action. Instead, they focus on comparing two quantities to determine how much more or how much less one amount is than another.
When teaching comparison problems, I guide students to look closely at the two amounts being compared and then decide what is being asked: Are we finding the difference, one of the amounts, or the total? Students use a Comparison template that models the relationship between the two amounts and the difference.
We also practice with the unknown in all positions:
Examples:
Difference unknown: Mia has 12 stickers. Jacob has 7 stickers. How many more stickers does Mia have?
Larger unknown: Mia has 5 more stickers than Jacob. Jacob has 7 stickers. How many stickers does Mia have?
Smaller unknown: Mia has 12 stickers. She has 5 more than Jacob. How many stickers does Jacob have?
As you can see, learning to recognize different word problem types gives students the tools they need to approach math with confidence. In my next blog post, I’ll be diving into multiplication problem formats…stay tuned! :)