The bar model is a useful visualization tool to interpret word problems, but there are more than one way to solve the problem after visualizing. Often, we jump straight to the methods we are used to without seeing that another approach might be easier. Take the choice of whether to trim or add for example, as illustrated by this question discussed recently in our 2nd grade class:
Melissa is three years older than Samuel. The sum of their ages is 15. How old is Melissa?
Solutions – Trial and Error, Trim and Add
Three ways of solving this were suggested in class.
Method 1 – Trial and Error
Find 2 numbers that make 15, where the difference is 3.
9 + 6 = 15
9 – 6 = 3, so Melissa is 9 years old.
Method 2 – Bar Model (Trim)
We first visualize the problem with a bar model:
We want to make Melissa and Samuel’s ages the same. To do that, we can subtract 3 away from the sum of ages (15), so we are left with 12. Then divide 12 evenly between Melissa and Samuel.
Samuel is 6 years old and Melissa is 9 years old!
Method 3 – Bar Model (Add)
We want to make Melissa and Samuel’s ages to be the same. To do that, we can ADD 3 to the sum of ages (15), so we have 18. Then divide 18 evenly between Melissa and Samuel.
Melissa is 9 years old!
Discussion
Many of us who are used to bar modelling are familiar with method 2. The first step to trying to solve the problem is to spot the simplest common quantity between the two bars. Sometimes we label them as 1 unit (or 1u in the above). So, in method 2. we trim the longer bar to the common unit, find the sum (15-3=12), then solve for what 1u is (6). Finally, we arrive at our answer by see the relationship between the common unit (6) and the answer we are looking for (6+3=9).
However, the question asks for the age of Melissa, not Samuel. So, making Melissa’s age the common unit might have save us one step, since we do not have to go through the last step to arrive at our answer. So, instead or trimming the longer bar, we add to the shorter one to fill the gap, as in Method 3
Method 3 is is in fact how we would solve it in a linear system of equation (or simultaneous equations) in higher grades, as the two equations we were presented with are:
- M + S = 15
- M – S = 3
To solve for M, we simply use the elimination method (equation 1 + equation 2) to get 2M = 18, then divide both sides by 2 to get M=9.
Conclusion
When using tools to help visualize problems, it is important not to loose sight of what we are trying to solve for and jump straight to the methods we are used to. To add or to trim (or even to guess) are all valid ways of solving this particular problem. Use similar word problems to discuss strategies with your students and let them come up with different ways they might solve it.
Related Resources
For more related resources, please see our Number Sense, Addition and Subtraction page.
an interesting method. i’ve never known it before. i hope i can get more of these enlightenments.